MODULI SPACES OF HIGHER SPIN CURVES AND INTEGRABLE HIERARCHIES

TYLER J. JARVIS, TAKASHI KIMURA, AND ARKADY VAINTROB

Abstract. We prove the genus zero part of the generalized Witten conjecture relating moduli spaces of spin curves to Gelfand-Dickey hierarchies. That is, we show that intersection numbers on the moduli space of stable r-spin curves assemble into a generating function which yields a solution of the semiclassical limit of the KdV_{r} equations. We formulate axioms for a cohomology class on this moduli space which allow one to construct a cohomological field theory of rank r − 1 in all genera. In genus zero it produces a Frobenius manifold which is isomorphic to the Frobenius manifold structure on the base of the versal deformation of the singularity A_{r−1}. We prove analogs of the puncture, dilaton, and topological recursion relations by drawing an analogy with the construction of Gromov-Witten invariants and quantum cohomology.

0. Introduction

The moduli space of stable curves of genus g with n marked points \overline{M}_{g,n} is a fascinating object. Mumford [37] introduced tautological cohomology classes associated to the universal curve C_{g,n} ^{}\overline{M}_{g,n}, and Witten conjectured [42] and Kontsevich [28] proved that the intersection numbers of certain tautological cohomology classes on the moduli space of stable curves \overline{M}_{g,n} have a generating function which satisfies the equations of the Korteweg-de Vries hierarchy (more precisely that it is a τ function of the KdV hierarchy satisfying some additional equations). This remarkable result provided an unexpected link between the algebraic geometry of these moduli spaces and integrable systems.

The spaces \overline{M}_{g,n} can be generalized in two ways. The first is by choosing a smooth projective variety V and considering the moduli space of stable maps into V of curves of genus g with n marked points, \overline{M}_{g,n}(V). When V is a point, \overline{M}_{g,n}(V) reduces to \overline{M}_{g,n}. The second is

Date: September 10, 1999. Research of the first author was partially supported by NSA grant MDA904-99-

1-0039.

Research of the second author was partially supported by NSF grant number DMS-9803427.

by considering moduli space of r-spin curves \overline{M}_{g,n}^{1/r}, introduced in [21, 22], which, upon forgetting the r-spin structure, reduces to \overline{M}_{g,n}. It is natural to ask if Kontsevich’s theorem admits a generalization to either of these cases.

The case of \overline{M}_{g,n}(V) remains mysterious. It gives the Gromov-Witten invariants of V (and their so-called gravitational descendants), which assemble into a generating function whose exponential is an analog of a τ function. In the case where V is a point, one recovers the τ function of the KdV hierarchy by Kontsevich’s theorem. More generally, there is a conjecture of Eguchi-Hori-Xiong [12] and of S.Katz which essentially states that this generating function is a highest weight vector for a particular representation of the Virasoro algebra. Presumably, there is some analog of an integrable system which gives rise to this Virasoro algebra action, should the conjecture hold.

On the other hand, the KdV hierarchy is known to be merely the first in a series of integrable hierarchies known as the KdV_{r} (or r-th Gelfand-Dickey) hierarchies, where r = 2, 3, 4, . . .. In the case r = 2 this becomes the usual KdV hierarchy. Each of these hierarchies has a formal solution corresponding to the unique τ function which satisfies an equation known as the string (or puncture) equation. In [40, 41] Witten formulated a generalization of his original conjecture, which states that for each r ≥ 2 there should exist moduli spaces and cohomology classes on them whose intersection numbers assemble into the τ function of the KdV_{r} hierarchy.

In this paper, we are motivated by analogy with the construction of Gromov-Witten invariants from the moduli space of stable maps to introduce axioms which must be satisfied by a cohomology class c^{1/r} (called the virtual class) on the moduli space of r-spin curves \overline{M}_{g,n}^{1/r} in order to obtain a cohomological field theory (CohFT) of rank (r −1) in the sense of Kontsevich-Manin [29]. This virtual class on \overline{M}_{g,n}^{1/r} should be regarded as an analog of the Gromov-Witten classes of a variety V, i.e. the pullbacks via the evaluation maps of elements in H^{•}(V).

We realize this virtual class in genus zero as the top Chern class of a tautological bundle over \overline{M}_{0,n}^{1/r} associated to the r-spin structure. This yields a Frobenius manifold structure [10, 20, 33] on the state space of the CohFT which is isomorphic to the Frobenius manifold associated to the versal deformation of the A_{r−1} singularity [10].

This is an indication of the existence of a kind of “mirror symmetry” between the moduli space of r-spin curves and singularities. According to Manin [34] “isomorphisms of Frobenius manifolds of different classes remain the most direct expression of various mirror phenomena.” Proving the generalized Witten conjecture for all genera would provide further evidence of this relationship.

As in the case of Gromov-Witten invariants, one can construct a potential function from the integrals of c^{1/r} on different components of \overline{M}_{g,n}^{1/r} to form the small phase space of the theory. The large phase space is constructed by introducing the tautological classes ψ associated to the universal curve C_{g,n}^{1/r} ^{}\overline{M}_{g,n}^{1/r}. It can be regarded as a parameter space for a family of CohFTs. A very large phase space (see [25, 26, 13, 35]) parametrizing an even larger space of CohFTs is obtained by considering tautological classes λ, associated to the Hodge bundles, and tautological classes μ, associated to the universal spin structure.

We show that the corresponding potential function satisfies analogs of the puncture and dilaton equations and also a new differential equation obtained from a universal relation involving the class μ_{1}. These relations hold in all genera. Topological recursion relations are also obtained from presentations of these classes in terms of boundary classes in low genera.

Finally, using the new relation on μ_{1}, we show that the genus zero part of the large phase space potential Φ_{0}(t) is completely determined by the geometry, and this potential agrees with the generalized Witten conjecture in genus zero.

Some of our constructions were foreshadowed by Witten, who formulated his conjecture even before the relevant moduli spaces and cohomology classes had been constructed, just as he had done in the case of the topological sigma model and quantum cohomology. We prove that his conjecture has a precise algebro-geometric foundation, just as in the case of Gromov-Witten theory. Witten also outlined a formal argument to justify his conjecture in genus zero. Our work shows that the formulas that he ultimately obtained for the large phase space potential function in genus zero are indeed correct, provided that the geometric objects involved are suitably interpreted. This is nontrivial even in genus zero because of the stacky nature of the underlying moduli spaces. We proceed further to prove relations between various tautological classes associated to the r-spin structures and derive differential equations for the potential function associated to them.

Notice also that one can introduce moduli spaces \overline{M}_{g,n}^{1/r}(V) of stable r-spin maps into a variety V, where one combines the data of both the stable maps and the r-spin structures. The analogous construction on these spaces yields a Frobenius manifold which combines Gromov-Witten invariants (and quantum cohomology) with the KdV_{r} hierarchies. Work in this direction is in progress [24].

In the first section of this paper, we review the moduli space of genus g, stable r-spin curves, \overline{M}_{g,n}^{1/r}, which was introduced in [21, 22]. We

also discuss the stratification of the boundary of \overline{M}_{g,n}^{1/r}. The boundary strata fall into two distinct categories—the so-called Neveu-Schwarz and Ramond types. Only the former has the factorization properties analogous to properties of the moduli space of stable maps.

In the second section, we introduce canonical morphisms, tautological bundles, tautological cohomology classes, and cohomology classes associated to the boundary strata of \overline{M}_{g,n}^{1/r} and derive a new relation involving the μ_{1} class. In the third section, we define a cohomological field theory (CohFT) in the sense of Kontsevich-Manin, its small phase space potential function, and the associativity (WDVV) equation. We then review the construction of Gromov-Witten invariants for the moduli space of stable maps and define the large and very large phase spaces in the Gromov-Witten theory. Motivated by this example, we explain how to construct a CohFT and the various potential functions from analogous intersection numbers on \overline{M}_{g,n}^{1/r}, assuming that the virtual class c^{1/r} exists.

In the fourth section, we state axioms which c^{1/r} must satisfy in order to obtain a CohFT. We show that these axioms give a complete CohFT with a flat identity, and we construct the class c^{1/r} in genus zero as well as in the case r = 2. In the fifth section, we obtain analogs of the string and dilaton equations (which essentially yield the L_{−1} and L_{0} Virasoro generators) for this r-spin CohFT, and a find a new equation based on the relation involving the μ_{1} class. We also prove the analog of topological recursion relations in genus zero. Finally in the sixth section, we determine the large phase space potential of this theory in genus zero and show that it yields a solution to (the semiclassical limit of) the KdV_{r} hierarchy, thereby proving the genus zero generalized Witten conjecture. We conclude with our own W -algebra conjecture, a KdV_{r} analog of a refinement of the Virasoro conjecture [12].

Acknowledgments . We would like to thank D. Abramovich, M. Adler, B. Dubrovin, J. Figueroa-O’Farrill, E. Getzler, A. Kabanov, Yu. Manin, P. van Moerbeke, A. Polishchuk, M. Rosellen, and J. Stasheff for useful exchanges. We would also like to thank Heidi Jarvis for help with typesetting this paper. A.V. thanks the Max-Planck-Institut für Mathematik in Bonn for hospitality and financial support.

1. The Moduli Space of r-spin Curves

1.1. Definitions.

Definition 1.1. Let (X, p_{1}, …, p_{n}) be a nodal, n-pointed algebraic curve, and let K be a rank-one, torsion-free sheaf on X. A d-th root of K of type m = (m_{1}, …, m_{n}) is a pair (E, b) of a rank-one, torsion-free sheaf E, and an O_{X}-module homomorphism

b : E^{⊗d} ^{}K ⊗ O_{X}(− ∑ m_{i}p_{i})

with the following properties:

• d · deg E = deg K − ∑ m_{i}

• b is an isomorphism on the locus of X where E is locally free

• for every point p ∈ X where E is not free, the length of the cokernel of b at p is d − 1.

For any d-th root (E, b) of type m, and for any m′ congruent to m mod d, we can construct a unique d-th root (E′, b′) of type m′ simply by taking E′ = E ⊗ O(1/d ∑(m_{i} − m′_{i})p_{i}). Consequently, the moduli of curves with d-th roots of a bundle K of type m is canonically isomorphic to the moduli of curves with d-th roots of type m′. Therefore, unless otherwise stated, we will always assume the type m of a d-th root has the property that 0 ≤ m_{i} < d for all i.

Unfortunately, the moduli space of curves with d-th roots of a fixed sheaf K is not smooth when d is not prime, and so we must consider not just roots of a bundle, but rather coherent nets of roots [21]. This additional structure suffices to make the moduli space of curves with a coherent net of roots smooth.

Definition 1.2. Let K be a rank-one, torsion-free sheaf on a nodal n-pointed curve (X, p_{1}, …, p_{n}). A coherent net of r-th roots of K of type m = (m_{1}, …, m_{n}) consists of the following data:

• For every divisor d of r, a rank-one torsion-free sheaf E_{d} on X;

• For every pair of divisors d′, d of r, such that d′ divides d, an O_{X}-module homomorphism

c_{d,d′} : E_{d}^{⊗d/d′} ^{}E_{d′}.

These data are subject to the following restrictions.

1. E_{1} = K and c_{1,1} = 1

2. For each divisor d of r and each divisor d′ of d, let m′′ = (m′′_{1}, …, m′′_{n}) be such that m′′_{i} is the unique non-negative integer less than d/d′, and congruent to m_{i} mod d. Then the homomorphism c_{d,d′} makes (E_{d}, c_{d,d′}) into a d/d′ root of E_{d′} of type m′′.

3. The homomorphisms {c_{d,d′}} are compatible. That is, the diagram

[figure: (E_{d}^{⊗d/d′})^{⊗d′/d′′}; (c_{d,d′})^{⊗d′/d′′}; E_{d′}^{⊗d′/d′′}; E_{d′′}; c_{d′,d′′}; c_{d,d′′}]

commutes for every d′′|d′|d|r.

If r is prime, then a coherent net of r-th roots is simply an r-th root of K. Moreover, if E_{d} is locally free of type m, then for d′|d if m ≡ m (mod d)′ and 0 ≤ m′_{i} < d′, then the sheaf E_{d′} is isomorphic to E_{d}^{⊗d/d′} ⊗ O(\frac{1}{d^{r}} ∑(m_{i} − m′_{i})p_{i}) and c_{d,d′} is uniquely determined up to an automorphism of E_{d′}.

Definition 1.3. An n-pointed r-spin curve of type m = (m_{1}, …, m_{n}) is an n-pointed, nodal curve (X, p_{1}, …, p_{n}) with a coherent net of r-th roots of ω_{X} of type m, where ω_{X} is the canonical (dualizing) sheaf of X. An r-spin curve is called smooth, if X is smooth, and it is called stable, if X is stable.

Example 1.4. Smooth 2-spin curves of type m = (0, 0, …, 0) correspond to classical spin curves (a curve and a theta-characteristic) together with an explicit isomorphism E_{2}^{⊗2} ^{∼}^{}ω.

Definition 1.5. An isomorphism of r-spin curves

(X, p_{1}, …, p_{n}, {E_{d}, c_{d,d′}}) ^{∼}^{}(X′, p′_{1}, …, p′_{n}, {E′_{d}, c′_{d,d′}})

of the same type m is an isomorphism of pointed curves

τ : (X, p_{1}, …, p_{n}) ^{∼}^{}(X′, p′_{1}, …, p′_{n})

and a family of isomorphisms {β_{d} : τ^{∗}E′_{d} ^{∼}^{}E_{d}}, with β_{1} being the canonical isomorphism τ^{∗}ω_{X′}(− ∑_{i} m_{i}p′_{i}) ^{∼}^{}ω_{X}(− ∑ m_{i}p_{i}), and such that the β_{d} are compatible with all the maps c_{d,d′} and τ^{∗}c′_{d,d′}.

Every r-spin structure on a smooth curve X is determined up to isomorphism by a choice of a line bundle E_{r}, such that E_{r}^{⊗r} ≅ ω_{X}(− ∑ m_{i}p_{i}). In particular, if X has no automorphisms, then the set of isomorphism classes of r-spin structures of a fixed type m is a principal Jac_{r}Xbundle, where Jac_{r}X is the group of r-torsion points of the Jacobian of X, thus there are r^{2g} such isomorphism classes.

Example 1.6. If g = 1 and m = 0 := (0, …, 0), then ω_{X} ≅ O_{X} and a smooth r-spin curve is just an r-torsion point of the elliptic curve X, again with an explicit isomorphism of E_{r}^{⊗r} ^{∼}^{}O_{X}. In particular, the stack of stable r-spin curves of genus one and type 0 forms a gerbe over the disjoint union of modular curves ∐_{d|r} X_{1}(d).

Definition 1.7. The moduli space of stable n-pointed, r-spin curves of genus g and type m is denoted \overline{M}_{g,n}^{1/r,m}. The disjoint union ∐_{\begin{matrix}m \\ 0≤m_{i}<r \end{matrix}} \overline{M}_{g,n}^{1/r,m}

is denoted \overline{M}_{g,n}^{1/r}. Remark 1.8. As mentioned above, no information is lost by restricting m to the range 0 ≤ m_{i} ≤ r − 1, since when m ≡ m′ mod r every r-spin curve of type m naturally gives an r-spin curve of type m′ simply by

E_{d} ↦ E_{d} ⊗ O(∑ \frac{m_{i} − m′_{i}}{d} p_{i}),

and thus \overline{M}_{g,n}^{1/r,m} is canonically isomorphic to \overline{M}_{g,n}^{1/r,m′}. 1.2. Basic properties of the moduli space.

In [21] it is shown that \overline{M}_{g,n}^{1/r} is a smooth Deligne-Mumford stack, finite over \overline{M}_{g,n}, with a projective coarse moduli space.

Throughout this paper we will denote the forgetful morphism by p : \overline{M}_{g,n}^{1/r} ^{}\overline{M}_{g,n}, and the universal curve by π : C_{g,n}^{1/r} ^{}\overline{M}_{g,n}^{1/r}. As in the case of the moduli space of stable curves, the universal curve possesses canonical sections σ_{i} : \overline{M}_{g,n}^{1/r} ^{}C_{g,n}^{1/r} for i = 1 … n.

There is one other canonical morphism associated to these spaces; namely, when d divides r, the morphism

[r/d] : \overline{M}_{g,n}^{1/r,m}  ^{}\overline{M}_{g,n}^{1/d,m′′} and [r/d] : \overline{M}_{g,n}^{1/r}  ^{}\overline{M}_{g,n}^{1/d},

which forgets all of the roots and homomorphisms in the net of r-th roots except those associated to divisors of d. Here, as above, m′′ is congruent to m mod d and has 0 ≤ m′′_{i} < d. In the case that the underlying curve is smooth, this is equivalent to replacing the line bundle E_{r} by its r/d-th tensor power (and tensoring with O(1/d ∑(m_{i}− m′′_{i})p_{i})).

As is shown in [21], for g > 1 the spaces \overline{M}_{g,n}^{1/r,m} are irreducible if gcd(r, m_{1}, …, m_{n}) is odd, and they are the disjoint union of two irreducible components if gcd(r, m_{1}, …, m_{n}) is even. In the case that r = 2, this is due to the well-known fact that even and odd theta characteristics on a curve cannot be deformed into one another [36]. In general, the two components that arise in the case that gcd(r, m_{1}, …, m_{n}) is even are just the preimages of the even and odd components of \overline{M}_{g,n}^{1/2,0}

under the map [r/2] : \overline{M}_{g,n}^{1/r,m} ^{}\overline{M}_{g,n}^{1/2,0}. These two components will

be denoted \overline{M}_{g,n}^{1/r,m,even} and \overline{M}_{g,n}^{1/r,m,odd} respectively.

When g = 0 the moduli space \overline{M}_{0,n}^{1/r,m} is either empty (if r does not divide ∑ m_{i} + 2) or is canonically isomorphic to \overline{M}_{0,n}. Note, however, that this isomorphism is not an isomorphism of stacks, since the automorphisms of elements of \overline{M}_{g,n}^{1/r,m} vary differently from the way

that automorphisms of the underlying curves vary. In any case, \overline{M}_{0,n}^{1/r,m} is always irreducible.

When g = 1 the space \overline{M}_{1,n}^{1/r,m} is the disjoint union of d irreducible components, where d is the number of divisors of gcd(r, m_{1}, …, m_{n}).

1.3. Boundary behavior.

1.3.1. Neveu-Schwarz and Ramond nodes. At any node q of a nodal curve X, an r-th root (E_{r}, b_{r}) of ω_{X}(− ∑ m_{i}p_{i}) is either locally free at q, with b_{r} an isomorphism at q, or it is torsion-free but not locally free at q. The locally free case is what Witten [40] calls the Ramond sector, and the non-locally free case he calls a generalized Neveu-Schwarz sector.

In the case that E_{r} is Ramond, all of the terms in the spin structure are completely determined (up to isomorphism) by the root (E_{r}, c_{r,1}), simply by the rule E_{d} = E_{r}^{⊗r/d} ⊗ O(q/d ∑(m_{i} − m′_{i})p_{i}), and c_{d,1} = c_{r,1} tensored with the identity homomorphism

O(∑(m_{i} − m′′_{i})p_{i}) ^{}O(∑(m_{i} − m′′_{i})p − i).

The Neveu-Schwarz sector is more complicated, in some sense, but this is compensated for by the fact that for the Neveu-Schwarz sector, the cohomology classes defined by boundary strata factor in a nice combinatorial way. Moreover, the hope of constructing a cohomological field theory from \overline{M}_{g,n}^{1/r} is based on the belief that one can construct a canonical cohomology class which vanishes on the Ramond sector. (This would follow from Axiom 4 in Section 4.1.)

Recall from [40] or [21, 22] that in the Neveu-Schwarz case, an r-th root (E_{r}, b_{r}) of ω_{X}(− ∑ m_{i}p_{i}) is uniquely determined by an r-th root (Ẽ_{r}, b̃_{r}) of ωX̃(− ∑ m_{i}p_{i} − m^{+}q^{+} − m^{−}q^{−}) on the normalization ν : X̃ ^{}X of X at the node q. Here q^{+} and q^{−} are the two points of X̃ lying over the point q of X, and m^{+} and m^{−} are non-negative integers^{∗} which sum to r − 2. The sheaf Ẽ_{r} is simply ν^{∗}E_{r} modulo torsion; and ν_{∗}Ẽ_{r} = E_{r}, with b_{r} induced from b̃_{r} by adjointness. In the case that m^{+}+ 1 and m^{−} + 1 are relatively prime, one can show (see [21]) that Ẽ and b̃ uniquely determine the entire net. But if gcd(m^{+}+1, m^{−}+1) = d, then although E_{r} still completely determines the Neveu-Schwarz roots, d divides r, and the d-th root (E_{d}, c_{d,1}) of the net is locally free (Ramond), as are all roots (E_{d′}, c_{d′,1}) such that d′|d. In particular, E_{d′} is simply E_{d}^{⊗d/d′}, tensored by the appropriate bundle O(1/d′ ∑ m′_{i} − m′′_{i})p_{i}). If the normalization X̃ at q has two connected components, then the dth root (E_{d}, c_{d,1}) is determined up to isomorphism by (E_{r}, b_{r}), but if X̃ is connected, then (E_{d}, c_{d,1}) is not determined by (E_{r}, b_{r}), since an additional gluing datum of a d-th root of unity is required to construct E_{d} from E_{r}^{⊗r/d} (see Section 1.5).

∗Note that u and v of the papers [21] and [22] are m^{+}+1 and m^{−}+1, respectively.

1.3.2. More detailed study of the Ramond sector (after Witten [40]). Let (E, b) be an r-th root of ω_{X}(− ∑ m_{i}p_{i}) which is Ramond at a node q of X. The restriction of E to q gives an exact sequence

0 ^{}𝔪_{q} ⊗ E ^{}E ^{}E|_{q} ^{}0,

where 𝔪 is the maximal ideal of the point q. The sheaf E′ := E ⊗ 𝔪_{q} is a rank-one, torsion-free sheaf of degree (2g − 2 − ∑ m_{i})/r − 1 on X and pulling E′ back to the normalization ν : X̃ ^{}X of X at q gives, after modding out torsion, E′′ := ν^{∗}E′/torsion with ν_{∗}E′′ = E′.

If x and y are local coordinates on X̃ near q^{+} and q^{−} respectively, then E′′ is locally generated by x(\frac{dx}{x})^{1/r} (respectively y(\frac{dy}{y})^{1/r}). Therefore, the homomorphism

b′′ : E′′^{⊗r} ^{}ν^{∗}ω_{X}(− ∑ m_{i}p_{i}) = ωX̃(− ∑ m_{i}p_{i} + q^{+} + q^{−}),

induced by b : E^{⊗r} ^{}ωX̃(− ∑ m_{i}p_{i}) factors through

ωX̃(− ∑ m_{i}p_{i} − (r − 1)q^{+} − (r − 1)q^{−}) ^{}ωX̃(− ∑ m_{i}p_{i} + q^{+} + q^{−}).

Thus the Ramond case can occur if and only if m^{+} = r − 1 and m^{−} = r − 1 satisfy the degree conditions:

deg_{X^{(j)}} ωX̃ − ∑ m_{i} − m^{+} − m^{−} ≡ 0 mod r

on every connected component X^{(j)} of X̃.

Similarly,

(ν^{∗}E)^{⊗r} ^{ν^{∗}b}^{}ν^{∗}ω_{X}(− ∑ m_{i}p_{i}) = ωX̃(− ∑ m_{i}p_{i} + q^{+} + q^{−}),

corresponds to a choice of m^{+} = m^{−} = −1. In this special case there is a residue map that canonically identifies E_{r}|_{q^{+}} and E_{r}|_{q^{−}} with ℂ.

Proposition 1.9. If (E_{r}, b) is an r-th root of ω_{X}(− ∑ m_{i}p_{i}) with m_{i} = −1 for some i, then there is a canonical isomorphism

R_{p_{i}} : E_{r}|_{p_{i}} ^{∼}^{}ℂ. (1)

An immediate consequence is the following corollary:

Corollary 1.10. If σ_{i} is the i-th section of the universal curve

π : C_{g,n}^{1/r,m} ^{}\overline{M}_{g,n}^{1/r,m}

with m = (m_{1}, …, m_{n}) and m_{i} = −1 for some i, then the pullback σ_{i}^{∗}(E_{r}) of the universal r-th root E_{r} is the trivial line bundle O\overline{M}{1/r,m \atop g,n}.

Proof. (of the proposition) Let z be a local parameter on X near p, so that the sheaf E_{r} is locally generated by an element (\frac{dz}{z})^{1/r}. Define the map R_{p} : E_{r}|_{p} ^{∼}^{}ℂ by R_{p}((a_{0} + a_{1}z + …)(\frac{dz}{z})^{1/r}) := a_{0}. It is easy to check that this definition is independent of local parameter, and hence defines a canonical isomorphism.

1.4. Dual graphs. Much of the information about the structure of the boundary can be encoded in terms of decorated dual graphs.

Recall that the dual graph of an n-pointed nodal curve X consists of the following elements:

• Vertices corresponding to the irreducible components of X; a vertex v is labeled with g(v), the (geometric) genus of the component;

• Edges corresponding to the nodes of the curve: an edge connects two vertices if and only if the corresponding node lies on the associated irreducible components;

• Tails, corresponding to marked points on a component, protrude from the corresponding vertex, and an ordering of the marked points corresponds to a labeling of the tails by the integers 1, …, n.

Definition 1.11. Let Γ be a graph with n tails and V (Γ) be its set of vertices. The genus of the graph Γ is defined as

g(Γ) = ∑_{v∈V (Γ)} g(v) + dimH^{1}(Γ).

A graph Γ is called stable if 2g(v) − 2 + n(v) > 0 for every v ∈ V (Γ) (in particular, it satisfies 2g(Γ) − 2 + n > 0 ). Here n(v) is the total number of half-edges at the vertex v (i.e., the ends of true edges as well as tails of Γ at v).

In the case of spin curves we also decorate the graph with additional data coming from the spin structure. In particular the type m = (m_{1}, …, m_{n}) gives a marking to each of the tails.

Definition 1.12. Fix a positive integer r. A decorated stable graph is a stable graph together with a marking of each half-edge (i.e. tail or half of a “real” edge) with a non-negative integer m < r, such that for each edge e the marks m^{+} and m^{−} of the two half-edges of e satisfy

m^{+} + m^{−} ≡ r − 2 mod r.

As discussed in Section 1.3, decorated stable graphs with n tails correspond to boundary strata in \overline{M}_{g,n}^{1/r}.

Definition 1.13. Let Γ be a connected stable graph (resp. a decorated stable graph) with n tails and of genus g . We denote by \overline{M}_{Γ} (resp. by \overline{M}_{Γ}^{1/r}) the closure in \overline{M}_{g,n} (resp. in \overline{M}_{g,n}^{1/r}) of the moduli space of stable curves (resp. r-spin curves) with the dual graph Γ.

If Γ is the disjoint union of connected subgraphs Γ^{(i)} for some index set \{ i \} then \overline{M}_{Γ} := ∏_{i} \overline{M}_{Γ^{(i)}} and similarly for \overline{M}_{Γ}^{1/r}.

1.5. Gluing.

In the case of moduli spaces of ordinary stable curves, if a graph Γ is obtained from another graph Γ̃ (not necessarily connected) by gluing together two tails of Γ̃ (thus producing a new edge), there is a natural gluing map

ρ : \overline{M}Γ̃ ^{}\overline{M}_{Γ} ^{⊂} ^{}\overline{M}_{g,n}. (2)

It glues together the corresponding punctures on a curve X ∈ \overline{M}Γ̃ and produces an additional node on ρ(X) ∈ \overline{M}_{Γ}.

A similar gluing operation sometimes exists for r-spin curves, but in this case we sometimes need to add some extra data.

1.5.1. Two irreducible components. Consider the case of a spin curve X with a single Neveu-Schwarz node whose normalization has two connected components: X^{+} of genus k, and X^{−} of genus g − k. The dual graph of X looks like this

Γ = .

The r-th root bundle E_{r} factors as E_{r} = E_{r}^{+} ⊕ E_{r}^{−} with E_{r}^{+} an r-

th root of ω_{X^{+}}(− ∑_{l=1}^{j} m_{i_{l}}p_{i_{l}} − m^{+}q^{+}) on X^{+} and E^{−} an r-th root of

ω_{X^{−}}(− ∑_{l=j+1}^{n} m_{i_{l}}p_{i_{l}} − m^{−}q^{−}) on X^{−}. Here m^{+} and m^{−} can easily be

calculated (for degree reasons) as the unique non-negative integers summing to r − 2 which make their respective canonical bundles divisible by r; namely,

(3)  m^{+} ≡ 2k − 2 − ∑_{l=1}^{j} m_{i_{l}} mod r and

(4)  m^{−} ≡ 2(g − k) − 2 − ∑_{l=j+1}^{n} m_{i_{l}} mod r.

Of course, since the degree of the original bundle was divisible by r, each of the two relations implies the other. The Neveu-Schwarz case occurs exactly when the congruences (3) and (4) can be solved for m^{+} and m^{−} in {0, 1, …, r − 2} (see [22]).

If gcd(m^{+} + 1, m^{−} + 1) = 1 then these data completely determine the spin structure on X and in a canonical way, so that in this case we have a well-defined gluing morphism

ρ : \overline{M}_{Γ_{1}⊔Γ_{2}}^{1/r} = \overline{M}_{Γ_{1}}^{1/r} × \overline{M}_{Γ_{2}}^{1/r} ^{}\overline{M}_{Γ}^{1/r} ^{⊂} ^{}\overline{M}_{g,n}^{1/r,m}. (5)

If, however, gcd(m^{+} + 1, m^{−} + 1) = d > 1, there is no longer a morphism of stacks as in (5). To construct a gluing morphism would require an isomorphism ϕ : E_{d}^{+}|_{q^{+}} ^{∼}^{}E_{d}^{−}|_{q^{−}} that makes E_{d} Ramond at the node. Since the graph Γ is a tree, the root (E_{d}, c_{d,1}) is determined up to (non-canonical) isomorphism by (E_{r}, c_{r,1}). Still, choosing one gluing datum ϕ : E_{d}^{+}|_{q^{+}} ^{∼}^{}E_{d}^{−}|_{q^{−}} does not give a morphism of stacks because an automorphism of the spin structure on X^{+} or on X^{−} changes the gluing datum ϕ and thus induces a different (but isomorphic) spin structure on the curve X

1.5.2. Irreducible curve with one node. In the case of an r-spin curve X with a single component and one node we have the dual graph

Γ_{irr} = .

This determines a spin structure on the normalized curve X̃, with the dual graph

Γ̃_{irr} = .

If d = gcd(m^{+} + 1, m^{−} + 1) = gcd(m^{+} + 1, r) = 1, then all roots E_{d} in the spin structure are of the Neveu-Schwarz type and we can define the gluing map

ρ : \overline{M}1/r ; Γ̃_{irr} ^{}\overline{M}_{Γ_{irr}}^{1/r} ^{⊂} ^{}\overline{M}_{g,n}^{1/r,m} (6)

in the obvious way.

But if d = gcd(m^{+} + 1, r) > 1, then, as mentioned above, an additional gluing datum is required to construct (E_{d}, c_{d,1}) from (E_{r}^{⊗r/d}, c_{r,1}). In this case, set u^{+} = \frac{m^{+}+1}{d} and u^{−} = \frac{m^{−}+1}{d} and define

E′_{d} := E_{r}^{⊗r/d} ⊗ O(u^{+}p^{+} + u^{−}p^{−})

where p^{+} and p^{−} are the inverse images under normalization of the node. This shows that E′_{d} is a d-th root of ωX̃(− ∑ m_{i}p_{i} + p^{+} + p^{−}), that is, m^{+} and m^{−} are both replaced with −1. To construct E_{d} from E′_{d} we need to choose an isomorphism

ϕ : E′_{d}|_{p^{+}} ^{}E′_{d}|_{p^{−}}

compatible with the canonical isomorphism

E_{d}^{′⊗d}|_{p^{+}} ^{∼}^{}ωX̃(p^{+} + p^{−}) ^{∼}^{}E_{d}^{′⊗d}|_{p^{−}},

and there are exactly d such isomorphisms.

Unlike in the case of the tree, now an automorphism of the spin structure on the normalized curve induces the same automorphism on both sides of the gluing datum ϕ, and thus it preserves ϕ. Consequently, we expect d = gcd(m^{+} + 1, r) gluing morphisms

ρ_{ϕ} : \overline{M}1/r ; Γ̃_{irr} ^{}\overline{M}_{Γ_{irr}}^{1/r} ^{⊂} ^{}\overline{M}_{g,n}^{1/r,m}.

indexed by the set of different choices of ϕ.

But in order to construct such a morphism, one needs to be able to define the glue ϕ in families, that is, we need to define isomorphisms

ϕ : E′_{d}|_{D^{+}} = σ^{+∗}(E′_{d}) ^{∼}^{}σ^{−∗}(E′_{d}) = E′_{d}|_{D^{−}}

where D^{+} and D^{−} are the zero loci of the two sections σ^{+} and σ^{−} to be glued. Such an isomorphism exists precisely because E′_{d} corresponds to the graph Γ̃_{irr} with m^{+} = m^{−} = −1. By Corollary 1.10, both E′_{d}|_{D^{+}} and E′_{d}|_{D^{−}} are trivial bundles on \overline{M}_{g−1,n+2}^{1/r,(m,−1,−1)}, and hence are isomorphic.

Thus if 𝔤 is the set of isomorphisms E′_{d}|_{D^{+}} ^{∼}^{}E′_{d}|_{D^{−}} of the universal sheaf E′_{d} = E_{r}^{⊗r/d} ⊗ O(u^{+}D^{+} + u^{−}D^{−}), we have a morphism

ρ_{Γ_{irr}} : \overline{M}1/r ; Γ̃_{irr} × 𝔤 ^{}\overline{M}_{Γ_{irr}}^{1/r} ^{⊂} ^{}\overline{M}_{g,n}^{1/r,m}.

2. Tautological cohomology classes

2.1. Definitions.

There are many natural cohomology classes in H^{•}(\overline{M}_{g,n}^{1/r,m}, ℚ); these include the classes induced by pullback from \overline{M}_{g,n}, as well as classes induced by replacing the canonical (relative dualizing) sheaf ω_{π} with E_{r} in the usual constructions of tautological cohomology classes on \overline{M}_{g,n}.

In particular, we have the Chern classes λ_{i} of the Hodge bundle π_{∗}ω. However, π_{∗}E_{r} is not especially well-behaved. Instead, we prefer to use π_{!}E_{r} (also called Rπ_{∗}E). Recall that for any coherent sheaf F on the universal curve π : C_{g,n}^{1/r,m} ^{}\overline{M}_{g,n}^{1/r,m} the element π_{!}F of K_{0}(\overline{M}_{g,n}^{1/r,m}) is the difference π_{∗}F − R^{1}π_{∗}F (π has relative dimension 1). Here R^{1}π_{∗}F is the sheaf whose fibre over a point p of the base \overline{M}_{g,n}^{1/r,m} is the vector space H^{1}(π^{−1}(p), F|_{π^{−1}(p)}). Serre duality shows that H^{1}(X, ω) is canonically isomorphic to ℂ, and so R^{1}π_{∗}ω is a trivial line bundle. Hence π_{∗}ω = π_{!}ω + O in K_{0}(\overline{M}_{g,n}^{1/r,m}). Therefore

c_{t}π_{∗}ω_{π} = c_{t}π_{!}ω_{π} = 1 + λ_{1}t + λ_{2}t^{2} + …

Tautological classes ν_{i} defined as components of the Chern character of the Hodge bundle

ch_{t}π_{∗}ω = 1 + ch_{t}π_{!}ω = g + ν_{1}t + ν_{2}t^{3} + ν_{3}t^{5} + … (7)

(The even components of ch_{t}π_{∗}ω vanish by Mumford’s theorem [37].) Similarly, we define classes μ_{i} as components of the Chern character of π_{!}E_{r}:

ch_{t}π_{!}E_{r} = D + μ_{1}t + μ_{2}t^{2} + …. (8)

Here −D is the Euler characteristic χ(E_{r}|_{C\overline{s}}) of E_{r} on any geometric fibre C\overline{s} of π, and by Riemann-Roch

D = \frac{1}{r} ((r − 2)(g − 1) + ∑_{i} m_{i}) . (9)

Serre duality shows that for any F we have π_{!}(Hom(F, ω_{π})) = π_{!}F, so π_{!}O = π_{!}ω. More importantly, for purposes of comparison with Witten’s calculations of [40], in the special case that R^{1}π_{∗}Hom(E_{r}, ω) = 0 (or equivalently, π_{∗}E_{r} = 0), the bundle V := π_{∗}Hom(E_{r}, ω) of [40] corresponds to π_{!}E_{r}.

In addition to the Hodge-like classes, there are those induced by the canonical sections σ_{i} of π : C_{g,n}^{1/r,m} ^{}\overline{M}_{g,n}^{1/r,m}. These are classes

ψ_{i} := c_{1}σ_{i}^{∗}(ω_{π}) and ψ̃_{i} := c_{1}σ_{i}^{∗}(E_{r})

(and also class ψ̃_{i}^{(d)} for each divisor d of r). When working in Pic \overline{M}_{g,n}^{1/r,m}, we will abuse notation and use ψ_{i} to indicate the line bundle σ_{i}^{∗}(ω_{π}), and ψ̃_{i} the line bundle σ_{i}^{∗}(E_{r}).

Finally, there are the boundary classes. In particular, if A ⊔ B is a partition of {1, …, n} into two subsets, we denote by α_{k;A} the class of the divisor associated to spin curves with the dual graph of the

form , with {i_{1}, …, i_{j}} = A and {i_{j+1}, …, i_{n}} = B. Of

course,

α_{k;A} = α_{g−k;B}

.

If g is greater than 1 and r and all of the m_{i} are even, then the moduli space has two components, \overline{M}_{g,n}^{1/r,m,even} and \overline{M}_{g,n}^{1/r,m,odd} If 2 ≤ k ≤ g−2, then α_{k;A} is the sum of four divisors — two on each irreducible component of the moduli space. In this case, α_{k;A}^{(2,2)} denotes the class in

Pic \overline{M}_{g,n}^{1/r,m,even} of the divisor , where both components

are endowed with an even spin structure; and α_{k;A}^{(1,1)} corresponds to both

components carrying an odd spin structure. Similarly in Pic \overline{M}_{g,n}^{1/r,m,odd}, we use α_{k;A}^{(1,2)} to denote the class where the component of genus k carries an odd spin structure, and the component of genus (g − k) carries an even one. The class α_{k;A}^{(2,1)} corresponds to the case where the component of genus k is even and the component of genus (g − k) is odd. Thus α_{k;A}^{(1,2)} = α_{g−k;B}^{(2,1)}.

Finally, denote by δ̃_{irr,m^{+}} the divisor associated to with

the spin structure inducing on the normalization. and the

divisor corresponding to the Ramond root will be denoted by δ̃_{irr,r−1}.^{†} δ̃_{irr,m^{+}} is not necessarily irreducible, since different gluings may induce distinct (even disjoint) divisors, all in δ̃_{irr,m^{+}}. And even or odd spin structures on the normalization will also give disjoint divisors. The classes δ̃_{irr,m^{+}}^{even} and δ̃_{irr,m^{+}}^{odd} are defined in the obvious way, provided they make sense. Of course,

†Beware that the integer m of δ̃_{irr,m} here differs by one from that used with γ in [23].

δ̃_{irr,m} = δ̃_{irr,r−2−m}.

2.2. Basic properties of the tautological classes.

The following Proposition describes relations between various elements in the Picard group of \overline{M}_{g,n}^{1/r,m}. It is a straightforward generalization of the corresponding result for the case m = 0 proved in [23].

Proposition 2.1.

• The forgetful map p : \overline{M}_{g,n}^{1/r,m} ^{}\overline{M}_{g,n} induces an injection

p^{∗} : Pic \overline{M}_{g,n} ⊗ ℚ ^{}Pic \overline{M}_{g,n}^{1/r,m} ⊗ ℚ

• Let δ_{k;A} denote the pullback to Pic \overline{M}_{g,n}^{1/r,m} of the class in Pic \overline{M}_{g,n} associated to the union of all strata in \overline{M}_{g,n} with the dual graph

Γ = , with A = {i_{1}, …, i_{j}}, then

δ_{k;A} = \frac{r}{gcd(m^{+} + 1, r)}α_{k;A}

where m^{+} is determined by k, m, and A, as in Section 1.5.1.

• Let δ_{irr} be the pullback of the divisor of all curves in \overline{M}_{g,n} with

the dual graph Γ_{irr} = , then

δ_{irr} = ∑_{r−2/2≤m<r} \frac{r}{gcd(r, m + 1)}δ̃_{irr,m}.

The fact that, for an r-th root (E, b) of ω(− ∑ m_{i}p_{i}), the map b is almost an isomorphism means that ψ̃_{i} and ψ_{i} are closely related. Indeed, the map b : E^{⊗r} ^{}ω(− ∑ m_{j}p_{j}) pulls back, via σ_{i}^{∗} to give

σ_{i}^{∗}b : (σ_{i}^{∗}E)^{⊗r} ^{}σ_{i}^{∗}ω(− ∑ m_{j}p_{j}).

Since im(σ_{i}) is disjoint from the nodes of X, and σ_{i}^{∗}b is an isomorphism even on the boundary strata where E fails to be locally free, we have the following result.

Proposition 2.2. The line bundles σ_{i}^{∗}ω and σ_{i}^{∗}(E_{d}) on the stack \overline{M}_{g,n}^{1/r,m} are related by

rσ_{i}^{∗}(E_{r}) = (m_{i} + 1)σ_{i}^{∗}(ω)

and

dσ_{i}^{∗}(E_{d}) = (m′′_{i} + 1)σ_{i}^{∗}(ω),

where m′′_{i} is the smallest non-negative integer congruent to m_{i} mod d. Therefore, in Pic \overline{M}_{g,n}^{1/r,m} ⊗ ℚ we have

ψ̃_{i} = \frac{m_{i} + 1}{r}ψ_{i}

and

ψ̃_{i}^{(d)} = (\frac{m′′_{i} + 1}{m_{i} + 1}) \frac{r}{d}ψ̃_{i}.

The proof follows immediately from the fact that σ_{i}^{∗}c_{r,1} is an isomorphism and from the following well-known fact.

Lemma 2.3. Let π : C_{g,n} ^{}\overline{M}_{g,n} be the universal n-pointed curve, and σ_{i} the i-th tautological section of π. If D_{i} is the divisor of C_{g,n} associated to σ_{i}, and if ω = ω_{π} is the canonical (relative dualizing) sheaf, then

σ_{i}^{∗}(O(−D_{j})) ≅ \begin{cases}σ_{i}^{∗}(ω|_{D_{i}}) = ψ_{i} & if i = j \\ O & if i ≠ j \end{cases}

Proof. When i ≠ j the bundle σ_{i}^{∗}(O(−D_{j})) is trivial because the sections are disjoint. In the case i = j the result follows from the fact that taking residues gives an isomorphism between ω_{π}(D_{i})|_{D_{i}} and O_{D_{i}}.

2.3. Non-trivial relations involving the class μ_{1}.

Proposition 2.4. Define the boundary divisors < Ẽ_{r}, 𝔈 > and δ by

< Ẽ_{r}, 𝔈 >= ∑_{k,A} \frac{(m^{+} + 1)(m^{−} + 1)}{u_{k;A}}α_{k;A}

+ ∑_{\frac{r}{2} −1≤m<r−1} \frac{(m + 1)(r − m − 1)}{v_{m}} δ̃_{irr,m}

and

δ = ∑ δ_{k;A} = ∑_{\frac{r}{2} −1≤m<r} \frac{r}{v_{m}} δ̃_{irr,m} + ∑_{k,A} \frac{r}{u_{k;A}} α_{k;A},

where v_{m} = gcd(m+1, r), u_{k;A} = gcd(m^{+}+1, r), and m^{+} is determined by k, A, and m via relation (3).

Then the following relation holds in Pic \overline{M}_{g,n}^{1/r,m}:

r < Ẽ_{r}, 𝔈 >= (2r^{2}−12r+12)λ_{1}−2r^{2}μ_{1}+(r−1)δ+ ∑_{1≤i≤n} m_{i}(r−2−m_{i})ψ_{i}.

The proof of the proposition is almost identical to its counterpart in [23, Theorem 4.3.3] except that E_{r} is not an r-th root of ω, but rather of ω(− ∑ m_{i}p_{i}). The only extra information necessary to prove the proposition is the content of the following two lemmas.

Lemma 2.5. Let π : C_{g,n} ^{}\overline{M}_{g,n} be the universal curve and

<, >: Pic C_{g,n} × Pic C_{g,n} ^{}Pic \overline{M}_{g,n},

be Deligne’s bilinear product defined by

< L, M >:= det(π_{!}(L ⊗ M) − π_{!}L − π_{!}M + π_{!}O).

If D is the image of a section σ : \overline{M}_{g,n} ^{}C_{g,n}, then for any line bundle L on C_{g,n} the product < L, O_{C}(D) > is equal to the restriction of L to D; that is, it is just σ^{∗}L.

This is proven in [8, Prop. 6.1.3], but it also follows from the fact that when the base is a smooth curve B, then deg< L, M > is exactly the usual intersection number (L.M). Since line bundles on \overline{M}_{g,n} are completely determined by their degree on smooth curves in \overline{M}_{g,n} [4], and since in the case of a smooth, one-parameter base the degrees agree, then the lemma is true in general.

Lemma 2.6. We have

< O(D_{i}), O(D_{j}) >= \begin{cases}−σ_{i}^{∗}ω_{π} = −ψ_{i} & if i = j \\ 0 & if i ≠ j. \end{cases}

This follows immediately from Lemmas 2.3 and 2.5.

Corollary 2.7. If g = 0 then classes λ_{1} and δ̃_{irr,m} vanish (for all m), and

2r^{2}μ_{1} = (r − 1)δ + ∑_{i} m_{i}(r − 2 − m_{i})ψ_{i} − r(< Ẽ, 𝔈 >)
= ∑_{A∈[n]} \frac{r}{u_{0;A}} [r − 1 − (m^{+} + 1)(m^{−} + 1)]α_{0;A}

+ ∑_{1≤i≤n} m_{i}(r − 2 − m_{i})ψ_{i}.

3. Cohomological field theory

In this section we begin with a review of the notion of cohomological field theory (CohFT) in the sense of Kontsevich and Manin [29]. This is an object which formalizes the factorization properties of the theory of topological gravity coupled to topological matter. The Gromov-Witten invariants associated to a smooth, projective variety V correspond to the physical situation where the matter sector arises from the topological sigma model [29, 38]. The analogous intersection numbers associated to the moduli space of r-spin curves have their physical origins in a different choice of the matter sector. Our goal in this section is to give a precise formulation of these notions in terms of the moduli spaces described above.

3.1. Axioms of CohFT.

Definition 3.1. A (complete) cohomological field theory (CohFT) of rank d (denoted by (H, η, Λ) or just (H, η)) is a d-dimensional vector space H with a metric η and a collection Λ := { Λ_{g,n} } of n-linear H^{•}(\overline{M}_{g,n})-valued forms on H

Λ_{g,n} ∈ H^{•}(\overline{M}_{g,n}) ⊗ H^{∗⊗n} = Hom(H^{⊗n}, H^{•}(\overline{M}_{g,n})) (10)

defined for stable pairs (g, n) and satisfying the following axioms C1– C3 (where {e_{0}, …, e_{d−1}} is a fixed basis of H, η^{μν} is the inverse of the matrix of the metric η in this basis, and the summation convention has been used). C1. The element Λ_{g,n} is invariant under the action of the symmetric

group S_{n}. C2. Let

(11) ρ_{tree} : \overline{M}_{Γ_{1}⊔Γ_{2}} = \overline{M}_{Γ_{1}} × \overline{M}_{Γ_{2}} ^{}\overline{M}_{Γ_{tree}} ^{⊂} ^{}\overline{M}_{g,n}

be the gluing map (2) corresponding to the stable graph

Γ_{tree} =

and the two graphs Γ_{1} and Γ_{2} obtained by cutting the edge of Γ_{tree}. Then the forms Λ_{g,n} satisfy the composition property

(12) ρ_{tree}^{∗}Λ_{g,n}(γ_{1}, γ_{2} …, γ_{n}) =
Λ_{k,j}(γ_{i_{1}}, …, γ_{i_{j}}, e_{μ})η^{μν} ⊗ Λ_{g−k,n−j}(e_{ν} γ_{i_{j+1}}, …, γ_{i_{n}})

for all γ_{i} ∈ H. C3. Let

(13)  ρ_{loop} : \overline{M}Γ̃ = \overline{M}_{g−1,n+2}  ^{}\overline{M}_{Γ_{loop}} ^{⊂} ^{}\overline{M}_{g,n}

be the gluing map (2) corresponding to the stable graph

Γ_{loop} = (14)

and the graph Γ̃ obtained by cutting the loop of Γ_{loop}. Then

ρ_{loop}^{∗} Λ_{g,n}(γ_{1}, γ_{2}, …, γ_{n}) = Λ_{g−1,n+2} (γ_{1}, γ_{2}, …, γ_{n}, e_{μ}, e_{ν}) η^{μν}. (15)

The pair (H, η) is called the state space of the CohFT.

An element e_{0} ∈ H is called a flat identity of the CohFT, if in addition, the following equations hold.

C4a. For all γ_{i} in H we have

Λ_{g,n+1}(γ_{1}, …, γ_{n}, e_{0}) = π^{∗}Λ_{g,n}(γ_{1}, …, γ_{n}), (16)

where π : \overline{M}_{g,n+1} → \overline{M}_{g,n} is the universal curve on \overline{M}_{g,n} and C4b.

∫_{\overline{M}_{0,3}} Λ_{0,3}(γ_{1}, γ_{2}, e_{0}) = η(γ_{1}, γ_{2}) (17)

A CohFT with flat identity is denoted by (H, η, Λ, e_{0}).

A genus g̃ CohFT on the state space (H, η) is the collection of forms { Λ_{g,n} }_{g ≤ g̃} that satisfy only those of the equations (12), (15), (16), and (17), where g ≤ g̃.

Remarks 3.2.

1. In general, the state space H of CohFT is ℤ_{2}-graded, but here for simplicity, we are assuming that H contains only even elements, since this is the only case that will arise in this paper.

2. The definition of a CohFT given above has an equivalent dual description in terms of homology. Consider the maps H_{•}(\overline{M}_{g,n}) → T^{n}H^{∗} given by [c] ↦ ∫_{[c]} Λ_{g,n}. These maps are called the correlators of the CohFT. A structure of a (complete) CohFT on (H, η) is equivalent to the requirement that these correlators endow (H, η) with the structure of an algebra over the modular operad \{ H_{•}(\overline{M}_{g,n}) \} in the sense of Getzler and Kapranov [18].

3. Clearly, the definition of a cohomological field theory extends from ℂ to more general ground rings K.

Let Γ be a stable graph, then there is a canonical composition map ρ_{Γ}

ρ_{Γ} : ∏_{v∈V (Γ)} \overline{M}_{g(v),n(v)} → \overline{M}_{Γ} → \overline{M}_{g,n} (18)

where V (Γ) denotes the set of vertices of Γ. Since the map ρ_{Γ} can be constructed from gluing morphisms (11) and (13), the forms Λ_{g,n} satisfy a restriction property

ρ_{Γ}^{∗}Λ_{g,n} = ρ_{Γ}^{−1}( ⨂_{v∈V (Γ)} Λ_{g(v),n(v)} ) (19)

where

ρ_{Γ}^{−1} : ⨂_{v∈V (Γ)} T^{n(v)} H^{∗} → T^{n} H^{∗}

contracts the factors T^{n}H^{∗} by means of the inverse of the metric η and successive application of equations (12) and (15) .

There is a parameter which can be introduced into the definition of a CohFT. This parameter can be regarded as a coupling constant in the theory.

Lemma 3.3. Let (H, η̃, Λ̃, e_{0}) be a CohFT with flat identity e_{0} and let λ be a nonzero parameter. If Λ = { Λ_{g,n} } where

Λ_{g,n} := λ^{2g−2} Λ̃_{g,n}

and

η := λ^{−2} η̃

then (H, η, Λ, e_{0}) is a CohFT with flat identity.

The proof is obvious.

Definition 3.4. The small phase space potential function of the CohFT (H, η̃, Λ̃) is a formal series Φ ∈ ℂ[[H]] given by

Φ(x) := ∑_{g = 0}^{∞} Φ_{g}(x), (20)

where

Φ_{g}(x) := ∑_{n} \frac{1}{n!} ∫_{\overline{M}_{g,n}} ⟨ Λ̃_{g,n} , x^{⊗n} ⟩ .

Here ⟨ ⋯ ⟩ denotes evaluation, the sum over n is understood to be over the stable range, and x = ∑_{α} x^{α} e_{α}, where { e_{α} } is a basis of H.

Remark 3.5. The small phase space potential function of the CohFT (H, η, Λ, e_{0}) associated to (H, η̃, Λ̃, e_{0}) as in Lemma 3.3 may be regarded as an element in λ^{−2}ℂ[[H, λ^{2}]].

All of the information of a genus zero CohFT is encoded in this potential.

Theorem 3.6. [29, 33] An element Φ_{0} in ℂ[[H]] is the potential of a rank d, genus zero CohFT (H, η) if and only if it contains only terms which are of cubic and higher order in the coordinates x^{0}, …, x^{d−1} (corresponding to a basis { e_{0}, … e_{d−1} } of H) and it satisfies the associativity, or WDVV (Witten-Dijkgraaf-Verlinde^{2}) equation

∂_{a}∂_{b}∂_{e}Φ_{0} η^{ef} ∂_{f}∂_{c}∂_{d}Φ_{0} = ∂_{b}∂_{c}∂_{e}Φ_{0} η^{ef} ∂_{f}∂_{a}∂_{d}Φ_{0},

where η^{ab} is the inverse matrix of the matrix of η in the basis {e_{a}}, ∂_{a} is derivative with respect to x^{a}, and the summation convention has been used.

Conversely, a genus zero CohFT structure on (H, η) is uniquely determined by its potential Φ_{0}, which must satisfy the WDVV equation.

A genus zero CohFT with flat identity is essentially equivalent to endowing the state space (H, η) with the structure of a formal Frobenius manifold [10, 20, 33]. The theorem follows from the work of Keel [27] who proved that H^{•}(\overline{M}_{0,n}) is generated by boundary classes and that all relations between boundary divisors arise from lifting the basic codimension one relation on \overline{M}_{0,4}.

Higher genus CohFTs are not completely determined by their potentials because H^{•}(\overline{M}_{g,n}) is not generated by its boundary classes. Nonetheless, Getzler [16] proved that the potential function of a genus one CohFT must satisfy an analogous (but much more complicated) equation arising from a codimension-two relation in H^{•}(\overline{M}_{1,4}).

3.2. Gromov-Witten invariants and their potentials.

Our construction of CohFTs from the moduli space of stable spin curves is guided by analogy with the moduli space of stable maps and Gromov-Witten invariants. Let us briefly review this construction. Let V be a smooth projective variety, H = H^{•}(V, ℂ), and η the Poincaré pairing. Let \overline{M}_{g,n}(V) be the moduli stack of stable maps into V of genus g with n marked points. The Gromov-Witten invariants of V are multilinear maps H^{⊗n} → ℂ given by

⟨ τ_{0}(γ_{1}) ⋯ τ_{0}(γ_{n}) ⟩_{g} = λ^{2g−2} ∫_{[\overline{M}_{g,n}(V)]^{vir}} ev_{1}^{∗} γ_{1} ∪ ⋯ ∪ ev_{n}^{∗} γ_{n} (21)

where [\overline{M}_{g,n}(V)]^{vir} is the virtual fundamental class of the moduli stack \overline{M}_{g,n}(V) and λ is a formal parameter.

The corresponding small phase space potential Φ(x) is defined by (20) where the genus g part is given by

Φ_{g}(x) = ∑_{n} λ^{2g−2} ∫_{[\overline{M}_{g,n}(V)]^{vir}} ev_{1}^{∗} x ∪ ⋯ ∪ ev_{n}^{∗} x,

x = ∑_{a} x^{a} e_{a}, and { e_{0}, …, e_{r} } is a basis for H such that e_{0} is the identity element.

If V is a convex variety then \overline{M}_{0,n}(V) is a smooth stack and its virtual fundamental class coincides with its topological fundamental class. In this situation, [15] shows that (H, η) forms a genus zero CohFT with potential Φ_{0}. This result can be generalized to higher genera and to more general varieties, as well.

Remark 3.7. In the usual definition of Gromov-Witten invariants, there is no factor of λ^{2g−2} but, rather, this factor is inserted into the potential function by hand. We have chosen our conventions so that this factor appears in the correlator but is not explicitly inserted into the potential function.

The gravitational descendants are defined by twisting the Gromov-Witten classes with the tautological ψ classes as follows:

⟨τ_{a_{1}}(γ_{1}) ⋯ τ_{a_{n}}(γ_{n})⟩_{g} := λ^{2g−2} ∫_{[\overline{M}_{g,n}(V)]^{vir}}ev_{1}^{∗}γ_{1} ∪ ψ_{1}^{a_{1}} ∪ ⋯ ∪ ev_{n}^{∗}γ_{n} ∪ ψ_{n}^{a_{n}} (22)

for all a_{1}, …, a_{n} = 0, 1, 2, … and γ_{1}, …, γ_{n} in H^{•}(V). This gives rise to the large phase space potential Φ(t) ∈ λ^{−2}ℂ[[t, λ^{2}]] where t = (t_{0}, t_{1}, …) and t_{n} = (t_{n}^{0}, …, t_{n}^{r}), which is defined by

Φ(t) := ∑_{g} Φ_{g}(t) (23)

and

Φ_{g}(t) := ∑ ⟨ τ_{a_{1}}(e_{α_{1}}) ⋯ τ_{a_{n}}(e_{α_{n}}) ⟩_{g} t_{a_{1}}^{α_{1}} ⋯ t_{a_{n}}^{α_{n}} \frac{1}{n!}. (24)

Setting t_{n}^{α} = 0 for n ≥ 1 and x^{α} = t_{0}^{α} reduces the large phase space potential Φ(t) to the small phase space potential Φ(x).

When V is a point, Kontsevich’s theorem gives that

Z(t) := exp(Φ(t))

is a τ function of the KdV hierarchy. In addition Kontsevich showed that Z(t) is a highest weight vector for the Virasoro Lie algebra, a condition which completely allows one to solve for these intersection numbers. The existence of a similar Virasoro algebra action has been conjectured by Eguchi, Hori, and Xiong [12] in the case where V is not a point. Evidence for this conjecture is mounting [19, 14].

A very large phase space has recently been introduced in [25, 35] for the case where V is a point and for more general varieties in [13] by including variables corresponding to the Hodge classes ν_{i} as well. These additional variables parametrize an even larger family of CohFTs than just the large phase space coordinates [25]. We will shortly introduce, in addition, variables associated to the r-spin structure (see (48)).

3.3. r-spin CohFT.

We perform an analogous construction of a very large phase space where the role of the moduli space of stable maps \overline{M}_{g,n}(V) is played by the moduli space of stable r-spin curves \overline{M}_{g,n}^{1/r}. Unlike the moduli space of general stable maps, the moduli space of stable r-spin curves is a smooth stack. Intersection theory is therefore simpler in this case than for the case of stable maps. However, the difficulty lies instead in the construction of the analogs of the Gromov-Witten classes.

In the next section, we shall introduce axioms which a collection of cohomology classes (called a virtual class) c_{g,n}^{1/r}(m) in H^{•}(\overline{M}_{g,n}^{1/r,m}) must satisfy in order to insure that the following result holds.

Theorem 3.8. Let (H^{(r)}, η̃) be a vector space of dimension r − 1 with a basis { e_{0}, …, e_{r−2} } and metric η̃ given by

η̃(e_{μ}, e_{ν}) := η̃_{μν} := \frac{1}{r} δ_{μ+ν,r−2}. (25)

Let c_{g,n}^{1/r}(m) be a virtual class in H^{•}(\overline{M}_{g,n}^{1/r,m}) satisfying Axioms 1 through

5 from the next section, and let p : \overline{M}_{g,n}^{1/r,m} → \overline{M}_{g,n} be the map which forgets the r-spin structure. Let Λ̃ := { Λ̃_{g,n} } be defined by

(26)  Λ̃_{g,n}^{(s,u)}(e_{m_{1}}, …, e_{m_{n}}) := p_{∗} (c_{g,n}^{1/r}(m) exp(s · μ + u · ν))

where these forms have values in the ring ℂ[[s, u]], then (H, η̃, Λ̃^{(s,u)}, e_{0}) is a CohFT satisfying Axiom C4a. Furthermore, let

Λ̃_{g,n} := Λ̃_{g,n}^{(0,0)} (27)

then (H, η̃, Λ̃, e_{0}) is a CohFT with flat identity. The latter will be called the r-spin CohFT.

Restricting the r-spin CohFT to genus zero shows that (H^{(r)}, η) is endowed with the structure of a Frobenius manifold.

This theorem is proven in Section 4.2.

Corollary 3.9. Let (H, η, Λ, e_{0}) be constructed from (H, η̃, Λ̃, e_{0}) above by setting

η(e_{μ}, e_{ν}) := η_{μν} := \frac{1}{rλ^{2}} δ_{μ+ν,r−2}, (28)

Λ_{g,n}^{(s,u)}(e_{m_{1}}, …, e_{m_{n}}) := λ^{2g−2} Λ̃_{g,n}^{(s,u)}(e_{m_{1}}, …, e_{m_{n}}), (29) and

Λ_{g,n} := Λ_{g,n}^{(0,0)}, (30)

then (H, η, Λ^{(s,u)}, e_{0}) is a CohFT satisfying Axiom C4a and (H, η, Λ, e_{0}) is a CohFT with flat identity.

Proof. This is a direct consequence of Lemma 3.3 and the previous theorem.

The classes c_{g,n}^{1/r}(m) are analogs of the Gromov-Witten classes in this theory. The analogs of the gravitational descendants (22) are given by

⟨ τ_{a_{1}}(e_{m_{1}}) ⋯ τ_{a_{n}}(e_{m_{n}}) ⟩_{g} := λ^{2g−2} ∫_{\overline{M}{1/r,m \atop g,n}} ψ_{1}^{a_{1}} ⋯ ψ_{n}^{a_{n}} c_{g,n}^{1/r}(m) (31)

and the large phase space potential function is defined by the equations (23) and (24). The small phase space potential function is defined by restricting (31) to correlators with a_{i} = 0.

We will see that the case where λ = \frac{1}{\sqrt{r}} corresponds to the generalized Witten conjecture. This corresponds to the metric

η(e_{m_{1}}, e_{m_{2}}) = δ_{m_{1}+m_{2},r−2} (32)

and the forms

Λ_{g,n}(e_{m_{1}}, …, e_{m_{n}}) := r^{1−g} p_{∗} c_{g,n}^{1/r}(m). (33)

Remark 3.10. The astute reader will notice in the sequel that, strictly speaking, the state space of this r-spin CohFT (33) should be instead, (Ĥ^{(r)}, η̂) where Ĥ^{(r)} is an r-dimensional vector space with basis {e_{0}, …, e_{r−1}} and a metric given by

η̂(e_{a}, e_{b}) = \begin{cases}1 & if a + b ≡ (r − 2) mod r \\ 0 & otherwise. \end{cases}

However, it follows from the axioms for c^{1/r} in Section 4.1 that the obvious orthogonal decomposition Ĥ^{(r)} = H^{(r)} ⊕ H′, where H′ is the trivial one-dimensional CohFT with basis { e_{r−1} }, is a direct sum of CohFTs. For this reason, we can (and will) restrict ourselves to the state space (H^{(r)}, η).

4. Virtual classes

To endow the pair (H^{(r)}, η) from Theorem 3.8 with the structure of a CohFT by equation (33), we must define cohomology classes c_{g,n}^{1/r}(m). We will call this collection of classes an r-spin virtual class. It should satisfy the axioms described below.

Throughout this section, we will restrict ourselves to the case where the coupling constant λ = \frac{1}{\sqrt{r}} unless otherwise stated. This is done purely for convenience as analogous results hold for general λ as well.

4.1. Axioms for the virtual class.

Definition 4.1. An r-spin virtual class is an assignment of a cohomology class to every genus g, stable, decorated graph with n-tails marked with the n-tuple m = (m_{1}, …, m_{n}) with 0 ≤ m_{i} ≤ r − 1

c_{Γ}^{1/r} ∈ H^{2D}(\overline{M}_{Γ}^{1/r}), (34)

where

D = \frac{1}{r}((r − 2)(g − 1) + ∑_{i=1}^{n} m_{i}) (35)

satisfying the axioms below. In the special case where Γ has one vertex and no edges, we denote c_{Γ}^{1/r} by c_{g,n}^{1/r}(m). Axiom 1a (Connected Graphs) Let Γ be a connected, genus g, stable decorated graph with n tails. Let E(Γ) denote the set of edges of Γ. For each edge e of Γ, let l_{e} := gcd(m_{e}^{+} + 1, r) where m_{e}^{+} is an integer decorating a half-edge of e. The classes c_{Γ}^{1/r} and c_{g,n}^{1/r}(m) are related by

c_{Γ}^{1/r} = ( ∏_{e ∈ E(Γ)} \frac{r}{l_{e}}) ĩ^{∗} c_{g,n}^{1/r}(m) ∈ H^{2D}(\overline{M}_{Γ}^{1/r}) (36)

where ĩ : \overline{M}_{Γ}^{1/r} ^{⊂} ^{}\overline{M}_{g,n}^{1/r,m} is the canonical inclusion map. Axiom 1b (Disconnected Graphs) Let Γ be stable, decorated graph which is the disjoint union of connected graphs Γ^{(d)} then

c_{Γ}^{1/r} = ⨂_{d} c_{Γ^{(d)}}^{1/r} ∈ H^{•}(\overline{M}_{Γ}^{1/r}).

Axiom 2 (Convexity): Consider the universal r-spin structure {E_{d}, c_{d,d′}} on the universal curve π : C_{g,n}^{1/r,m} ^{}\overline{M}_{g,n}^{1/r,m}. For each irreducible

(and connected) component of \overline{M}_{g,n}^{1/r} (denoted here by \overline{M}_{g,n}^{1/r,(d)} for

some index d), if π_{∗}E_{r} = 0 on \overline{M}_{g,n}^{1/r,(d)}, then c_{g,n}^{1/r}(m) restricted to \overline{M}_{g,n}^{1/r(d)} is c_{D}(−R^{1}π_{∗}E_{r}), the top Chern class of the bundle with fibers H^{1}(E_{r})^{∗}.

Axiom 3 (Cutting edges): Given any genus g stable decorated graph with n tails marked with m, we have a diagram

[figure: \overline{M}Γ̃ ×_{\overline{M}_{Γ}} \overline{M}_{Γ}^{1/r}; μ̃; \overline{M}_{Γ}^{1/r}; ⊂; ĩ; \overline{M}_{g,n}^{1/r}; \overline{M}1/r ; Γ̃; p_{1}; \overline{M}Γ̃; p; μ; p_{2}; \overline{M}_{Γ}; p; ⊂; i; \overline{M}_{g,n}; p] (37)

where \overline{M}Γ̃ is the stack of stable curves with graph Γ̃, the graph obtained by cutting all edges of Γ, and \overline{M}1/r ; Γ̃ is the stack of stable spin curves with graph Γ̃ (still marked with m^{±} on each half edge).

p_{1} is the following morphism: The fibre product consists of triples of a spin curve (X/T, {E_{d}, c_{d,d′}}), a stable curve X̃/T, and a morphism ν : X̃ ^{}X making X̃ into the normalization of X. Also the dual graphs of X and X̃ are Γ and Γ̃, respectively. The associated spin curve in \overline{M}1/r ; Γ̃ is simply (X̃/T, ν^{∗}\{E_{d}, c_{d,d′}\}).

We require that

p_{1∗}μ̃^{∗}c_{Γ}^{1/r} = r^{|E(Γ)|}c1/r ; Γ̃

where E(Γ) is the set of edges of Γ that are cut in Γ̃. Axiom 4 (Vanishing): If Γ contains a tail marked with m_{i} = r−1,

then c_{Γ}^{1/r} = 0. Axiom 5 (Forgetting tails): Let Γ̃ be a stable graph with a tail t_{i}

marked by m_{i} = 0, and Γ the stable graph obtained by removing the tail t_{i}, and

π : \overline{M}1/r ; Γ̃ ^{}\overline{M}_{Γ}^{1/r}

the forgetful morphism, then

c1/r ; Γ̃ = π^{∗}c_{Γ}^{1/r}.

Remarks 4.2.

1. The factor of ∏_{e} \frac{r}{l_{e}} in Axiom 1 arises from the fact that the right hand square in equation (37) is not quite Cartesian. Rather, because of ramification of p over \overline{M}_{Γ}, we have that for any cohomology class c on \overline{M}_{g,n}^{1/r},

i^{∗}p_{∗} c = (∏_{e} \frac{r}{l_{e}}) p_{∗}ĩ^{∗} c. (38)

2. The astute reader will notice in Axiom 3 that, unlike the case of a tree, if the spin structure is Ramond and Γ contains a loop, the dimensions D_{Γ} and DΓ̃ of the virtual classes c_{Γ}^{1/r} and c1/r ; Γ̃ are different. Thus Axiom 3 actually requires the vanishing of both p_{1∗}μ̃^{∗}c_{Γ}^{1/r} and c1/r ; Γ̃ in this case. Of course for the Ramond case, Axiom 4 already requires the vanishing of c1/r ; Γ̃, since the cut halfedges are both marked with r − 1. Thus for any graph (tree or otherwise) in the Ramond case Axiom 3 amounts essentially to the requirement that p_{1∗}μ̃^{∗}c_{Γ}^{1/r} vanish.

3. Although the vanishing of H^{0} is often called concavity, the Serre dual Hom(E_{r}, ω), corresponding to Witten’s sheaf V in [40] is convex (H^{1} vanishes) exactly when E_{r} is concave. Moreover, π_{!}V = π_{!}E_{r}. Therefore, we use the term convex to describe the case when π_{∗}E_{r} = 0.

4. One might think that the class c_{D}π_{!}E_{r} would be a good candidate for a virtual class, since it coincides with c^{1/r} in the convex case. However, this is not the case (see Section 4.4).

5. Witten has described in [40, §1.3] an analytic construction of a class that he calls the “top Chern class”, but it is not clear that this class satisfies the above axioms. Witten’s index-like construction is reminiscent of the analytic construction of a virtual fundamental class of the moduli space of stable maps in the theory of Gromov-Witten invariants. Ideally one should be able to construct c^{1/r} by methods similar to those used in algebraic constructions of the fundamental class.

6. Although, as explained in Remark 1.8, the restriction 0 ≤ m_{i} < r does not change the moduli space \overline{M}_{g,n}^{1/r}, it does give a different choice of c^{1/r}. Indeed, replacing m_{i} by m_{i} + r changes the dimension of c^{1/r} by 1. Thus on a given moduli space \overline{M}_{g,n}^{1/r} there are potentially several (but still only finitely many, for dimension reasons) choices of c^{1/r} and the corresponding CohF T. However, without the restriction 0 ≤ m_{i} < r the corresponding metric η is not necessarily invertible, and several other unusual considerations also arise. These issues will be treated in a forthcoming paper [24]. In the remainder of this paper we will assume 0 ≤ m_{i} < r except where explicitly stated otherwise.

4.2. Verification of the CohFT axioms for (H^{(r)}, η).

In this section we give a proof of Theorem 3.8, first for the case where s = u = 0, and then in general.

4.2.1. The case s = u = 0. Let c_{g,n}^{1/r} be a cohomology class on \overline{M}_{g,n}^{1/r} satisfying the axioms of Section 4.1. We will show that the collection of forms {Λ} given by (33) satisfies the CohFT axioms C1–C4. Axiom C1 clearly holds by the definition of {Λ}.

Let ρ = ρ_{tree} be the gluing map (11). Condition (12) of Axiom C2 is equivalent

ρ^{∗} p_{∗}c_{g,n}^{1/r}(m)r^{1−g} = (39)

∑_{a,b=0}^{r−2} r^{1−k} p_{∗}c_{k,j+1}^{1/r}(m_{i_{1}}, …, m_{i_{j}}, a) ⊗ r^{1−(g−k)} p_{∗} c_{g−k,n−j+1}^{1/r}(b, m_{i_{j+1}}, …, m_{i_{n}j})η^{ab}

for all 0 ≤ m_{i} ≤ r − 1.

Consider the stable decorated graph

Γ =

and denote the graph obtained from cutting its edge by Γ̃ = Γ_{1} ⊔ Γ_{2} where

Γ_{1} =  and Γ_{2} =  .

Since the spaces \overline{M}_{Γ_{i}}^{1/r} are non-empty for 0 ≤ m^{±} ≤ r − 1 only when m^{±} are determined by the conditions (3) and (4), the sum in the righthand side of equation (39) has only one non-vanishing term. By the definition of the metric η (32), Axiom C2 reduces to the following

ρ^{∗}p_{∗}c_{g,n}^{1/r}(m)r^{1−g} = \begin{cases}(p × p)_{∗}(c_{Γ_{1}}^{1/r}r^{1−k} ⊗ c_{Γ_{2}}^{1/r}r^{1−(g−k)}), & if 0 ≤ m^{+} ≤ r − 2 \\ 0 & if m^{+} = r − 1 \end{cases} (40)

with m^{−} = r − 2 − m^{+}.

In other words, we must show that

ρ^{∗}p_{∗}c_{g,n}^{1/r}(m) = r(p_{∗}c_{Γ_{1}}^{1/r} ⊗ p_{∗}c_{Γ_{2}}^{1/r}) = r · p_{2∗}(c1/r ; Γ̃) (41)

where we have the diagram

[figure: \overline{M}Γ̃ ×_{\overline{M}_{Γ}} \overline{M}_{Γ}^{1/r}; χ̃; \overline{M}_{Γ}^{1/r}; ĩ; \overline{M}_{g,n}^{1/r}; \overline{M}1/r ; Γ̃; p_{1}; \overline{M}Γ̃; p; χ; p_{2}; \overline{M}_{Γ}; p; i; \overline{M}_{g,n}; p].

But if l = gcd(m^{+} + 1, r) then

ρ^{∗}p_{∗}c_{g,n}^{1/r}(m) = χ^{∗}i^{∗}p_{∗}c_{g,n}^{1/r}
= χ^{∗}p_{∗}ĩ^{∗}(r/l)c_{g,n}^{1/r}
(by the def. of c_{Γ}^{1/r}) = χ^{∗}p_{∗}c_{Γ}^{1/r}
= p_{∗}χ̃^{∗}c_{Γ}^{1/r}
= p_{2∗}p_{1∗}χ̃^{∗}c_{Γ}^{1/r}
(by Axiom 3) = (p_{2∗}c1/r ; Γ̃)r

This gives (39) and, therefore, Axiom C2 is verified. The statement (15) of Axiom C3 is equivalent by (10) and (32) to

ρ^{∗}p_{∗}c_{g,n}^{1/r}(m) = r ∑_{m^{+}=0}^{r−2} p_{∗} c_{g−1,n+2}^{1/r}(m_{1}, …, m_{n}, m^{+}, m^{−}) , (42)

where ρ = ρ_{loop} is the gluing map (13) and

m^{−} = \begin{cases}r − 2 − m^{+} & if 0 ≤ m^{+} ≤ r − 2 \\ r − 1 & if m^{+} = r − 1. \end{cases}

Let

Γ̃_{m^{+}} = and Γ_{m^{+}} =

be stable decorated graphs and Γ̃ and Γ = Γ_{loop} be the corresponding underlying graphs. We have the commuting diagram

[figure: ⨆_{(r−2)/2≤m^{+}<r} F_{Γ,m^{+}}; χ̃; ⨆_{(r−2)/2≤m^{+}<r} \overline{M}_{Γ,m^{+}}^{1/r}; ĩ; \overline{M}_{g,n}^{1/r}]

⨆_{(r−2)/2≤m^{+}<r} \overline{M}1/r ; Γ̃ \frac{}{p_{1}}

[figure: \overline{M}Γ̃; p; χ; p_{2}; \overline{M}_{Γ}; p; i; \overline{M}_{g,n}.; p]

where F_{Γ,m^{+}} := \overline{M}Γ̃ ×_{\overline{M}_{Γ}} \overline{M}_{Γ,m^{+}}^{1/r}. Therefore, if l_{e} = gcd(m^{+} + 1, r) then

ρ^{∗}p_{∗}c_{g,n}^{1/r} = χ^{∗}p_{∗} ∑_{m^{+}}(ĩ^{∗}c_{g,n})r/l_{e}
= χ^{∗}p_{∗} ∑_{m^{+}} c_{Γ,m^{+}}^{1/r}
= p_{∗}χ̃^{∗} ∑_{m^{+}} c_{Γ,m^{+}}^{1/r}
= p_{2∗}p_{1∗}χ̃^{∗} ∑_{m^{+}} c_{Γ,m^{+}}^{1/r}
= r · p_{2∗} ∑_{m^{+}} c_{Γ̃,m^{+}}^{1/r}.

This proves axiom C3.

To prove Axiom C4a, consider the Cartesian square

[figure: \overline{M}_{g,n+1}^{1/r,(m_{1},... ,m_{n},0)}; π̃; \overline{M}_{g,n}^{1/r,m}; \overline{M}_{g,n+1}; p; π; \overline{M}_{g,n}; p]

Then by (10) and Axiom 5 (forgetting tails) we have the required equation (16)

π^{∗}Λ_{g,n}(e_{m}) = r^{1−g} π^{∗} p_{∗}c_{g,n}^{1/r}(m) = r^{1−g} p_{∗}π̃^{∗}c_{g,n}^{1/r}(m)
= r^{1−g} c_{g,n+1}^{1/r}(m_{1}, …, m_{n}, 0) = Λ_{g,n+1}(e_{m_{1}}, …, e_{m_{n}}, e_{0}).

Finally, a direct calculation yields Axiom 4b (see Proposition 6.1).

4.2.2. The general case. The proof of Axioms C1 and C4a are the same. We need only prove axioms C2 and C3. First consider the Cartesian square

[figure: C_{Γ}; î; C; \overline{M}_{Γ}^{1/r}; π; ĩ; \overline{M}_{g,n}^{1/r}; π]

where π is the universal curve. If E_{r} is the rth root from the universal spin structure on C, since π is projective, and \overline{M}_{g,n}^{1/r} carries an ample invertible line bundle [21, 3.1.1], Lemma 4.3, below, on regular imbeddings holds, so that ch_{i}(π_{!}î^{∗}E_{r}) = ch_{i}(ĩ^{∗}π_{!}E_{r}), but î^{∗}E_{r} is the rth root from the universal spin structure C_{Γ}. Now, if F is the fibre product

F := \overline{M}Γ̃ ×_{\overline{M}_{Γ}} \overline{M}_{Γ}^{1/r}, (43)

then we have

CΓ̃ \underline{}Ĉ_{F}

[figure: C_{F}; θ; C_{Γ}]

\overline{M}1/r ; Γ̃^{}^{_{}p_{1}}  F j_{}\overline{}\overline{M}_{Γ}^{1/r}^{}

where CΓ̃ is defined to be the universal curve over \overline{M}1/r ; Γ̃ and Ĉ_{F} and C_{F} are the fibred products F ×\overline{M}1/r ; Γ̃ CΓ̃ and F ×\overline{M}{1/r \atop Γ} C_{Γ}, respectively.

The morphism θ is just normalization, and in fact, if Ĉ_{Γ} ^{θ_{Γ}}^{}C_{Γ} is the normalization of C_{Γ}, then we have the following fibred diagram (all rectangles are Cartesian). ^{‡}

[figure: CΓ̃; p̂_{1}; Ĉ_{F}; χ̂; Ĉ_{Γ}]

[figure: C_{F}; θ; C_{Γ}; θ_{Γ}]

[figure: \overline{M}1/r ; Γ̃; π; p_{1}; F; π; χ̃; \overline{M}_{Γ}^{1/r}; π]

Moreover, if EΓ̃, E_{Γ}, Ê_{Γ}, E_{F}, and Ê_{F} are the rth roots of the universal spin structures on CΓ̃, C_{Γ}, Ĉ_{Γ}, C_{F}, and Ĉ_{F}, respectively, we have Ê_{F} = p̂_{1}^{∗}EΓ̃. Also, since θ is finite, (π ∘ θ)_{!}Ê_{F} = π_{!}E_{F}. Since χ̃ is flat, we have χ̃^{∗}π_{!}E_{Γ} = π_{!}χ̃^{∗}E_{Γ} = π_{!}E_{F} = (π ∘ θ)_{!}Ê_{F}. Also, since p_{1} is flat p_{1}^{∗}π_{!}EΓ̃ = (π ∘ θ)_{!}p_{1}^{∗}EΓ̃ = (π ∘ θ)_{!}Ê_{F}. Furthermore, p_{1}^{∗} π_{!} = (π ∘ θ)_{!} p_{1}^{∗}. If ĩ : \overline{M}_{Γ} ^{}\overline{M}_{g,n}^{1/r,m} is the inclusion map then the above implies that

(44) χ̃^{∗} ĩ^{∗}ch_{i}(π_{!} E) = χ̃^{∗} ch_{i}(π_{!} E_{Γ}) = ch_{i}(π_{!} χ̂^{∗} E_{Γ}) = ch_{i}((π ∘ θ)_{!} p̂_{1}^{∗} EΓ̃) = ch_{i}(p_{1}^{∗} π_{!} EΓ̃) = p_{1}^{∗} ch_{i}(π_{!} EΓ̃).

The previous equation, the fact that ν_{i} on \overline{M}_{g,n}^{1/r,m} is the lift of ν_{i} on \overline{M}_{g,n} and the projection formula yields the desired result.

It remains to prove the lemma about regular imbeddings.

Lemma 4.3. Let i be a regular imbedding, and let

[figure: Y′; ĩ; X′; Y; π′; i; X; π]

be a fibre (Cartesian) square with π and π′ proper and flat, and E a coherent sheaf on X′, flat over X. If X and Y both carry an ample invertible sheaf, then the Chern character behaves well with respect to base change, that is i^{∗}chπ_{!}E = chπ′_{!}ĩ^{∗}E. Moreover, if the class of π_{!}E in the Grothendieck group is represented by a locally free sheaf, then the Chern classes behave well with respect to base change; that is, i^{∗}c_{j}π_{!}E = c_{j}π′_{!}ĩ^{∗}E for every j.

‡The maps χ̃ and θ were called μ̃ and ν respectively elsewhere in the paper but we have renamed them here in order to avoid confusion with the cohomology classes μ_{i} and ν_{i}.

Proof. Proposition 5.13 of [39] states that in this case if i has finite Tor dimension, and if O_{X′} and O_{Y} are Tor-independent over X, then Li^{∗}π_{!} = π′_{!}Lĩ^{∗}, where Li^{∗} and Lĩ^{∗} are the left derived functors of i^{∗} and ĩ^{∗}, respectively. But since E is flat over X, we have Lĩ^{∗}E := ∑_{j=0}^{∞}(−1)^{j}T or_{j}^{X}(E, O_{Y}) = E ⊗_{O_{X}} O_{Y} = ĩ^{∗}E_{j} and since Li^{∗} commutes with the Chern character in general and with Chern classes for locally free sheaves, the lemma will hold provided the conditions of finite Tor dimension and Tor independence hold. Tor independence means simply that T or_{j}^{O_{X}}(O_{X′}, O_{Y}) = 0 for all j > 0, but that follows from the flatness of π. Finite Tor dimension means that there is an integer N such that for every coherent O_{X}-module F, the O_{Y} -modules T or_{j}^{O_{X}}(O_{Y} , F) vanish for j > N. This follows from the fact that i is a regular imbedding. (e.g. one may assume X = Spec A and Y = Spec A/(x) for some regular element x ∈ A. This gives the free resolution 0 ^{}A ^{(x)}^{}A ^{}A/(x) ^{}0 of O_{Y}, and shows i has finite Tor dimension).

This finishes the proof of Theorem 3.8.

4.3. The genus-zero case.

As we explained in Remark 4.2.3, we call the spin structure {E_{d}, c_{d,d′}} on the universal curve C_{Γ}^{1/r} convex if π_{∗}E_{r} is identically zero. This occurs, for example, when g = 0, as is shown in the following proposition.

Proposition 4.4. Let X be a nodal curve of genus zero with n punctures and markings (m_{1}, …, m_{n}), such that −1 ≤ m_{i} ≤ r − 1 for all i and m_{i} ≥ 0 for all i except at most one. Then, if (E, b) is an r-th root of ω_{X}(− ∑ m_{i}p_{i}), we have H^{0}(X, E) = 0.

Proof. The degree of E is an integer and is equal to −(2 + ∑ m_{i})/r. Thus ∑ m_{i} ≥ r − 2, and the degree of E is strictly negative. Therefore, when X is irreducible, E has no global sections. When X is not irreducible, but E is locally free (Ramond) at each node, the same argument holds. If E is Neveu-Schwarz at some nodes, then normalization ν : X̃ ^{}X at the nodes of X where E is not locally free gives E = ν_{∗}F, where F is locally free on X̃. Restricting F to X̃ we obtain an r-th root of ωX̃(− ∑ m̃_{i}p̃_{i}), where the points p̃_{i} are either marked points or inverse images of nodes and thus the collection m̃_{i} still meets the hypotheses of the proposition, but now F is locally free on each component, and hence has no global sections. Since ν is finite, H^{0}(X, E) = H^{0}(X̃, F) = 0.

The previous proposition shows that if a class c^{1/r} on \overline{M}_{0,n}^{1/r} satisfying Axioms 1–5 exists, then by Axiom 2 it must be given by the top Chern class of the index bundle. In this case it does indeed satisfy the required properties.

Theorem 4.5. Define cohomology classes on \overline{M}_{0,n}^{1/r} by

c_{0,n}^{1/r}(m) = c_{D}π_{!}E_{r} = (−1)^{D}c_{D}(R^{1}π_{∗}E_{r}), (45)

where

D = \frac{1}{r}(2 − r + ∑_{i=1}^{n} m_{i})

Then the collection of classes c_{Γ}^{1/r} defined by (36) for decorated stable graphs of genus zero satisfies Axioms 1–5.

Proof. It is clear from the construction of the classes c_{Γ}^{1/r} that they satisfy Axiom 2 (convexity).

Axiom 1 follows from the fact that E_{r,Γ} = E_{r,Γ_{1}}⊕E_{r,Γ_{2}} and c_{D(Γ)} = c_{top}, the top-dimensional Chern class of π_{!}E_{r,Γ}, is simply the product of the top-dimensional classes of π_{!}E_{r,Γ_{1}} and π_{!}E_{r,Γ_{2}}.

Now we will show that Axiom 3 holds. If E_{r} is the rth root from the universal spin structure on C_{g,n}^{1/r} → \overline{M}_{g,n}^{1/r} then since g = 0, E_{r} is convex and repeating the argument in the proof of Axiom 3 in Section 4.2.2 with the Chern character replaced by the top Chern class yields

c_{Γ}^{1/r} = (r^{|E|}/ ∏_{e ∈ E} l_{e}) c_{D}(−R^{1}π_{∗}E_{r})

where E denotes the edge set of Γ.

Also, we can compute the degree of p_{1} : F ^{}\overline{M}1/r ; Γ̃ from the diagram

F  \overline{χ̃ ^{}}\overline{M}_{Γ}^{1/r}

\overline{M}1/r ; Γ̃ \frac{}{p_{1}}

[figure: \overline{M}Γ̃; p̃; p_{2}; \overline{M}_{Γ}; p_{Γ}]

The morphism \overline{M}Γ̃ ^{}\overline{M}_{Γ} has degree 1, since Γ is a tree, and therefore χ̃ also has degree 1. The morphism p_{Γ} has degree ∏_{e} l_{e}r^{−|V (Γ)|} as can be seen from the fact that the coarse moduli space map induced by p_{Γ} has degree 1, and there are r^{|V (Γ)|}/∏_{e} l_{e} automorphisms of a generic spin structure. Thus the map p̃ also has degree ∏_{e} l_{e}r^{−|V (Γ)|}, the map p_{2} has degree r^{−|V (Γ)|}, so p_{1} has degree exactly ∏_{e} l_{e}.

Now we can compute p_{1∗}χ̃^{∗}c_{Γ}^{1/r} using equation 44 (replacing ch_{i} with c_{D}) to obtain

p_{1∗}(r^{|E|}/ ∏_{e} l_{e})χ̃^{∗}c_{D}(−R^{1}π_{∗}E_{Γ}) = (r^{|E|}/∏_{e} l_{e})p_{1∗}c_{D}(p_{1}^{∗}(−R^{1}π_{∗}EΓ̃))
= (r^{|E|}/∏_{e} l_{e})p_{1∗}p_{1}^{∗}c_{D}(−R^{1}π_{∗}EΓ̃)
= r^{|E|}c_{D}(−R^{1}π_{∗}EΓ̃)
= r^{|E|}c1/r ; Γ̃

as desired.

All that remains to check is Axiom 4 (vanishing). Let p be a point corresponding to a tail marked by m = r − 1. Tensoring E_{r} with the exact sequence

0 ^{}O ^{}O(p) ^{}O(p)|_{p} ^{}0

gives the exact sequence

0 ^{}E_{r}  ^{}E′  ^{}E′|_{p}  ^{}0,

where E′ = E_{r} ⊗O(p). Thus E′ corresponds to a root with p_{i} marked by m′ = −1. Since R^{1}π_{∗}(E′|_{p}) = 0, and since the residue isomorphism (1)

R_{p} : π_{∗}(E′|_{p}) ^{∼}^{}O shows that π_{∗}(E′|_{p}) is a trivial bundle, we have π_{!}(E′) = O + π_{!}(E_{r}). By Proposition 4.4 the sheaves E_{r} and E′ are both convex; thus π_{!}E_{r} = −R^{1}π_{∗}E_{r} and π_{!}E′ = −R^{1}π_{∗}E′ are both locally free and have the same Chern classes in all dimensions. But the vector bundle π_{!}E′ has dimension D′ = D − 1, and so c^{1/r} = c_{D}(π_{!}E_{r}) = 0. This gives Axiom 4.

4.4. The case \overline{M}_{1,1}^{1/r,0}. In this section, we will calculate the virtual

class on \overline{M}_{1,1}^{1/r,0} from the axioms. Let Γ = Γ_{loop} be the graph with one tail, one node of genus zero, and one loop. Let Γ_{m^{+}} be Γ with one half edge of the loop marked with m^{+}, the other half-edge marked with m^{−}. The stack \overline{M}_{1,1}^{1/r,0} is a disjoint union ∐_{d|r} \overline{M}_{1,1}^{1/r,(d)}, where the component indexed by d has a generic geometric point corresponding to a smooth spin curve (X, E_{r}, c_{r,1}) with E_{r}^{⊗d} isomorphic to O_{X}. That is, E_{r} is a d-torsion point in the Jacobian of X.

Since E_{r} has global sections if and only if d = 1, the case of d > 1 is convex. D = 0, and so c_{1,1}^{1/r,(d)} = 1 for d > 1. Moreover, consider the graph Γ_{r−1}. By Axiom 3, p_{1∗}μ̃^{∗} = cΓ̃_{r−1} where Γ̃_{r−1} is Γ_{r−1} with its edge cut. Since one of the halves of the cut edge will be labeled by r − 1, by Axiom 4, it must vanish. Therefore, the Ramond case of Γ_{r−1} with trivial glue (i.e. E_{r} = O_{X}) yields

c_{Γ_{r−1}}^{1/r,(1)} = i^{∗}c_{1,1}^{1/r,(1)} = −(r − 1), (46)

since the remaining Ramonds all have i^{∗}c_{1,1}^{1/r,(d)} = +1, and there are r − 1 of them. Notice that this differs from the top Chern class of the index bundle c_{D}(π_{!} E_{r}) = c_{0}(π_{!}E_{r}) = 1.

Since D = 0, this means c_{1,1}^{1/r,(1)} is also −(r−1). Now p^{(1)} : \overline{M}_{1,1}^{1/r,(1)} →

\overline{M}_{1,1} has degree 1/r, and p^{(d)} : \overline{M}_{1,1}^{1/r,(d)} → \overline{M}_{1,1} has degree [d^{2} ∏_{p|d}(1− 1/p^{2})]/r. The latter is 1/r times the number of points of exact order d inside of the Jacobian of the underlying curve. Thus

p_{∗}c_{1,1}^{1/r} = 1/r ∑_{d>1} d^{2} ∏_{p|d}(1 − 1/p^{2}) − (r − 1)/r

= \frac{r^{2} − 1}{r} − \frac{r − 1}{r} = r − 1.

Therefore,

⟨ τ_{1,0} ⟩_{1} = ∫_{\overline{M}{1/r,0 \atop 1,1}} ψ_{1}c_{1,1}^{1/r,0} = ∫_{\overline{M}_{1,1}} ψ_{1}p_{∗}c_{1,1}^{1/r,0} = (r − 1) ∫_{\overline{M}_{1,1}} ψ_{1}

= \frac{r − 1}{24}.

This is consistent with the prediction from CohFT stated in equation (57).

4.5. The case r = 2.

In this Section we will show that in the case of theta-characteristics (i.e. when r = 2) there exists a unique virtual class c^{1/2} satisfying the axioms of Section 4.1.

Theorem 4.6. If r = 2, then the collection of classes

c_{g,n}^{1/2}(m) ∈ H^{•}(\overline{M}_{g,n}^{1/2,m}),

satisfies Axioms 1–5 of Section 4.1 if and only if c_{g,n}^{1/2}(m) = 0 for m ≠ 0, and for m = 0 the class c_{g,n}^{1/2}(0) belongs to H^{0}(\overline{M}_{g,n}^{1/2,0}) and is given by

(47) c_{g,n}^{1/2}(0) = \begin{cases}1 & on \overline{M}_{g,n}^{1/2,0,even} \\ −1 & on \overline{M}_{g,n}^{1/2,0,odd} \end{cases}

Proof. Let us show first that the conditions of the theorem are necessary. If m ≠ 0, then by Axiom 4, c_{g,n}^{1/2}(m) must vanish, therefore we can assume that m = 0. In this case the dimension (35) of the class c_{g,n}^{1/2} is equal to zero, and since \overline{M}_{g,n}^{1/2,0} has two connected components \overline{M}_{g,n}^{even} and \overline{M}_{g,n}^{odd}, it will be sufficient to find c_{Γ}^{1/2} for two single graphs

Γ_{0} and Γ_{1}, such that the intersections \overline{M}_{Γ_{0}}^{1/2} ∩ \overline{M}_{g,n}^{even} and \overline{M}_{Γ_{1}}^{1/2} ∩ \overline{M}_{g,n}^{odd} are non-empty.

Let Γ_{0} be the graph with one genus-zero vertex, n tails, and g Neveu-Schwarz (i.e. all half-edges are decorated with zeroes) loops. In this case, \overline{M}1/2 ; Γ̃_{0} = \overline{M}Γ̃_{0} ×_{\overline{M}_{Γ_{0}}} \overline{M}_{Γ_{0}}^{1/2} in equation (37), so by Axiom 3

(cutting edges) the class c_{Γ_{0}}^{1/2} pulls back to c1/2 ; Γ̃_{0}, where Γ̃_{0} is the graph with one vertex of genus zero and n+2g tails. Since g = 0 the universal square root E of ω on the universal curve over \overline{M}_{0,n+2g}^{1/2,0} is convex by Proposition 4.4. Therefore, if E is the edge set of Γ_{0}

c_{g,n}^{1/2,0,even} = \frac{1}{2^{|E|}} c_{Γ_{0}}^{1/2,0,even} = c1/2 ; Γ̃_{0} = c_{0}(−R^{1}π_{∗}E) = 1

where the first equality follows from Axiom 1, the second from Axiom 3, and the third from Axiom 2.

To find c_{odd}^{1/2}, consider the graph Γ_{1} with a single vertex of genus one, n tails, and g − 1 Neveu-Schwarz loops. Axiom 3 again shows that c_{Γ_{1}}^{1/2} pulls back to c1/2 ; Γ̃_{1}, where Γ̃_{1} has a single vertex of genus one and 2g − 2 + n tails. Since m = 0, Axiom 5 (forgetting tails) shows that c_{odd}^{1/2} is a pullback from \overline{M}_{1,1}^{1/2,0,odd} = \overline{M}_{1,1}^{1/2,(1)}, and c_{1,1}^{1/2,(1)} = −1 by equation (46).

Now let us show that the classes c_{g,n}^{1/2}(m) defined above for r = 2 indeed satisfy Axioms 1–5.

Axiom 2 (convexity) holds when m = 0, since in this case the class has dimension 0, and if E is convex (and therefore, even) R^{1}π_{∗}E = 0 and c^{1/2} = c_{top} = 1 as required.

If m ≠ 0 then E is not convex on the universal curve over \overline{M}_{g,n}^{1/2,m} for any g > 0. In particular, consider the degenerate curve of genus g which has all n marked points on a common component E of genus 0, and which has only one other irreducible component C of genus g. For degree reasons the node must be Neveu-Schwarz, and so E corresponds to E_{E} ⊕E_{C}, for E_{E} a square root of type m of ω_{E} = O_{E}(−2−∑ m_{i}), and E_{C} is a theta-characteristic on C and hence in general does have global sections. Thus E has non-zero global sections, hence the universal square root is not convex.

When g = 0 and m ≠ 0, E is convex by Proposition 4.4. By Theorem 4.5 (and Axiom 4) the class c_{D}(−R^{1}π_{∗}E) vanishes, and so agrees with our definition c_{g,n}^{1/2}(m).

Axiom 1 holds because of the simple observation that the parity of a root E over a curve with the graph Γ_{1} ⊔Γ_{2} is equal to the sum modulo 2 of the parities of the restrictions of E to the components corresponding to Γ_{1} and Γ_{2}

To prove Axiom 3, we may assume that m = 0, and it is sufficient to check the case that Γ has only one edge. We have by definition c_{Γ}^{1/2,even} = 2 and c_{Γ}^{1/2,odd} = −2 for a tree. Let F be defined as in equation (43). The canonical morphism p_{1} : F ^{}\overline{M}1/2 ; Γ̃ is actually an isomorphism if Γ is a tree, and so we get p_{1∗}μ̃^{∗}c_{Γ}^{1/2} = 2 · c1/2 ; Γ̃ since the parity of E_{2} does not change when restricting to the normalization.

In the case of a loop, there are two cases: First when m^{+} = 1, F is isomorphic to two copies of \overline{M}1/2 ; Γ̃ (because of the two choices of gluing data—see Section 1.3.1). Consequently, p_{1} has degree 2 if m^{+} = 1.

In the second case m^{+} = 0, and F is isomorphic to \overline{M}1/2 ; Γ̃, so p_{1} has degree 1. Also, c_{Γ}^{1/2} is 2ĩ^{∗}c_{g,n}^{1/2} if m^{+} = 0, and ĩ^{∗}c_{g,n}^{1/2} if m^{+} = 1.

Thus, when m^{+} = 0, p_{1∗}μ̃^{∗}c_{Γ}^{1/2} = 2c1/2 ; Γ̃ as desired. When m^{+} = 1, the two gluings give different parities. Since parity is deformation invariant, this can be seen by degenerating to the special case of the curve X, whose partial normalization X̃ at one node q consists of two irreducible components, joined at a single node q. One component C is of genus g −1, and contains the marked points p_{1}, …, p_{n}. The other component Ẽ is of genus zero and contains marked points q^{+} and q^{−}. Degree reasons force the node to be Neveu-Schwarz, and so Ẽ is simply a direct sum E_{C} ⊕ EẼ. Moreover, since ẼẼ is a square root of ωẼ = OẼ(−2) of type (0, −1, −1), EẼ must be trivial (EẼ ≅ OẼ). Gluing EẼ via +1 and −1 yields the trivial bundle O_{E} = E_{E}^{+} and another non-trivial bundle E_{E}^{−} of degree zero, respectively. Consequently, h^{0}(E^{+}) = h^{0}(E_{C} ⊕ O) = 1+h^{0}(E_{C} ⊕E_{E}^{−}). Since the parities of E^{+} and E^{−} are simply the parities of h^{0}(E^{+}) and h^{0}(E^{−}), respectively, E^{+} and E^{−} have different parities.

The different parities of the two gluings give p_{1∗}μ̃^{∗}c_{Γ}^{1/2} = 0 = c1/2 ; Γ̃, since Γ̃ has a tail marked with +1 = m^{+}. This completes the proof of Axiom 3.

Axiom 5 is true because the projection \overline{M}_{g,n+1}^{1/2,(m_{1},... ,m_{n},0)} ^{}\overline{M}_{g,n}^{1/2,m} respects the parity of components. Axiom 4 is satisfied by the definition of the classes c^{1/2}.

5. Intersection numbers and recursion relations

In this section, we use relations between boundary classes and tautological classes in order to derive recursion relations between intersection numbers on the moduli space of stable r-spin curves.

Throughout this section, we assume the existence of the virtual class c^{1/r} satisfying Axioms 1–5 of Section 4.1

This class was shown to exist in genus zero for arbitrary r, and in arbitrary genus for r = 2.

Let (H^{(r)}, η, Λ, e_{0}) be the r-spin CohFT with the standard basis { e_{0} …, e_{r−2} } described in Corollary 3.9.

We will find it convenient to introduce a very large phase potential in this section. For all v_{1}, …, v_{n} in H^{(r)}, define

⟨⟨ τ_{a_{1}}(v_{1}) ⋯ τ_{a_{n}}(v_{n}) ⟩⟩_{g} := ⟨ τ_{a_{1}}(v_{1}) ⋯ τ_{a_{n}}(v_{n}) exp(t · τ + s · μ + u · ν) ⟩_{g} where

t · τ = ∑_{\begin{matrix}0 ≤ m ≤ r−2 \\ a ≥ 0 \end{matrix}} τ_{a,m} t_{a}^{m}

and τ_{a,m} = τ_{a}(e_{m}). Here

u · ν = ∑_{i ≥ 1} u_{i} ν_{i} s · μ = ∑_{i ≥ 1} s_{i}μ_{i},

where the classes ν_{i} and μ_{i} are defined by (7) and (8) as components of the Chern characters of the Hodge bundle and its spin analog. These expressions should be understood as formal power series in variables t_{a}^{m}, u_{i}, v_{i}. The correlators are defined by

⟨ τ_{a_{1}}(e_{m_{1}}) ⋯ τ_{a_{n}}(e_{m_{n}}) ⟩_{g} :=
λ^{2g−2} ∫_{\overline{M}{1/r,m \atop g,n}} ψ_{1}^{a_{1}} ⋯ ψ_{n}^{a_{n}} c_{g,n}^{1/r}(m) exp(s · μ + u · ν)

where m = (m_{1}, …, m_{n}). In particular, the very large phase space potential is

(48) Φ(t, s, u) = ∑_{g = 0}^{∞} Φ_{g}(t, s, u),

where

Φ_{g}(t, s, u) = ⟨⟨ ⟩⟩_{g}.

The other two potentials are restrictions of Φ(t, s, u). The large phase space potential Φ(t) = Φ(t, 0, 0) corresponds to setting all the s and u variables to zero. The small phase space potential is

Φ(x) := Φ(x^{0}, …, x^{r−2})

where x^{i} := t_{0}^{i} and all other variables are set to zero. The function Φ(x) is the potential of the cohomological field theory p_{∗}c^{1/r}. It will also be useful to define

(49) Z(t, s, u) := exp(Φ(t, s, u)).

5.1. The Euler vector field.

We begin with a differential equation arising from the grading. The dimensions of the moduli spaces and cohomology classes induce a grading on the potential function.

Definition 5.1. The Euler vector field E is the differential operator

E = ∑_{\begin{matrix}a ≥ 0 \\ 0 ≤ m ≤ r−2 \end{matrix}} (a − 1 + \frac{m}{r})t_{a}^{m} \frac{∂}{∂t_{a}^{m}}

+ ∑_{a ≥ 1} (as_{a} \frac{∂}{∂s_{a}} + (2a − 1)u_{a} \frac{∂}{∂u_{a}})

Proposition 5.2. The very large phase space potential Φ(t, u, s) satisfies the grading equation

E Φ = (1 + \frac{1}{r})λ \frac{∂}{∂λ} Φ (50)

Proof. This follows from the definition of the potential, the dimensions of the cohomology classes, and the dimensions of the moduli spaces \overline{M}_{g,n}^{1/r,m}. It encodes the fact that intersection numbers between cohomology classes vanish if the classes do not have proper the dimension

This equation encodes the fact that the potential function is invariant under the rescaling

t_{a}^{m} ↦ ϵ^{a−1+ \frac{m}{r}} t_{a}^{m}, u_{a} ↦ ϵ^{2a−1} u_{a}, s_{a} ↦ ϵ^{a} s_{a}, and λ ↦ ϵ^{−1− \frac{1}{r}} λ.

This grading shows that our small phase space potential function cannot arise as the small phase space potential associated to the Gromov-Witten invariants of a smooth, projective variety. This is because the elements in H^{(r)} have fractional dimension with respect to this Euler vector field, whereas cohomology classes of a space always have integral dimensions.

5.2. The string equation and its cousins.

We begin by proving the analog of the string equation, the dilaton equation, and a new equation arising from our identity on the μ_{1} class.

Theorem 5.3. The following identities are satisfied:

⟨ τ_{0,0} τ_{0,m_{1}} τ_{0,m_{2}} exp(s · μ + u · ν) ⟩_{0} = η_{m_{1},m_{2}} (51)

and

⟨ τ_{0,0}τ_{a_{1},m_{1}} ⋯ τ_{a_{n},m_{n}} exp(s · μ + u · ν) ⟩_{g} =

∑_{i = 1}^{n} ⟨ τ_{a_{1},m_{1}} ⋯ τ_{a_{i}−1,m_{i}} ⋯ τ_{a_{n},m_{n}} exp(s · μ + u · ν) ⟩_{g},

where we assume that the terms in the sum containing τ_{a,m} with a < 0 vanish.

HIGHER SPIN CURVES AND INTEGRABLE HIERARCHIES 43

These two equations are equivalent to the string (or puncture) equation L_{−1} Z = 0 (52) where

L_{−1} := − \frac{∂}{∂t_{0}^{0}} + ∑_{0 ≤ m_{1},m_{2} ≤ r−2} \frac{1}{2} η_{m_{1},m_{2}} t_{0}^{m_{1}}t_{0}^{m_{2}}
+ ∑_{\begin{matrix}0 ≤ m ≤ r−2 \\ a ≥ 0 \end{matrix}} t_{a+1}^{m} \frac{∂}{∂t_{a}^{m}}

Similarly, the following identities are satisfied:

⟨ τ_{1,0} exp(s · μ + u · ν) ⟩_{1} = \frac{r − 1}{24} (53)

and

⟨ τ_{1,0} τ_{a_{1},m_{1}} ⋯ τ_{a_{n},m_{n}} exp(s · μ + u · ν) ⟩_{g} =
(2g − 2 + n)⟨ τ_{a_{1},m_{1}} ⋯ τ_{a_{n},m_{n}} exp(s · μ + u · ν) ⟩_{g}

These two equations are equivalent to the dilaton equation D Z = 0 (54) where

D = −\frac{∂}{∂t_{1}^{0}} + ∑_{\begin{matrix}0 ≤ m ≤ r−2 \\ a ≥ 0 \end{matrix}} t_{a}^{m} \frac{∂}{∂t_{a}^{m}} + λ \frac{∂}{∂λ} + \frac{r − 1}{24} (55)

Finally, let L_{0} be the differential operator

L_{0} := − (1 + \frac{1}{r}) \frac{∂}{∂t_{1}^{0}} + ∑_{\begin{matrix}0 ≤ m ≤ r−2 \\ a ≥ 0 \end{matrix}}(a + \frac{m + 1}{r}) t_{a}^{m} \frac{∂}{∂t_{a}^{m}} + \frac{r^{2} − 1}{24r}
+ ∑_{a ≥ 1} ((2a − 1) u_{a} \frac{∂}{∂u_{a}} + a \frac{∂}{∂s_{a}})

then the following equation holds: L_{0} Z = 0. (56)

Furthermore, [L_{0}, L_{−1}] = L_{−1}. When restricted to the large phase space (u = s = 0), the operators L_{0} and L_{−1} become the usual generators L_{0} and L_{1} in the Virasoro Lie algebra.

Proof. The arguments are lifts of analogous arguments on the moduli space of stable curves. Recall that on the moduli space of stable curves, \overline{M}_{g,n}, the ψ_{i} classes (where i = 1, …, n) satisfy the ψ_{i} = π^{∗} ψ_{i} + D_{i,n+1} where D_{i,n+1} is the image of the i-th canonical section and π : \overline{M}_{g,n+1} → \overline{M}_{g,n}. Let p be the forgetful morphism \overline{M}_{g,n}^{1/r,m} → \overline{M}_{g,n},

then since the class ψ_{i} on \overline{M}_{g,n}^{1/r,m} is the pullback via p of the ψ_{i} class

on \overline{M}_{g,n}, one can lift the same formula to \overline{M}_{g,n}^{1/r,m} to obtain ψ_{i} =

π̃^{∗} ψ_{i} + D_{i,n+1} where this equation is now regarded as being on \overline{M}_{g,n}^{1/r,m}, D_{i,n+1} is the pullback via p of the divisors with the same name on \overline{M}_{g,n}, and π̃ is the forgetful morphism \overline{M}_{g,n+1}^{1/r,m⊔0} → \overline{M}_{g,n}^{1/r,m}.

Suppose that (g, n + 1) ≠ (0, 3), (1, 1). Using the lifting formula and canceling trivial terms, we obtain

ψ_{j}^{a} = π̃^{∗} ψ_{j}^{a} + D_{j,n+1} π̃^{∗}(ψ_{j}^{a−1}).

Since π̃^{∗}c^{1/r} = c^{1/r}, we have

⟨ τ_{0,0} τ_{a_{1},m_{1}} ⋯ τ_{a_{n},m_{n}} ⟩_{g} = λ^{2g−2} ∫_{\overline{M}{1/r,m \atop g,n+1}} ψ_{1}^{a_{1}} … ψ_{n}^{a_{n}} c^{1/r}
= ∑_{1 ≤ i ≤ n} λ^{2g−2} ∫_{\overline{M}{1/r,m \atop g,n+1}} D_{i,n+1} ψ_{1}^{a_{1}} ⋯ ψ_{i}^{a_{i}−1} ⋯ ψ_{n}^{a_{n}} c^{1/r}

where the right hand side is understood to vanish if an exponent is negative. Integration over the fiber of \overline{M}_{g,n+1}^{1/r,m⊔0} → \overline{M}_{g,n}^{1/r,m} yields the desired result. Inclusion of the additional μ and ν classes into the correlators does not change the argument since π̃^{∗} μ_{i} = μ_{i} and similarly for ν_{i}.

Finally, the exceptional cases follow from dimensional considerations and the fact that on \overline{M}_{0,3}^{1/r,m}, c^{1/r} is the identity element in cohomology provided that m_{1} + m_{2} + m_{3} = r − 2,.

The dilaton equation is proven by a similar analysis, where the exceptional case can be computed by using the explicit presentation for ψ_{1} on \overline{M}_{1,1}^{1/r,0} to obtain

(57)  ⟨ τ_{1,0} ⟩_{1} = \frac{1}{24} η^{m_{+},m_{−}} ⟨ τ_{0,0} τ_{0,m_{+}} τ_{0,m_{−}} ⟩_{0} = \frac{r − 1}{24}.

Finally, the equation L_{−1} Z = 0 is obtained by combining the dilaton equation and the grading equation (50).

The new relation for the μ_{1} class yields new equations between correlators.

Theorem 5.4. Let ξ : H^{(r)} → H^{(r)} be defined by

ξ_{m_{+}}^{m_{−}} := \frac{m_{+}m_{−}}{r^{2}} δ_{m_{+} + m_{−},r−2}

relative to the standard basis and ξ^{m_{+},m_{−}} := η^{m_{+},m} ξ_{m}^{m_{−}}. The following differential equation holds:

\frac{∂Φ}{∂s_{1}} = \frac{r^{2} − 6r + 6}{r} \frac{∂Φ}{∂u_{1}} + ∑_{\begin{matrix}a ≥ 0 \\ 0 ≤ m_{+},m_{−} ≤ r−2 \end{matrix}} \frac{1}{2}t_{a}^{m_{+}} ξ_{m_{+}}^{m_{−}} \frac{∂Φ}{∂t_{a+1}^{m−}}

− ∑_{m_{+},m_{−}} \frac{1}{2} \frac{∂Φ}{∂t_{0}^{m_{+}}} ξ^{m_{+},m_{−}} \frac{∂Φ}{∂t_{0}^{m_{−}}} − ∑_{m_{+},m_{−}} \frac{1}{4} \frac{∂^{2}Φ}{∂t_{0}^{m_{+}}∂t_{0}^{m_{−}}} ξ^{m_{+},m_{−}}

where the summation over m_{+} and m_{−} runs over 0, …, r − 2. This is equivalent to the following relations between the correlators:

⟨⟨ τ_{a_{1},m_{1}} ⋯ τ_{a_{n},m_{n}} μ_{1} ⟩⟩_{g} = \frac{r^{2} − 6r + 6}{r^{2}} ⟨⟨ τ_{a_{1},m_{1}} ⋯ τ_{a_{n},m_{n}} ν_{1} ⟩⟩_{g}

+ ∑_{\begin{matrix}1 ≤ i ≤ n \\ m′_{i} \end{matrix}} \frac{1}{2} ξ_{m_{i}}^{m′_{i}} ⟨⟨ τ_{a_{1},m_{1}} ⋯ τ_{a_{i}+1,m′_{i}} ⋯ τ_{a_{n},m_{n}} ⟩⟩_{g}

− ∑_{\begin{matrix}I_{+} ⊔ I_{−} = [n] \\ g_{+}+g_{−}=g \\ m_{+},m_{−} \end{matrix}} \frac{1}{2} ⟨⟨ ( ∏_{i ∈ I_{+}} τ_{a_{i},m_{i}})τ_{0,m_{+}} ⟩⟩_{g_{+}} ξ^{m_{+},m_{−}} ⟨⟨ τ_{0,m_{−}} ( ∏_{j ∈ I_{−}} τ_{a_{j},m_{j}}) ⟩⟩_{g_{−}}

− ∑_{m_{+},m_{−}} \frac{1}{4} ⟨⟨ τ_{a_{1},m_{1}} ⋯ τ_{a_{n},m_{n}} τ_{0,m_{+}}τ_{0,m_{−}} ⟩⟩_{g−1} ξ^{m_{+},m_{−}},

where we use the notation

[n] = {1, 2, …, n}.

The class ν_{1} which appears above is precisely λ_{1}. This class vanishes on \overline{M}_{g,n}^{1/r,m} for g = 0.

Proof. The proof follows from the facts that p_{∗}(c^{1/r} exp(sμ)) forms a CohFT, ν and ψ are lifts of the analogous classes on the moduli space of stable curves, and Proposition 2.4.

5.3. Topological recursion relations.

Topological recursion relations are relations between correlators which arise from presentations of tautological classes in terms of boundary classes in low genera.

Theorem 5.5. The following topological recursion relations hold in genus zero:

⟨⟨ τ_{a_{1}+1,m_{1}} ⋯ τ_{a_{n},m_{n}} ⟩⟩_{0} =

∑_{\begin{matrix}I_{+} ⊔ I_{−} = [n−2] \\ m_{+},m_{−} \end{matrix}} ⟨⟨ τ_{a_{1},m_{1}}( ∏_{i ∈ I_{+}} τ_{a_{i},m_{i}})τ_{0,m_{+}} ⟩⟩_{0} η^{m_{+},m_{−}} ⟨⟨ τ_{0,m_{−}}τ_{a_{2},m_{2}}τ_{a_{3},m_{3}} ∏_{j ∈ I_{−}} τ_{a_{j},m_{j}} ⟩⟩_{0}

which is equivalent to the differential equation

\frac{∂^{3}Φ_{0}}{∂t_{a_{1}+1}^{m_{1}}∂t_{a_{2}}^{m_{2}}t_{a_{3}}^{m_{3}}} = ∑_{m_{+},m_{−}} \frac{∂^{2}Φ_{0}}{∂t_{a_{1}}^{m_{1}}∂t_{0}^{m_{+}}} η^{m_{+},m_{−}} \frac{∂^{3}Φ_{0}}{∂t_{0}^{m_{−}} ∂t_{a_{2}}^{m_{2}} ∂t_{a_{3}}^{m_{3}}}

Proof. On \overline{M}_{0,n}^{1/r,m}, the class ψ_{1} can be written in terms of boundary classes as

ψ_{1} = ∑_{\begin{matrix}I_{+} ⊔ I_{−} = [n] \\ n−1,n∈I_{+} \\ 1 ∈ I_{−} \end{matrix}} δ_{0;I_{+}}.

This equation is obtained from lifting the analogous relation on \overline{M}_{0,n}. The classes ψ_{i} can be written similarly by applying an element of the permutation group S_{n}. The recursion relation follows from this presentation and the restriction properties of the ψ_{i} to the boundary strata.

Theorem 5.6. The following topological recursion relation holds in genus one:

⟨⟨ τ_{a_{1}+1,m_{1}}τ_{a_{2},m_{2}} ⋯ τ_{a_{n},m_{n}} ⟩⟩_{1} = \frac{1}{24}⟨⟨ τ_{a_{1},m_{1}}τ_{a_{2},m_{2}} ⋯ τ_{a_{n},m_{n}}τ_{0,m_{+}}τ_{0,m_{−}} ⟩⟩_{0}η^{m_{+},m_{−}}

+ ∑_{I_{+}⊔I_{−}={2,... ,n}} ⟨⟨ τ_{a_{1},m_{1}} ( ∏_{j ∈ I_{+}} τ_{a_{j},m_{j}}) τ_{0,m_{+}} ⟩⟩_{0} η^{m_{+},m_{−}} ⟨⟨ τ_{0,m_{−}}( ∏_{i ∈ I_{−}} τ_{a_{i},m_{i}}) ⟩⟩_{1}

It is equivalent to

\frac{∂Φ_{1}}{∂t_{a_{1}+1}^{m_{1}}} = \frac{1}{24} ∑_{m_{+} , m_{−}} \frac{∂^{3}Φ_{0}}{∂t_{a_{1}}^{m_{1}}∂t_{0}^{m_{+}}∂t_{0}^{m_{−}}} η^{m_{+},m_{−}}

+ ∑_{m_{+},m_{−}} \frac{∂^{2}Φ_{0}}{∂t_{a_{1}}^{m_{1}}∂t_{0}^{m_{+}}} η^{m_{+},m_{−}} \frac{∂Φ_{1}}{∂t_{0}^{m_{−}}}

The topological recursion relation for ν_{1} = λ_{1} from [25]

⟨⟨ τ_{a_{1},m_{1}} ⋯ τ_{a_{n},m_{n}}ν_{1} ⟩⟩_{1} =

\frac{1}{24} ∑_{m_{+},m_{−}} η^{m_{+},m_{−}}⟨⟨ τ_{a_{1},m_{1}} ⋯ τ_{a_{n},m_{n}}τ_{0,m_{+}}τ_{0,m_{−}} ⟩⟩_{0}

can be written as

\frac{∂Φ_{1}}{∂u_{1}} = \frac{1}{24} ∑_{m_{+},m_{−}} η^{m_{+},m_{−}} \frac{∂^{2}Φ_{0}}{∂t_{0}^{m_{+}}∂t_{0}^{m_{−}}} .

Proof. The proof of the first topological recursion relation arises from the relation on \overline{M}_{1,n}^{1/r,m}

ψ_{1} = \frac{1}{12} δ_{irr} + ∑_{\begin{matrix}I_{+}⊔I_{−}=[n] \\ n−1,n∈I_{+} \\ 1∈I_{−} \end{matrix}} δ_{0;I_{+}}

which is obtained from lifting the analogous relation from \overline{M}_{1,n}. The action of S_{n} yields ψ_{i}. This combined with the restriction properties of the ψ classes yields the desired result. The second comes from the presentation of λ_{1} on \overline{M}_{1,n}^{1/r,m}

λ_{1} = \frac{1}{12} δ_{irr}

and the restriction properties of λ_{1} = ν_{1}.

These two relations allow one to completely reduce the large phase space potential to the small phase space potential in genus zero and one. Combined with the previous equations, one can compute Φ(t, s, u) in genus zero and one when one sets s_{i} = u_{i} = 0 for all i ≥ 1.

The genus one potential satisfies a generalization of the WDVV equation due to Getzler [16] which arises from relations between codimensiontwo boundary classes on \overline{M}_{1,4}. Using this equation, Dubrovin-Zhang [11] showed that if the Frobenius manifold is semisimple, then the genus-one potential is determined by the Frobenius structure. Since the Frobenius structure on (H^{(r)}, η) associated to Φ_{0}(x) is isomorphic to the Frobenius structure on the base of versal deformation of the A_{r−1} singularity and, therefore, is semisimple [10], the genus 1 potential Φ_{1}(x) is completely determined by Φ_{0}(x). However, in this case, the small phase potential in genus greater than zero must vanish due to dimensional considerations.

There exist presentations of products of ψ classes in terms of boundary classes [17, 7] which give rise to topological recursion relations in genus 2, but in general, they do not allow one to reduce the computation of the large phase space potential to the computation of the small phase space potential.

6. The Relationship to the KdV_{r} Hierarchy

In this section, we show that the genus zero, large phase space potential in the previous section corresponds to the τ function of the KdV_{r} hierarchy.

Throughout this section, We shall restrict ourselves to the large phase space potential Φ(t) where the other variables u_{a} and s_{a} are all set to zero. We will also restrict ourselves to the case where λ = \frac{1}{\sqrt{r}} since this is the value of the coupling constant which is relevant to the generalized Witten conjecture.

We follow the outline provided by Witten [40] and are able to rigorously prove the validity of his computations now that the relevant moduli spaces and classes have been constructed.

6.1. The large phase space potential in genus zero.

The following proposition rigorously demonstrates the formulas from [40] but the idea of our proof is quite different, as it uses our new relation for the μ_{1} class.

Proposition 6.1. The following identities hold:

⟨ τ_{0,m_{1}} τ_{0,m_{2}} τ_{0,m_{3}} ⟩_{0} = δ_{m_{1} + m_{2} + m_{3},r−2}

and

⟨ τ_{0,m_{1}} … τ_{0,m_{4}} ⟩_{0} = \frac{1}{r} Min_{1≤i≤4}(m_{i}, r − 1 − m_{i})

where Min is minimum value.

Proof. \overline{M}_{0,n}^{1/r,m} is nonempty if and only if (2 + ∑_{i} m_{i})/r ∈ ℤ where m = (m_{1}, …, m_{n}) and m_{i} = 0 … r − 1 for all i. The genus zero correlators are given by

⟨ τ_{0,m_{1}} … τ_{0,m_{n}} ⟩_{0} = r ∫_{\overline{M}{1/r,m \atop 0,n}} c^{1/r}

where c^{1/r} = c_{D}(−R^{1} π_{∗} E_{r}) is the (top) Chern class of degree

D = −1 + \frac{2}{r} + \frac{1}{r} ∑_{i} m_{i}.

The class c^{1/r} vanishes unless m_{i} = 0, … r − 2 for all i by Theorem 4.5. Furthermore, the correlator can only be nonzero if D = n − 3.

If n = 3 then the dimensionality condition becomes m_{1} + m_{2} + m_{3} = r − 2, in which case c^{1/r} is the identity. This proves the first part of the proposition.

If n = 4 then the dimensionality condition becomes

m_{1} + … + m_{4} = 2r − 2.

If this condition is satisfied then c^{1/r} = μ_{1}. The correlator is

⟨ τ_{0,m_{1}} … τ_{0,m_{4}} ⟩_{0} = r ∫_{\overline{M}{1/r,m \atop 0,4}} μ_{1}.

The right hand side can be computed using the relation for the class μ_{1} in Proposition 2.4, which becomes, in genus zero,

μ_{1} = ∑_{1≤i≤n} \frac{m_{i}(r − 2 − m_{i})}{2r^{2}}ψ_{i}

+ ∑_{I_{+}⊂[n]} \frac{r − 1 − (m_{+} + 1)(m_{−} + 1)}{2r^{2}} δ_{0;I_{+}}

where m_{+} and m_{−} are uniquely determined by the divisor δ_{0:I_{+}}. Let δ_{ij,kl} denote the divisor δ_{0;\{ i,j \}} on \overline{M}_{0,4}^{1/r,m}.

Plugging in this formula, one obtains (after doing a little case by case analysis to write m_{+} and m_{−} in terms of m_{1}, …, m_{4})

∫_{\overline{M}{1/r,m \atop 0,4}} μ_{1} = ∑_{i = 1}^{4} \frac{m_{i}(r − 2 − m_{i})}{2r^{2}} ∫_{\overline{M}{1/r,m \atop 0,4}} ψ_{i}

+ \frac{r − 1 − (χ_{12,34} + 1)(r − 1 − χ_{12,34})}{2r^{2}} ∫_{\overline{M}{1/r,m \atop 0,4}} δ_{12,34}

+ \frac{r − 1 − (χ_{13,24} + 1)(r − 1 − χ_{13,24})}{2r^{2}} ∫_{\overline{M}{1/r,m \atop 0,4}} δ_{13,24}

+ \frac{r − 1 − (χ_{14,23} + 1)(r − 1 − χ_{14,23})}{2r^{2}} ∫_{\overline{M}{1/r,m \atop 0,4}} δ_{14,23}

where χ_{ij,kl} := Min(m_{i} + m_{j}, m_{k} + m_{l}). Since each δ_{ij,kl} is Poincaré dual to the (topological) homology class represented by a point, one has

r ∫_{\overline{M}{1/r,m \atop 0,4}} δ_{ij,kl} = 1.

Similarly, each class ψ_{i} can be represented by δ_{ij,kl} for some i, j, k, l. Therefore, one obtains

r ∫_{\overline{M}{1/r,m \atop 0,4}} μ_{1} = ∑_{i = 1}^{4} \frac{m_{i}(r − 2 − m_{i})}{2r^{2}}

+ \frac{r − 1 − (χ_{12,34} + 1)(r − 1 − χ_{12,34})}{2r^{2}}

+ \frac{r − 1 − (χ_{13,24} + 1)(r − 1 − χ_{13,24})}{2r^{2}}

+ \frac{r − 1 − (χ_{14,23} + 1)(r − 1 − χ_{14,23})}{2r^{2}}.

The right hand side of this equation can be shown to be equal to \frac{1}{r} Min_{1≤i≤4}(m_{i}, r − 1 − m_{i}), an elementary but not obvious identity.

Proposition 6.2. [40] The genus zero potential Φ_{0}(t) is completely determined by ⟨ τ_{a_{1},m_{1}} τ_{a_{2},m_{2}} τ_{a_{3},m_{3}} τ_{a_{4},m_{4}} ⟩_{0} and the fact that

⟨ τ_{a_{1},m_{1}} τ_{a_{2},m_{2}} τ_{a_{3},m_{3}} ⟩_{0} = δ_{m_{1}+m_{2}+m_{3},r−2}.

Proof. The large phase space, genus zero potential Φ_{0}(t) is completely determined by its values on the small phase space by the topological recursion relations. Let Φ_{0}(x) denote the small phase space potential. Φ_{0}(x) must satisfy the WDVV equation since c^{1/r} yields a CohFT. Furthermore, the grading equation (50) shows that the small phase space potential is a polynomial in the variables { x^{0}, …, x^{r−2} } of degree of at most r + 1. One then performs an induction on the degree of the polynomial to show that Φ_{0}(x) is uniquely determined by the above data and the WDVV equation. The proof is straightforward.

6.2. The KdV_{r} hierarchy and the generalized Witten conjecture.

In this section, we state the generalized Witten conjecture in the language of Gelfand-Dickey (see [32] for a nice overview). Let D := \frac{i}{\sqrt{r}} \frac{∂}{∂x} and consider the space of (formal) differential operators

V := { L := D^{r} − ∑_{i = 0}^{r−2} u_{i}(x) D^{i} }

where u_{i}(x) are formal functions of x. A differential operator L ∈ V has an r-th root L^{1/r} in the space of (formal) pseudodifferential operators, i.e. there exists a pseudodifferential operator

L^{1/r} = D + ∑_{i > ,0} w_{i} D^{−i}

where each w_{i} is a differential polynomial in u_{i} such that (L^{1/r})^{r} = L.

A given pseudodifferential operator A has a maximal piece which is a differential operator, denoted by A_{+}. The KdV_{r} (or the r-th Gelfand-Dickey) hierarchy is an infinite family of formal commuting flows on V satisfying the Gelfand-Dickey equations

i \frac{∂L}{∂t_{a}^{m}} = [(L^{n+(m+1)/r})_{+}, L] \frac{k_{n,m}}{\sqrt{r}} (58)

where

k_{n,m} := \frac{(−1)^{n}r^{n+1}}{(m + 1)(r + m + 1) ⋯ (nr + m + 1)}

and the brackets denotes the commutator in the ring of differential operators.

For all m = 0, …, r − 2, let

v_{m} := −\frac{r}{m + 1} res( L^{(m+1)/r}) (59)

where res A picks off the coefficient of D^{−1} in the pseudodifferential operator A. The v_{m} can be written as differential polynomials in the u_{i}’s and conversely.

The KdV_{r} equations reduce in genus zero to an integrable hierarchy which can be regarded as the semiclassical (or dispersionless) limit of the KdV_{r} hierarchy [10, 40]. This hierarchy is obtained by replacing all appearances of the differential operator D by a formal variable p and the commutator of differential operators [L, M] is replaced by the Poisson bracket \frac{i}{\sqrt{r}} \{ L̃, M̃ \} where

\{ L̃, M̃ \} := \frac{∂L̃}{∂p} \frac{∂M̃}{∂x} − \frac{∂L̃}{∂x} \frac{∂M̃}{∂p} .

The semiclassical limit of the KdV_{r} hierarchy is

\frac{∂L̃}{∂t_{a}^{m}} = \{(L̃^{n+(m+1)/r})_{+}, L̃\} \frac{k_{n,m}}{r} . (60)

As before, one can define v_{m} using equation (59) and the u_{i}’s can be written as differential polynomials in the v_{m}’s and conversely. Putting everything together, we have proven the following.

Theorem 6.3. (The generalized Witten conjecture in genus zero) Let

x := t_{0}^{0} and v_{m} := \frac{∂^{2}Φ_{0}(t)}{∂t_{0}^{0}∂t_{0}^{m}} (61)

for m = 0, …, r − 2 then it solves equation (60). This is the unique solution of the hierarchy satisfying the string equation [40].

Proof. Our large phase space potential in genus zero Φ_{0}(t) agrees with Witten’s expression, and he proves [40] that this generating function is a solution of this hierarchy using a straightforward computation.

Corollary 6.4. The Frobenius manifold structure on (H^{(r)}, η), which exists by Theorems 3.8 and 4.5, is isomorphic to the Frobenius structure on the base of the versal deformation of the A_{r−1} singularity.

The generalized Witten conjecture in all genera is the following.

Conjecture 6.5. (The generalized Witten conjecture) The generalization of equation (61) to all genera solves the KdV_{r} hierarchy (equation (58)).

This solution is the unique one which satisfies the string equation. The differential operator L constructed from these v_{m} is therefore computable and it determines the large phase space potential.

Remarks 6.6.

1. The generalized Witten conjecture as it is stated here should be viewed as a refinement of the original Witten’s formulation [40], since it is not clear that his construction yields a class with the desired factorization properties.

2. The original Witten’s conjecture proved by Kontsevich is a particular case of the Conjecture 6.5 corresponding to the case r = 2. Indeed, for r = 2 the KdV_{r} hierarchy is the original KdV hierarchy. On the other hand, our analysis in Section 4.5 shows that for r = 2 and m = 0 we have c^{1/r} = ±1 depending on the parity of the theta characteristic (2-spin structure), and c^{1/r} = 0 when m ≠ 0. Therefore, the potential Φ(t) in this case coincides with the potential given by \overline{M}_{g,n} (cf. [40]).

3. According to general philosophy of mirror symmetry (cf. [34]), the coincidence of Frobenius structures given by Corollary 6.4 is a genus zero manifestation of some mirror phenomenon relating the moduli space of r-spin curves and singularities of type A_{r−1}. The generalized Witten conjecture should improve understanding of this relationship.

Finally, the unique τ function of the KdV_{r} hierarchy satisfying the string equation is annihilated by a series of differential operators (which forms a W_{r}^{+}-algebra) [3] part of which forms a subalgebra isomorphic to (half of) the Virasoro algebra. This yields an alternate formulation of the generalized Witten conjecture.

Conjecture 6.7. (W-algebra conjecture) There exist a collection of differential operators forming a W_{r}^{+} algebra (in which the generators { L_{n} }_{n ≥ −1} of the Virasoro algebra forms a subset) which annihilates and completely determines Z(t).

This can be regarded as the KdV_{r} analog of a refinement of the Virasoro conjecture [12]. When r = 2, this conjecture reduces to the usual Virasoro highest weight condition.

There is clearly much work to be done to be able to prove the conjecture in all genera.

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Department of Mathematics, Brigham Young University, Provo, UT 84602, USA

  E-mail address: jarvis@math.byu.edu

Department of Mathematics, 111 Cummington Street, Boston University, Boston, MA 02215, USA

  E-mail address: kimura@math.bu.edu

Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany and Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003, USA

  E-mail address: vaintrob@math.nmsu.edu