Relations between the correlators

of the topological sigma–model coupled to gravity

M. Kontsevich, Yu. Manin

Abstract

We prove a new recursive relation between the correlators ⟨τ_{d_{1}}γ_{1} … τ_{d_{n}}γ_{n}⟩_{g,β}, which together with known relations allows one to express all of them through the full system of Gromov–Witten invariants in the sense of Kontsevich–Manin and the intersection indices of tautological classes on \overline{M}_{g,n}, effectively calculable in view of earlier results due to Mumford, Kontsevich, Getzler, and Faber. This relation shows that a linear change of coordinates of the big phase space transforms the potential with gravitational descendants to another function defined completely in terms of the Gromov–Witten correspondence and the intersection theory on V^{n} × \overline{M}_{g,n}. We then extend the formalism of gravitational descendants from quantum cohomology to more general Frobenius manifolds and Cohomological Field Theories.

0. Introduction.

This note furnishes a list of relations between the correlators

⟨τ_{d_{1}}γ_{1} … τ_{d_{n}}γ_{n}⟩_{g,β}

where γ_{i} ∈ H^{∗}(V), V a smooth projective manifold, which allows one to express all of them through the following data:

(i) The (full) quantum cohomology of V in the sense of [KM], consisting of the maps I_{g,n,β}^{V} : H^{∗}(V^{n}) → H^{∗}(\overline{M}_{g,n}). Notice that quantum cohomology directly contains all the stable range correlators with d_{i} = 0 (cap I_{g,n}^{V} with the fundamental class of the moduli space), but generally for g > 0 cannot be reconstructed from these correlators only.

(ii) The intersection indices of tautological classes on \overline{M}_{g,n}, effectively calculable in view of the known results of Mumford, Kontsevich, Getzler, and Faber (cf. [F]).

Our main trick consists in introducing the generalized correlators which we denote ⟨τ_{d_{1},e_{i}}γ_{1} … τ_{d_{n},e_{n}}γ_{n}⟩_{g,β} and in deriving for them a general recursion relation. This is the content of the Theorem 1.2 which is the central result of §1. In the remaining part of §1 we collect some further (and well known) recursion formulas for the reader’s convenience: cf. [W1], [W2], [DijW], [DijVV], [EHX1], [EHX2]. Taken together, they provide transparent computation algorithms.

In §2 we apply these formulas to the comparison of two generating functions: the standard one involving the gravitational descendants (e_{i} = 0), and the new one, involving the modified gravitational descendants which in the notation above correspond to d_{i} = 0. We prove that the two functions are related by an invertible linear transformation T of the big phase space, common for all genera, and defined entirely in terms of two–point genus zero correlators with descendants at one point. A further calculation shows that T is also entirely determined by the three–point primary correlators. This might shed some light to the problem of Virasoro constraints, cf. [EHX1], [EHX2]. In fact, any system of differential equations for any generating function of the correlators with descendants equally well serves for the modified correlators after the coordinate change defined by T. On the other hand, the transition to the modified correlators almost decouples the a–indices from the d–indices: cf. formula (27) below.

Finally, in §3 we extend the new formalism of the gravitational descendants from quantum cohomology to the more general Frobenius manifolds and Cohomological Field Theories.

§1. Generalized correlators

1.1. The setting. The mathematical definition of the conventional correlators in the notation of [BM] is

⟨τ_{d_{1}}γ_{1} … τ_{d_{n}}γ_{n}⟩_{g,β} :=
∫_{J_{g,n}(V,β)} c_{1}(L_{1;g,n}(V, β))^{d_{1}} ∪ ev_{1}^{∗}(γ_{1}) ∪ ⋯ ∪ c_{1}(L_{n;g,n}(V, β))^{d_{n}} ∪ ev_{n}^{∗}(γ_{n}). (1)

where J_{g,n}(V, β) ∈ A_{∗}(\overline{M}_{g,n}(V, β)) is the virtual fundamental class, the line bundle L_{i} = L_{i;g,n}(V, β), i = 1, …, n has the geometric fiber T_{x_{i}}^{∗}C at the point [(C, x_{1}, …, x_{n}, f : C → V)], and ev_{i} sends this point to f(x_{i}). Recall also that β varies in the semigroup of the effective algebraic classes of H_{2}(V, Z)/(tors).

Put ψ_{i} := c_{1}(L_{i;g,n}(V, β)).

In the stable range 2g − 2 − n > 0 we have the absolute stabilization map st : \overline{M}_{g,n}(V, β) → \overline{M}_{g,n}, and the respective bundles L_{i} on \overline{M}_{g,n}. Put ϕ_{i} := st^{∗}(c_{1}(L_{i})).

Our generalized correlators, by definition, are:

⟨τ_{d_{1},e_{1}}γ_{1} … τ_{d_{n},e_{n}}γ_{n}⟩_{g,β} :=

∫_{J_{g,n}(V,β)} ψ_{1}^{d_{1}}ϕ_{1}^{e_{1}} ∪ ev_{1}^{∗}(γ_{1}) ∪ … ∪ ψ_{n}^{d_{n}}ϕ_{n}^{e_{n}} ∪ ev_{n}^{∗}(γ_{n}).  (2)

Since \overline{M}_{0,2}(V, 0) = ∅, we have

⟨τ_{d_{1}}γ_{1} τ_{d_{2}}γ_{2}⟩_{0,0} = 0.  (3)

Furthermore, in the stable range we have

⟨∏_{i=1}^{n} τ_{d_{i},0}γ_{i}⟩_{g,β} = ⟨∏_{i=1}^{n} τ_{d_{i}}γ_{i}⟩_{g,β}

1.2. Theorem. If 2g − 2 + n > 0, then for any j with d_{j} ≥ 1 we have

⟨∏_{i=1}^{n} τ_{d_{i},e_{i}}γ_{i}⟩_{g,β} = ⟨∏_{i=1}^{n} τ_{d_{i}−δ_{ij},e_{i}+δ_{ij}}γ_{i}⟩_{g,β}

+ ∑_{a, β_{1}+β_{2}=β} ±⟨τ_{d_{j}−1}γ_{j} τ_{0}Δ^{a}⟩_{0,β_{1}}⟨τ_{0,e_{j}}Δ_{a} ∏_{i: i≠j} τ_{d_{i},e_{i}}γ_{i}⟩_{g,β_{2}}.  (4)

Here (Δ_{a}), (Δ^{a}) are Poincaré dual bases of H^{∗}(V), and the sign arises from permuting γ_{j} with γ_{i} for all i < j.

1.3. Corollary. For g = 0, n = 3, d_{1} ≥ 1 we have:

⟨τ_{d_{1}}γ_{1}τ_{d_{2}}γ_{2}τ_{d_{3}}γ_{3}⟩_{0,β} = ∑_{a, β_{1}+β_{2}=β}⟨τ_{d_{1}−1}γ_{1} τ_{0}Δ^{a}⟩_{0,β_{1}}⟨τ_{0}Δ_{a} τ_{d_{2}}γ_{2} τ_{d_{3}}γ_{3}⟩_{0,β_{2}}. (4a)

In fact, ϕ_{i} = 0 here, so one should put e_{i} = 0 in (4), and the first summand will vanish.

This is a well known identity.

Sketch of proof. Consider the morphism of universal curves s̃t : C_{g,n}(V, β) → C_{g,n} covering st. It induces the morphism of relative 1–forms sheaves ω → ω(V, β), at least at the complement of singular points of the fiber. Restricting the latter to the j–th section (j ∈ S being fixed), we get the morphism st^{∗}(L_{j;g,n}) → L_{j;g,n}(V, β) on \overline{M}_{g,n}(V, β). It is a local isomorphism everywhere except for the points in this stack over which the j–th section lies on the component of fiber which gets contracted by s̃t. These points constitute the union of boundary strata \overline{M}(V, σ(β_{1}, β_{2})) where σ(β_{1}, β_{2}) is one–edge, two–vertex n–graph with one vertex of genus 0, class β_{1}, with tail j, and another of genus g, class β_{2}, with tails ≠ j. Naively, one would expect that all these boundaries are divisors, and over them sections of st^{∗}(L_{j;g,n}) have an extra zero of the first order. Hence in (2) we could replace one factor ψ_{j} by ϕ_{j} + ∑_{β_{1}+β_{2}=β}[\overline{M}(V, σ(β_{1}, β_{2}))]. Then the restriction to the boundary would give (4). A more precise reasoning uses the pullback property of the virtual fundamental classes J(V, σ) similar to Lemma 10 of [B]. The details will treated in [M2].

Clearly, these relations allow us to reduce all the generalized (in particular, the conventional ones) correlators to those with β = 0, to the conventional ones in the unstable range and to the generalized ones with all d_{i} = 0 in the stable range. Using (2) and the projection formula, one can rewrite the latter in the form

⟨τ_{0,e_{1}}γ_{1} … τ_{0,e_{n}}γ_{n}⟩_{g,β} :=
∫_{I_{g,n}(V,β)} c_{1}(pr_{2}^{∗}(L_{1}))^{e_{1}} ∪ pr_{1}^{∗}(γ_{1}) ∪ … ∪ c_{1}(pr_{2}^{∗}(L_{n}))^{e_{n}} ∪ pr_{1}^{∗}(γ_{n}).

where this time the integration refers to V^{n} × \overline{M}_{g,n}, I = (ev, st)_{∗}J is the Gromov– Witten correspondence, and pr_{i} are the two projections. Hence the correlators in the stable range with d_{i} = 0 are calculable if we know the full (not just top) Gromov–Witten invariants. We will call the expressions above the modified correlators.

Notice that for β = 0 we have ψ_{i} = ϕ_{i}, hence τ_{d,e} = τ_{d+e}, so that (4) gives no new information and is tautologically true because of (3). So we will remind what happens in the case β = 0, dim V > 0 separately.

1.4. The mapping to a point case. Recall that \overline{M}_{g,n}(V, 0) is canonically isomorphic to \overline{M}_{g,n} × V, and with this identification,

[\overline{M}_{g,n}(V, 0)]^{virt} = J_{g,n}(V, 0) = c_{G}(E ⊠ T_{V}) ∩ [\overline{M}_{g,n} × V],  (5)

where E = R^{1}π_{∗}O_{C}, π : C → \overline{M}_{g,n} is the universal curve, and G = g dim V. Consider the Chern classes and Chern roots of E and T_{V}:

c_{t}(E) = ∏_{i=1}^{g}(1 + a_{i}t) = ∑_{i=0}^{g}(−1)^{i}λ_{i;g,n}t^{i},

where λ_{i} are Mumford’s tautological classes defined as Chern classes of π_{∗}(ω_{π}),

c_{t}(T_{V}) = ∏_{j=1}^{δ}(1 + v_{j}t) = ∑_{j=0}^{δ} c_{j}(V)t^{j}, δ = dim V.

Then we get

c_{G}(E ⊠ T_{V}) = ∏_{i=1}^{g} ∏_{j=1}^{δ}(a_{i} ⊠ 1 + 1 ⊠ v_{j}) = ∏_{j=1}^{δ} ∑_{i=0}^{g}(−1)^{i}λ_{i;g,n} ⊠ v_{j}^{g−i}

= ∑_{(i_{1},...,i_{δ})}(−1)^{i_{1}+···+i_{δ}}λ_{i_{1};g,n} … λ_{i_{δ};g,n} ⊠ v_{1}^{g−i_{1}} … v_{δ}^{g−i_{δ}}

= (−1)^{G} ∑_{0≤i_{1}≤···≤i_{δ}≤g} λ_{i_{1};g,n} … λ_{i_{δ};g,n} ⊠ m_{g−i_{1},...,g−i_{δ}}(c_{0}(V), …, c_{δ}(V)). (6)

Here m_{g−i_{1},...,g−i_{δ}} is the symmetric function obtained by symmetrization of the obvious monomial in −v_{j} and expressed via the Chern classes of V.

Furthermore, L_{i;g,n}(V, 0) is the lift of L_{i;g,n} wrt the projection \overline{M}_{g,n} ×V → \overline{M}_{g,n} and ev_{i} is the projection \overline{M}_{g,n} × V → V. Hence we get

⟨τ_{d_{1}}γ_{1} … τ_{d_{n}}γ_{n}⟩_{g,0} =

= (−1)^{G} ∑_{0≤i_{1}≤···≤i_{δ}≤g} (∫_{\overline{M}_{g,n}} λ_{i_{1};g,n} … λ_{i_{δ};g,n}ψ_{1;g,n}^{d_{1}} … ψ_{n;g,n}^{d_{n}}

× ∫_{V} m_{g−i_{1},...,g−i_{δ}}(c_{0}(V), …, c_{δ}(V))γ_{1} … γ_{n}) ,  (7)

where ψ_{i;g,n} = c_{1}(L_{i;g,n}).

The generalized correlators give nothing new: τ_{d,e} = τ_{d+e}.

Most of the correlators (7) vanish for dimensional reasons. Here is the list of those that may remain.

1.4.1. Proposition. The correlators (7) identically vanish except for the following cases.

a) g = 0, n ≥ 3, ∑ d_{i} = n − 3, ∑ |γ_{i}| = 2δ, where γ ∈ H^{|γ|}(V), δ = dimV :

⟨τ_{d_{1}}γ_{1} … τ_{d_{n}}γ_{n}⟩_{0,0} = \frac{(d_{1} + … d_{n})!}{d_{1}! … d_{n}!} ∫_{V} γ_{1} … γ_{n}.  (8)

b) g = 1, n ≥ 1, ∑ d_{i} = n (resp. n − 1), ∑ |γ_{i}| = 0, (resp. 2):

⟨τ_{d_{1}}1 … τ_{d_{n}}1⟩_{1,0} = deg c_{δ}(V) ∫_{\overline{M}_{1,n}} ψ_{1;1,n}^{d_{1}} … ψ_{n;1,n}^{d_{n}},  (9)

⟨τ_{d_{1}}γ τ_{d_{2}}1 … τ_{d_{n}}1⟩_{1,0} = −(c_{δ−1}(V), γ) ∫_{\overline{M}_{1,n}} λ_{1,1,n}ψ_{1;1,n}^{d_{1}} … ψ_{n;1,n}^{d_{n}}  (10)

for |γ| = 2.

c) g ≥ 2, n ≥ 0, ∑ |γ_{i}|/2 ≤ δ ≤ 3, ∑(d_{i} + |γ_{i}|/2) = (g − 1)(3 − δ) + n.

In particular, the g ≥ 2, β = 0 correlators vanish for dim V ≥ 4.

Proof. First of all, E = E_{g,n} is lifted from \overline{M}_{≥2,0}, \overline{M}_{1,1} or \overline{M}_{0,3}.

For g = 0, E is the zero bundle, and J_{0,n}(V, 0) = [\overline{M}_{0,n} × V]. Formula (8) follows from this and from the known expression for g = 0, V = a point correlators:

∫_{\overline{M}_{0,n}} ψ_{1;0,n}^{d_{1}} … ψ_{n;0,n}^{d_{n}} = \frac{(d_{1} + ⋯ + d_{n})!}{d_{1}! … d_{n}!}.  (11)

For g = 1, (6) becomes

c_{δ}(E ⊠ T_{V}) = c_{δ}(V) ⊠ 1 − c_{δ−1}(V) ⊠ λ_{1,1,n}

from which (9) and (10) follow.

Finally, for g ≥ 2 one sees that the virtual fundamental class can be non–zero only if the virtual dimension for n = 0 is non–negative, which means that dim V ≤ 3. The remaining inequalities follow from the dimension matching.

One can further specialize (7) and write formulas similar to (8)–(10) separately for curves, surfaces and threefolds, g ≥ 2.

1.5. Unstable range case. If 2g − 2 + n ≤ 0, we cannot use the absolute stabilization morphism as in 1.2 and 1.3 because \overline{M}_{g,n} is empty, whereas for β ≠ 0, the stack \overline{M}_{g,n}(V, β) may well be non–empty. Always assuming this (otherwise the relevant correlators vanish), we will use instead the forgetful morphism \overline{M}_{g,n+1}(V, β) → \overline{M}_{g,n}(V, β) to produce recursion.

1.5.1. Proposition. All the unstable range correlators can be calculated through the genus zero and one primary (d_{i} = 0) stable range correlators, and the β = 0 correlators.

Proof. We will be considering the cases (g, n) = (0, 2), (0, 1), (0, 0), (1, 0) in this order, reducing each in turn to the previously treated ones.

1.5.2. Lemma. Let γ_{0} be a divisor class on V or more generally, a class in H^{2}(V). Then we have

⟨γ_{0} τ_{d_{1}}γ_{1} … τ_{d_{n}}γ_{n}⟩_{g,β} = (γ_{0}, β) ⟨τ_{d_{1}}γ_{1} … τ_{d_{n}}γ_{n}⟩_{g,β}
+ ∑_{k: d_{k}≥1}⟨τ_{d_{1}}γ_{1} … τ_{d_{k}−1}(γ_{0} ∪ γ_{k}) … τ_{d_{n}}γ_{n}⟩_{g,β}  (12)

(We omit sometimes τ_{0} in notation).

This is a generalization of the Divisor Axiom in [KM] following from the properties of J(V, β).

To treat the two–point correlators with, say d_{1} > 0, we first use (12) and write for some γ_{0} with (γ_{0}, β) ≠ 0:

⟨τ_{d_{1}}γ_{1} τ_{d_{2}}γ_{2} ⟩_{0,β} =

\frac{1}{(γ_{0}, β)} (⟨γ_{0} τ_{d_{1}}γ_{1} τ_{d_{2}}γ_{2} ⟩_{0,β} − ⟨τ_{d_{1}−1}(γ_{0} ∪ γ_{1}) τ_{d_{2}}γ_{2}⟩_{0,β} − ⟨τ_{d_{1}}γ_{1} τ_{d_{2}−1}(γ_{0} ∪ γ_{2})⟩_{0,β}) .

(13)

The last two terms in (13) contain only two–point correlators with smaller sum d_{1} + d_{2} − 1. To the first term we apply the Corollary 1.3:

⟨γ_{0} τ_{d_{1}}γ_{1} τ_{d_{2}}γ_{2}⟩_{0,β} = ∑_{a, β_{1}+β_{2}=β}⟨τ_{d_{1}−1}γ_{1} Δ_{a}⟩_{0,β_{1}}⟨Δ^{a} γ_{0} τ_{d_{2}}γ_{2}⟩_{0,β_{2}}.  (14)

The right hand side contains only two–point correlators with smaller sum d_{1} − 1 and three–point correlators with maximum one τ_{d}, d ≠ 0. If necessary, we can again apply (14) to the three–point correlators there, again reducing the order of the gravitational descendants involved.

Iterating this procedure, we will arrive to the expressions containing only primary correlators. Finally, the two–point primary correlators can be reduced to the three– point stable range ones:

⟨γ_{1}γ_{2}⟩_{0,β} = \frac{1}{(γ_{0}, β)}⟨γ_{0}γ_{1}γ_{2}⟩_{0,β}.  (15)

For later use, we register the following explicit reduction of some two–point correlators to the three–point ones following from (13):

⟨τ_{d}γ_{1} τ_{0}γ_{2}⟩_{0,β} = ∑_{j=1}^{d+1}(−1)^{j+1}(γ_{0}, β)^{−j}⟨γ_{0} τ_{d+1−j}γ_{1} τ_{0}(γ_{0}^{j−1} ∪ γ_{2})⟩_{0,β}.  (15a)

Clearly, one can invoke (12) in the same way in order to calculate the one–point and zero–point correlators. Alternatively, one can exploit the following identity, called the dilaton equation:

1.5.3. Lemma. We have

⟨τ_{1}1 τ_{d_{1}}γ_{1} … τ_{d_{n}}γ_{n}⟩_{g,β} = (2g − 2 + n) ⟨τ_{d_{1}}γ_{1} … τ_{d_{n}}γ_{n}⟩_{g,β}

This again follows from the axioms for J(V, β) stated in [BM] and proved in [B].

1.6. Correlators for zero–dimensional V. This case is covered by the Witten–Kontsevich theory and additional relations summarized in [F].

§2. Generating functions on the big phase space

2.1. The big phase space. The conventional gravitational potential is a generating series for the correlators (1) considered as a formal function on the extended phase (super)space ⊕_{d=0}^{∞}H^{∗}(V)[d]. The d–th copy of H^{∗}(V) accomodates τ_{d}γ’s. Thus the symbol τ_{d} acquires an independent meaning as the linear operator identifying H^{∗}(V) = H^{∗}(V)[ 0 ] with H^{∗}(V)[d] or even shifting each H^{∗}(V)[e] to H^{∗}(V)[e + d] so that we can write τ_{d} = τ_{1}^{d}.

For convenience choose a basis {Δ_{a} | a = 0, … r} of H^{∗}(V, C). Denote by {x_{d,a}} the dual coordinates to {τ_{d}Δ_{a}} and by Γ = ∑_{a,d} x_{d,a}τ_{d}Δ_{a} the generic even element of the extended phase superspace. As usual, x_{d,a} has the same Z_{2}–parity as Δ_{a}, and the odd coordinates anticommute. The formal functions we will be considering are formal series in weighted variables, where the weight of x_{d,a} is d.

We need the universal character B(V) → Λ : β ↦ q^{β} with values in the Novikov ring Λ which is the completed semigroup ring of B(V) eventually localized with respect to the multiplicative system q^{β}. It is topologically spanned by the monomials q^{β} = q_{1}^{β_{1}} … q_{m}^{b_{m}} where β = (b_{1}, …, b_{m}) in a basis of the numerical class group of 1–cycles, and (q_{1}, …, q_{m}) are independent formal variables. We will not need the genus expansion parameter because our main formula does not mix genera.

We now put formally

F_{g}(x) = ∑_{β} q^{β}⟨e^{Γ}⟩_{g,β} = ∑_{β} q^{β} ∑_{n} \frac{⟨Γ^{⊗n}⟩_{g,β}}{n!}

= ∑_{n,(a_{1},d_{1}),...,(a_{n},d_{n})} ϵ(a) \frac{x_{d_{1},a_{1}} … x_{d_{n},a_{n}}}{n!} ∑_{β} q^{β}⟨τ_{d_{1}}Δ_{a_{1}} … τ_{d_{n}}Δ_{a_{n}}⟩_{g,β}  (16)

where ϵ is the standard sign in superalgebra. We define F_{g}^{st}(x) by the same formula in which the last summation is restricted to the stable range of (g, n) that is, n ≥ 3 for g = 0 and n ≥ 1 for g = 1.

We will introduce the generating function G_{g}(x) for modified correlators by the same formula as F_{g}^{st} in which every τ_{d} in the stable range correlators is replaced by τ_{0,d} :

G_{g}(x) = ∑_{n,(a_{1},d_{1}),...(a_{n},d_{n})} ϵ(a) \frac{x_{d_{1},a_{1}} … x_{d_{n},a_{n}}}{n!} ∑_{β} q^{β}⟨τ_{0,d_{1}}Δ_{a_{1}} … τ_{0,d_{n}}Δ_{a_{n}}⟩_{g,β}.

(17)

We will prove that the two functions are connected by a linear change of coordinates of the big phase space.

2.2. Theorem. We have for all g ≥ 0

F_{g}^{st}(x) = G_{g}(y)  (18)

where

y_{c,b} = x_{c,b} + ∑_{(a,d),d≥c+1} ∑_{β} q^{β}x_{d,a}⟨τ_{d−c−1}Δ_{a} τ_{0}Δ^{b}⟩_{0,β}.  (19)

Proof. For d ≥ 1, define the linear operators

U_{d} : H^{∗}(V, Λ) → H^{∗}(V, Λ)

by the formula

U_{d}(γ) := ∑_{a,β} q^{β}⟨τ_{d−1}γ τ_{0}Δ_{a}⟩_{0,β}Δ^{a}  (20)

and put U_{0}(γ) = γ.

The formula (4) means that in the stable range and for d ≥ 1 the correlator of any element of the form

τ_{d,e}γ − τ_{d−1,e+1}γ − τ_{0,e}(U_{d}(γ))

with any product of others τ_{d_{i}}γ_{i} vanishes; the same is true for d = 0 by the definition of U_{0}. Hence by induction, in any stable range correlator we can replace any expression τ_{d,0}γ by ∑_{j=0}^{d} τ_{0,j}(U_{d−j}(γ)) without changing the value of the correlator. In particular,

F_{g}^{st}(x) = ∑_{n,β} \frac{q^{β}}{n!} ⟨∏_{i=1}^{n} ∑_{a_{i},d_{i}} x_{d_{i},a_{i}}τ_{d_{i}}Δ_{a_{i}}⟩_{g,β}

= ∑_{n,β} \frac{q^{β}}{n!} ⟨∏_{i=1}^{n} ∑_{a_{i},d_{i}} x_{d_{i},a_{i}} ∑_{j_{i}=0}^{d_{i}} τ_{0,j_{i}}(U_{d_{i}−j_{i}}(Δ_{a_{i}}))⟩_{g,β}

= ∑_{n,β} \frac{q^{β}}{n!} ⟨∏_{i=1}^{n} ∑_{c_{i},b_{i}} y_{c_{i},b_{i}}τ_{0,c_{i}}Δ_{b_{i}}⟩_{g,β} = G_{g}(y).

To obtain the last equality, use (20) in order to represent each sum in the correlator product as a linear combination of terms τ_{0,c}Δ_{b}. The straightforward calculation of coefficients furnishes (19).

Remark. The operator T defined by y = T (x) is a linear transformation of the big phase space with coefficients in Λ defined entirely in terms of genus zero two–point correlators. It is invertible, because (19) shows that it is the sum of identity and the operator which strictly raises the gravitational weight c. Hence we may define the corrected version of G_{g}(x) by G̃_{g}(x) := F_{g}(T^{−1}(x)). Equivalently, we can extend the modified correlators to the unstable range keeping the natural functional equations.

One can also use these formulas in order to give independent meaning to the symbols τ_{0,d} as linear operators on the infinite sum of the Λ–modules H^{∗}(V, Λ)[d].

2.3. Expressing T through the three–point primary correlators. Formulas (16) and (19) make the following definition natural:

⟨τ_{d_{1}}γ_{1} … τ_{d_{n}}γ_{n}⟩_{g} := ∑_{β} q^{β}⟨τ_{d_{1}}γ_{1} … τ_{d_{n}}γ_{n}⟩_{g,β}.  (21)

We will write simply ⟨… ⟩ when g = 0. These correlators are Λ–polylinear functions on the Λ–module ⊕_{d≥0}H^{∗}(V, Λ)[d]. Setting in (14) d_{2} = 0, multiplying by q^{β} and summing, we obtain:

⟨γ_{0} τ_{d}γ_{1} γ_{2}⟩ = ∑_{a}⟨τ_{d−1}γ_{1} Δ_{a}⟩⟨Δ^{a} γ_{0} γ_{2}⟩  (22)

Put

γ_{0} · γ_{2} := ∑_{a} Δ_{a}⟨Δ^{a}γ_{0} γ_{2}⟩  (23)

(this is essentially the product in “small” quantum cohomology where the structure constants are the third derivatives of the genus zero potential restricted to H^{2}).

Then we can rewrite (22) as

⟨γ_{0} τ_{d}γ_{1} γ_{2}⟩ = ⟨τ_{d−1}γ_{1} γ_{0} · γ_{2}⟩.  (24)

Now let l be any linear function on H_{2}(V, Λ). It defines the derivation ∂_{l} : Λ → Λ, ∂_{l}q^{β} := l(β) q^{β}. We extend it to formal series over Λ coefficientwise. If γ_{0} is an ample divisor class considered as a linear function on H_{2}, we write ∂_{γ_{0}} for this derivation. Turning now to the equation (15a), multiply it by q^{β} and sum over all β. The left hand side of (15a) vanishes for β = 0, and the right hand side does not make sense, so we get:

⟨τ_{d}γ_{1} γ_{2}⟩ =

∑_{j=1}^{d+1}(−1)^{j+1}∂_{γ_{0}}^{−j}[⟨γ_{0} τ_{d+1−j}γ_{1} τ_{0}(γ_{0}^{j−1} ∪ γ_{2})⟩ − ⟨γ_{0} τ_{d+1−j}γ_{1} τ_{0}(γ_{0}^{j−1} ∪ γ_{2})⟩_{0,0}].

To interpret this, notice that since (γ_{0}, β) ≠ 0 for all algebraic effective non–zero 2–homology classes on V, ∂_{γ_{0}}^{−1}F makes sense for any series F whose coefficients are correlators not involving the β = 0 ones. As the result of this “integration” we take the series again not involving the β = 0 terms.

Actually, in view of (8), the β = 0 terms vanish unless j = d + 1. Separating this summand and replacing the remaining triple correlators with the help of (24), we get the following result.

2.3.1. Proposition. The matrix coefficients of T can be expressed inductively through the triple primary correlators, that is, Gromov–Witten invariants, of genus zero: for d ≥ 1

⟨τ_{d}γ_{1} γ_{2}⟩ = ∑_{j=1}^{d}(−1)^{j+1}∂_{γ_{0}}^{−j}⟨τ_{d−j}γ_{1} γ_{0} · (γ_{0}^{j−1} ∪ γ_{2})⟩+

(−1)^{d}∂_{γ_{0}}^{−(d+1)}[⟨γ_{0} γ_{1} γ_{0}^{d} ∪ γ_{2}⟩ − ⟨γ_{0} γ_{1} γ_{0}^{d} ∪ γ_{2}⟩_{0,0}].  (25)

§3. Coupling of Frobenius manifolds

and Cohomological Field Theories to topological gravity

3.1. Coupling of Frobenius manifolds to topological gravity. The restriction Φ(x) to the small phase space (x_{d,a} = 0 for d > 0) of the genus zero potential F_{0}(x) from (16) satisfies the so called Associativity Equations and defines on H^{∗}(V, Λ) the structure of the formal Frobenius manifold, or the tree level quantum cohomology of V. The notion of Frobenius manifold was axiomatized and studied by B. Dubrovin in [D]. There are many interesting examples which do not come from quantum cohomology. In the sec. 6 of [D] Dubrovin sets to reconstruct the whole potential with gravitational descendants from its small phase space part. Our previous discussion shows how one can do it for quantum cohomology potentials. In this subsection we show how to do this for a wider class of formal Frobenius manifolds which are not supposed to come from quantum cohomology. Our approach considerably differs from that of [D]. It would be important to relate it to the integrable hierarchies as in [D].

We will divide our discussion into two steps.

First, we will introduce the modified potential with gravitational descendants which reduces to G_{0}(x) in the quantum cohomology case.

Second, we will discuss the additional conditions needed to define the analog of the linear transformation T and the conventional potential with gravitational descendants F_{0}(x) := G_{0}(T (x)).

3.1.1. The big phase space and the modified potential. We will use the formalism of Frobenius manifolds as it was presented in [M1].

Let Λ be a Q–algebra (playing role of the Novikov ring), H a free Z_{2}–graded Λ–module of finite rank (in the quantum cohomology case H = H^{∗}(V, Λ) ), η a symmetric non–degenerate pairing on H replacing the Poincaré form. To keep intact as much notation as possible, we introduce formally the big phase space as linear infinite dimensional formal supermanifold ⊕_{d≥0}H[d] with basis τ_{d}Δ_{a} and coordinates x_{d,a} as in sec. 6 above. Put x_{a} = x_{0,a}, x = {x_{a}}. By definition, a Frobenius potential on (H, η) is a formal series Φ(x) ∈ Λ[[x]] whose third derivatives Φ_{ab}^{c} (with one index raised by η) form the structure constants of the commutative, associative Λ[[x]]–module spanned by ∂_{a} := ∂/∂x_{a}. Finally, any such triple M = (H, η, Φ) is called a formal Frobenius manifold (over Λ).

The primary correlators of M are by definition the symmetric polylinear functions H^{⊗n} → Λ, n ≥ 3, whose values on the tensor products of τ_{0}Δ_{a} are essentially the coefficients of Φ written as in (16):

Φ(x) = ∑_{n,a_{1},...,a_{n}} ϵ(a) \frac{x_{a_{1}} … x_{a_{n}}}{n!} ⟨τ_{0}Δ_{a_{1}} … τ_{0}Δ_{a_{n}}⟩.  (26)

In the case of quantum cohomology this agrees with our notation (21). Notice that the Associativity Equations do not constrain the terms of Φ of degree ≤ 2. In this subsection we will use only correlators with ≥ 3 arguments.

In order to extend the potential Φ to a formal function on the big phase space which in the quantum cohomology case will coincide with G_{0}, we will use the Second Reconstruction Theorem of [KM], proved in [KMK] and [M1]:

3.1.2. Proposition. For any Frobenius manifold M as above, there exists a unique sequence of Λ–linear maps I_{n}^{M} : H^{⊗n} → H^{∗}(\overline{M}_{0,n}, Λ), n ≥ 3, satisfying the folowing properties:

(i) S_{n}–invariance and compatibility with restriction to boundary divisors (cf.

[KM] or [M1], p. 101).

(ii) The top degree term of I_{n}^{M} capped with the fundamental class is the correlator of M with n arguments.

Moreover, in the quantum cohomology case

I_{n}^{M} = ∑_{β} q^{β}I_{0,n,β}^{V}

where I_{0,n,β}^{V} are the genus zero Gromov–Witten invariants discussed in [KM].

We now define the modified M–correlators with gravitational descendants by

⟨τ_{0,d_{1}}Δ_{a_{1}} … τ_{0,d_{n}}Δ_{a_{n}}⟩ := ∫_{[\overline{M}_{0,n}]} I_{n}^{M}(τ_{0}Δ_{a_{1}} ⊗ ⋯ ⊗ τ_{0}Δ_{a_{n}})ϕ_{1}^{d_{1}} … ϕ_{n}^{d_{n}}  (27)

where ϕ_{i} are defined in 1.1. Finally put

G_{0}^{M}(x) = ∑_{n,(a_{1},d_{1}),...(a_{n},d_{n})} ϵ(a) \frac{x_{d_{1},a_{1}} … x_{d_{n},a_{n}}}{n!} ⟨τ_{0,d_{1}}Δ_{a_{1}} … τ_{0,d_{n}}Δ_{a_{n}}⟩  (28)

where this time x denotes coordinates on the big phase space. Clearly, if M is quantum cohomology, we have reproduced (17).

The expressions (27) are universal polynomials in the coefficients of Φ and η^{ab} depending only on the superrank of H and (a_{i}, d_{i}). They can be calculated using some results of [Ka].

To explain this, recall that H_{∗}(\overline{M}_{0,n}) is spanned by the classes of the boundary strata \overline{M}_{0,τ} indexed by trees whose tails are labelled by \{1, …, n\}. Any cohomology class is uniquely defined by its values on these classes. For I_{n}^{M} these values are given in [KMK], (0.7). For ϕ_{1}^{d_{1}} … ϕ_{n}^{d_{n}} they are products of multinomial coefficients over all vertices of τ: put on each flag d_{i} if this is a tail with label i, 1 otherwise, and divide the factorial of the sum of labels at each vertex by the product of factorials of labels.

It remains to calculate the cup product of the described classes. This problem was solved in [Ka]. Admittedly, the explicit formula is rather complicated.

3.1.3. Higher genus case. If I_{n}^{M} = I_{0n}^{M} is extended to a Cohomological Field Theory I_{gn}^{M}, as defined in [KM], one can use the evident version of formula (26) in order to define the modified correlators and functions G_{g}(x) of any genus. However, unlike the genus zero case, a CohFT cannot be reconstructed only from its primary correlators.

3.2. The operator T on the big phase space. If we want to extrapolate the construction of T from the case of quantum cohomology to more general Frobenius manifolds, we encounter several difficulties. The basic problem is that the inductive formula (25) for the coefficients of T involves some additional structures, not required in the general definition of fromal Frobenius manifolds. Namely, we need submodules H_{2} and H^{2} in H, a semigroup in H_{2} with indecomposable zero accomodating β, the ring Λ with derivatives ∂_{γ_{0}}. All of these structures must satisfy several conditions, ensuring in particular the independence of the right hand side of (25) from the choice of γ_{0}.

The following seems to be the most straightforward way to describe the additional restrictions starting with the more conventional data on M = (H, η, Φ).

(i) Assume that M is endowed with the flat identity e and an Euler vector field E, such that ad E is semisimple on H. Assume that the spectrum D, (d_{a}) belongs to Λ (see [M1], Ch.1, §2 for precise definitions).

(ii) Denote by H^{2} ⊂ H the submodule of H coresponding to the zero eigenvalue of ad E. Assume that it is a free direct submodule. Denote by H_{2} ⊂ H the submodule of H coresponding to the eigenvalue −D of ad E. Assume that it is a free direct submodule, and that η makes H_{2} strict dual to H^{2}.

(iii) Assume that a semigroup B ⊂ H_{2} with indecomposable zero and finite decomposition is given such that Φ(x) can be expanded into a formal Fourier series with respect to the part of the coordinates dual to a basis of H_{2}, with coefficients vanishing outside B. Denote by Ψ the part corresponding to β ≠ 0. Assume finally that Φ = Ψ + c where c is a cubic form, EΨ = (D + d_{0})Ψ (without additional terms of degree ≤ 2, cf. [M1], Ch.1, (2.7)) and E_{1}c = (D+d_{0})c where E_{1} is the projection of E to the orthogonal complement to H^{2}.

These structures allow us to imitate the constructions of §2, starting with β– decomposition of the primary correlators, and to define T via (25). For more details, see [M3].

Notice that the cup product on H and the ⟨… ⟩_{0,0} correlators are defined using the constant terms of the relevant Fourier decomposition. The independence of (25) from the choice of γ_{0} follows from the postulated properties.

Acknowledgements. One of the authors (Yu.M.) is thankful to Ezra Getzler for stimulating discussions which prompted him to focus on this part of a larger project [M2], and to Kai Berend for his explanations about the proof of Lemma 10 in [B] and the restriction formula for virtual fundamental classes.

References

[B] K. Behrend. Gromov–Witten invariants in algebraic geometry. Inv. Math., 127 (1997), 601–617.

[BF] K. Behrend, B. Fantechi. The intrinsic normal cone. Inv. Math., 128 (1997), 45–88.

[BM] K. Behrend, Yu. Manin. Stacks of stable maps and Gromov–Witten invariants. Duke Math. J., 85:1 (1996), 1–60.

[DijVV] R. Dijkgraaf, H. Verlinde, E. Verlinde. Loop equations and Virasoro constraints in non–perturbative two–dimensional quantum gravity. Nucl. Phys. B348 (1991), 435–456.

[DijW] R. Dijkgraaf, E. Witten. Mean field theory, topological field theory, and multimatrix models. Nucl. Phys. B342 (1990), 486–522.

[D] B. Dubrovin. Geometry of 2D topological field theories. In: Springer LNM, 1620 (1996), 120–348

[EHX1] T. Eguchi, K. Hori, Ch.–Sh. Xiong. Gravitational quantum cohomology. Preprint UT–753, hep–th/9605225.

[EHX2] T. Eguchi, K. Hori, Ch.–Sh. Xiong. Quantum cohomology and Virasoro algebra. Preprint UT–769, hep–th/9703086.

[F] C. Faber. Algorithms for computing intersection numbers on moduli spaces of curves, with an application to the class of the locus of Jacobians. Preprint, 1997.

[Ka] R. Kaufmann. The intersection form in H^{∗}(\overline{M}_{0n}) and the explicit Künneth formula in quantum cohomology. Int. Math. Res. Notices, 19 (1996), 929–952.

[KM] M. Kontsevich, Yu. Manin. Gromov-Witten classes, quantum cohomology, and enumerative geometry. Comm. Math. Phys., 164:3 (1994), 525–562.

[KMK] M. Kontsevich, Yu. Manin (with Appendix by R. Kaufmann). Quantum cohomology of a product. Inv. Math., 124 (1996), f. 1–3, 313–340.

[M1] Yu. Manin Frobenius manifolds, quantum cohomology, and moduli spaces (Chapters I, II, III). Preprint MPI 96–113, 1996.

[M2] Yu. Manin. Algebraic geometric introduction to the gravitational quantum cohomology (in preparation).

[M3] Yu. Manin. Frobenius manifolds of quantum cohomology type and tensor products (in preparation)

[W1] E. Witten. On the structure of the topological phase of two–dimensional gravity. Nucl. Phys. B340 (1990), 281–332.

[W2] E. Witten. Two–dimensional gravity and intersection theory on moduli space. Surveys in Diff. Geometry, 1 (1991), 243–310.