The Chiral de Rham Complex and Positivity of the Equivariant Signature of the Loop Space

V. Gorbounov, F. Malikov^{1}

Abstract

In this note we show that the positivity property of the equivariant signature of the loop space, first observed in [MS1] in the case of the even-dimensional projective spaces, is valid for Picard number 2 toric varieties. A new formula for the equivariant signature of the loop space in the case of a toric spin variety is derived.

0. Introduction The equivariant signature of the loop space, an example of an elliptic type genus, is, in particular, a rule which assigns to a manifold X a power series in q:

X ↦ sign(q, LX) = b_{0} + b_{1}q + b_{1}q^{2} + · · · .

The intuition behind its definition (a well-known theorem in fact, see sect. 2) is that sign(q, LX) is the equivariant index of a certain elliptic operator and therefore the coefficients b_{j} are integers, not necessarily positive since they are equal to the difference between the dimensions of two vector spaces. The cohomological computations of [MS1] fairly unexpectedly showed that in the case of the evendimensional complex projective space ℙ^{2n} the series sign(q, Lℙ^{2n}) has positive coefficients. Moreover, there is a graded vector space naturally associated to ℙ^{2n} so that b_{j} essentially equals the dimension of the j-th homogeneous component. Let us formulate this result more precisely.

Associated with any smooth manifold X there is a sheaf of vertex algebras Ω_{X}^{ch}, the chiral de Rham complex [MSV]. It is graded:

Ω_{X}^{ch} = ⊕_{j≥0}(Ω_{X}^{ch})_{j},

and so are its cohomology groups. The following equality of formal power series was proven in [MS1] (cf. Theorem 4.1 below)

sign(q, Lℙ^{2n}) = 2 ∑_{j=0}^{∞} dimH^{0}(ℙ^{2n}, (Ω_{ℙ^{2n}}^{ch})_{j})q^{j} − 1.  (0.1)

One can say that the vector space H^{0}(ℙ^{2n}, Ω_{ℙ^{2n}}^{ch}), a vertex algebra in fact, provides a realization of sign(q, Lℙ^{2n}).

The present note came out of the discussion of an earlier version of [MS1], and (0.1) appearing in the final version of [MS1] is a result of this discussion. We further use the Borisov-Libgober formula for the equivariant signature of the loop space in the case of a toric variety [BL] to extend the positivity result to the Picard number 2 toric varieties. (Recall that the projective spaces exhaust the class of Picard number 1 smooth toric varieties.) The very nature of this calculation does not allow us to conclude whether there is a graded vector space such that it is naturally associated to the manifold in question and realizes this signature. The existence of such a vector space, be it a vertex algebra or not, remains an open question.

1partially supported by an NSF grant

A similar computation combined with Witten’s rigidity theorem gives the following formula for the equivariant signature of the loop space in the case of a spin toric variety X of complex dimension d (cf. Theorem 6.2):

sign(q, LX) = signX \frac{ϵ^{− \frac{d}{4}}}{2^{d}} ,  (0.2)

where signX is the signature of X and ϵ a well-known modular form defined by (2.2) below.

The r.h.s. of this formula is not unfamiliar to vertex algebra specialists. Let V be the vertex algebra of 2d free bosons coupled to 2d free fermions. V is naturally graded:

V = ⊕_{j=0}^{∞}V_{j}

so that

∑_{j=0}^{∞} dimV_{j}q^{j} = ϵ^{− \frac{d}{4}} .  (0.3)

Having compared (0.1) to (0.2-0.3) one may perhaps be so bold as to conjecture that the vertex algebra V is isomorphic to H^{0}(X, Ω_{X}^{ch}). If correct this conjecture will be a natural extension of the realization result (0.1) to the spin-case.

Acknowledgments. We are grateful to F.Hirzebruch and D.Zagier for their interest in this work. Part of this work was completed when we were visiting the Max-Planck-Institut für Mathematik in Bonn. We are grateful to the Institut for the outstanding working conditions.

1. Genera. In our brief review of the relevant genera we shall follow the book [HBJ].

Let R be a commutative ring, Ω^{∗} the cobordism ring. Recall that Ω^{∗} ⊗ ℚ = ℚ[P^{2}, P^{4}, ...], where P^{2n} is the cobordism class of the complex projective space of dimension 2n. According to Hirzebruch, a genus is a ring homomorphism

g : Ω^{∗} ⊗ ℚ → R.  (1.1)

An invertible even formal power series Q(x) = a_{0} + a_{1}x^{2} + … with coefficients in R defines a genus, to be denoted g_{Q}, as follows. Let x_{1}, … x_{n} be formal variables and p_{i} the i-th elementary symmetric function in x_{i}^{2}. Then

Q(x_{1})Q(x_{2}) … Q(x_{n}) = a_{0}^{n}(1 + K_{1}(p_{1}) + K_{2}(p_{1}, p_{2}) + …)

where K_{i}(p_{1}, …, p_{i}) is a (uniquely determined) homogeneous polynomial of degree 2i. For a manifold X of dimension 4n define

g_{Q}(X) = a_{0}^{n}K_{n}(p_{1}, … p_{n})[X],  (1.2)

where p_{i}(M), i = 1, ..., n, is the i-th the Pontryagin class of X and [X] is the fundamental class of X. In fact the rule Q(x) → g_{Q} sets up a 1-1 correspondence.

2. The elliptic genus and the equivariant signature of the loop space. The notion of an elliptic genus is due to Ochanine[O]. A genus is called elliptic if it is associated to a series Q(x) such that:

if f(x) = \frac{x}{Q(x)}, then

(f′)^{2} = 1 − 2δf^{2} + ϵf^{4},  (2.1)

for some parameters δ and ϵ.

All solutions to (2.1) can be constructed as follows. Fix a lattice L in ℂ with generators ω_{1}, ω_{2} and let 𝔭_{L}(z) be the corresponding Weierstraßfunction. Then

f(x) = \frac{1}{\sqrt{𝔭_{L}(x) − 𝔭_{L}(ω_{1}/2)}}

is a solution to (2.1) with

δ = −\frac{2}{3}𝔭_{L}(ω_{1}/2),

ϵ = [𝔭_{L}(ω_{1}/2) − 𝔭_{L}(ω_{2}/2)][𝔭_{L}(ω_{1}/2) − 𝔭_{L}((ω_{1} + ω_{2})/2].

Therefore each elliptic genus equals (cf. (1.2))

g_{x\sqrt{𝔭_{L}(x)−𝔭_{L}(ω_{1}/2)}}(.),

for some L and ω_{1}, and g_{x\sqrt{𝔭_{L}(x)−𝔭_{L}(ω_{1}/2)}}(.), as a function of L and ω_{1}, can be considered the universal elliptic genus.

Introducing the modular parameter q = exp (2πiω_{2}/ω_{1}) one sees that all the defined expressions are naturally identified with functions of q. In particular, δ and ϵ become the standard generators of the ring M_{∗}(2) of modular forms on Γ_{0}(2) so that δ ∈ M_{2}(2), ϵ ∈ M_{4}(2). Needed for (2.6) below is the following formula for ϵ:

ϵ = (2 ∏_{n=1}^{∞} \frac{(1 + q^{n})^{2}}{(1 − q^{n})^{2}} )^{−4},  (2.2)

Likewise, for any X, g_{x\sqrt{𝔭_{L}(x)−𝔭_{L}(ω_{1}/2)}}(X) is a function of q. We shall emphasize this by changing (and unburdening) the notation as follows:

och(q, X) = g_{x\sqrt{𝔭_{L}(x)−𝔭_{L}(ω_{1}/2)}}(X).

In fact, for a manifold X of real dimension 4k, och(q, X) is a weight 2k polynomial in δ counted with weight 2 and ϵ counted with weight 4. Therefore och(q, X) is a weight 2k modular form on Γ_{0}(2).

We will be more interested in a closely related genus, sign(q, LX), proposed by Witten and called the formal equivariant signature of the loop space [HBJ]. By definition

sign(q, LX) = g_{Q},  (2.3a)

where

Q(x) = x\frac{1 + e^{−x}}{1 − e^{−x}} Π_{n=1}^{∞} \frac{(1 + q^{n}e^{−x})(1 + q^{n}e^{x})}{(1 − q^{n}e^{−x})(1 − q^{n}e^{x})}.  (2.3b)

This genus can be equivalently defined to be the signature of X twisted by the bundle

W = ⊗_{n=1}^{∞}S_{q^{n}}T X_{ℂ} ⊗_{n=1}^{∞} Λ_{q^{n}}T X_{ℂ},  (2.4)

where we habitually set

Λ_{t}E = ∑_{i=0}^{∞} t^{i}Λ^{i}E,  (2.5a)

S_{t}E = ∑_{i=0}^{∞} t^{i}S^{i}E.  (2.5b)

Yet another possibility is to define sign(q, LX) to be the index of the elliptic operator d + d^{∗} acting on the global sections of the bundle (2.4).

We conclude this section by noting that sign(q, LX) is related to the elliptic genus as follows:

sign(q, LX) = och(q, X)ϵ^{− \frac{k}{2}} ,  (2.6)

where ϵ is defined by (2.2) and dimX = 4k.

3. The chiral de Rham complex and the equivariant signature of the loop space. Defined in [MSV] for any smooth complex manifold X of complex dimension n there is a sheaf Ω_{X}^{ch}(X) of vertex algebras over X. Morally, Ω_{X}^{ch}(X) is a semi-infinite de Rham complex on a “small” loop space with coefficients in distributions supported on the submanifold of analytically contractible loops. This vague assertion was made a theorem in [KV] after overcoming considerable technical difficulties.

Ω_{X}^{ch}(X) is not a sheaf of O_{X}-modules but it possesses a filtration such that the associated graded sheaf grΩ_{X}^{ch} is. In fact, grΩ_{X}^{ch} is associated to a holomorphic vector bundle and this vector bundle was explicitly described in [BL] as follows:

grΩ_{X}^{ch} = ⊗_{n=1}^{∞}{S_{q^{n}}T X ⊗ S_{q^{n}}T X^{∗} ⊗ Λ_{yq^{n−1}}T^{∗}X ⊗ Λ_{y^{−1}q^{n}}T X},  (3.1)

where we use the notation introduced in (2.5a,b). Thus grΩ_{X}^{ch}(X) is bi-graded so that the component of weight (i, j) is the coefficient of q^{j}y^{i}. In fact, the sheaf Ω_{X}^{ch} is itself bi-graded :

Ω_{X}^{ch} = ⊕_{i∈ℤ,j≥0}(Ω_{X}^{ch})_{j}^{i},  (3.2)

and this bi-grading descends to the graded object. In particular, Ω_{X}^{i}, i.e. the sheaf of holomorphic i-forms , canonically identifies with (Ω_{X}^{ch})_{0}^{i}:

Ω_{X}^{i} ⟶^{∼} (Ω_{X}^{ch})_{0}^{i},  (3.3.)

which partially justifies the name “ a chiral de Rham complex”.

Consider the Euler character Eu(Ω_{X}^{ch})(q, y), which by definition is a formal Laurent power series in q, y such that the coefficient of y^{i}q^{j} is the Euler characteristic of the component (Ω_{X}^{ch})_{j}^{i}. Define Eu(Ω_{X}^{ch})(q) to be Eu(Ω_{X}^{ch})(q, 1) Applications of Ω_{X}^{ch} to elliptic genera are based on the following observation due to [BL]:

Eu(Ω_{X}^{ch})(q) = sign(q, LX).  (3.4)

Let us prove (3.4) for the sake of completeness. Since the Euler characteristic does not change under the passage to the graded object, we can write

Eu(Ω_{X}^{ch})(q) = Eu(grΩ_{X}^{ch})(q) = ∫_{X} ch(grΩ_{X}^{ch})tdX,  (3.5)

where the 2nd equality follows from the Riemann-Roch Theorem, and tdX is the Todd genus of X. We now compute ch(grΩ_{X}^{ch}) by using (3.1) and the multiplicativity of ch to the effect that

ch(grΩ_{X}^{ch}) = ∏_{n=1}^{∞}{ch(S_{q^{n}}T X)ch(S_{q^{n}}T X^{∗})ch(Λ_{q^{n−1}}T^{∗}X)ch(Λ_{q^{n}}T X)}. (3.6)

As is well known for a k dimensional manifold X

chΛ_{q^{n−1}}T X = Π_{i=1}^{k}(1 + q^{n−1}e^{x_{i}}), chΛ_{q^{n−1}}T X^{∗} = Π_{i=1}^{k}(1 + q^{n−1}e^{−x_{i}}),

chS_{q^{n}}T X = Π_{i=1}^{k} \frac{1}{(1 − q^{n}e^{x_{i}})}, chS_{q^{n}}T X^{∗} = Π_{i=1}^{k} \frac{1}{(1 − q^{n}e^{−x_{i}})},

tdX = Π_{i=1}^{k} \frac{x_{i}}{1 − e^{−x_{i}}} .

Plugging these in the right hand side of (3.6) we readily see that the integrand in the right hand side of (3.5) is Q(x_{1})Q(x_{2}) ⋯ Q(x_{k}), where Q(x) is the series (2.3b). Therefore (3.4) is identical to the definition (1.2) with Q(x) given by (2.3b). □

Note that Eu(Ω_{X}^{ch})(0) is equal to the signature of the manifold X.

4. Positivity of the equivariant signature of the loop space.

Introduce the formal character

chH^{i}(X, Ω_{X}^{ch}) = ∑_{j=0}^{∞} dimH^{i}(X, (Ω_{X}^{ch})_{j})q^{j},

where we ignore the upper-index grading, cf. (3.2). (It follows easily from (3.1) that the dimensions entering the formal character are all finite.)

Theorem 4.1. The following character formula is valid

2chH^{0}(ℙ^{2n}, Ω_{ℙ^{2n}}^{ch}) − 1 = sign(q, Lℙ^{2n}).

Corollary 4.2. The series sign(q, Lℙ^{2n}) has positive coefficients and the coefficients against positive powers of q are even.

Corollary 4.2 is an immediate consequence of Theorem 4.1 whereas Theorem 4.1 is a simple consequence of certain properties of the chiral de Rham complex over projective spaces discovered in [MS1]. We shall list these properties here and reproduce from [MS1] a simple computation leading to Theorem 4.1.

The definition of the Euler characteristic and (3.5) give

sign(q, Lℙ^{2n}) = ∑_{i=0}^{2n}(−1)^{i}chH^{i}(ℙ^{2n}, Ω_{ℙ^{2n}}^{ch}).  (4.1)

We now make use of the following

Theorem [MS1]. The natural embedding of sheaves Ω_{ℙ^{n}}^{∗} ↪ Ω_{ℙ^{n}}^{ch} due to (3.3) provides an isomorphism

H^{i}(ℙ^{n}, Ω_{ℙ^{n}}^{∗}) ⟶^{∼} H^{i}(ℙ^{n}, Ω_{ℙ^{n}}^{ch}).

for 0 < i < n.

This assertion reduces (4.1) to

sign(q, Lℙ^{2n}) = chH^{0}(ℙ^{2n}, Ω_{ℙ^{2n}}^{ch}) + chH^{2n}(ℙ^{2n}, Ω_{ℙ^{2n}}^{ch}) − 1.  (4.2)

To conclude it remains to notice that due to the chiral Poincaré duality [MS2]

chH^{0}(ℙ^{2n}, Ω_{ℙ^{2n}}^{ch}) = chH^{2n}(ℙ^{2n}, Ω_{ℙ^{2n}}^{ch}).

□

Remark The vector space H^{0}(ℙ^{n}, Ω_{ℙ^{n}}^{ch}) is a vertex algebra. Thanks to Theorem 4.1, the known [HBJ] modular properties of the equivariant signature of the loop space say that chH^{0}(ℙ^{n}, Ω_{ℙ^{n}}^{ch}) is a modular function when n = 0 mod 4. The modular properties of characters have been the hallmark of vertex algebra theory, see for example [Z].

5. Extending the positivity result

5.1 It seems natural to ask in what generality the positivity result (Corollary 4.2) holds true. Since according to [BR] every cobordism class contains a non-singular toric variety, not every non-singular toric variety X has positive sign(q, LX). We will show nevertheless that apart from projective spaces there is a class of toric varieties with this property. To do so, we will have to rely on the calculations from [BL] instead of sheaves of vertex algebras.

When talking about toric varieties we shall keep to the following notation. Let e_{i}, 1 ≤ i ≤ d, be the standard basis of ℤ^{d}; thus the j-th component of e_{i} is δ_{ij}. Define the inner product

ℤ^{d} × ℤ^{d} → ℤ, x, y ↦ x · y

by the requirement that

e_{i} · e_{j} = δ_{ij}.

This identifies ℤ^{d} and hence ℝ^{d} with their duals.

By Σ we denote a complete, regular fan in ℝ^{d}. This means, in particular, that each cone C^{∗} ∈ Σ is spanned by part of a basis of ℤ^{d}, and we denote this spanning set by |C^{∗}|.

Associated to Σ there is a smooth compact toric variety of complex dimension d to be denoted X_{Σ} or simply X if no confusion is likely to arise.

The following formula holds true [BL]

sign(q, LX) = ∑_{m∈ℤ^{d}} ∑_{C^{∗}∈Σ}(−1)^{codimC^{∗}}( ∏_{n∈|C^{∗}|} \frac{1}{1 + q^{m·n}})ϵ^{−d/4}.  (5.1)

Note that to make sense out of this expression one has to expand each factor

\frac{1}{1 + q^{m·n}}

at q = 0 and then convince oneself that the sum of thus arising power series with respect to m ∈ ℤ^{d} makes sense as a formal power series.

Remark The formula in Theorem 5.5 in [BL] where we borrowed (5.1) contains an extra factor (−1)^{d/2}. We drop it so as to conform to the standard notation.

5.2 Now we extend the result of Theorem 4.1 to a larger class of toric variates. Recall that the Picard number of a toric variety is the difference between the dimension of the variety and the number of one dimensional cones. Each smooth Picard number one toric variety is a projective space. The Picard number two toric varieties, which can be viewed as generalized Hirzebruch’s surfaces, were classified in [Kl]. Let us formulate this result.

We give ourselves a triple of integers d, s, r such that 1 < d, 1 < s < d + 1, r = d − s + 1, and an increasing sequence of non-negative integers a_{1}, … a_{r}. Define the following vectors in ℤ^{d}:

u_{i} = e_{i}, 0 < i < r + 1;

u_{r+1} = − ∑_{i=1}^{r} u_{i};

v_{j} = e_{r+j}, 0 < j < s;

v_{s} = ∑_{i=1}^{r} a_{i}e_{i} − ∑_{j=1}^{s−1} v_{j};

We set U = {u_{1}, …, u_{r+1}}, V = {v_{1}, …, v_{s}} and let C_{ij}^{∗} ⊂ ℝ^{d}, 0 < i < r + 2, 0 < j < s + 1, be the cone spanned by U ∪ V \ {u_{i}, v_{j}}). One checks that there is a uniquely determined regular complete fan such that {C_{ij}^{∗}, 0 < i < r + 2, 0 < j < s + 1} is the set of d-dimensional cones. We denote this fan by ∑_{d}(a_{1}, … a_{r}) and the corresponding toric variety by X_{d}(a_{1}, … a_{r}).

Theorem 5.1 [Kl] Every compact smooth toric variety of complex dimension d with d + 2 generators is isomorphic to precisely one of the varieties X_{d}(a_{1}, … a_{r}).

Theorem 5.2

1. For d even

sign(q, Lℙ^{d}) = ∑_{m∈ℤ^{d}} \frac{2}{(1 + q^{−m_{1}...−m_{d}}) ∏_{i=1}^{d}(1 + q^{m_{i}})}ϵ^{−d/4},  (5.2)

where m = (m_{1} … m_{d}). The series in the RHS has positive coefficients.

2. For a smooth toric variety X of even complex dimension d with Picard number 2, sign(q, LX) is 0 if s is even. Otherwise

sign(q, LX) = ∑_{m∈ℤ^{d}} \frac{2 + 2q^{m·(v_{1}+...v_{s})}}{∏_{i=1}^{r+1}(1 + q^{m·u_{i}}) ∏_{j=1}^{s}(1 + q^{m·v_{j}})}ϵ^{−d/4}  (5.3)

The series in the RHS has positive coefficients.

Remark. Statement 1. of Theorem 5.2 is an alternative to the chiral de Rham complex approach of sect. 4. But note that formula (5.2) is of a different nature than that in Theorem 4.1, and the comparison of (5.2) and Theorem 4.1 may give rise to non-trivial combinatorial identities.

Proof. Both of the statements follow from (5.1) and the explicit description of the fans of the toric varieties in question.

1. Let k_{1} = e_{1}, …, k_{d} = e_{d}, k_{d+1} = −e_{1} … − e_{d}. The fan defining ℙ^{d} consists of the cones spanned by all proper subsets of the set {k_{1}, …, k_{d+1}}.

Now observe that (5.1) rewrites as follows:

sign(q, LX) = ϵ^{−d/4} ∑_{m∈M} < m, ∑_{C^{∗}∈Σ}(−1)^{codimC^{∗}}( ∏_{k∈|C^{∗}|} \frac{1}{1 + q^{k}}) >,  (5.4)

where q^{k} is an element of the group ring of ℤ^{d} and < m, q^{k} >= q^{m·k}.

The expression

∑_{C^{∗}∈Σ}(−1)^{codimC^{∗}}( ∏_{k∈|C^{∗}|} \frac{1}{1 + q^{k}}),  (5.5)

(appearing in the r.h.s. of (5.4)) in the case of the projective space simplifies as follows:

∑_{B}(−1)^{d−#B} \frac{1}{∏_{k∈B}(1 + q^{k})},  (5.6)

where the summation is performed over all proper subsets B ⊊ {k_{1}, …, k_{d+1}} and #B is the number of elements in B. Converting (5.6) to the common denominator we obtain

\frac{∑_{B}(−1)^{d−#B} ∏_{k∈B̄}(1 + q^{k})}{∏_{i=1}^{d+1}(1 + q^{k_{i}})},  (5.7)

where B̄ denotes the complement of B. The multiple use of the binomial identuty

∑_{i=0}^{n}(−1)^{i}C_{n}^{i} = 0 if n > 0  (5.8)

allows us to collect the like terms in the numerator of (5.7). The result is

\frac{1 + (−1)^{d}q^{k_{1}+...+k_{n+1}}}{∏_{i=1}^{d+1}(1 + q^{k_{i}})} .  (5.9)

Since k_{1} + k_{2} + ⋯ + k_{d+1} = 0, it is zero if d is odd (as it should !) and

\frac{2}{∏_{i=1}^{d+1}(1 + q^{k_{i}})}  (5.10)

otherwise. Plugging (5.10) in the r.h.s. of (5.4) and performing pairing with m ∈ ℤ^{d} we obtain (5.2), as desired. It is clear that every term in the denominator of (5.2) cancels against the appropriate term in ϵ^{−d/4}. Indeed, for a fixed m ∈ ℤ^{d} the denominator of (5.2) contains at most d + 1 factors (1 + q^{n}) for each n ∈ ℤ whereas ϵ^{−d/4}, see (2.2), contains 2d such factors in the numerator. Therefore all the coefficients in the series sign(q, Lℙ^{d}) are positive.

2. Proof of the second statement is similar. Theorem 5.1 combined with (5.1) gives the following analogue of (5.4):

sign(q, LX) = ϵ^{−d/4} ∑_{m∈M} < m, ∑_{I,J} \frac{(−1)^{d−#I−#J}}{∏_{i∈I}(1 + q^{u_{i}}) ∏_{j∈J}(1 + q^{v_{j}})} >,  (5.11)

where I and J are proper subsets of {1, …, r + 1} and {1, …, s} respectively and the vectors u_{i}, v_{j} are those defined in the beginning of 5.2.

Converting the sum ∑_{I,J} in (5.11) to the common denominator we obtain

∑_{I,J} \frac{(−1)^{d−#I−#J} ∏_{i∈Ī}(1 + q^{u_{i}}) ∏_{j∈J̄}(1 + q^{v_{j}})}{∏_{i=1}^{r+1}(1 + q^{u_{i}}) ∏_{j=1}^{s}(1 + q^{v_{j}})}  (5.12)

Now observe that the numerator of (5.12) is the product of two factors analogous to the numerator of (5.7) – one is the numerator of (5.7) with B replaced with {u_{i}, i ∈ I}, another is also with B replaced with {v_{j}, j ∈ J}. Therefore identity (5.8), which allowed us to pass from (5.7) to (5.9), allows us to analogously rewrite (5.12) as follows:

\frac{(1 + (−1)^{r}q^{u_{1}+u_{2}+···+u_{r+1}})(1 + (−1)^{s−1}q^{v_{1}+v_{2}+···+v_{s}})}{∏_{i=1}^{r+1}(1 + q^{u_{i}}) ∏_{j=1}^{s}(1 + q^{v_{j}})}.  (5.13)

Since by definition (see the beginning of 5.1)

∑_{i=1}^{r+1} u_{i} = 0,

(5.13) vanishes if r is odd and equals

\frac{2(1 + q^{v_{1}+v_{2}+···+v_{s}})}{∏_{i=1}^{r+1}(1 + q^{u_{i}}) ∏_{j=1}^{s}(1 + q^{v_{j}})}  (5.14)

otherwise. (Note that (−1)^{s−1} has disappeared because, d being even, the relation d = r + s − 1 forces r and s to have different parity.)

Plugging (5.14) in the r.h.s. of (5.11) and performing pairing with m ∈ ℤ^{d} we obtain desired formula (5.3).

To show that series (5.3) has positive coefficients we observe that for a fixed m ∈ ℤ^{d} the denominator of (5.3) contains at most r + s + 1, that is, d + 2 factors (1 + q^{n}) for each n ∈ ℤ whereas ϵ^{−d/4}, see (2.2), contains 2d such factors in the numerator. Therefore, having carried out the cancellations we make (5.3) into a sum of power series with positive coefficients. □

6. Toric spin varieties

Recall Witten’s rigidity theorem [W] proved in [BT]. Let a torus T^{n} act on a manifold X. This action lifts to an action on the holomorphic bundle grΩ_{X}^{ch}, see (3.1) Therefore each cohomology group H^{i}(X, grΩ_{X}^{ch}) becomes a T^{n}-module – a direct sum of the torus characters t ↦ t^{m}, t ∈ T^{n}, m ∈ ℤ^{n}, in fact. Formula (3.4) then implies that sign(q, LX) is a formal sum of the torus characters and one can think of sign(q, LX) as a function of t ∈ T^{n} with values in ℂ[[q]].

Theorem 6.1 [BT] If X is a spin manifold equipped with an action of a torus T^{n}, then sign(q, LX) is a constant function of t ∈ T^{n}.

Theorem 6.2 For a toric spin variety X of complex dimension d,

sign(q, LX) = signX \frac{ϵ^{− \frac{d}{4}}}{2^{d}} .

Proof. Any toric variety X carries the natural action of a torus; hence sign(q, LX) is a formal sum of the torus characters. Formula (5.1) sharpens accordingly [BL]:

sign(q, LX) = ∑_{m∈M} t^{m} ∑_{C^{∗}∈Σ}(−1)^{codimC^{∗}}( ∏_{i=1,...dimC^{∗}} \frac{1}{1 + q^{m·n_{i}}})ϵ^{−d/4}. (6.1)

Theorem 6.1 implies that only t^{0} may appear in the r.h.s. of (6.1) with non-zero coefficient. Therefore

sign(q, LX) = Cϵ^{−d/4}  (6.2)

for some constatnt C. To compute C recall that

sign(0, LX) = signX.

Therefore, having specialized (6.2) to q = 0 we obtain

signX = C2^{d},

and Theorem 6.2 follows. □

Remarks.

1) According to [HS] sign(q, LX) = signX if X is a homogeneous space and a spin manifold at the same time. Therefore Theorem 6.2 is an extension of this result to spin toric varieties.

2) Theorem 6.2 suggests that perhaps the vertex algebra of 2d bosons coupled to 2d fermions provides a natural realization of sign(q, LX) in the case of a toric spin manifold as discussed in greater detail in the introduction .

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