THE CHARACTER OF THE INFINITE WEDGE REPRESENTATION

SPENCER BLOCH AND ANDREI OKOUNKOV

A man’s character is his fate.

– Heraclitus, 540-480 B.C.

Abstract. We study the character of the infinite wedge projective representation of the algebra of differential operators on the circle. We prove quasi-modularity of this character and also compute certain generating functions for traces of differential operators which we call correlation functions. These correlation functions are sums of determinants built from genus 1 theta functions and their derivatives.

Contents

0. Introduction 2 1. The Infinite Wedge Representation 9 2. Elliptic transformation of the character Ω 14 3. Quasimodular forms 17 4. Quasimodularity for characters Ω and V 20 5. Preliminaries on partitions 24 6. The formula for correlation functions 26 7. Beginning of the proof of Theorem 6.1 31 8. Difference equations for the correlation functions 34 9. Singularities of correlation functions 39 10. Difference equations for the RHS in (6.5) 43 11. Singularities of the RHS in (6.5) 47 12. Conclusion of the proof of Theorem 6.1 52 13. Another Example 53 References 56

The authors are supported in part by the NSF.

0. Introduction

The purpose of this paper is to study the character associated to a certain basic projective representation, the infinite wedge representation [9], of the Lie algebra D of differential operators on the circle. Concretely, we write

D_{n} := (t\frac{d}{dt})^{n} , n ≥ 0 (0.1)

acting as differential operators on the ring R = ℂ[t, t^{−1}] of Laurent polynomials. (Here D_{0} = 1 is the identity operator. We occasionally write D = D_{1}.) D has a vector space basis t^{m}D_{n}, m ∈ ℤ, n ≥ 0.

The Witt algebra of derivations of R, W = Der(R) ⊂ D is spanned by t^{m}D, m ∈ ℤ. These algebras are graded, with t^{m}D_{n} having weight m. One has a central extension [10] 0 → ℂ · c → D̂ → D → 0 inducing the Virasoro central extension Ŵ of W, and these central extensions are also graded, with c having degree 0. Certain graded highest weight representations of Ŵ arise in connection with conformal field theory and have been of considerable interest to physicists. These representations have the property that for a suitable choice D̃ ∈ Ŵ lifting D, the character

Trace(qD̃)

is well-defined (i.e. the action of D̃ is semi-simple with finite eigenspaces) and is the q-expansion of a modular form [16]. Notice that a different choice of lifting of D to Ŵ multiplies the character by q^{a} for some constant a. In particular, there is at most one lifting for which the character is modular.

For representations of D̂ one may consider the Abelian subalgebra

H := ℂD_{0} ⊕ ℂD_{1} ⊕ ℂD_{2} ⊕ … ⊂ D. (0.2)

Suppose the action of H is semi-simple with finite dimensional simultaneous eigenspaces. If we choose liftings D̃_{n} ∈ D̂ of D_{n} we may define the character

Ω(q_{0}, q_{1}, …) := Trace (qD̃_{0} ; 0qD̃_{1} ; 1qD̃_{2} ; 2 ⋯) . (0.3)

It is also frequently convenient to write q_{r} = e^{2πiτ_{r}} so the character becomes (with abuse of notation)

(0.4) Ω(τ_{0}, τ_{1}, …) := Trace (exp (2πi ∑_{r≥0} D̃_{r}τ_{r})) .

Assuming the eigenspaces of D_{1} are themselves finite dimensional, we may specialize the τ_{r} ↦ 0 for r ≠ 1 and develop Ω(τ_{0}, τ_{1}, …) in a formal Taylor series expansion in τ_{0}, τ_{2}, … with coefficients functions in τ_{1}:

Ω(τ_{0}, τ_{1}, …) = ∑_{A} ω_{A}(τ_{1})τ^{A}/A! . (0.5)

Here A = (a_{0}, a_{2}, a_{3}, …) with almost all a_{j} = 0, and τ^{A}/A! is multiindex notation. Finally, D̃_{0} is central in D̂ so its eigenspaces are stable under the D̂ action. We write

(0.6) V (q_{1}, q_{2}, …) = Coeff. of q_{0}^{0} in Ω(q_{0}, q_{1}, …)

for the character of the D̃_{0} = 0 eigenspace. Again, by abuse of notation, we also write V (τ_{1}, τ_{2}, …) and we expand in a series

V = ∑_{B=(b_{2},b_{3},...)} v_{B}(τ_{1})τ^{B}/B! , (0.7)

where, by construction,

\frac{v_{B}(τ_{1})}{(2πi)^{b_{2}+b_{3}+...}} = Trace|_{D̃_{0}=0} (qD̃_{1} ; 1 ∏_{k=2}^{∞} \binom{}{D_{k}}^{b_{k}}) . (0.8)

Here the trace is taken in the D̃_{0} = 0 subspace.

In the case of the infinite wedge representation, these characters have the following shape

(0.9) Ω(q_{0}, q_{1}, q_{2}, …) =

q_{1}^{−ξ(−1)}q_{3}^{−ξ(−3)} ⋯ ∏_{r≥0}(1 + q_{0}q_{1}^{r+ \frac{1}{2}} q_{2}^{(r+ \frac{1}{2} )^{2}} ⋯)(1 + q_{0}^{−1}q_{1}^{r+ \frac{1}{2}} q_{2}^{−(r+ \frac{1}{2} )^{2}} ⋯)

(0.10) V (q_{1}, q_{2}, …) = q_{1}^{−ξ(−1)}q_{3}^{−ξ(−3)} ⋯

× ∑_{(m|n)} q_{1}^{∑_{j}(m_{j}+ \frac{1}{2} )+(n_{j}+ \frac{1}{2} )}q_{2}^{∑_{j}(m_{j}+ \frac{1}{2} )^{2}−(n_{j}+ \frac{1}{2} )^{2}}q_{3}^{∑_{j}(m_{j}+ \frac{1}{2} )^{3}+(n_{j}+ \frac{1}{2} )^{3}} ⋯

Here

ξ(s) := ∑_{n≥1} (n − \frac{1}{2})^{−s} = (2^{s} − 1)ζ(s). (0.11)

Also, the sums are over all

(m|n) = (m_{1}, m_{2}, …, m_{a}|n_{1}, …, n_{a}) (0.12)

m_{a} > m_{a−1} > ⋯ > m_{1} ≥ 0, n_{a} > n_{a−1} > ⋯ > n_{1} ≥ 0 .

Such data are Frobenius coordinates for partitions λ (cf. [13], p. 3.) In particular, the exponents in (0.10) are functions on the set of partitions. We define

p_{r}(λ) := ∑_{j} (m_{j} + \frac{1}{2})^{r} + (−1)^{r+1} (n_{j} + \frac{1}{2})^{r} . (0.13)

Thus p_{0}(λ) = 0 and p_{1}(λ) = |λ| is the number being partitioned.

The formulas (0.9) and (0.10) appear (without the anomaly terms) in [4] and [1]. The anomaly terms depend on a choice of liftings of the D_{n} to D̂. For representations of the Virasoro algebra, this lifting is determined by modularity. The correct analog, quasimodularity, for representations of D̂ was suggested by Dijkgraaf [3] based on an interpretation of V (τ_{1}, τ_{2}, 0, 0, …) as a generating function counting covers of a fixed elliptic curve E with given degree and genus. From this point of view, a mirror symmetry argument suggested an interpretation in terms of (nonholomorphic) modular forms on the dual moduli space. More precisely, note that in (0.5) and (0.7) the variable τ_{1} plays a distinguished role. The analogue of modularity involves the behavior of Ω or V under the transformations

τ_{1} ↦ \frac{aτ_{1} + b}{cτ_{1} + d}, τ_{j} ↦ \frac{τ_{j}}{(cτ_{j} + d)^{j+1}} , j ≠ 1 , (0.14)

for (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ, where Γ ⊂ SL_{2}(ℤ) is some subgroup of finite index.

Given C = (c_{0}, c_{2}, c_{3}, …) (resp. C = (c_{2}, c_{3}, …)) with almost all c_{i} = 0 define the weight of C

wt(C) := ∑(i + 1)c_{i} . (0.15)

To say that the ω_{A}(τ_{1}) (resp. the v_{B}(τ_{1})) were modular of weight wt(A) + w (resp. wt(B) + w) for some w would imply that Ω (resp. V) was modular of weight w under the transformation (0.14). This doesn’t happen in the examples we know. Instead, the ω_{A} and v_{B} are quasimodular of these weights. The notion of quasimodular form is developed in [11], where a rigorous proof of quasimodularity for the ω_{A} (resp. v_{B}) is given for A = (a_{0}, a_{2}, 0, 0, …) (resp. B = (b_{2}, 0, 0, …)). By definition, the ring of quasimodular forms is the graded algebra generated over the ring of modular forms by the Eisenstein series of weight 2

G_{2}(q) := −\frac{B_{2}}{4} + ∑_{n=1}^{∞} ∑_{d|n} dq^{n}.

Unlike the ring of modular forms, this ring is closed under the derivation q \frac{d}{dq}. One can canonically associate to a quasimodular form a certain nonholomorphic (almost holomorphic in the terminology of op. cit.) modular form.

Definition 0.1. A series F = ∑_{A} f_{A}(τ_{1})τ^{A}/A! for A = (a_{0}, a_{2}, …) is quasimodular of weight w if each f_{A}(τ_{1}) is quasimodular of weight w + wt(A).

Theorem 0.2. Ω(τ_{0}, τ_{1}, …) (resp. V (τ_{1}, τ_{2}, …)) is quasimodular of weight 0 (resp. weight −\frac{1}{2}).

Although the anomaly factors q_{2r+1}^{−ξ(−2r−1)} are uniquely determined by quasimodularity, they can also be understood solely in representationtheoretic terms. Roughly speaking, if one tries to construct the infinite wedge representation as a representation of D (rather than D̂) one is lead to meaningless infinite constants of the form ∑_{n≥1} (n − \frac{1}{2})^{r} , r ≥ 1. Regularizing these sums via analytic continuation of ξ(s) and using the fact that ξ(−2r) = 0 for r ≥ 0 leads to the stated values.

The analogy between (0.9) and the triple product formulas for the genus 1 theta functions continues with the following elliptic transformation formula for Ω(τ_{0}, τ_{1}, …).

Theorem 0.3. Define the transformation T by

T (τ_{j}) = τ_{j} − \binom{j + 1}{1}τ_{j+1} + \binom{j + 2}{2}τ_{j+2} − …,

and set q′_{i} = exp(2πiT (τ_{i})). Then we have

Ω(q′_{0}, q′_{1}, q′_{2}, …) = q_{0}q_{1}^{− \frac{1}{2}} q_{2}^{+ \frac{1}{3}} q_{3}^{− \frac{1}{4}} ⋯ Ω(q_{0}, q_{1}, …)

The above results constitute a pleasant but perhaps not terribly surprising generalization of the work in [3] and [11]. The character V possesses some other hidden structure and relation to the genus 1 theta functions, which seems to us quite surprising and new.

The idea is to replace the bulky generating function (0.10) for the quantities (0.8) by other generating functions, where the auxiliary variables are attached not to the exponents b_{i} in (0.8) but to the indices k of D̃_{k}. These new generating functions admit a neat evaluation in terms of genus 1 theta functions.

We call these generating functions n-point correlation functions because both by their definition (as averages of product of n generating series) and by their analytic structure (determinants built from theta functions and their derivatives) these functions closely resemble correlation functions in QFT. However, we were not able to find any precise connection and in the present text we work with these functions using solely the classical methods of analysis.

Concretely, the definition of these n-point functions is the following. Let f(λ) be a function on partitions, and define

⟨f⟩_{q} := ∑_{λ} f(λ)q^{|λ|}/ ∑_{λ} q^{|λ|} . (0.16)

That is, ⟨f⟩_{q} is the expectation of the function f provided the the probability of each partition λ is proportional to q^{|λ|}. Then, the equations (0.8), (0.10) and (0.12) can be restated as follows:

Trace|_{D̃_{0}=0} (qD̃_{1} ; 1 ∏_{k=2}^{∞} \binom{}{D_{k}}^{b_{k}}) = η(q_{1}) ⟨∏_{k=2}^{∞} (p_{k}(λ) − ξ(−k))^{b_{k}}⟩_{q_{1}} . (0.17)

One checks (see section 5) that

p_{k}(λ) − ξ(−k) = (t\frac{d}{dt})^{k} (∑_{i=1}^{∞} t^{λ_{i}−i+1/2} − \frac{1}{log t})|_{t=1} . (0.18)

By definition, set for n = 1, 2, 3, …

F (t_{1}, …, t_{n}; q) = ⟨∏_{k=1}^{n} (∑_{i=1}^{∞} t_{k}^{λ_{i}−i+ \frac{1}{2}} )⟩_{q} , (0.19)

that is, F (t_{1}, …, t_{n}; q) is the expectation of the product of n generating series for the quantities (0.18). By linearity, all quantities (0.17) satisfying b_{2} + b_{3} + ⋯ ≤ m for some m are encoded in the functions (0.19) with n ≤ m.

Definition 0.4. The functions (0.19) are called n-point correlation functions.

The equation (0.18) can be restated as follows. Write t_{k} = e^{u_{k}} and define a differential operator

δ(u) := \frac{1}{u} + (2πi)^{−1} ∑_{r=1}^{∞} \frac{∂}{∂τ_{r}} u^{r}/r! . (0.20)

Then

⟨ ∏_{k=1}^{n} ( ∑_{i=1}^{∞} t_{k}^{λ_{i}−i+ \frac{1}{2}} )⟩_{q_{1}} = η(q_{1})δ(u_{1}) ∘ … ∘ δ(u_{n})V |_{τ_{2}=τ_{3}=...=0}. (0.21)

This is also proved in section 5.

The form of the singular term 1/u and in (0.20) and the corresponding term in (0.18) very much depends on our choice of anomaly factors in the character.

Write q = q_{1}, write F (t_{1}, …, t_{n}) for the n-point function, and write Θ(t) = Θ(t; q) for the following theta function:

Θ(t) := η(q)^{−3} ∑_{n∈ℤ}(−1)^{n}q^{\frac{(n+ \frac{1}{2} )^{2}}{2}} x^{n+ \frac{1}{2}} = (q)_{∞}^{−2}(x^{\frac{1}{2}} − x^{− \frac{1}{2}} )(qx)_{∞}(q/x)_{∞} .

Let Θ^{(p)}(t) = (t\frac{d}{dt})^{p}Θ(t). Then our main result on the n-point function is following:

Theorem 0.5.

F (t_{1}, …, t_{n}) = ∑_{σ∈𝔖(n)} \frac{det (\frac{Θ^{(j−i+1)}(t_{σ(1)} ⋯ t_{σ(n−j)})}{(j − i + 1)!})}{Θ(t_{σ(1)}) Θ(t_{σ(1)}t_{σ(2)}) … Θ(t_{σ(1)} ⋯ t_{σ(n)})} . (0.22)

Here σ runs through all permutations 𝔖(n) of {1, …, n}, the matrices in the numerator have size n × n, and we define 1/(−n)! = 0 if n ≥ 1.

For the 1-point function, this is simply

⟨ ∑_{i=1}^{∞} t^{λ_{i}−i+ \frac{1}{2}} ⟩_{q} = \frac{1}{Θ(t)} ,

because

Θ′(1) = 1 .

For n = 3 the equation (0.22) becomes:

(0.23) F (t_{1}, t_{2}, t_{3}) =

\frac{1}{Θ(t_{1}t_{2}t_{3})} ∑_{σ∈𝔖(3)}det ( \begin{matrix}\frac{Θ′ (t_{σ(1)}t_{σ(2)})}{Θ (t_{σ(1)}t_{σ(2)})} & \frac{1}{2} \frac{Θ′′ (t_{σ(1)})}{Θ (t_{σ(1)})} & \frac{Θ′′′(1)}{3!} \\ 1 & \frac{Θ′ (t_{σ(1)})}{Θ (t_{σ(1)})} & 0 \\ & 1 & 1 \end{matrix} )

The traces (0.17) can be computed from (0.22) using repeated differentiation, L’Hospital’s rule and formulas for the derivatives Θ^{(p)}(1; q) in terms of the Eisenstein series:

Θ^{(3)}(1; q) = −6G_{2}(q) ,
Θ^{(5)}(1; q) = −10G_{4}(q) + 60G_{2}(q)^{2} ,
Θ^{(7)}(1; q) = −14G_{6}(q) + 420G_{4}(q)G_{2}(q) − 840G_{2}(q)^{3} ,

and so on, see (6.7). Note that all even order derivatives of the odd function Θ(x) vanish at x = 1.

Sections 6-12 are devoted to the proof of (0.22). An essential ingredient of this proof are the following q-difference equations for the functions F (t_{1}, …, t_{n}) which is established in section 8:

Theorem 0.6. For n = 1, 2, … we have

F (qt_{1}, t_{2}, …, t_{n}) = −q^{1/2}t_{1} … t_{n} ×

∑_{s=0}^{n−1}(−1)^{s} ∑_{1<i_{1}<···<i_{s}≤n} F (t_{1}t_{i_{1}}t_{i_{2}} ⋯ t_{i_{s}}, …, t̂_{i_{1}}, …, t̂_{i_{s}}, …) .

These equations are analogs of elliptic transformations for the npoint functions F (t_{1}, …, t_{n}) .

Here is another point of view about these things. Let A be an associative algebra with 1, and let ρ : A → End(F) be a representation of the associated Lie algebra (with commutator bracket). We do not assume ρ compatible with the associative algebra structures. Let D ∈ A be given and assume ρ(D) is semisimple with finite dimensional eigenspaces. Let F_{a} ⊂ F be the eigenspace of ρ(D) with eigenvalue a. Since [D, D^{n}] = 0, ρ(D^{n}) stabilizes F_{a}, and it makes sense to consider for a variable u the expression

f_{ρ}(u, z) := χ_{ρ}(z)^{−1} ∑_{a} (Tr_{F_{a}} ∑_{n≥0} ρ(D^{n})u^{n}/n!)e^{az} . (0.24)

where χ_{ρ}(x) := Tre^{xρ(D)} is the trace of ρ. If in fact ρ(D^{n}) = ρ(D)^{n} for all n, we find

f_{ρ}(u, z) = χ_{ρ}(z)^{−1} ∑_{a} Tr_{F_{a}}e^{at}e^{az} = χ_{ρ}(u + z)/χ_{ρ}(z) . (0.25)

More generally, letting n = (n_{1}, …, n_{r}) run through r-tuples of nonnegative integers and replacing ρ(D^{n}) with ∏_{i} ρ(D^{n_{i}}) and u^{n}/n! with ∏ u_{i}^{n_{i}}/n_{i}! we may define f_{ρ}(u_{1}, …, u_{r}; z). When ρ is compatible with associative multiplication, we have f_{ρ} = χ_{ρ}(u_{1} + … + u_{r} + z)/χ_{ρ}(z).

When ρ is a projective representation, ρ(D^{n}) is well defined only up to adding α_{n} · Id. Such a modification replaces f_{ρ}(u, z) with f_{ρ}(u, z) + (∑_{n} α_{n}u^{n}/n!). If H is spanned by the powers of D as in (0.2) we find writing χ_{ρ}(τ_{0}, τ_{1}, τ_{2}, …) for the full character that

(0.26) f_{ρ}(u_{1}, …, u_{r}; τ_{1}) =

χ_{ρ}(τ_{1})^{−1} ∑_{n_{1},... ,n_{r}≥0} (∏_{i=1}^{r} \frac{∂}{∂τ_{n_{i}}} ) χ_{ρ}(τ_{0}, τ_{1}, τ_{2}, …)|_{τ_{0}=τ_{2}=...=0} ∏_{i} u_{i}^{n_{i}}/n_{i}! .

For the representation (0.6), this is (0.21).

Finally, section 13 proves quasimodularity and computes the n-point function for a representation of a subalgebra of D studied in [2].

The first author would like to acknowledge considerable inspiration from conversations and correspondence with V. Kac and D. Zagier. It was Zagier who suggested that these characters might be quasimodular.

1. The Infinite Wedge Representation

Material in this section is quite well known. References are [7], [8], [15], [4]. We work throughout with vector spaces, Lie algebras, and representations of Lie algebras over the field ℂ. Recall that a (projective) representation of a Lie algebra L is a Lie algebra homomorphism L → End(F); (resp. L → End(F)/ℂ · Id) for a vector space F. A projective representation gives rise to a central extension of Lie algebras via pullback:

0 ⟶ ℂ · c ⟶ L̃ ⟶  L  ⟶ 0
↓  ↓  ↓
0 ⟶ ℂ · Id ⟶ End(F) ⟶ End(F)/ℂ · Id ⟶ 0 .

(1.1)

Let V be a vector space with basis v_{n}, n ∈ ℤ. Define the finite matrices

A_{0} := ⨁_{i,j∈ℤ}ℂE_{ij} ⊂ End(V) . (1.2)

where E_{ij}(v_{i}) = v_{j} and E_{ij}(v_{k}) = 0; k ≠ i. Let A ⊃ A_{0} be the larger space of all matrices supported in a bounded strip. An infinite matrix M = ∑_{infinite} a_{ij}E_{ij} lies in A if and only if there exists a constant c such that a_{ij} = 0 if |i − j| > c. For example, Id ∈ A − A_{0}. Suppose given a representation α_{0} of A_{0} which extends to a projective representation α of A:

A_{0}  ↪^{ι} A
↓α_{0} ↙σ ↓α
End(F) ↠ End(F)/ℂ · Id .

(1.3)

Let σ be a lifting of α which is a map of vector spaces, not necessarily compatible with Lie algebra structures and not necessarily extending α_{0}. Assume, however, that σ is continuous in the sense that for any f ∈ F and any x = ∑_{infinite} x_{ij}ι(E_{ij}) ∈ A there exists N = N(x, f) such that x_{ij}σι(E_{ij})(f) = 0 for |i| + |j| > N, and

σ(x)(f) = ∑_{|i|+|j|≤N} x_{ij}σι(E_{ij})(f) .

Notice that such a σ is unique up to a finite modification. More precisely, if σ′ is another map with the same property, σι(E_{ij}) = σ′ι(E_{ij}) for almost all i, j. We obtain in this way a map

ϵ := α_{0} − σ ∘ ι : A_{0} → ℂ · Id . (1.4)

The functional ϵ is uniquely determined (up to a finite modification) by α_{0}, α.

Definition 1.1. Let S be a collection of sequences of complex numbers c = {c_{ij}}_{i,j∈ℤ}. A regularization scheme for S is a map r : S → ℂ such that if {c_{ij}}, {c′_{ij}} ∈ S differ for only finitely many pairs i, j, then

r({c_{ij}}) = r({c′_{ij}}) + ∑_{i,j}(c_{ij} − c′_{ij}).

(We will usually assume S is saturated in the sense that if c ∈ S and c′ differs from c for only finitely many i, j then c′ ∈ S.)

Assume now we have α, α_{0}, and that there exists a σ as in (1.3) above. Let Â be the corresponding central extension of A. Let S, r be as in definition (1.1). Suppose given x = ∑ x_{ij}ι(E_{ij}) ∈ A, such that {x_{ij}ϵ(E_{ij})} ∈ S, where ϵ is defined in (1.4). Then

σ(x) + r({x_{ij}ϵ(E_{ij})}) · Id ∈ End(F)

is well-defined independent of the choice of σ. Thus we have a lifting x̂ ∈ Â depending on the regularization scheme r but not on the choice of lifting σ.

Example 1.2. For the infinite wedge representation,

ϵ(E_{ij}) = \begin{cases}1 & i = j ≤ 0 , \\ 0 & else . \end{cases} (1.5)

The elements x we want to lift have the form x = ∑_{m∈ℤ} (m − \frac{1}{2})^{k} E_{mm} for k = 0, 1, 2, …, so the sequences we need to regularize are

\{(m − \frac{1}{2})^{k}\}_{m≤0} .

The natural way to do this is by analytic continuation of the function

ξ(s) = ∑_{n≥1} (n − \frac{1}{2})^{−s} (1.6)

We define r(\{(m − \frac{1}{2})^{k}\}_{m≤0}) = (−1)^{k}ξ(−k) .

The infinite wedge representation F := Λ^{∞}V is the vector space with basis

v_{I} = v_{i_{m}} ∧ v_{i_{m−1}} ∧ v_{i_{m−2}} ∧ …, (1.7)

where i_{k−1} < i_{k} and i_{k} = k for k ≪ 0. By definition m is the charge of v_{I} and

energy(v_{I}) := ∑_{k=m}^{k=−∞}(i_{k} − k) .

For example, the elements

|m⟩ := v_{m} ∧ v_{m−1} ∧ v_{m−2} ∧ … (1.8)

have charge m and energy 0. The wedge here is formal, but we can use the expected alternating linearity to make sense for A ∈ A_{0} of expressions like

Av_{I} := ∑_{k=m}^{k=−∞} v_{i_{m}} ∧ … ∧ Av_{i_{k}} ∧ v_{i_{k−1}} ∧ …. (1.9)

We obtain a representation α_{0} : A_{0} → End(F). To extend α_{0} to A, we would have to define expressions like

∑_{p∈ℤ}a_{p}E_{p,p+r}(v_{I}) . (1.10)

When r ≠ 0 it is easy to see that E_{p,p+r}v_{I} = 0 for |p| >> 0, so (1.10) makes sense. Similarly E_{p,p}v_{I} = 0 for p >> 0. Define σ : A_{0} → End(F) by

σ(E_{ij}) = \begin{cases}α_{0}(E_{ij}) & i ≠ jor i = j > 0 \\ α_{0}(E_{ii}) − Id & i ≤ 0 \end{cases} (1.11)

It is straightforward to check that for any r and any I there exists an N such that σ(E_{p,p+r})(v_{I}) = 0 for |p| ≥ N, so σ extends to a map A → End(F) as in (1.3). Define α to be the composition

A →^{σ} End(F) → End(F)/ℂ · Id .

Lemma 1.3. The map α is a projective representation of A.

Proof. The assertion is there exists a bilinear map a : A × A → ℂ with

[σ(x), σ(y)] = σ([x, y]) + a(x, y)Id.

One defines

a(E_{ij}, E_{kℓ}) = \begin{cases}1 & i = ℓ ≤ 0 and j = k ≥ 1 , \\ −1 & i = ℓ ≥ 1 and j = k ≤ 0 , \\ 0 & else . \end{cases}

Because of the conditions on the signs of the indices, a extends to A×A as desired.

Given u ∈ ℂ, we have an action as differential operators of D on ℂ[t, t^{−1}]t^{u}. Identifying this space with V via t^{n+u} ↦ v_{n}, we get a mapping of associative algebras (and hence a fortiori of Lie algebras)

δ_{u} : D → A ⊂ End(V) ,

δ_{u}(t^{p}D^{r}) = ∑_{n∈ℤ}(n + u)^{r}E_{n,n+p} .

We define a projective representation

ρ_{u} := α ∘ δ_{u} : D → End(F)/ℂ · Id ,
ρ := ρ_{− \frac{1}{2}} .

We will customarily view ρ as a representation on a central extension, ρ : D̂ → End(F). Our objective now is to compute the character of the representation ρ_{u}. Define operators ψ_{−r}^{∗}, r ≥ 0 and ψ_{−r}, r > 0 on F by

ψ_{−r}(v_{I}) := v_{r} ∧ v_{I}; ψ_{−r}^{∗}(… ∧ v_{−r} ∧ …) = … ∧ v̂_{−r} ∧ ….

A basis for F can be written

ψ_{−i_{1}} ⋯ ψ_{−i_{a}}ψ_{−j_{1}}^{∗} ⋯ ψ_{−j_{b}}^{∗}|0⟩ , 0 < i_{1} < i_{2} < … < i_{a}, 0 ≤ j_{1} < j_{2} < … < j_{b} .

We have

σ ∘ δ_{u}(D_{r})(ψ_{−i_{1}} ⋯ ψ_{−i_{a}}ψ_{−j_{1}}^{∗} ⋯ ψ_{−j_{b}}^{∗}|0⟩) =

σ (\begin{matrix}. & . & . \\ & & & (n + u)^{r} \\ & & & & . & . & . \end{matrix}) (ψ_{−i_{1}} ⋯ ψ_{−i_{a}}ψ_{−j_{1}}^{∗} ⋯ ψ_{−j_{b}}^{∗}|0⟩) =

( ∑_{ℓ=1}^{a}(i_{ℓ} + u)^{r} − ∑_{m=1}^{b}(−j_{m} + u)^{r})(ψ_{−i_{1}} ⋯ ψ_{−i_{a}}ψ_{−j_{1}}^{∗} ⋯ ψ_{−j_{b}}^{∗}|0⟩).

For example, writing q_{r} = exp(2πiτ_{r}) we get

exp (2πi ∑_{r≥0} τ_{r} · σ ∘ δ_{u}(D_{r}))(ψ_{−n}|0⟩) = q_{0}q_{1}^{n+u}q_{2}^{(n+u)^{2}} ⋯,

exp (2πi ∑_{r≥0} τ_{r} · σ ∘ δ_{u}(D_{r}))(ψ_{−n}^{∗}|0⟩) = q_{0}^{−1}q_{1}^{n+u}q_{2}^{−(n+u)^{2}} · · · .

On all of F we find

Tr exp (2πi ∑_{r≥0} τ_{r} · σ ∘ δ_{u}(D_{r})) =

∏_{n≥0} (1 + q_{0}q_{1}^{n+u+1}q_{2}^{(n+u+1)^{2}} ⋯)(1 + q_{0}^{−1}q_{1}^{n−u}q_{2}^{−(n−u)^{2}} ⋯).

We now specialize to the case u = −\frac{1}{2}. The reason why this particular value of u yields a quasimodular character is not clear, but it may have to do with the fact that for u = p + \frac{1}{2} ∈ ℤ + \frac{1}{2} the space V = ℂ[t, t^{−1}]t^{u} has a nondegenerate, symmetric bilinear form

(t^{n+u}, t^{m+u}) := res_{t=0}t^{n+m+2u−1}dt.

Further, if we consider the involution

σ : D → D; σ(t^{a}D^{b}) := −t^{a}(−D − a)^{b},

we find

σ^{2} = I; σ[x, y] = [σ(x), σ(y)]; (x(a), b) + (a, σ(x)(b)) = 0.

With reference to example (1.2), we note that ξ(s) = (2^{s} − 1)ζ(s) vanishes at s = 0, −2, −4, … We therefore define the character of the infinite wedge representation of D to be

(1.12) Ω(q_{0}, q_{1}, …) = Ω(τ_{0}, τ_{1}, …) =

q_{1}^{−ξ(−1)}q_{3}^{−ξ(−3)} ⋯ ∏_{n≥0} (1+q_{0}q_{1}^{n+ \frac{1}{2}} q_{2}^{(n+ \frac{1}{2} )^{2}} ⋯)(1+q_{0}^{−1}q_{1}^{n+ \frac{1}{2}} q_{2}^{−(n+ \frac{1}{2} )^{2}} ⋯) .

Note that δ_{u}(D_{0}) = ∑_{i∈ℤ} E_{ii} is the charge operator

δ_{u}(D_{0})ψ_{−i_{1}} ⋯ ψ_{−i_{a}}ψ_{−j_{1}}^{∗} ⋯ ψ_{−j_{b}}^{∗}|0⟩ = (a − b)ψ_{−i_{1}} ⋯ ψ_{−i_{a}}ψ_{−j_{1}}^{∗} ⋯ ψ_{−j_{b}}^{∗}|0⟩.

Operators in D preserve the charge, so the charge eigenspaces are stable under ρ_{u}. For u = −\frac{1}{2}, the characters of the subrepresentation of charge n is the coefficient of q_{0}^{n} in (1.12). Of particular interest is the charge 0 part. Using (0.10) and (0.13), this can be written

(1.13) V (q_{1}, q_{2}, …) = V (τ_{1}, τ_{2}, …) :=

q_{1}^{−ξ(−1)}q_{3}^{−ξ(−3)} ⋯ {Coeff. of q_{0}^{0} in Ω(q_{0}, q_{1}, …)} =

q_{1}^{−ξ(−1)}q_{3}^{−ξ(−3)} ⋯ ∑_{λ} q_{1}^{p_{1}(λ)}q_{2}^{p_{2}(λ)} · · · .

Here the sum is over all partitions λ.

2. Elliptic transformation of the character Ω

In this section we want to expose some interesting analogies between Ω(q_{0}, q_{1}, q_{2}, …) (1.12) and the classical genus 1 theta function, which we will write

θ(q_{0}, q_{1}) := ∑_{n∈ℤ}q_{0}^{n}q_{1}^{n^{2}/2}. (2.1)

The triple product formula (cf. [14], p. 70) implies

(2.2) θ(q_{0}, q_{1}) = ∏_{m≥1}(1 − q_{1}^{m}) ∏_{n≥0}(1 + q_{0}q_{1}^{n+ \frac{1}{2}} )(1 + q_{0}^{−1}q_{1}^{n+ \frac{1}{2}} ) =

η(q_{1})Ω(q_{0}, q_{1}, 1, 1, …).

Here η(q_{1}) = q1^{\frac{1}{24}} ∏_{m≥1}(1 − q_{1}^{m}) is the classical eta function, and we have used the fact that the first anomaly factor in Ω is given by

q_{1}^{−ξ(−1)} = q_{1}^{−(2^{−1}−1)ζ(−1)} = q_{1}^{− \frac{1}{24}} .

We want to generalize to Ω the elliptic transformation law

θ(q_{0}q_{1}^{−1}, q_{1}) = q_{0}q_{1}^{− \frac{1}{2}} θ(q_{0}, q_{1}). (2.3)

For this, we define infinite matrices

U := (\begin{matrix}0 & −1 & 0 & 0 & … \\ 0 & 0 & −2 & 0 & … \\ 0 & 0 & 0 & −3 & … \\ ⋮ & ⋮ & ⋮ & ⋮ \end{matrix}) (2.4)

and

T := exp(U) = (\begin{matrix}1 & −1 & 1 & −1 & 1 & … \\ 0 & 1 & −\binom{2}{1} & \binom{3}{2} & −\binom{4}{3} & … \\ 0 & 0 & 1 & −\binom{3}{1} & \binom{4}{2} & … \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \end{matrix}) . (2.5)

Define for j ≥ 0

τ′_{j} := T (τ_{j}) = τ_{j} − \binom{j + 1}{1}τ_{j+1} + \binom{j + 2}{2}τ_{j+2} − …,

q′_{j} := T (q_{j}) = q_{j}q_{j+1}^{−\binom{j+1}{1}}q\binom{j+2}{2}_{j+2} · · · .

Theorem 2.1. Ω(q′_{0}, q′_{1}, q′_{2}, …) = q_{0}q_{1}^{− \frac{1}{2}} q_{2}^{+ \frac{1}{3}} q_{3}^{− \frac{1}{4}} ⋯ Ω(q_{0}, q_{1}, …).

Lemma 2.2. For n ≥ 2 we have

∑_{i=1}^{n−1}(−1)^{i}\binom{n}{i}ξ(i − n) = \frac{(−1)^{n+1}}{n + 1} + \frac{(−1)^{n}}{2^{n}} .

Proof of lemma. Recall one has Bernoulli numbers B_{n}, n ≥ 0 satisfying B_{0} = 1, B_{1} = −\frac{1}{2}, ζ(1 − n) = −\frac{B_{n}}{n} for n ≥ 2. Substituting and using the identity \frac{n+1}{j+1} \binom{n}{j} = \binom{n+1}{j+1}, the desired formula becomes

∑_{k=1}^{n}(−1)^{k−1}\binom{n + 1}{k}(1 − 2^{1−k})B_{k} =^{?} \frac{n + 1}{2^{n}} − 1 . (2.6)

Consider the Bernoulli polynomials

B_{N}(x) := ∑_{k=0}^{N} \binom{N}{k}B_{k}x^{N−k} ,

which may be defined by the generating function

∑_{N=0}^{∞} B_{N}(x)t^{N}/N! = \frac{te^{xt}}{e^{t} − 1} .

As a consequence we get

B_{N}(x + 1) − B_{N}(x) = Nx^{N−1} . (2.7)

Summing for −1 ≥ x ≥ −p we get for N = n + 1 and p ≥ 1 the equivalent identities

(n + 1) ∑_{ℓ=−1}^{−p} ℓ^{n} + B_{n+1}(−p) − B_{n+1}(0) = 0 ,

(n + 1) ∑_{ℓ=−1}^{−p} ℓ^{n} + (−p)^{n+1} + ∑_{k=1}^{n} B_{k}\binom{n + 1}{k}(−p)^{n+1−k} = 0 . (2.8)

(We are grateful to V. Kac for suggesting (2.8).) It is straightforward to deduce (2.6) from (2.8) taking p = 1, 2 .

Proof of theorem. The transformation T satisfies

T (q_{0})T (q_{1})^{s}T (q_{2})^{s^{2}} ⋯ = q_{0}q_{1}^{s−1}q_{2}^{(s−1)^{2}} ⋯,
T (q_{0})^{−1}T (q_{1})^{s}T (q_{2})^{−s^{2}} ⋯ = q_{0}^{−1}q_{1}^{s+1}q_{2}^{−(s+1)^{2}} · · · .

Note the second identity follows from the first, replacing s by −s and inverting. The first identity is left for the reader. It follows that

(2.9) ; \frac{∏_{n≥0} (1 + T (q_{0})T (q_{1})^{n+ \frac{1}{2}} ⋯)(1 + T (q_{0})^{−1}T (q_{1})^{(n+ \frac{1}{2} )}T (q_{2})^{−(n+ \frac{1}{2} )^{2}} ⋯)}{∏_{n≥0} (1 + q_{0}q_{1}^{n+ \frac{1}{2}} ⋯)(1 + q_{0}^{−1}q_{1}^{(n+ \frac{1}{2} )}q_{2}^{−(n+ \frac{1}{2} )^{2}} ⋯)}

= \frac{1 + q_{0}q_{1}^{− \frac{1}{2}} q2^{\frac{1}{4}} q_{3}^{− \frac{1}{8}} ⋯}{1 + q_{0}^{−1}q1^{\frac{1}{2}} q_{2}^{− \frac{1}{4}} ⋯} = q_{0}q_{1}^{− \frac{1}{2}} q2^{\frac{1}{4}} · · · .

As a consequence of the lemma we have

\frac{T (q_{1})^{−ξ(−1)}T (q_{3})^{−ξ(−3)} ⋯}{q_{1}^{−ξ(−1)}q_{3}^{−ξ(−3)} ⋯} = q_{2}^{\frac{1}{3} − \frac{1}{4}} q_{3}^{− \frac{1}{4} + \frac{1}{8}} q_{4}^{\frac{1}{5} − \frac{1}{16}} · · · . (2.10)

The proof follows by combining (2.9) and (2.10).

Recall we have defined V (q_{1}, q_{2}, …) to be the coefficient of q_{0}^{0} in Ω(q_{0}, q_{1}, …).

Theorem 2.3. We have the following series expansion for Ω:

Ω(q_{0}, q_{1}, q_{2}, …) = ∑_{n=−∞}^{n=∞} V (T^{−n}(q_{1}), T^{−n}(q_{2}), …)q_{0}^{n}q_{1}^{n^{2}/2}q_{2}^{n^{3}/3} · · · .

Proof. First note

T (q_{0})^{n}T (q_{1})^{n^{2}/2}T (q_{2})^{n^{3}/3} ⋯ = q_{0}q_{1}^{− \frac{1}{2}} q2^{\frac{1}{3}} ⋯ q_{0}^{n−1}q_{1}^{(n−1)^{2}/2} · · · . (2.11)

Write Ω = ∑_{−∞}^{∞} V_{n}(q_{1}, …)q_{0}^{n}q_{1}^{n^{2}/2} · · ·. Then

q_{0}q_{1}^{− \frac{1}{2}} q2^{\frac{1}{3}} ⋯ ∑_{n} V_{n−1}(q)q_{0}^{n−1}q_{1}^{(n−1)^{2}/2} ⋯ =

q_{0}q_{1}^{− \frac{1}{2}} q2^{\frac{1}{3}} ⋯ Ω(q) = Ω(T (q)) =

∑ V_{n}(T (q))T (q_{0})^{n}T (q_{1})^{n^{2}/2} ⋯ =

q_{0}q_{1}^{− \frac{1}{2}} q2^{\frac{1}{3}} ⋯ ∑_{n} V_{n}(T (q))q_{0}^{n−1}q_{1}^{(n−1)^{2}/2} · · · .

Taking coefficients of q_{0}^{n−1} we get V_{n}(T (q)) = V_{n−1}(q). The formula follows since V_{0} = V.

Example 2.4. Consider the formula in the theorem with q_{j} ↦ 1, j ≥ 2. We have

T (q)|_{q_{j}↦1,j≥2} = q_{1}

and by (0.13)

V (q_{1}, 1, 1, …) = q_{1}^{− \frac{1}{24}} ∑_{λ} q_{1}^{p_{1}(λ)} = q_{1}^{− \frac{1}{24}} ∏_{m≥1}(1 − q_{1}^{m})^{−1} = η(q_{1})^{−1}.

The assertion of the theorem is then

η(q_{1})^{−1} ∑_{n∈ℤ}q_{0}^{n}q_{1}^{n^{2}/2} = q1^{\frac{−1}{24}} ∏_{m≥0}(1 + q_{0}q_{1}^{m+ \frac{1}{2}} )(1 + q_{0}^{−1}q_{1}^{m+ \frac{1}{2}} ). (2.12)

Multiplying through by η(q_{1}) yields the triple product formula (2.2).

3. Quasimodular forms

In this section, we recall the theory of quasimodular forms as developed in [11]. All results are due to Kaneko and Zagier and are recalled here solely for the convenience of the reader.

We fix a subgroup of finite index Γ ⊂ Γ_{1} := SL(2, ℤ). A holomorphic modular form of weight k (for Γ) is a holomorphic function f(τ) on the upper half-plane H = {τ = x + iy | y > 0} satisfying

f (\frac{aτ + b}{cτ + d}) = (cτ + d)^{k}f(τ), (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ . (3.1)

Note for some ℓ ≥ 1 we have ( \begin{matrix}1 & ℓ \\ 0 & 1 \end{matrix} ) ∈ Γ, so f may be expanded in a Fourier series

f(τ) = ∑ a_{n} exp(2πinτ/ℓ) = ∑ a_{n}q^{n/ℓ} . (3.2)

We assume that f is holomorphic at i∞, i.e. a_{n} = 0 for n < 0 and the Fourier series converges for |q| < 1. The holomorphic modular forms constitute a graded ring

M_{∗}(Γ) := ⊕M_{k}(Γ)

graded by the weight k.

An almost holomorphic function on H will be a function

F (τ) = ∑_{m=0}^{N} f_{m}(τ)y^{−m} (3.3)

on H where y = Im(τ) and each f_{m} has a Fourier expansion as in (3.2). An almost holomorphic modular form of weight k is an almost holomorphic function f(τ) satisfying the weight k modularity property (3.1).

Example 3.1. The classical Eisenstein series (B_{k} Bernoulli number as in Lemma (2.2))

G_{k}(q) := \frac{−B_{k}}{2k} + ∑_{n=1}^{∞} ( ∑_{d|n} d^{k−1})q^{n}, k = 2, 4, 6, … (3.4)

is modular of weight k for Γ = Γ_{1} and k ≥ 4. On the other hand, G_{2} satisfies the transformation

G_{2} (\frac{aτ + b}{cτ + d}) = (cτ + d)^{2}G_{2}(τ) − \frac{c(cτ + d)}{4πi} . (3.5)

Notice, however, that

[Im (\frac{aτ + b}{cτ + d})]^{−1} − (cτ + d)^{2}y^{−1} = −2ic(cτ + d) . (3.6)

It follows that the function

G_{2}^{∗}(τ) := G_{2}(τ) + \frac{y^{−1}}{8π} (3.7)

is an almost holomorphic modular form of weight 2 for Γ_{1}.

Definition 3.2. A quasimodular form of weight k is a holomorphic function f(τ) on H such that there exists an almost holomorphic modular form F = ∑_{m=0}^{N} f_{m}y^{−m} of weight k with f_{0} = f.

Example 3.3. G_{2} is quasimodular of weight 2. Indeed, one can take F = G_{2}^{∗}.

Let us write AHM_{∗}(Γ) (resp. QM_{∗}(Γ)) for the graded ring of almost holomorphic (resp. quasimodular) forms. (Kaneko and Zagier write M̂ and M̃, but this makes it difficult to remember which is which.)

Proposition 3.4. The assignment F = ∑_{j=0}^{M} y^{−j}f_{j} ↦ f_{0} defines an isomorphism of graded rings, AHM_{∗}(Γ) ≅ QM_{∗}(Γ).

Proof. Note this is well defined, i.e. ∑_{j=0}^{M} y^{−j}f_{j}(τ) ≡ 0 for holomorphic f_{j} if and only if all the f_{j} are zero. To see this one can e.g. apply the differential operator \frac{iM}{2} y^{−1}+ \frac{d}{dτ̄} and argue by induction on M. The map in question is surjective by definition, so it suffices to show injectivity. Suppose for some r ≥ 1 and f_{r} ≠ 0 that F = y^{−r}f_{r} + … + y^{−M}f_{M} is almost holomorphic modular of weight k. Let A = ( \begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ and write j = cτ + d. Using (3.6) we get

(j^{2}y^{−1} − 2icj)^{r}f_{r}(Aτ) + … + (j^{2}y^{−1} − 2icj)^{M}f_{M}(Aτ) =

j^{k}y^{−r}f_{r}(τ) + … + j^{k}y^{−M}f_{M}(τ).

Now identify coefficients of powers of y^{−1}

(3.8)  f_{M}(Aτ) = j^{k−2M}f_{M}(τ) ,
f_{M−1}(Aτ) − 2icj\binom{M}{1}f_{M}(Aτ) = j^{k−2M+2}f_{M−1}(τ) ,
f_{M−2}(Aτ) − 2icj\binom{M − 1}{1}f_{M−1}(Aτ) + (2icj)^{2}\binom{M}{2}f_{M}(Aτ)
= j^{k−2M+4}f_{M−2}(τ) ,
⋮  ⋮

f_{r}(Aτ) − 2icj\binom{r + 1}{1}f_{r+1}(Aτ)

+(2icj)^{2}\binom{r + 2}{2}f_{r+2}(Aτ) − … = j^{k−2r}f_{r}(τ) .

The terms not involving y give

(−2icj)^{r}f_{r}(Aτ) + … + (−2cij)^{M}f_{M}(Aτ) = 0. (3.9)

Solve (3.8) recursively:

f_{M}(Aτ) = j^{k−2M}f_{M}(τ) ,
f_{M−1}(Aτ) = 2ic\binom{M}{1}j^{k−2M+1}f_{M}(τ) + j^{k−2M+2}f_{M−1}(τ) ,
f_{M−2}(Aτ) = … + j^{k−2M+4}f_{M−2}(τ) ,
⋮  ⋮
f_{r}(Aτ) = … + j^{k−r}f_{r}(τ) .

Finally, substituting in (3.9) yields

(3.10) (cτ + d)^{k−r}c^{r}f_{r}(τ) + α_{r+1}(cτ + d)^{k−r−1}c^{r+1}f_{r}(τ) + …

+ α_{M}(cτ + d)^{k−M}c^{M}f_{M}(τ) = 0.

Here the α_{j} are constants independent of c and d. Varying A ∈ Γ yields a contradiction. Indeed, the map Γ → ℂ^{2}, A ↦ (c, d) has Zariski dense image, so in the above identity, c and d can be taken to be independent variables. The coefficient of c^{r}d^{k−r} is nontrivial.

Proposition 3.5. (i) M_{∗}(Γ) ⊂ QM_{∗}(Γ). (ii) QM_{∗}(Γ) = M_{∗}(Γ) ⊗ ℂ[G_{2}]. (iii) QM_{∗}(Γ) is stable under the operator (of degree 2) D := \frac{d}{dτ}. We

have

QM_{k}(Γ) = ⊕_{0≤i≤k/2}D^{i}M_{k−2i}(Γ) ⊕ ℂ · D^{k/2−1}G_{2}.

Proof. (i) is clear. To prove (ii), we claim first that the map

M_{∗}(Γ) ⊗ ℂ[G_{2}^{∗}] → AHM_{∗}(Γ) (3.11)

is an isomorphism, where G_{2}^{∗} is as in (3.7). Indeed, for F = f_{0} + … + y^{−M}f_{M} almost holomorphic modular of weight k, it follows from the first line of (3.8) that f_{M} is holomorphic modular of weight k − 2M. We have

F − f_{M} · (8πG_{2}^{∗})^{M} = g_{0} + … + y^{−M+1}g_{M−1}

is almost holomorphic modular of weight k. Surjectivity of (3.11) follows by induction on M. Injectivity is straightforward, keeping track of powers of y^{−1}. Assertion (ii) now follows from proposition (3.4). Finally, (iii) is left for the reader.

4. Quasimodularity for characters Ω and V

The purpose of this section is to prove

Theorem 4.1. The series Ω(τ_{0}, τ_{1}, …) and V (τ_{1}, τ_{2}, …) are quasimodular of weights 0 and −\frac{1}{2} respectively.

We focus first on Ω. Recall (definition (3.2)) a series F (τ_{0}, τ_{1}, τ_{2}, …) is said to be quasimodular of weight k if it can be expanded in a formal Taylor series

∑_{J=(j_{0},j_{2},j_{3},...)} B_{J}(τ_{1})τ^{J}/J!

with B_{J}(τ_{1}) quasimodular of weight

k + wt(J) := k + j_{0} + 3j_{2} + 4j_{3} + …. (4.1)

Since we have not specified a group Γ, there will be no harm e.g. in replacing τ_{1} by 2τ_{1}. Define

Φ(q) = Ω(−q_{0}, q_{1}, q_{2}, …).

One has Ω(q) = Φ(q_{0}^{2}, q_{1}^{2}, q_{2}^{2}, …)/Φ(q), so, since the space of quasimodular forms is a ring, it will suffice to prove Φ(q) is quasimodular. We have

Φ(1, x_{1}, 1, 1, …) = x_{1}^{− \frac{1}{24}} ∏_{r≥0}(1 − x_{1}^{r+ \frac{1}{2}} )^{2} = (η(x^{1/2})/η(x))^{2},

which is modular of weight 0. It therefore suffices to show

G := log (Φ(x)/Φ(1, x_{1}, 1, 1, …))

THE CHARACTER OF THE INFINITE WEDGE REPRESENTATION 21

is quasimodular. We have (with ξ(s) as in (1.6))

(4.2) G = −2πi( ∑_{n≥1} ξ(−2n − 1)τ_{2n+1}) +

∑_{r≥0} [ log(1 − x_{0}x_{1}^{n+ \frac{1}{2}} x_{2}^{(n+ \frac{1}{2} )^{2}} ⋯) + log(1 − x_{0}^{−1}x_{1}^{n+ \frac{1}{2}} x_{2}^{−(n+ \frac{1}{2} )^{2}} ⋯) −

2 log(1 − x_{1}^{n+ \frac{1}{2}} )] .

We expand the final sum in (4.2)

B :=

− ∑_{n≥0 , l≥1} \frac{x_{1}^{(n+ \frac{1}{2} )l}}{l} (exp (2πil (τ_{0} + (n + \frac{1}{2})^{2} τ_{2} + (n + \frac{1}{2})^{3} τ_{3} + …))

+ exp (2πil (−τ_{0} − (n + \frac{1}{2})^{2} τ_{2} + (n + \frac{1}{2})^{3} τ_{3} − …)) − 2) .

Define coefficients α_{m}(J), where J = (j_{0}, j_{2}, j_{3}, …) and j_{k} ≥ 0, by

\frac{∂^{J}}{∂τ^{J}} B|_{τ_{0}=τ_{2}=...=0} = ∑_{m=1}^{∞} α_{m}(J)x_{1}^{m/2} . (4.3)

Note α_{m}(0) = 0. For J ≠ 0, write |J| = j_{0} + j_{2} + . . .. We get

α_{m}(J) = −(2πi)^{|J|} ∑_{n≥0 , l≥1} l^{|J|−1}(1 + (−1)^{wt(J)}) (n + \frac{1}{2})^{wt(J)−|J|} x_{1}^{(n+ \frac{1}{2} )l}. (4.4)

Thus, α_{m}(J) = 0 for wt(J) odd. For even weight, we find

(4.5) \frac{∂^{J}}{∂τ^{J}} B|_{τ_{0}=τ_{2}=...=0} = −2(2πi)^{|J|}2^{|J|−wt(J)} ∑_{m≥1} m^{|J|−1}x_{1}^{m/2} ∑_{d|m; dodd} d^{wt(J)−2|J|+1}

= −2(2πi)^{|J|}2^{|J|−wt(J)} ∑_{m≥1} m^{wt(J)−|J|}x_{1}^{m/2} ∑_{l|m; \frac{m}{l} odd} l^{2|J|−wt(J)−1} .

The Eisenstein series of weight k and level 1 was given in (3.4). As in [11], we work with the level two Eisenstein series

F_{k}^{(1)}(q) := G_{k}(q^{\frac{1}{2}} ) − G_{k}(q) = ∑_{n=1}^{∞} ( ∑_{d|n,2 |̸d}(n/d)^{k−1})q^{n/2} ,

F_{k}^{(2)} := G_{k}(q^{\frac{1}{2}} ) − 2^{k−1}G_{k}(q) = (1 − 2^{k−1})ζ(1 − k)/2 + ∑_{n=1}^{∞} ( ∑_{d|n,2 |̸d} d^{k−1})q^{n/2}.

Suppose first that the inequality k(J) := wt(J) − 2|J| + 2 > 1 holds. It follows from the first equality in (4.5) that

\frac{∂^{J}}{∂τ^{J}} B|_{τ_{0}=τ_{2}=...=0} = −4πi(2^{−k(J)+1})(∂/∂τ_{1})^{|J|−1}(F_{k(J)}^{(2)}(x_{1}) − F_{k(J)}^{(2)}(0)) . (4.6)

Note k(J) is even, so the remaining possibility is k(J) ≤ 0. In this case, the second equality in (4.5) implies

\frac{∂^{J}}{∂τ^{J}} B|_{τ_{0}=τ_{2}=...=0} = −2(2πi)^{2−k(J)}(∂/∂τ_{1})^{wt(J)−|J|}(F_{2−k(J)}^{(1)}(x_{1})) . (4.7)

Since the F_{k}^{(i)} are quasimodular, it follows that the \frac{∂^{J}}{∂τ^{J}} B|_{τ_{0}=τ_{2}=...=0} are quasimodular except possibly (because of the constant term on the right) in in the case k(J) ≥ 2; |J| = 1 where the constant term is not correct. This only happens if J = (0, …, 0, 1, 0, …) where j = 1 for some j ≥ 3 odd and all the other entries are zero. In this case k(J) = j+1 and it follows from (4.6) and the identity ξ(s) = (2^{s}−1)ζ(s) that

\frac{∂}{∂τ_{j}} B|_{τ=0} = (−4πi)2^{−j}(F_{j+1}^{(2)}(x_{1}) + (2^{j} − 1)ζ(−j)/2) =

(−4πi)2^{−j}F_{j+1}^{(2)}(x_{1}) + 2πiξ(−j) .

The last constant term exactly cancels the −2πiξ(−j) which appears in the first sum on the right in (4.2). This completes the proof of quasimodularity for Ω.

It remains to prove quasi-modularity for V (x_{1}, …). The proof parallels that in [11]. We begin by expanding

V (τ_{1}, τ_{2}, …) = ∑_{K=(k_{2},k_{3},...)} A_{K}(τ_{1})τ^{K}/K! , (4.8)

V (T^{−n}(τ)) = ∑_{m=0}^{∞} ∑_{K}(∂/∂τ_{1})^{m}A_{K}(τ_{1})(T^{−n}(τ_{1}) − τ_{1})^{m}(m!)^{−1}(T^{−n}(τ))^{K}/K! , (4.9)

Ω(τ) = ∑_{n=−∞}^{∞} ∑_{J=(j_{0},j_{2},j_{3},...)} ∑_{m=0}^{∞} ∑_{K}(∂/∂τ_{1})^{m}A_{K}(τ_{1})(T^{−n}(τ_{1}) − τ_{1})^{m}(m!)^{−1} × (4.10)

×(T^{−n}(τ))^{K}(K!)^{−1}a_{J}n^{wt(J)}τ^{J}(J!)^{−1}x_{1}^{n^{2}/2} .

Here ∑_{J} a_{J}n^{wt(J)}τ^{J}(J!)^{−1} is the power series expansion of x_{0}^{n}x_{2}^{n^{3}/3}x_{3}^{n^{4}/4} ⋯, with a_{J} independent of n.

One verifies the expansion

T^{−n}(τ_{j}) = τ_{j} + \binom{j + 1}{1}nτ_{j+1} + \binom{j + 2}{2}n^{2}τ_{j+2} + …. (4.11)

Fix m ≥ 0 and K = (k_{2}, k_{3}, …). Let P (τ) = τ_{2}^{p_{2}}τ_{3}^{p_{3}} … and Q_{i}(τ) = τ_{i}^{q_{ii}}τ_{i+1}^{q_{i,i+1}} …; i ≥ 2 be monomials with deg(P) = ∑ p_{j} = m and deg(Q_{i}) = k_{i}. Define the weights by wt(P) = ∑(j +1)p_{j} and wt(Q_{i}) = ∑_{j≥i}(j + 1)q_{ij}. As a consequence of (4.11), P appears in (T^{−n}(τ_{1}) − τ_{1})^{m}(m!)^{−1} with a coefficient c_{P,m}n^{wt(P)−2m}, where c_{P,m} is constant independent of n. Similarly, Q := ∏ Q_{i} appears in (T^{−n}(τ))^{K}(K!)^{−1} with coefficient c_{{Q_{i}},K}n^{∑_{i} wt(Q_{i})−(i+1)k_{i}}. Now fix J = (j_{0}, j_{2}, …). We obtain a contribution to the coefficient of the monomial P (τ)Q(τ)τ^{J} in (4.10) of the form

c_{J,K,m,\{Q_{i}\},P} \frac{∂^{m}}{∂τ_{1}^{m}} A_{K}(τ_{1}) ∑_{n=−∞}^{∞} n^{wt(P)−2m+∑_{i}(wt(Q_{i})−(i+1)k_{i})+wt(J)}x_{1}^{n^{2}/2}. (4.12)

If the exponent of n is odd, this cancels. Assume this exponent equals 2r for r ≥ 0 an integer. Then (4.12) can be rewritten

b_{J,K,m,\{Q_{i}\},P} \frac{∂^{m}}{∂τ_{1}^{m}} A_{K}(τ_{1})\frac{∂^{r}}{∂τ_{1}^{r}} θ_{00}(τ_{1}) , (4.13)

where θ_{00}(τ_{1}) = ∑ exp(πin^{2}τ_{1}) is a modular form of weight 1/2. Since differentiation preserves quasi-modularity and increases weight by 2, it follows that, assuming A_{K}(τ_{1}) is quasimodular of weight wt(K) − \frac{1}{2}, the expression in (4.13) is quasi-modular of weight wt(P)+∑ wt(Q_{i})+ wt(J).

We will prove A_{K}(τ_{1}) is quasimodular of weight wt(K) − 1/2 by induction on wt(K). Write

θ_{00}(x) = ∑_{−∞}^{∞} x^{n^{2}/2} = ∏_{m≥1}(1 − x^{m}) ∏_{m≥0}(1 + x^{m+ \frac{1}{2}} )^{2} =

η(x)Ω(1, x, 1, 1 …) .

Substituting x_{j} = 1; j ≠ 1 in Theorem (2.3) yields

Ω(1, x_{1}, 1, …) = V (x_{1}, 1, …) ∑ x_{1}^{n^{2}/2} = V (x_{1}, 1, …)θ_{00}(x_{1}) ,
V (x_{1}, 1, …) = η(x_{1})^{−1} .

This is modular of weight −1/2 as desired. Now fix an index set M = (m_{2}, m_{3}, …) with wt(M) > 0 and assume A_{K}(τ_{1}) is quasimodular of weight wt(K)−1/2 for all K with wt(K) < wt(M). Consider the coefficient B_{M}(τ_{1}) of τ^{M} in the expansion of Ω(τ) (4.10). We know that B_{M} is quasimodular of weight wt(M). It is a sum of terms (4.13), of which all but one involve A_{K} with wt(K) < wt(M) and are quasimodular of weight wt(M) by our inductive hypothesis. The one remaining term, which appears with coefficient (M!)^{−1}, is A_{M}(τ_{1})θ_{00}(τ_{1}) (take J = 0 = m, P = 1, Q_{i} = τ_{i}^{m_{i}}.) It follows that A_{M}(τ_{1}) is quasimodular of weight wt(M) − 1/2 as desired.

5. Preliminaries on partitions

Let λ = λ_{1} ≥ λ_{2} ≥ … ≥ λ_{ℓ} > 0 = λ_{ℓ+1} = … be a partition. ℓ = ℓ(λ) is the length of the partition, and |λ| = ∑ λ_{i} is the number being partitioned.

Let f(λ) be a function on the set of partitions. Assuming f does not grow too rapidly with |λ| we may consider for |q| ≪ 1 the ratio

⟨f⟩_{q} := \frac{∑_{λ} f(λ)q^{|λ|}}{∑_{λ} q^{|λ|}} = (q)_{∞} ∑_{λ} f(λ)q^{|λ|}. (5.1)

Here, we write for q given and n ≤ ∞,

(a)_{n} = (1 − a)(1 − aq)(1 − aq^{2}) ⋯ (1 − aq^{n}). (5.2)

Fix an integer n ≥ 1 and variables t_{1}, …, t_{n}. We will be interested in the following “n-point correlation function”

F (t_{1}, …, t_{n}) := ⟨ ∏_{k=1}^{n} ( ∑_{i=1}^{∞} t_{k}^{λ_{i}−i+ \frac{1}{2}} )⟩_{q}. (5.3)

To understand the relation between F and the character V (q), we take a small detour, beginning with some basic ideas from the theory of partitions. The diagram of a partition λ is simply the array of dots (or squares) with λ_{1} dots in the first row, λ_{2} dots in the second row, etc. Here is 4, 4, 3, 2, 1.

\begin{matrix}• & • & • & • \\ • & • & • & • \\ • & • & • \\ • & • \\ • \end{matrix}

The transposed partition λ′ is obtained by flipping along the diagonal

\begin{matrix}• & • & • & • & • \\ • & • & • & • \\ • & • & • \\ • & • \end{matrix}

For λ_{i} ≥ i let m_{i} = λ_{i} − i be the number of dots to the right of the i-th point on the diagonal, and let n_{i} be the number of points below it. (so n_{i}(λ) = m_{i}(λ′).) Note m_{1} > m_{2} > … ≥ 0 and similarly for the n_{i}. The m_{i} and n_{i} are called Frobenius coordinates of λ and written (m_{i}, …, m_{p}|n_{1}, …, n_{p}). For example, Frobenius coordinates for 4, 4, 3, 2, 1 are (3, 2, 0|4, 2, 0). Frobenius coordinates are related to the λ_{i} via the following generating function identity.

Lemma 5.1. Let λ be a partition of length ℓ with Frobenius coordinates (m_{i}, …, m_{p}|n_{1}, …, n_{p}). Then

∑_{i}(t^{λ_{i}−i+ \frac{1}{2}} − t^{−i+ \frac{1}{2}} ) = ∑_{k}(t^{m_{k}+ \frac{1}{2}} − t^{−(n_{k}+ \frac{1}{2} )}).

Proof. It is clear that the m_{k} correspond to the nonnegative λ_{i} − i. Thus it will suffice to identify the terms with negative exponents on the two sides. This follows from the identity of sets of numbers

{λ_{p+1} − (p + 1), …, λ_{ℓ} − ℓ, −n_{1} − 1, …, −n_{p} − 1} = {−ℓ, …, −2, −1}, ,

which, in turn, follows by comparing the negative elements from the identity

{λ_{1} − 1, …, λ_{ℓ} − ℓ, −λ′_{1}, …, −λ′_{k} + k − 1} = {−ℓ, …, k − 1} , ∀k ≥ λ_{1} ,

established in [13], Chapter I, (1.7).

Substituting t = exp(u) and expanding in the previous lemma yields identities

(5.4) ∑_{i} (λ_{i} − i + \frac{1}{2})^{r} + (−1)^{r+1} (i − \frac{1}{2})^{r} =

∑_{j} (m_{j} + \frac{1}{2})^{r} + (−1)^{r+1} (n_{j} + \frac{1}{2})^{r} =: p_{r}(λ).

For example, |λ| = p_{1}(λ). Recall we have already encountered the p_{r}(λ) in the expansion for the character V (q):

V (q_{1}, q_{2}, q_{3}, …) = q_{1}^{−ξ(−1)}q_{3}^{−ξ(−3)} ⋯ ∑_{λ} ∏_{r≥1} q_{r}^{p_{r}(λ)}. (5.5)

To see the connection more specifically, write q_{n} = exp(2πiτ_{n}) and consider the differential operator

δ = δ(u) := u^{−1} + (2πi)^{−1} ∑_{n=1}^{∞} \frac{u^{n}}{n!} \frac{∂}{∂τ_{n}} . (5.6)

Proposition 5.2. We have with F as in (5.3) and t = exp(u)

F (t) = η(q_{1})δV |_{τ_{2}=τ_{3}=...=0}.

26 SPENCER BLOCH AND ANDREI OKOUNKOV

Proof. Consider the Bernoulli polynomials, defined by

\frac{u exp(xu)}{exp(u) − 1} = ∑_{k=0}^{∞} B_{k}(x)u^{k}/k!; B_{0}(x) = 1, B_{1}(x) = x − \frac{1}{2} (5.7)

One has ([6], theorem 1, p. 43)

ξ(1 − m) = \frac{−B_{m}(\frac{1}{2})}{m}. (5.8)

Combining these identities yields

− ∑_{n=1}^{∞} ξ(−n)u^{n}/n! = \frac{1}{t^{\frac{1}{2}} − t^{− \frac{1}{2}}} − u^{−1} = ∑_{i=1}^{∞} t^{−i+ \frac{1}{2}} − u^{−1}. (5.9)

On the other hand, using lemma 5.1 and the identity

η(q_{1})^{−1} = q_{1}^{−ξ(−1)} ∑_{λ} q_{1}^{|λ|}, (5.10)

one computes

η(q_{1})δV |_{τ_{2}=...=0} = u^{−1} − ∑ ξ(−n)u^{n}/n! +

+ ( ∑_{λ} q_{1}^{|λ|})^{−1} ∑_{λ} q_{1}^{|λ|} ∑_{i=1}^{∞}(t^{λ_{i}−i+ \frac{1}{2}} − t^{−i+ \frac{1}{2}} ) =

= u^{−1} + ( ∑_{λ} q_{1}^{|λ|})^{−1} ∑_{λ} q_{1}^{|λ|} ∑_{i=1}^{∞} t^{λ_{i}−i+ \frac{1}{2}} − u^{−1} =

= ⟨ ∑_{i=1}^{∞} t^{λ_{i}−i+ \frac{1}{2}} ⟩_{q_{1}} = F (t).

More generally, one finds

η(q_{1})δ(u_{1}) ∘ … ∘ δ(u_{n})V |_{τ_{2}=...=0} = ⟨ ∏_{k=1}^{n} ( ∑_{i=1}^{∞} t_{k}^{λ_{i}−i+ \frac{1}{2}} )⟩_{q_{1}} = F (t_{1}, …, t_{n}). (5.11)

This is also equivalent to the formula (0.18) from the introduction.

6. The formula for correlation functions In this section we give a detailed statement of our results on the

generating function F (t_{1} …, t_{n}) defined in (5.3). To simplify notation, we write Θ(x) = Θ(x; q) for the following theta function

(6.1)  Θ(x) = Θ_{11}(x; q) = η^{−3}(q) ∑_{n∈ℤ}(−1)^{n}q^{\frac{(n+1/2)^{2}}{2}} x^{n+1/2}

= (q)_{∞}^{−2}(x^{1/2} − x^{−1/2})(qx)_{∞}(q/x)_{∞} . (6.2)

The function Θ(x) is odd,

Θ(x^{−1}) = −Θ(x),

and satisfies the difference equation

Θ(q^{m}x) = (−1)^{m}q^{−m^{2}/2}x^{−m}Θ(x) , m ∈ ℤ . (6.3)

Define

Θ^{(k)}(x) := (x\frac{∂}{∂x})^{k} Θ_{11}(x; q) , Θ′ = Θ^{(1)}, etc. (6.4)

where x\frac{∂}{∂x} is the natural invariant vector field on the group ℂ^{∗} = ℂ \ 0. We shall prove the following

Theorem 6.1.

F (t_{1}, …, t_{n}) = ∑_{σ∈𝔖(n)} \frac{det (\frac{Θ^{(j−i+1)}(t_{σ(1)} ⋯ t_{σ(n−j)})}{(j − i + 1)!})_{i,j=1}^{n}}{Θ(t_{σ(1)}) Θ(t_{σ(1)}t_{σ(2)}) … Θ(t_{σ(1)} ⋯ t_{σ(n)})} (6.5)

Here σ runs through all permutations 𝔖(n) of {1, …, n},the matrices in the numerator have size n × n, and we define 1/(−n)! = 0 if n ≥ 1.

In particular, since Θ′(1) = 1 and Θ′′(1) = 0, we have

F (t_{1}) = \frac{1}{Θ(t_{1})} ,
F (t_{1}, t_{2}) = \frac{1}{Θ(t_{1}t_{2})} (\frac{Θ′(t_{1})}{Θ(t_{1})} + \frac{Θ′(t_{2})}{Θ(t_{2})}) .

For n = 3 the formula (6.5) looks as follows:

(6.6) F (t_{1}, t_{2}, t_{3}) =

\frac{1}{Θ(t_{1}t_{2}t_{3})} ∑_{σ∈𝔖(3)}det ( \begin{matrix}\frac{Θ′ (t_{σ(1)}t_{σ(2)})}{Θ (t_{σ(1)}t_{σ(2)})} & \frac{1}{2} \frac{Θ′′ (t_{σ(1)})}{Θ (t_{σ(1)})} & \frac{Θ′′′(1)}{3!} \\ 1 & \frac{Θ′ (t_{σ(1)})}{Θ (t_{σ(1)})} & 0 \\ & 1 & Θ′(1) \end{matrix} )

= \frac{1}{Θ(t_{1}t_{2}t_{3})} ( ∑_{1≤i≠j≤3} \frac{Θ′(t_{i})}{Θ(t_{i})} \frac{Θ′(t_{i}t_{j})}{Θ(t_{i}t_{j})} − ∑_{i=1}^{3} \frac{Θ′′(t_{i})}{Θ(t_{i})} + Θ′′′(1)) .

The determinants in (6.5), having only one non-zero diagonal below the main diagonal, have a nice combinatorial expansion, see the formula (7.7) below.

One has the following simple

Lemma 6.2.

Θ^{(2m+1)}(1; q) = (2m + 1)! ∑_{k_{1}+2k_{2}+···=m} \frac{(−2)^{k_{1}+k_{2}+...}}{k_{1}! k_{2}! ⋯} ∏_{i} (\frac{G_{2i}(q)}{(2i)!})^{k_{i}} . (6.7)

Observe that the sum in (6.7) is over all partitions 1^{k_{1}}2^{k_{2}}3^{k_{3}} … of the number m and k_{1} + k_{2} + … is the length of such a partition. In particular,

Θ^{(3)}(1; q) = −6G_{2}(q) ,
Θ^{(5)}(1; q) = −10G_{4}(q) + 60G_{2}(q)^{2} ,
Θ^{(7)}(1; q) = −14G_{6}(q) + 420G_{4}(q)G_{2}(q) − 840G_{2}(q)^{3} .

This lemma will be proved below.

Remark 6.3. The equality in Theorem 6.1 may look like an equality of multivalued functions but in fact the only ambiguity is the factor

\sqrt{t_{1} ⋯ t_{n}}

which appears on both sides of of the theorem and, therefore, can be ignored.

Remark 6.4. Alternatively, one may consider the (n + 1) × (n + 1) matrix

( \begin{matrix}Θ(t_{1} ⋯ t_{n}) & Θ′(t_{1} ⋯ t_{n−1}) & … & \frac{1}{n!}Θ^{(n)}(0) \\ 0 & Θ(t_{1} ⋯ t_{n−1}) & … & \frac{1}{(n−1)!}Θ^{(n−1)}(0) \\ ⋮ & ⋮ & … & ⋮ \\ 0 & 0 & … & Θ′(0) \\ 0 & 0 & … & 0 \end{matrix} ) (6.8)

(Note the bottom right hand entry is Θ(0) = 0.) This matrix has rank n, so there will be a unique column vector

( v_{n}(t_{1}, …, t_{n}) ; v_{n−1}(t_{1}, …, t_{n−1}) ; ⋮ ; v_{1}(t_{1}) ; 1)

which is killed by multiplication on the left by the matrix (6.8). Then

F (t_{1}, …, t_{r}) = ∑_{σ} v_{r}(t_{σ(1)}, …, t_{σ(r)}).

Proof of Lemma 6.2. Let f(a) be some function of a variable a and let b be another variable. We have:

e^{f(a+b)−f(a)} = exp (∑_{m=1}^{∞} \frac{f^{(m)}(a)}{m!}b^{m})

= ∏_{m=1}^{∞} exp (\frac{f^{(m)}(a)}{m!}b^{m})

= ∏_{m=1}^{∞} ∑_{k=0}^{∞} (\frac{f^{(m)}(a)}{m!})^{k} \frac{b^{mk}}{k!}

= ∑_{s=0}^{∞} ∑_{k_{1}+2k_{2}+3k_{3}+···=s} \frac{b^{s}}{k_{1}! k_{2}! k_{3}! ⋯} ∏_{i=1}^{∞} (\frac{f^{(i)}(a)}{i!})^{k_{i}} .

In other words, we have

(6.9) e^{−f(a)} \frac{d^{s}}{da^{s}} e^{f(a)} =

s! ∑_{k_{1}+2k_{2}+3k_{3}+···=s} \frac{1}{k_{1}! k_{2}! k_{3}! ⋯} (\frac{f′}{1!} )^{k_{1}} (\frac{f′′}{2!} )^{k_{2}} (\frac{f′′′}{3!} )^{k_{3}} · · · .

We wish to apply the above formula to the triple product formula for the theta function and use the fact that

(x\frac{d}{dx})^{m} log ((qx)_{∞}(q/x)_{∞})|_{x=1} = \begin{cases}−2G_{m}(q) − B_{m}/m , & m is even \\ 0 , & m is odd . \end{cases}

The derivative Θ^{(2m+1)}(1; q) is quasi-modular of weight 2m. It is, therefore, sufficient, to compute the term

(2m + 1) (x\frac{d}{dx}(x^{1/2} − x^{−1/2})) (x\frac{d}{dx})^{2m} (qx)_{∞}(q/x)_{∞}|_{x=1} ,

which gives the contribution of the maximal weight. Using (6.9) we obtain that the above term equals the RHS of (6.7) modulo terms of lower weight. This establishes (6.7).

Before diving into the proof of Theorem 6.1 in the next section, we give as a warmup the proof for the 1-point function.

Theorem 6.5.

⟨ ∑_{i} t^{λ_{i}−i+ \frac{1}{2}} ⟩_{q} = \frac{1}{Θ(t)}.

Lemma 6.6.

⟨t^{λ_{k}}⟩_{q} = \frac{(q)_{∞}}{(1 − q) ⋯ (1 − q^{k−1})(1 − q^{k}t)(1 − q^{k+1}t) ⋯}

Proof. The notation is as in (5.2). Recall in section (5) we defined the transpose λ′ of a partition λ. The lemma is straightforward once one observes that

⟨t^{λ_{k}}⟩_{q} = ⟨t^{λ′_{k}}⟩_{q} = ⟨t^{#{i|λ_{i}≥k}}⟩_{q}.

Recall the following Heine’s q-analog of the Gauss _{2}F_{1}-summation. Given any a, b, c satisfying |c| < |ab| we have

∑_{n=0}^{∞} \frac{(a)_{n}(b)_{n}}{(c)_{n}(q)_{n}} (\frac{c}{ab})^{n} = \frac{(c/a)_{∞}(c/b)_{∞}}{(c)_{∞}(c/ab)_{∞}} , (6.10)

see for example [5], Section 1.5. Recall that the symbol (a)_{n} was defined in (5.2).

Proof of theorem 6.5.

⟨ ∑_{i} t^{λ_{i}−i+ \frac{1}{2}} ⟩_{q} = ∑_{k=1}^{∞} t^{\frac{1}{2} −k} \frac{(q^{k})_{∞}}{(q^{k}t)_{∞}} =

= t^{− \frac{1}{2}} \frac{(q)_{∞}}{(qt)_{∞}} ∑_{r=0}^{∞} t^{−r} \frac{(qt)_{i}}{(q)_{i}} = \frac{(q)_{∞}^{2}}{(t^{\frac{1}{2}} − t^{− \frac{1}{2}} )(qt)_{∞}(q/t)_{∞}} = \frac{1}{Θ(t)}.

7. Beginning of the proof of Theorem 6.1

Our strategy for proving theorem 6.1 will be based on the following simple implication

(\begin{matrix}f(x) is holomorphic on ℂ \ 0 and \\ f(qx) = f(x) \end{matrix} ) ⇒ (f(x) is a constant) .

We shall show that both sides of theorem 6.1 satisfy the same difference equation and have the same singularities.

We now introduce some useful notation. Given indices 1 < i_{1} < ⋯ < i_{n} . set

⟨\begin{matrix}i_{1}, …, i_{r} \\ t_{1}, …, t_{r} \end{matrix}⟩ : = ⟨∏_{k=1}^{n} t_{k}^{λ_{i_{k}} −i_{k}+1/2}⟩_{q} (7.1)

= ∏_{k=1}^{n} t_{k}^{1/2−i_{k}} ⟨∏_{k=1}^{n} t_{k}^{λ′_{i_{k}}} ⟩_{q}

= (q)_{∞} \frac{t_{1}^{1/2−i_{1}} … t_{r}^{1/2−i_{r}}}{(q)_{i_{1}−1}(q^{i_{1}}t_{1})_{i_{2}−i_{1}}(q^{i_{2}}t_{1}t_{2})_{i_{3}−i_{2}} … (q^{i_{r}}t_{1} ⋯ t_{r})_{∞}} ,

The notation (a)_{n} is as in (5.2), and the last equality in (7.1) is a straightforward generalization of lemma (6.6). For brevity, write

⟨\begin{matrix}i \\ t \end{matrix}⟩ := ⟨\begin{matrix}i_{1}, …, i_{r} \\ t_{1}, …, t_{r} \end{matrix}⟩ .

Set

(7.2)  H(t_{1}, …, t_{n}) := ∑_{i_{1}<···<i_{n}} ⟨\begin{matrix}i \\ t \end{matrix}⟩ ,
(7.3)  G(t_{1}, …, t_{n}) := ∑_{σ∈𝔖(n)}H(t_{σ(1)}, …, t_{σ(n)}) .

By Π_{n} denote the set of all partitions of the set {1, …, n}. An element π of Π_{n}

π = {π_{1}, …, π_{ℓ}} ∈ Π_{n} , π_{i} ⊂ {1, …, n} ,

is by definition an unordered collection of subsets of {1, …, n} such that

π_{i} ∩ π_{j} = ∅ , i ≠ j ,
π_{1} ∪ ⋯ ∪ π_{ℓ} = {1, …, n} .

Note the difference between partitions of the set {1, …, n} and partitions of the number n. The number ℓ is called the length of π and is denoted by ℓ(π). The subsets π_{i} are called the blocks of π.

Given a partition π ∈ Π_{n} set

G^{π}(t_{1}, …, t_{n}) := G ( ∏_{k∈π_{1}} t_{k} , …, ∏_{k∈π_{ℓ(π)}} t_{k}) .

For example, if n = 3 and π = {{1, 2}, {3}} then

G^{π}(t_{1}, t_{2}, t_{3}) = G(t_{1}t_{2}, t_{3}) .

With this notation we have

F (t_{1}, …, t_{n}) = ∑_{π∈Π_{n}} G^{π}(t) . (7.4)

Set also

F^{π}(t_{1}, …, t_{n}) := F ( ∏_{k∈π_{1}} t_{k} , …, ∏_{k∈π_{ℓ(π)}} t_{k}) , π ∈ Π_{n} . (7.5)

We shall also need the set Γ_{n} of all compositions of the set {1, …, n}. By definition, the set Γ_{n} consists of all ordered collections

γ = (γ_{1}, …, γ_{ℓ}) , γ_{i} ⊂ {1, …, n} ,

such that

γ_{i} ∩ γ_{j} = ∅ , i ≠ j ,
γ_{1} ∪ ⋯ ∪ γ_{ℓ} = {1, …, n} .

The number ℓ is called again the length of γ and is denoted by ℓ(γ). The set Γ_{n} is in a natural one-to-one correspondence with the set Υ_{n} of all partial flags

∅ = υ_{1} ⊊ υ_{2} ⊊ ⋯ ⊊ υ_{ℓ} ⊊ υ_{ℓ+1} = {1, …, n}

in the set {1, …, n}, namely, we set

υ(γ)_{k} = γ_{1} ∪ ⋯ ∪ γ_{k−1} . (7.6)

For any partition π or composition γ of an n-element set of length ℓ define its sign as

(−1)^{π} = (−1)^{γ} = (−1)^{n+ℓ} .

Then, in particular, the sign of a permutation equals the sign of the corresponding partition into disjoint cycles.

Denote by U(t_{1}, …, t_{n}) the RHS of (6.5), and set

T (t_{1}, …, t_{n}) := Θ(t_{1} … t_{n})U(t_{1}, …, t_{n})

= ∑_{σ∈𝔖(n)} \frac{1}{Θ(t_{σ(1)})Θ(t_{σ(1)}t_{σ(2)}) … Θ(t_{σ(1)} ⋯ t_{σ(n−1)})} ×

× det (\frac{Θ^{(j−i+1)}(t_{σ(1)} ⋯ t_{σ(n−j)})}{(j − i + 1)!}) .

Expanding the determinants one obtains

T (t_{1}, …, t_{n}) = ∑_{γ∈Γ_{n}}(−1)^{γ}Θ^{(#γ_{1})}(1) ∏_{k=2}^{ℓ(γ)} \frac{Θ^{(#γ_{k})} (∏_{i∈υ(γ)_{k}} t_{i})}{Θ (∏_{i∈υ(γ)_{k}} t_{i})} , (7.7)

where we used the notation (7.6) for the partial sums of a composition γ. Define the functions T^{π}(t), π ∈ Π_{n}, as in (7.5).

We will use two subsets of Π_{n}. By Π_{n}^{∘} denote the set of all partitions π such that π has at most one block of cardinality > 1 which, in addition, contains the number 1. By Π_{n}^{•} denote the set of all partitions π that have {1} as a block. We have, for example,

{{1, 2}, {3}} ∈ Π_{3}^{∘} ,
{{1}, {2, 3}} ∈ Π_{3}^{•} .

Denote by

Atom({1, …, n}) = {{1}, …, {n}} ∈ Π_{n}

the partition into 1-element subsets. It is clear that

Π_{n}^{∘} ∩ Π_{n}^{•} = Atom({1, …, n}) .

Define the subsets Γ_{n}^{∘}, Γ_{n}^{•} ⊂ Γ_{n} in the same way as for partitions of {1, …, n}.

We shall need the following identities Lemma 7.1.

∑_{π∈Π_{n}}(−1)^{π}ℓ(π)! = ∑_{γ∈Γ_{n}}(−1)^{γ} = 1 (7.8)

∑_{π∈Π_{n}}(−1)^{ℓ(π)}(ℓ(π) − 1)! = 0 . (7.9)

Proof. We shall prove (7.8). The proof of (7.9) is similar. Define f : ∏_{n} → ∏_{n−1} in the obvious way, by simply omitting n from a partition. We argue by induction on n. Note

#f^{−1}(π) = ℓ(π) + 1.

More precisely, f^{−1}(π) contains ℓ(π) partitions of length ℓ(π) and one partition of length ℓ(π) + 1. We compute

∑ ; π∈∏_{n}(−1)^{n+ℓ(π)}ℓ(π)! =

∑ ; π∈∏_{n−1}(−1)^{n}((−1)^{ℓ(π)}ℓ(π)ℓ(π)! + (−1)^{ℓ(π)+1}(ℓ(π) + 1)!) =

∑ ; π∈∏_{n−1}(−1)^{n−1+ℓ(π)}ℓ(π)! = 1.

8. Difference equations for the correlation functions

The aim of this section is the following

Theorem 8.1.

F (qt_{1}, t_{2}, …, t_{n}) = −q^{1/2}t_{1} ⋯ t_{n} ( ∑_{π∈Π_{n}^{∘}}(−1)^{π}F^{π}(t_{1}, …, t_{n})) .

Recall that, by definition,

∑_{π∈Π_{n}^{∘}}(−1)^{π}F^{π}(t_{1}, …, t_{n}) =

∑_{s=0}^{n−1}(−1)^{s} ∑_{1<i_{1}<···<i_{s}≤n} F (t_{1}t_{i_{1}}t_{i_{2}} ⋯ t_{i_{s}}, …, t̂_{i_{1}}, …, t̂_{i_{s}}, …) .

We begin with some auxiliary propositions.

Proposition 8.2. For any k = 1, …, n

H(t_{1}, …, qt_{k}, …, t_{n}) = − q^{1/2}t_{1} ⋯ t_{n}H(t_{1}, …, t_{n})

+ \frac{qt_{k}}{1 − qt_{k}} H(t_{1}, …, t_{k−1}, qt_{k}t_{k+1}, t_{k+2}, …, t_{n})

− \frac{1}{1 − qt_{k}} H(t_{1}, …, t_{k−2}, qt_{k−1}t_{k}, t_{k+1}, …, t_{n}) .

Here if k = 1 (or k = n) the third (second) summand should be omitted.

Corollary 8.3.

G(qt_{1}, t_{2}, …, t_{n}) = − q^{1/2}t_{1} ⋯ t_{n}G(t_{1}, …, t_{n}) (8.1)

− ∑_{k=2}^{n} G(qt_{1}t_{k}, t_{2}, …, t_{k−1}, t_{k+1}, …, t_{n})

(8.2)  = − q^{1/2}t_{1} ⋯ t_{n} ( ∑_{π∈Π_{n}^{∘}}(−1)^{π}(n − ℓ(π))! G^{π}(t)) .

Lemma 8.4. Let u and v be two variables. For any integers a < b

∑_{a<i<b} \frac{(qv)^{1/2−i}}{(q^{a}u)_{i−a}(q^{i}uv)_{b−i}} = \frac{1}{1 − qv} (\frac{(qv)^{3/2−b}}{(q^{a}u)_{b−a−1}} − \frac{(qv)^{1/2−a}}{(q^{a+1}uv)_{b−a−1}} ) . (8.3)

In particular, for a = 1 and u = 1 we obtain

∑_{1≤i<b} \frac{(qv)^{1/2−i}}{(q)_{i−1}(q^{i}v)_{b−i}} = \frac{1}{1 − qv} \frac{(qv)^{3/2−b}}{(q)_{b−2}} . (8.4)

Proof. Follows from the following particular case of the q-Gauss summation formula (6.10): for any c and d one has

∑_{i=c}^{∞}(qv)^{1/2−i} \frac{(q^{d}uv)_{i−c}}{(q^{d}u)_{i−c}} = −(qv)^{3/2−c} \frac{1 − q^{d−1}u}{1 − qv} . (8.5)

In the rest of the section we shall write a ≪ b if a + 1 < b.

Lemma 8.5. We have

H(t_{1}, …, qt_{k}, …, t_{n}) =

= ∑_{i_{1}<···<i_{k}≪i_{k+1}<···<i_{n}} q^{1/2−i_{k}}(1 − q^{i_{k}}t_{1} ⋯ t_{k})t_{k+1} ⋯ t_{n} ⟨\begin{matrix}i_{1}, …, i_{n} \\ t_{1}, …, t_{n} \end{matrix}⟩ (8.6)

= ∑_{i_{1}<···<i_{k−1}≪i_{k}<···<i_{n}} q^{3/2−i_{k}}(1 − q^{i_{k}−1}t_{1} ⋯ t_{k−1})t_{k} ⋯ t_{n} ⟨\begin{matrix}i_{1}, …, i_{n} \\ t_{1}, …, t_{n} \end{matrix}⟩ (8.7)

Proof. Recall that by definition

H(t_{1}, …, qt_{k}, …, t_{n}) = ∑_{i_{1}<···<i_{n}} ⟨\begin{matrix}i_{1}, …, i_{n} \\ t_{1}, …, qt_{k}, …, t_{n} \end{matrix}⟩ .

We have the two following identities:

⟨\begin{matrix}i_{1}, …, i_{n} \\ t_{1}, …, qt_{k}, …, t_{n} \end{matrix}⟩ =

(8.8) = q^{1/2−i_{k}}(1 − q^{i_{k}}t_{1} ⋯ t_{k})t_{k+1} ⋯ t_{n} ⟨\begin{matrix}i_{1}, …, i_{k}, i_{k+1} + 1, …, i_{n} + 1 \\ t_{1}, …, t_{n} \end{matrix}⟩

(8.9) = q^{1/2−i_{k}}(1 − q^{i_{k}}t_{1} ⋯ t_{k−1})t_{k} ⋯ t_{n} ⟨\begin{matrix}i_{1}, …, i_{k−1}, i_{k} + 1, …, i_{n} + 1 \\ t_{1}, …, t_{n} \end{matrix}⟩

Replacing in (8.8) the indices i_{k+1} + 1, …, i_{n} + 1 by i_{k+1}, …, i_{n} we obtain (8.6). Replacing in (8.9) the indices i_{k}+1, …, i_{n}+1 by i_{k}, …, i_{n} we obtain (8.7).

Proof of proposition (8.2). Consider the general case 1 < k < n. The two other cases k = 1 and k = n are similar. Rewrite the sum (8.6) as follows

(8.10) ∑_{i_{1}<···<i_{k}≪i_{k+1}<···<i_{n}} q^{1/2−i_{k}}(1 − q^{i_{k}}t_{1} ⋯ t_{k})t_{k+1} ⋯ t_{n} ⟨\begin{matrix}i_{1}, …, i_{n} \\ t_{1}, …, t_{n} \end{matrix}⟩

= ∑_{i_{1}<···<i_{n}}(…) − ∑_{i_{k+1}=i_{k}+1^{i_{1}<···<i_{k}≪i_{k+2}<···<in}}(…) ,

and denote the two summands in the RHS of (8.10) by Σ_{1} and Σ_{2}. The formula (8.6) together with the following identity

q^{1/2−i_{k}}(1 − q^{i_{k}}t_{1} ⋯ t_{k})t_{k+1} ⋯ t_{n} ⟨\begin{matrix}i_{1}, …, i_{k}, i_{k} + 1, i_{k+2}, …, i_{n} \\ t_{1}, …, t_{n} \end{matrix}⟩ =

q^{1/2−i_{k}}(1−q^{i_{k}}t_{1} ⋯ t_{k+1})t_{k+2} ⋯ t_{n} ⟨\begin{matrix}i_{1}, …, i_{k−1}, i_{k}, & i_{k+2}, …, i_{n} \\ t_{1}, …, t_{k−1}, t_{k}t_{k+1}, & t_{k+2}, …, t_{n} \end{matrix}⟩

implies that

Σ_{2} = H(t_{1}, …, qt_{k}t_{k+1}, …, t_{n}) .

Set

Σ_{3} : = Σ_{1} + q^{1/2}t_{1} ⋯ t_{n}H(t_{1}, …, t_{n})

= t_{k+1} ⋯ t_{n} ∑_{i_{1}<···<i_{n}} q^{1/2−i_{k}}⟨\begin{matrix}i \\ t \end{matrix}⟩ . (8.11)

Using Lemma 8.4 we can evaluate one of the nested sums in (8.11) as follows:

∑_{i_{k−1}<i_{k}<i_{k+1}} \frac{(qt_{k})^{1/2−i_{k}}}{(q^{i_{k−1}}t_{1} ⋯ t_{k−1})_{i_{k}−i_{k−1}}(q^{i_{k}}t_{1} ⋯ t_{k})_{i_{k+1}−i_{k}}}

= \frac{1}{1 − qt_{k}} (\frac{(qt_{k})^{3/2−i_{k+1}}}{(q^{i_{k−1}}t_{1} ⋯ t_{k−1})_{i_{k+1}−i_{k−1}−1}} − \frac{(qt_{k})^{1/2−i_{k−1}}}{(q^{i_{k−1}+1}t_{1} ⋯ t_{k})_{i_{k+1}−i_{k−1}−1}} ) .

Then sum Σ_{3} becomes the following difference

\frac{t_{k} ⋯ t_{n}}{1 − qt_{k}} ∑_{···<i_{k−1}≪i_{k+1}<...} q^{3/2−i_{k+1}}(1 − q^{i_{k+1}−1}t_{1} ⋯ t_{k−1}) ⟨\begin{matrix}…, i_{k−1}, i_{k+1}, … \\ …, t_{k−1}, t_{k}t_{k+1}, … \end{matrix}⟩

− \frac{t_{k+1} ⋯ t_{n}}{1 − qt_{k}} ∑_{···<i_{k−1}≪i_{k+1}<...} q^{1/2−i_{k−1}}(1 − q^{i_{k−1}}t_{1} ⋯ t_{k}) ⟨\begin{matrix}…, i_{k−1}, i_{k+1}, … \\ …, t_{k−1}t_{k}, t_{k+1}, … \end{matrix}⟩ .

By Lemma 8.5 this yields

Σ_{3} = \frac{1}{1 − qt_{k}} H(t_{1}, …, t_{k−1}, qt_{k}t_{k+1}, …, t_{n})

− \frac{1}{1 − qt_{k}} H(t_{1}, …, qt_{k−1}t_{k}, t_{k+1}, …, t_{n}) .

Thus,

H(t_{1}, …, qt_{k}, …, t_{n}) =Σ_{3} − q^{1/2}t_{1} ⋯ t_{n}H(t_{1}, …, t_{n}) − Σ_{2}
= − q^{1/2}t_{1} ⋯ t_{n}H(t_{1}, …, t_{n})

+ \frac{qt_{k}}{1 − qt_{k}} H(t_{1}, …, t_{k−1}, qt_{k}t_{k+1}, t_{k+2}, …, t_{n})

− \frac{1}{1 − qt_{k}} H(t_{1}, …, t_{k−2}, qt_{k−1}t_{k}, t_{k+1}, …, t_{n}) .

This concludes the proof.

Proof of corollary 8.3. We will prove (8.1); equation (8.2) follows easily from (8.1). Substitute the formula from proposition 8.2 into (7.3) and consider the coefficient of a summand

H(…, qt_{1}t_{j}, …) , j = 2, …, n .

Such a summand arises from the expansion of

H(…, qt_{1}, t_{j}, …) and H(…, t_{j}, qt_{1}, …) .

The sum of the coefficients equals

\frac{qt_{1}}{1 − qt_{1}} − \frac{1}{1 − qt_{1}} = −1 .

This proves (8.1).

Proposition 8.6.

F (qt_{1}, t_{2}, …, t_{n}) = −q^{1/2}t_{1} ⋯ t_{n} ( ∑_{π∈Π_{n}^{•}} G^{π}(t)) . (8.12)

Proof. Substitute (8.2) into (7.4). Let c_{π} denote the coefficient of the summand G^{π}(t) in this expansion. It is clear that

c_{π} = −q^{1/2}t_{1} ⋯ t_{n} , π ∈ Π_{n}^{•} .

Suppose that π ∉ Π_{n}^{•} and show that c_{π} = 0. Since F (qt_{1}, t_{2}, …, t_{n}) is symmetric in variables t_{2}, …, t_{n} we can assume that

π = {{1, 2, …, m}, π_{2}, …, π_{l}} , 1 < m ≤ n ,

where {π_{2}, …, π_{l}} is a partition of the set {m + 1, …, n}.

Let σ be a partition of {1, …, m}

σ = {σ_{1}, …, σ_{s}} ∈ Π_{m} .

Set

π ∧ σ = {σ_{1}, …, σ_{s}, π_{2}, …, π_{l}} ∈ Π_{n} .

It is easy to see that the term G^{π}(t) arises precisely from the expansion of the terms of the form

G^{π∧σ}(qt_{1}, t_{2}, …, t_{n}) , σ ∈ Π_{m} .

By (8.2) it arises with the coefficient

q^{1/2}t_{1} ⋯ t_{n}(−1)^{ℓ(σ)}(ℓ(σ) − 1)! .

By (7.9), the sum of these coefficients over all σ ∈ Π_{m} equals zero. This concludes the proof.

Proof of theorem 8.1. We will show that the RHS in theorem 8.1 equals the RHS of (8.12). Substitute (7.4) into RHS of theorem 8.1 and let c_{π} denote the coefficient of the summand G^{π}(t) in this expansion. Again, it is clear that

c_{π} = −q^{1/2}t_{1} ⋯ t_{n} , π ∈ Π_{n}^{•} .

and we have to show that

c_{π} = 0 , π ∉ Π_{n}^{•} .

Again, we can suppose that

π = {{1, 2, …, m}, π_{2}, …, π_{l}} , 1 < m ≤ n ,

where {π_{2}, …, π_{l}} is a partition of the set {m + 1, …, n}.

For any set S denote by Atom(S) the partition of S into one-element blocks. The summand G^{π}(t) arises in the RHS of from the expansion of terms

F^{{{S},Atom({1,...,n}\S)}}(t) , 1 ∈ S ⊂ {1, …, m} ,

and therefore the coefficient c_{π} equals

c_{π} = −q^{1/2}t_{1} ⋯ t_{n} ∑_{1∈S⊂{1,...,m}}(−1)^{|S|+1}

= −q^{1/2}t_{1} ⋯ t_{n} ∑_{k=0}^{m−1}(−1)^{k}\binom{m − 1}{k}

= 0 .

This concludes the proof.

9. Singularities of correlation functions

In this section we consider the singularities of the function F (t_{1}, …, t_{n}). The series (7.4) converges uniformly on compact subsets of the set

{|t_{i}| > 1 , i = 1, …, n} \ ⋃_{S⊂{1,...,n}} ⋃_{m=1}^{∞} {q^{m} ∏_{k∈S} t_{k} = 1}

and has simple poles on the divisors

{q^{m} ∏_{k∈S} t_{k} = 1} , S ⊂ {1, …, n} .

The difference equation from theorem 8.1 gives a meromorphic continuation of the function F (t) onto (a double cover of) the domain (ℂ \ 0)^{n}.

By symmetry, it suffices to consider the divisors

q^{m}t_{1}t_{2} … t_{k} = 1 , k ≤ n, m = 1, 2, …. (9.1)

We shall prove the following

Theorem 9.1. We have

F (t_{1}, …, t_{n}) = (−1)^{m} \frac{q^{m^{2}/2}m^{k−1}}{q^{m}t_{1} … t_{k} − 1} \frac{F (t_{k+1}, …, t_{n})}{(t_{k+1} … t_{n})^{m}} + …, (9.2)

where dots stand for terms regular at the divisor (9.1) and we assume that

F (t_{k+1}, …, t_{n}) = 1 , k = n .

A curious property of (9.2) is that the residue

Res_{{q^{m}t_{1}t_{2}...t_{k}=1}} F (t_{1}, …, t_{n}) (9.3)

does not depend on the variables t_{1}, …, t_{k}.

Introduce the following notation. Given a function f(i_{1}, …, i_{n}) set

; ∑_{1≤i_{1}≤···≤i_{k}≤m}f(i_{1}, …, i_{n}) = ∑_{1≤i_{1}≤···≤i_{n}≤m} \frac{f(i_{1}, …, i_{n})}{# Stab_{S(n)}\{i_{1}, …, i_{n}\}} .

If the function f is symmetric, we have

; ∑_{1≤i_{1}≤···≤i_{k}≤m}f(i_{1}, …, i_{n}) = \frac{1}{n!} ∑_{i_{1},...,i_{n}=1}^{m} f(i_{1}, …, i_{n}) . (9.4)

With this notation we have

F (t) = ∑_{of t_{1}, …, t_{n}^{over all permutations}} ; ∑_{1≤i_{1}≤···≤i_{n}} ⟨\begin{matrix}i \\ t \end{matrix}⟩ .

It is clear that the evaluation of the residue (9.3) boils down to the evaluation of the following sum

(9.5) ∑_{of t_{1}, …, t_{k}^{permutations}}

; ∑_{1≤i_{1}≤···≤i_{k}≤m} \frac{t_{1}^{1/2−i_{1}} … t_{k}^{1/2−i_{k}}}{(q)_{i_{1}−1}(q^{i_{1}}t_{1})_{i_{2}−i_{1}}(q^{i_{2}}t_{1}t_{2})_{i_{3}−i_{2}} … (q^{i_{k}}t_{1} … t_{k})_{m−i_{k}}} ,

where the variables t_{1}, …, t_{k} are subject to constraint

q^{m}t_{1} ⋯ t_{k} = 1 .

This evaluation follows from a curious rational function identity which shall now be established. Set

\begin{cases}i_{1}, …, i_{k} \\ t_{1}, …, t_{k} \end{cases}\}_{m} := \frac{1}{(q)_{i_{1}−1}(q^{i_{1}}t_{1})_{i_{2}−i_{1}}(q^{i_{2}}t_{1}t_{2})_{i_{3}−i_{2}} … (q^{i_{k}}t_{1} … t_{k})_{m−i_{k}}} .

Then we have the following

Theorem 9.2. Let m, k ∈ ℕ and suppose that

q^{m}t_{1}t_{2} … t_{k} = 1 .

Then

(9.6)  ∑_{of t_{1}, …, t_{k}^{cyclic permutations}}   ; ∑_{1≤i_{1}≤···≤i_{k}≤m} \begin{cases}i_{1}, …, i_{k} \\ t_{1}, …, t_{k} \end{cases}\}_{m} = \frac{m^{k−1}}{(k − 1)!} .

In particular, this sum is independent of variables q, t_{1}, …, t_{k}.

Proof of Theorem 9.2. Let us use q, t_{1}, …, t_{k−1} as independent variables. First, show that LHS of (9.6) does not depend on t_{1}, …, t_{k−1}. Since it is a rational function bounded on infinity it suffices to show that it does not have any poles. The only poles it can possibly have are simple poles on divisors

t_{s}t_{s+1} … t_{r−1}t_{r} = q^{−l} , s ≤ r ≤ k − 1, 1 ≤ l ≤ m .

By symmetry, it suffices to consider the pole on

t_{1} … t_{r} = q^{−l} , r ≤ k − 1, 1 ≤ l ≤ m . (9.7)

This pole arises from two type of summands in (9.6). First, it arises from the summands

\begin{cases}i_{1}, …, i_{k} \\ t_{1}, …, t_{k} \end{cases}\}_{m} , i_{r} ≤ l < i_{r+1} , (9.8)

and it also arises from summands

\begin{cases}i_{1}, …, i_{k} \\ t_{r+1}, …, t_{k}, t_{1}, …, t_{r} \end{cases}\}_{m} , i_{k−r} ≤ m − l < i_{k−r+1} . (9.9)

We shall match each summand of the form (9.8) to a summand of the form (9.9) in such a way that the poles in each pair will cancel out. Namely, it is easy to see that the sum

\begin{cases}i_{1}, …, i_{k} \\ t_{1}, …, t_{k} \end{cases}\}_{m} + \begin{cases}i_{r+1} − l, …, i_{k} − l, i_{1} + m − l, …, i_{r} + m − l \\ t_{r+1}, …, t_{k}, t_{1}, …, t_{r} \end{cases}\}_{m} , (9.10)

where

1 ≤ i_{1} ≤ ⋯ ≤ i_{r} ≤ l < i_{r+1} ≤ ⋯ ≤ i_{k} ≤ m ,

is regular on the divisor (9.7); notice that the inequalities

1 ≤ i_{1} ≤ ⋯ ≤ i_{r} < i_{r+1} ≤ ⋯ ≤ i_{k} ≤ m

imply that

# Stab{i_{1}, …, i_{k}} = # Stab{i_{r+1}−l, …, i_{k}−l, i_{1}+m−l, …, i_{r}+m−l} .

This proves that the LHS of (9.6) is regular on (9.7).

Therefore, the LHS of (9.6) does not depend on t_{1}, …, t_{k} and we can substitute

t_{1} = t_{2} = ⋯ = t_{k−1} = 1 , t_{k} = q^{−m} .

Then the LHS of (9.6) becomes

; ∑_{1≤i_{1}≤···≤i_{k}≤m} ∑_{r=1}^{k} \frac{1}{(q)_{i_{r}}(q^{i_{r}−m})_{m−i_{r}}} . (9.11)

Since the inner sum in (9.11) is symmetric in i_{1}, …, i_{k} we can use (9.4) to obtain

\frac{1}{k!} ∑_{i_{1},...,i_{k}=1}^{m} ∑_{r=1}^{k} \frac{1}{(q)_{i_{r}}(q^{i_{r}−m})_{m−i_{r}}}

= \frac{m^{k−1}}{(k − 1)!} ∑_{i=1}^{m} \frac{1}{(q)_{i}(q^{i−m})_{m−i}}

= \frac{m^{k−1}}{(k − 1)!} \frac{1}{(q^{1−m})_{m−1}} ∑_{i=1}^{m} \frac{(q^{1−m})_{i−1}}{(q)_{i}}

= \frac{m^{k−1}}{(k − 1)!} ,

where the last equality follows from the q-binomial theorem. This concludes the proof.

Replacing in (9.6) the variables q, t_{1}, …, t_{k} by their reciprocals one obtains the following

Corollary 9.3. Suppose that q^{m}t_{1}t_{2} … t_{k} = 1. Then the sum

∑_{of t_{1}, …, t_{k}^{cyclic permutations}}  ; ∑_{1≤i_{1}≤···≤i_{k}≤m} \frac{t_{1}^{1/2−i_{1}} … t_{k}^{1/2−i_{k}}}{(q)_{i_{1}−1}(q^{i_{1}}t_{1})_{i_{2}−i_{1}}(q^{i_{2}}t_{1}t_{2})_{i_{3}−i_{2}} … (q^{i_{k}}t_{1} … t_{k})_{m−i_{k}}} (9.12)

equals

(−1)^{m−1} \frac{q^{m^{2}/2}m^{k−1}}{(k − 1)!} ,

for all m and k.

This corollary gives the evaluation of the sum (9.5) and this proves (9.2)

10. Difference equations for the RHS in (6.5)

In this section we show that the functions U(t_{1}, …, t_{n}) satisfy the very same difference equations as the functions F (t) do.

Since we shall use only the difference equation (6.3) for the theta function Θ(x), let us consider following general situation.

Let t_{0} be an auxiliary variable. We shall eventually let

t_{0} → 1 .

Modify temporarily the definition (7.6) as follows

υ(γ)_{k} = {0} ∪ γ_{1} ∪ ⋯ ∪ γ_{k−1} . (10.1)

Suppose that a function

r(x; m) , m = 0, 1, 2, …,

satisfies the two following properties:

r(x; 0) = 1 (10.2)

identically and

r(qx; m) = ∑_{i=0}^{m}(−1)^{i}\binom{m}{i} r(x; m − i) . (10.3)

Consider the function

R(t_{1}, …, t_{n} | t_{0}) = ∑_{γ∈Γ_{n}}(−1)^{γ} ∏_{k=1}^{ℓ(γ)} r (∏_{i∈υ(γ)_{k}}t_{i}; #γ_{k}) , (10.4)

where we use the modified definition (10.1). We want to show that Theorem 10.1. Suppose a function r(x; m) satisfies (10.2) and (10.3). Then

R(qt_{1}, t_{2}, …, t_{n} | t_{0}) = ∑_{π∈Π_{n}^{∘}}(−1)^{π}R^{π}(t_{1}, …, t_{n} | t_{0}) , (10.5)

where the function R(t_{1}, …, t_{n} | t_{0}) was defined in (10.4)

Here, as usual,

R^{π}(t_{1}, …, t_{n} | t_{0}) := R ( ∏_{k∈π_{1}} t_{k} , …, ∏_{k∈π_{ℓ(π)}} t_{k} | t_{0}) , π ∈ Π_{n} . (10.6)

Proof. Given a composition

γ = (γ_{1}, …, γ_{l}) ,

let m = m(γ) denote the number of the block that contains 1. Substitute (10.3) into (10.4). We obtain

R(qt_{1}, t_{2}, …, t_{n} | t_{0}) = ∑_{γ∈Γ_{n}} ∑_{s_{m+1}=0}^{#γ_{m+1}} ⋯ ∑_{s_{l}=0}^{#γ_{l}} [γ_{1}, …, γ_{m} |\begin{matrix}γ_{m+1}, …, γ_{l} \\ s_{m+1}, …, s_{l} \end{matrix} ] , (10.7)

where

[γ_{1}, …, γ_{m} |\begin{matrix}γ_{m+1}, …, γ_{l} \\ s_{m+1}, …, s_{l} \end{matrix} ] (10.8)

stands for the following product

(−1)^{n+ℓ(γ)+s_{m+1}+···+s_{l}} ∏_{k=1}^{m} r (∏_{i∈υ(γ)_{k}}t_{i}; #γ_{k}) ×

∏_{k=m+1}^{l} \binom{#γ_{i}}{s_{i}}r (∏_{i∈υ(γ)_{k}}t_{i}; #γ_{k} − s_{i}) .

Let us divide all summands (10.8) into 3 following types according to occurrence of certain patters in the sequence s_{m+1}, …, s_{l}. The summands of the form

[γ_{1}, …, γ_{m} |\begin{matrix}γ_{m+1}, & …, & γ_{k}, & … \\ #γ_{m+1}, & …, & #γ_{k}, & 0, …, 0 \end{matrix}] , m + 1 ≤ k ≤ l , (10.9)

will be called type I summands. If (10.8) is not of type I then let k, m + 1 ≤ k ≤ l, be the minimal number such that

(0 < s_{k} < #γ_{k}) or (s_{k} = 0 and s_{k+1} = #γ_{k+1}) .

We shall say that (10.8) is of type II (type III) if the first (second) parenthesis contains a true statement.

We shall show that the the type II summands cancel with the type III summand while the type I summands produce the RHS of (10.5).

The cancelation of the type II and type III summands follows from the following identity. Suppose that

0 < s_{k} < #γ_{k} .

Then

(10.10) [γ_{1}, …, γ_{m} |\begin{matrix}…, γ_{k}, … \\ …, s_{k}, … \end{matrix}] +

∑_{#δ=s_{k}^{δ⊂γ_{k}}} [γ_{1}, …, γ_{m} |\begin{matrix}…, & γ_{k} \ δ, & δ, & … \\ …, & 0, & #δ, & … \end{matrix}] = 0 .

To see (10.10) notice that all \binom{γ_{k}}{s_{k}} summands in the sum over subsets δ ⊂ γ_{k} are equal and proportional to the first summand in (10.10).

Now consider a type I summand (10.9). Set

δ = γ_{m+1} ∪ ⋯ ∪ γ_{k} . (10.11)

We want to fix a subset

δ ⊂ {2, …, n}

and compute the sum of all type I summands (10.9) satisfying (10.11). Let us consider the nontrivial case

δ ≠ ∅ .

First sum over all

(γ_{m+1}, …, γ_{k}) ∈ Γ(δ) ,

where Γ(δ) stands for the set of all compositions of the set δ. We have

(10.12) [γ_{1}, …, γ_{m} |\begin{matrix}γ_{m+1}, & …, & γ_{k}, & γ_{k+1}, … \\ #γ_{m+1}, & …, & #γ_{k}, & 0, …, 0 \end{matrix}] =

(−1)^{k−m+1} [γ_{1}, …, γ_{m} |\begin{matrix}δ & γ_{k+1}, … \\ #δ, & 0, …, 0 \end{matrix}] .

Since (10.12) depends only on the parity of number of parts in the composition (10.11) we can use (7.8) to obtain

∑_{(γ_{m+1},...,γ_{k})∈Γ(δ)} [γ_{1}, …, γ_{m} |\begin{matrix}γ_{m+1}, & …, & γ_{k}, & γ_{k+1}, … \\ #γ_{m+1}, & …, & #γ_{k}, & 0, …, 0 \end{matrix}] =

(−1)^{#δ+1} [γ_{1}, …, γ_{m} |\begin{matrix}δ, & γ_{k+1}, … \\ #δ, & 0, …, 0 \end{matrix}] .

Now sum over the remaining blocks

(γ_{1}, …, γ_{m}, γ_{k+1}, …, γ_{l}) .

It follows from the definition (10.4), (10.6) that

(−1)^{#δ+1} ∑_{(γ_{1},...,γ_{m},γ_{k+1},...,γ_{l})} [γ_{1}, …, γ_{m} |\begin{matrix}δ, & γ_{k+1}, … \\ #δ, & 0, …, 0 \end{matrix}]

= (−1)^{n+ℓ(σ)}R^{σ}(t | t_{0}) ,

where σ = σ(δ) is the following element of Π_{n}^{∘}

σ = {1 ∪ δ, Atom({2, …, n} \ δ)} .

Recall that Atom({2, …, n}\δ) denotes the partition of the set {2, …, n}\ δ into 1-element blocks.

Thus, the sum of all type I summands in (10.7) equals the RHS of (10.5). This concludes the proof of the theorem.

Corollary 10.2.

T (qt_{1}, t_{2}, …, t_{n}) = ∑_{π∈Π_{n}^{∘}}(−1)^{n+ℓ(π)}T^{π}(t_{1}, …, t_{n}) . (10.13)

Proof. Take

r(x; m) = \frac{Θ^{(m)}(x)}{Θ(x)} .

By virtue of (6.3) this function satisfies (10.3) and it obviously satisfies (10.2). We have

T (t_{1}, …, t_{n}) = Res_{t_{0}=1} R(t_{1}, …, t_{n} | t_{0}) .

Taking the residue in (10.5) we obtain (10.13).

Corollary 10.3.

U(qt_{1}, t_{2}, …, t_{n}) = −q^{1/2}t_{1} ⋯ t_{n} ( ∑_{π∈Π_{n}^{∘}}(−1)^{n+ℓ(π)}U^{π}(t_{1}, …, t_{n})) . (10.14)

11. Singularities of the RHS in (6.5)

In this section we shall prove that U(t_{1}, …, t_{n}) has exactly the same singularities as F (t_{1}, …, t_{n}).

Theorem 11.1. For k = 1, …, n we have

U(t_{1}, …, t_{n}) = (−1)^{m} \frac{q^{m^{2}/2}m^{k−1}}{q^{m}t_{1} ⋯ t_{k} − 1} \frac{U(t_{k+1}, …, t_{n})}{(t_{k+1} ⋯ t_{n})^{m}} + …, (11.1)

where dots stand for terms regular at the divisor

q^{m}t_{1} ⋯ t_{k} = 1

and we assume that

U(t_{k+1}, …, t_{n}) = 1 , k = n .

Let us again point out that (11.1) implies that the residue

Res_{q^{m}t_{1}...t_{k}=1} U(t_{1}, …, t_{n})

is independent of t_{1}, ⋯, t_{k}. In the proof we shall use a curious identity good for any odd smooth function which we shall state as a separate Theorem 11.3.

It is clear that since the function U(t) satisfies the very same difference equation as F (t) does it suffices to consider the case

m = 0 .

That is, we have to show that

U(t_{1}, …, t_{n}) = \frac{1}{t_{1} − 1}U(t_{2}, …, t_{n}) + …, t_{1} → 1 , (11.2)

and that

(11.3) U(t) is regular on {t_{1} ⋯ t_{k} = 1} for 1 < k ≤ n

It follows from the very definition of the function U(t) (look, for example, at the formula (6.6)) that

U(t_{1}, …, t_{n}) = U(t_{1}, …, t_{k})U(t_{k+1} … t_{n}) + …

= \frac{T (t_{1}, …, t_{k})}{t_{1} ⋯ t_{k} − 1}U(t_{k+1} … t_{n}) + …,

where dots stand for terms regular on the divisor

t_{1} ⋯ t_{k} = 1 .

Since

T (1) = 1 .

the Theorem 4.1 will follow from the following Theorem 11.2. We have

T (t_{1}, …, t_{n})|_{t_{1}···t_{n}=1} = 0 (11.4)

provided n > 1.

Proof. Till the end of the proof the variables t_{1}, …, t_{n} will be always subject to constraint

t_{1} ⋯ t_{n} = 1 . (11.5)

Induct on n. Since

T (t_{1}, t_{2}) = \frac{Θ′(t_{1})}{Θ(t_{1})} + \frac{Θ′(t_{2})}{Θ(t_{2})}

the case n = 2 is clear.

First, show that

(11.6) T (t_{1}, …, t_{n}) is a constant.

By the difference equation (10.13) and the induction hypothesis

T (qt_{1}, t_{2}, …, t_{n}) = T (t_{1}, …, t_{n}) + (−1)^{n+1} .

Similarly

T (qt_{1}, t_{2}, …, t_{n}) = T (qt_{1}, q^{−1}t_{2}, …, t_{n}) + (−1)^{n+1} .

It follows that

T (qt_{1}, q^{−1}t_{2}, …, t_{n}) = T (t_{1}, …, t_{n}) .

Hence, by virtue of the strategy enunciated at the beginning of section 7, the claim (11.6) will follow from the following claim

(11.7) T (t_{1}, …, t_{n}) is regular ,

which will now be established.

By symmetry it suffices to show that T (t_{1}, …, t_{n}) is regular on the divisor

t_{1} … t_{k} = 1 , 1 ≤ k < n − 2 . (11.8)

Again, from the definition (7.7) it is clear that

(11.9) T (t_{1}, …, t_{n}) =

(\frac{1}{Θ(t_{1} … t_{k})} + \frac{1}{Θ(t_{k+1} … t_{n})}) T (t_{1} … t_{k})T (t_{k+1} … t_{n}) + …

where dots stand for a function regular on (11.8). By (11.5) the sum in parentheses in (11.9) vanishes and this proves (11.6) and (11.7).

Thus, what is left is to show that LHS of (11.4) vanishes at some point. This follows from the general identity established below in Theorem 11.3 where one has to substitute

f(x) = Θ(x) .

This concludes the proof of the theorem.

Let f(x) be an odd function

f(x^{−1}) = −f(x) .

Set

f^{(k)}(x) := (x\frac{∂}{∂x})^{k} f(x) , k ∈ ℕ .

Consider the following function

Φ(t_{1}, …, t_{n}) := ∑_{γ∈Γ_{n}} Ψ_{γ}(t) , (11.10)

where

Ψ_{γ}(t) = (−1)^{ℓ(γ)}f^{(#γ_{1})}(1) ∏_{i=2}^{ℓ(γ)} \frac{f^{(#γ_{i})} (∏_{j∈υ(γ)_{i}}t_{j})}{f (∏_{j∈υ(γ)_{i}}t_{j})} .

Note that Ψ_{γ}(t) = 0 if #γ_{1} is even. We want to prove the following

Theorem 11.3. Suppose that n > 1 and the variables t_{1}, …, t_{n} are subject to constraint

t_{1} ⋯ t_{n} = 1 .

Then for any odd function f(x) with a simple zero at x = 1

f(1) = 0 , f′(1) ≠ 0

we have

Φ(t_{1}, …, t_{n}) → 0 , t_{1} → 1 , (11.11)

where the function Φ(t) was defined in (11.10).

Proof. Let us divide all summands in (11.10) into 3 following types: 1. γ_{ℓ(γ)} = {1},

2. γ_{1} = {1},

3. others. Note that all type 3 summands are regular at t_{1} = 1.

Let γ = (γ_{1}, …, γ_{ℓ−1}, {1}) be a type 1 composition. Then the composition

γ′ = ({1}, γ_{1}, …, γ_{ℓ−1})

is of type 2. Consider the sum

Ψ_{γ}(t) + Ψ_{γ′}(t) . (11.12)

There are two possible cases: #γ_{1} is even and #γ_{1} is odd. If

#γ_{1} is even

then the first summand in (11.12) is zero and the second one is regular at t_{1} = 1. In this case we obtain

lim_{t_{1}→1} (Ψ_{γ}(t) + Ψ_{γ′}(t)) = (−1)^{ℓ}f^{(#γ_{1}+1)}(1) ∏_{i=2}^{ℓ−1} \frac{f^{(#γ_{i})} (∏_{j∈υ(γ)_{i}}t_{j})}{f (∏_{j∈υ(γ)_{i}}t_{j})} . (11.13)

It is easy to see that (11.13) will exactly cancel with the type 3 summand corresponding to the composition

({1} ∪ γ_{1}, …, γ_{ℓ−1}) . (11.14)

Note that if #γ_{1} is odd then the contribution of the composition (11.14) is zero.

Now consider the sum (11.12) in the case

#γ_{1} is odd .

We have

Ψ_{γ}(t) = (−1)^{ℓ+1}f^{(#γ_{1})}(1) \frac{f′(t_{1})}{f(t_{1})} ∏_{i=2}^{ℓ−1} \frac{f^{(#γ_{i})} (∏_{j∈υ(γ)_{i}}t_{j})}{f (∏_{j∈υ(γ)_{i}}t_{j})}

Ψ_{γ′}(t) = (−1)^{ℓ}f′(1) \frac{f^{(#γ_{1})}(t_{1})}{f(t_{1})} ∏_{i=2}^{ℓ−1} \frac{f^{(#γ_{i})} (t_{1} · ∏_{j∈υ(γ)_{i}}t_{j})}{f (t_{1} · ∏_{j∈υ(γ)_{i}}t_{j})}

Observe that

f′(1) \frac{f^{(#γ_{1})}(t_{1})}{f(t_{1})} − f^{(#γ_{1})}(1) \frac{f′(t_{1})}{f(t_{1})} → 0 , t_{1} → 1 (11.15)

THE CHARACTER OF THE INFINITE WEDGE REPRESENTATION 51

because (11.15) is regular and odd. Therefore

lim_{t_{1}→1} (Ψ_{γ}(t) + Ψ_{γ′}(t)) =

(−1)^{ℓ}f^{(#γ_{1})}(1) lim_{t_{1}→1} \frac{f′(t_{1})}{f(t_{1})} (∏_{i=2}^{ℓ−1} \frac{f^{(#γ_{i})} (t_{1} · ∏_{j∈υ(γ)_{i}}t_{j})}{f (t_{1} · ∏_{j∈υ(γ)_{i}}t_{j})}−

∏_{i=2}^{ℓ−1} \frac{f^{(#γ_{i})} (∏_{j∈υ(γ)_{i}}t_{j})}{f (∏_{j∈υ(γ)_{i}}t_{j})}) .

By L’Hospital rule this limit equals

(−1)^{ℓ}f^{(#γ_{1})}(1) (∑_{k=2}^{ℓ−1} (\frac{f^{(#γ_{k}+1)} (∏_{j∈υ(γ)_{k}}t_{j})}{f (∏_{j∈υ(γ)_{k}}t_{j})}−

− \frac{f^{(#γ_{k})} (∏_{j∈υ(γ)_{k}}t_{j}) f′ (∏_{j∈υ(γ)_{k}}t_{j})}{f (∏_{j∈υ(γ)_{k}}t_{j})^{2}} ) ∏_{i≠k^{i=2}}^{ℓ−1} \frac{f^{(#γ_{i})} (∏_{j∈υ(γ)_{i}}t_{j})}{f (∏_{j∈υ(γ)_{i}}t_{j})}) .

It is easy to see that the summand

(−1)^{ℓ}f^{(#γ_{1})}(1) \frac{f^{(#γ_{k}+1)} (∏_{j∈υ(γ)_{k}}t_{j})}{f (∏_{j∈υ(γ)_{k}}t_{j})} ∏_{i≠k^{i=2}}^{ℓ−1} \frac{f^{(#γ_{i})} (∏_{j∈υ(γ)_{i}}t_{j})}{f (∏_{j∈υ(γ)_{i}}t_{j})}

cancels with the contribution of the type 3 composition (γ_{1}, …, {1} ∪ γ_{k}, …, γ_{ℓ−1}) , 2 ≤ k ≤ l − 1 . (11.16) Similarly the summand

(−1)^{ℓ+1}f^{(#γ_{1})}(1) \frac{f^{(#γ_{k})} (∏_{j∈υ(γ)_{k}}t_{j}) f′ (∏_{j∈υ(γ)_{k}}t_{j})}{f (∏_{j∈υ(γ)_{k}}t_{j})^{2}} ×

∏_{i≠k^{i=2}}^{ℓ−1} \frac{f^{(#γ_{i})} (∏_{j∈υ(γ)_{i}}t_{j})}{f (∏_{j∈υ(γ)_{i}}t_{j})} .

cancels with the contribution of the type 3 composition (γ_{1}, …, {1}, γ_{k}, …, γ_{ℓ−1}) , 2 ≤ k ≤ l − 1 . (11.17)

It is clear that the compositions of the form (11.14), (11.16), and (11.17) exhaust the set of type 3 compositions. This concludes the proof.

12. Conclusion of the proof of Theorem 6.1

Induct on n. Suppose that n = 1. By Theorems 8.1 and 9.1 the function

Θ(t_{1})F (t_{1})

is holomorphic on ℂ \ 0, invariant under the transformation

t_{1} ↦ qt_{1}

and equal to 1 for t_{1} = 1. It follows that

F (t_{1}) = \frac{1}{Θ(t_{1})} .

Suppose that n > 1 and consider the function

Θ(t_{1} ⋯ t_{n})F (t_{1}, …, t_{n}) − T (t_{1}, …, t_{n}) . (12.1)

By induction hypothesis, we have

Θ(t_{1} ⋯ t_{n})F^{π}(t_{1}, …, t_{n}) − T^{π}(t_{1}, …, t_{n}) ,

provided

ℓ(π) < n .

Therefore by Theorem 1.1 and Corollary 3.2 the function is invariant under the transformation

t_{1} ↦ qt_{1} .

By symmetry it is invariant under all transformations

t_{i} ↦ qt_{i} , i = 1, …, n .

By Theorems 9.1 and 11.1 the function (12.1) is holomorphic and vanishes if

t_{1} ⋯ t_{n} = 0 .

Therefore (12.1) equals zero. This concludes the proof.

13. Another Example

Another representation of a subalgebra of the algebra D of differential operators on ℂ[t, t^{−1}] consisting of differential operators which are skew-adjoint in a suitable sense, was studied in [2]. This algebra contains Virasoro and also the odd powers D, D^{3}, D^{5}, … of D = t\frac{d}{dt}. The D^{2n+1} act semisimple with finite eigenspaces, and the resulting character is

Ψ(τ_{1}, τ_{3}, …) = q_{1}^{ζ(−1)/2}q_{3}^{ζ(−3)/2} ⋯ ∏_{n=1}^{∞}(1 − q_{1}^{n}q_{3}^{n^{3}} ⋯)^{−1} (13.1)

In this section, we will show that the Taylor expansion for Ψ is quasimodular of weight -1/2 (cf. (4.1)), and we will calculate the n-point function

(13.2) F_{n}(τ_{1}, z_{1}, …, z_{n}) :=

∑_{k_{1},...,k_{n}≥1} \frac{∂^{n}}{∂τ_{2k_{1}−1} ⋯ ∂τ_{2k_{n}−1}} (Ψ)|_{τ_{3}=···=0} \frac{z_{1}^{2k_{1}−1} ⋯ z_{n}^{2k_{n}−1}}{(2k_{1} − 1)! ⋯ (2k_{n} − 1)!}.

As before, we write

q_{r} = exp(2πiτ_{r}); r ≥ 1.

We compute

\frac{1}{2πi} \frac{∂}{∂τ_{2j−1}} log Ψ = ζ(1 − 2j)/2 + ∑_{m,n=1}^{∞} m^{2j−1}q_{1}^{nm}q_{3}^{nm^{3}} ⋯

\frac{1}{(2πi)^{r}} \frac{∂^{r}}{∂τ_{2j_{1}−1} ⋯ ∂τ_{2j_{r}−1}} log Ψ = ∑_{m,n=1}^{∞} m^{2(∑_{k} j_{k}−r)−1}(nm)^{r−1}q_{1}^{nm}q_{3}^{nm^{3}} ⋯

Note these expressions depend only on r and ∑_{k=1}^{r} j_{k}. Define

h_{r,2(j_{1}+...+j_{r})}(τ_{1}, τ_{3}, …) := \frac{1}{(2πi)^{r}} \frac{∂^{r}}{∂τ_{2j_{1}−1} ⋯ ∂τ_{2j_{r}−1}} log Ψ, (13.3)

g_{r,2(j_{1}+...+j_{r})}(τ_{1}) := h_{r,2(j_{1}+...+j_{r})}(τ_{1}, 0, 0, …) = \frac{∂^{r−1}}{∂τ_{1}^{r−1}} G_{2(j_{1}+...+j_{r}−r+1)}(τ_{1}) (13.4)

where G_{2p}(τ) is the Eisenstein series of weight 2p as in (3.4). Note g_{r,s} is quasimodular of weight 2a. For A = (a_{3}, a_{5}, …) write wt(A) = 4a_{3} + 6a_{5} + … (compare (4.1)). The coefficient of τ^{A}/A! in the Taylor expansion for log Ψ is g_{r,wtA}(τ_{1}). Also

Ψ|_{τ_{3}=···=0} = η^{−1}(τ_{1}) (13.5)

Thus, by (4.1) we conclude that log(Ψη) is quasimodular of weight 0. Exponentiating yields

Proposition 13.1. Ψ is quasimodular of weight −1/2.

We now consider the n-point function (13.2)

Lemma 13.2. Let S = {j_{1}, …, j_{n}} be a set of positive integers. Let Π(S) denote the set of all partitions μ = {μ_{1}, …, μ_{ℓ}} of S. For any finite set ϕ let #ϕ denote the number of elements in ϕ. For a subset ϕ ⊂ S let |ϕ| be the sum over the elements. Then

\frac{1}{(2πi)^{n}} \frac{∂^{n}}{∂τ_{2j_{1}−1} ⋯ ∂τ_{2j_{n}−1}} Ψ = Ψ ∑_{μ∈Π(S)} ∏_{k=1}^{#μ} h_{#μ_{k},2|μ_{k}|}

Proof. By induction on n. For n = 1 this is just the logarithmic derivative. Suppose now n ≥ 2 and the assertion holds for n − 1. Note

\frac{1}{2πi} \frac{1}{∂τ_{2j_{n}−1}} h_{n−1,2(j_{1}+...+j_{n−1})} = h_{n,2(j_{1}+...+j_{n})}

Let T = {j_{1}, …, j_{n−1}}. We have inductively

\frac{1}{(2πi)^{n}} \frac{∂^{n}}{∂τ_{2j_{1}−1} ⋯ ∂τ_{2j_{n}−1}} Ψ = \frac{1}{2πi} \frac{∂}{∂τ_{2j_{n}−1}} (Ψ ∑_{μ∈Π(T )} ∏_{k=1}^{#μ} h_{#μ_{k},2|μ_{k}|}) =

= Ψh_{1,2j_{n}} ∑_{μ∈Π(T )} ∏_{k=1}^{#μ} h_{#μ_{k},2|μ_{k}|}+Ψ ∑_{μ∈Π(T )} ∑_{p=1}^{#μ} h_{#μ_{p}+1,2(|μ_{p}|+j_{n})} ∏_{k≠p} h_{#μ_{k},2|μ_{k}|}

= Ψ ∑_{μ∈Π(S)} ∏_{k=1}^{#μ} h_{#μ_{k},2|μ_{k}|}.

Define

G_{n}(τ, z) := ∑_{r=1}^{∞} \frac{∂^{n−1}}{∂τ^{n−1}} G_{2r}(τ)z^{2r+n−2}/(2r + n − 2)! (13.6)

One has, with Θ as in (6.1)

2G_{1}(τ, 2πiz) := \frac{1}{2πi}(−\frac{d}{dz} log Θ(z) + \frac{1}{z}) (13.7)

To see this, let σ(z, τ) be the elliptic sigma function as defined e.g. in [12], p. 247. One has

\frac{−1}{2πi} \frac{d}{dz} log Θ = \frac{−1}{2πi} \frac{d}{dz} log σ + 2G_{2}(τ)(2πiz) =

\frac{−1}{2πiz} + 2G_{2}(τ)(2πiz) + 2G_{4}(τ)(2πiz)^{3}/3! + 2G_{6}(τ)(2πiz)^{5}/5! + …

Combining (13.2), (13.4), and (13.5) we get

(13.8) F(τ, z_{1}, …, z_{n}) = η(τ)^{−1} ∑_{k_{1},... ,k_{n}≥1} \frac{z_{1}^{2k_{1}−1} ⋯ z_{n}^{2k_{n}−1}}{(2k_{1} − 1)! ⋯ (2k_{n} − 1)!} ×

∑_{μ∈Π\{k_{1},...,k_{n}\}} ∏_{p=1}^{#μ} \frac{∂^{#μ_{p}−1}}{∂τ^{#μ_{p}−1}} G_{2(|μ_{p}|−#μ_{p}+1)}(τ).

For a function f(z_{1}, …, z_{n}) let ϵf denote the “oddification” of f, e.g.

ϵf(z_{1}, z_{2}) = \frac{1}{4}(f(z_{1}, z_{2}) − f(−z_{1}, z_{2}) − f(z_{1}, −z_{2}) + f(−z_{1}, −z_{2}))

Note

ϵ((z_{1} + … + z_{n})^{r}/r!) = ∑_{ℓ_{1}+...+ℓ_{n}=(r+n)/2} \frac{z_{1}^{2ℓ_{1}−1} ⋯ z_{n}^{2ℓ_{n}−1}}{(2ℓ_{1} − 1)! ⋯ (2ℓ_{n} − 1)!}

It follows that

ϵG_{n}(τ, z_{1} + … + z_{n}) = ∑_{r=1}^{∞} \frac{∂^{n−1}}{∂τ^{n−1}} G_{2r}(τ) ×

∑_{k_{1}+...+k_{n}=r+n−1} \frac{z_{1}^{2k_{1}−1} ⋯ z_{n}^{2k_{n}−1}}{(2k_{1} − 1)! ⋯ (2k_{n} − 1)!}

We conclude

Proposition 13.3. The n-point function is given by

F(τ, z_{1}, …, z_{n}) = η(τ)^{−1} ∑_{μ∈Π_{n}} ∏_{k=1}^{k=#μ} ϵG_{#μ_{k}}(∑_{i∈μ_{k}} z_{i})

Proof. We have an obvious identification

Π_{n} = Π{1, …, n} = Π{k_{1}, …, k_{n}}

(Note {k_{1}, …, k_{n}} is treated formally, i.e. we do not identify the k_{i} if they are equal.) Write κ = {k_{1}, …, k_{n}} and let μ_{p,κ} ⊂ κ be the corresponding subset. Given μ ∈ Π_{n} define

F_{μ}(τ, z_{1}, …, z_{n}) = η(τ)^{−1} ∑_{k_{1},... ,k_{n}≥1} \frac{z_{1}^{2k_{1}−1} ⋯ z_{n}^{2k_{n}−1}}{(2k_{1} − 1)! ⋯ (2k_{n} − 1)!} ×

∏_{p=1}^{#μ} \frac{∂^{#μ_{p}−1}}{∂τ^{#μ_{p}−1}} G_{2(|μ_{p,κ}|−#μ_{p}+1)}(τ).

so F = ∑_{Π_{n}} F_{μ}. It will suffice to show

F_{μ} = η(τ)^{−1} ∏_{p=1}^{p=#μ} ϵG_{#μ_{p}}(∑_{i∈μ_{p}} z_{i}). (13.9)

But given κ, the term z_{1}^{2k_{1}−1} ⋯ z_{n}^{2k_{n}−1} occurs exactly once in (13.9) and has the correct coefficient in τ.

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Department of Mathematics, University of Chicago, Chicago, IL 60637

  E-mail address: bloch@@math.uchicago.edu, okounkov@@math.uchicago.edu