Fermionic approach to weighted Hurwitz numbers and topological recursion

TL;DR

This paper introduces a fermionic framework for weighted Hurwitz numbers and topological recursion, deriving new formulas and spectral curves that unify various special cases and connect to quantum geometry.

Contribution

It provides a novel fermionic representation for generating functions, correlators, and spectral curves in the theory of weighted Hurwitz numbers and topological recursion.

Findings

Derived recursion relations for adapted bases and duals

Established Christoffel-Darboux formula for pair correlator

Connected quantum and classical spectral curves with topological recursion

Abstract

A fermionic representation is given for all the quantities entering in the generating function approach to weighted Hurwitz numbers and topological recursion. This includes: KP and 2D Toda $\tau$-functions of hypergeometric type, which serve as generating functions for weighted single and double Hurwitz numbers; the Baker function, which is expanded in an adapted basis obtained by applying the same dressing transformation to all vacuum basis elements; the multipair correlators and the multicurrent correlators. Multiplicative and differential recursion relations are deduced for the adapted bases and their duals, and a Christoffel-Darboux type formula is derived for the pair correlator. The quantum and classical spectral curves linking this theory with the topological recursion programme are derived, as well as the generalized cut and join equations. The results are detailed for four special cases: the simple single and double Hurwitz numbers, the weakly monotone case, corresponding to signed enumeration of coverings, the strongly monotone case, corresponding to Belyi curves and the simplest version of quantum weighted Hurwitz numbers.

Equations

$$\mathbf{I}=h_1,\cdots h_k,h_i\in \mathrm{cyc}({\mu }^{(i))}).$$

$${\ell }^{\ast }(\mu ):=|\mu |-\ell (\mu )$$

$$\chi =2-2g=2N-\sum _{i=1}^k{\ell }^{\ast }({\mu }^{(i)}),$$

$$\mathbf{c}:=(c_1,c_2,\dots ),$$

$$G(z)$$

$$\tilde{G}(z)$$

$$g_i=e_i(\mathbf{c}),{\tilde{g}}_i=h_i(\mathbf{c}).$$

$$\mathbb{K}=\mathbb{Q}[g_1,g_2,g_3,\dots ]$$

$$H_G^d(\mu ,\nu )$$

$$H_{\tilde{G}}^d(\mu ,\nu )$$

$$H_G^d(\mu )$$

$$W_G({\mu }^{(1)},\dots ,{\mu }^{(k)}):=m_{\lambda }(\mathbf{c})=\frac{1}{|\mathrm{aut}(\lambda )|}\sum _{\sigma \in S_k}\sum _{1\le i_1<\cdots <i_k}{c}_{i_{\sigma (1)}}^{{\lambda }_1}\cdots c_{i_{\sigma (k)}}^{{\lambda }_k},$$

$$W_{\tilde{G}}({\mu }^{(1)},\dots ,{\mu }^{(k)}):=f_{\lambda }(\mathbf{c})=\frac{(-1{)}^{{\ell }^{\ast }(\lambda )}}{|\mathrm{aut}(\lambda )|}\sum _{\sigma \in S_k}\sum _{1\le i_1\le \cdots \le i_k}{c}_{i_{\sigma (1)}}^{{\lambda }_1},\cdots c_{i_{\sigma (k)}}^{{\lambda }_k},$$

$$|\mathrm{aut}(\lambda )|=\prod _{i=1}^d{m}_i(\lambda )!,$$

$$H({\mu }^{(1)},\dots ,{\mu }^{(k)}):=0$$

$$H_G^d(\mu ,\nu )=0\mathrm{if}|\mu |\ne |\nu |,H_{\tilde{G}}^d(\mu ,\nu )=0\mathrm{if}|\mu |\ne |\nu |.$$

$$H_G^d(\mu ,\nu )$$

$$H_{\tilde{G}}^d(\mu ,\nu )$$

$$H_G^d(\mu ,\nu ),H_{\tilde{G}}^d(\mu ,\nu )\in \mathbb{K}.$$

$${\tilde{H}}_G^d(\mu ,\nu )$$

$${\tilde{H}}_{\tilde{G}}^d(\mu ,\nu )$$

$${\tilde{H}}_G^d(\mu )$$

$$r_{\lambda }^{(G,\beta )}$$

$$r_{\lambda }^{(\tilde{G},\beta )}$$

$$r_j^G(\beta ):=G(j\beta ),r_j^{\tilde{G}}(\beta ):=\tilde{G}(j\beta ).$$

$${\tau }^{(G,\beta ,\gamma )}(\mathbf{t},\mathbf{s})$$

$${\tau }^{(\tilde{G},\beta ,\gamma )}(\mathbf{t},\mathbf{s})$$

$$t_i$$

$$p_{\mu }(\mathbf{t})$$

$${\mathcal{F}}^{(G,\beta ,\gamma )}(\mathbf{t},\mathbf{s})$$

$${\mathcal{F}}^{(\tilde{G},\beta ,\gamma )}(\mathbf{t},\mathbf{s})$$

$$\mathcal{H}=L^2(S^1)={\mathcal{H}}_{+}\oplus {\mathcal{H}}_{-}$$

$${\mathcal{H}}_{+}:=\overline{\mathrm{span}\{z^i{\}}_{i\ge 0}},{\mathcal{H}}_{-}:=\overline{\mathrm{span}\{z^{-i}{\}}_{i>0}}.$$

$$\{e_i:=z^{-i-1}{\}}_{i\in \mathbf{Z}}$$

$$(f,g):=\underset{z=0}{\mathrm{res}}(z^{-1}f(z)g(1/z))\text{“}=\text{”}\frac{1}{2\pi i}{\oint }_{z=0}f(z)g(1/z)\frac{dz}{z},$$

Full text

1. 1 Weighted Hurwitz numbers and generating $\tau$-functions

2. 1.1 Weighted Hurwitz numbers

3. 1.2 KP and $2D$-Toda $\tau$-functions of hypergeometric type as generating functions

4. 1.3 Summary of results

5. 2 Infinite Grassmannian, Baker functions, adapted bases

6. 2.1 The formal infinite Grassmannian, $\tau$-functions and Baker functions

7. 2.2 Adapted bases

8. 3 Fermionic representations

9. 3.1 Fermionic Fock space

10. 3.2 Fermionic construction of $\tau$-functions, Baker function and adapted bases

11. 3.3 Multiplicative recursion relations for general $\hat{g}$

12. 3.4 Euler derivative relations for general $\hat{g}$

13. 3.5 The $n$-pair correlation function and Christoffel-Darboux type kernel

14. 3.6 Multicurrent correlator for general $\hat{g}$

15. 4 Specialization to the case of hypergeometric $\tau$-functions

16. 4.1 Convolution products

17. 4.2 Adapted basis, $\tau$-function and Baker function for the hypergeometric case

18. 4.3 An alternative realization of convolution actions

19. 4.4 Recursion relations in the hypergeometric case

20. 4.5 The spectral curve and Kac-Schwarz operators

21. 4.5.1 The quantum and classical spectral curve

22. 4.5.2 Kac-Schwarz operators

23. 4.6 The pair correlation function in the hypergeometric case

24. 4.7 Finiteness and generating function for Christoffel-Darboux matrix

25. 4.8 The multicurrent correlators as generating functions for weighted Hurwitz numbers

26. 5 Bose-Fermi correspondence and “cut and join” operators

27. 5.1 Bose-Fermi correspondence and cut and join representation of the $\tau$-function

28. 5.2 Cut and join equations

29. 6 Examples

30. 6.1 Simple (single and double) Hurwitz numbers [39, 35]

31. 6.2 Three branch points (Belyi curves): strongly monotonic paths in the Cayley graph [9, 29, 22]

32. 6.3 Signed Hurwitz numbers: weakly monotonic paths in the Cayley graph [16, 20, 22]

33. 6.4 Simple quantum Hurwitz numbers [20, 22]

34. A Fermionic representation of multipair correlators

35. A.1 Fermionic representation of $\tilde{K}_{2}^{g}(z,w)$

36. A.2 The $n$-pair correlator $\tilde{K}^{g}_{2n}({\bf z},{\bf w})$

37. A.3 Expansion of $\tilde{K}_{2}(z,w)$ in the adapted basis

1 Weighted Hurwitz numbers and generating $\tau$-functions

It has been known since the work of Pandharipande and Okounkov [39, 35] that certain special classes of $\tau$-functions of the KP and $2D$-Toda hierarchies can serve as generating functions for various types of weighted Hurwitz numbers. It also is known that the computation of these enumerative geometrical/combinatorial invariants may be embedded into the framework of Topological Recursion [10, 12, 13, 33]. Fermionic representations have played a key rôle in simplifying the calculations and clarifying the origin of these constructions. The present work is part of an ongoing series (see [1, 2]) devoted to the inclusion of the generating function approach to weighted Hurwitz numbers of the most general type into the framework of Topological Recursion.

1.1 Weighted Hurwitz numbers

For a set of partitions $\{\mu^{(i)}\}_{i=1,\dots,k}$ of weight $|\mu^{(i)}|=N$, the Hurwitz numbers $H(\mu^{(1)},\dots,\mu^{(k)})$ are defined geometrically [25, 26] as the number of inequivalent $N$-fold branched coverings $\mathcal{C}\rightarrow\mathbf{P}^{1}$ of the Riemann sphere having $k$ branch points with ramification profiles given by the partitions $\{\mu^{(1)},\dots,\mu^{(k)}\}$, normalized by the inverse $1/|\operatorname{aut}(\mathcal{C})|$ of the order of the automorphism group of the covering. An equivalent combinatorial/group theoretical definition [14, 15, 42] is that $H(\mu^{(1)},\dots,\mu^{(k)})$ is ${1/N!}$ times the number of distinct factorizations of the identity element $\mathbf{I}\in S_{N}$ in the symmetric group in $N$ elements as a product of $k$ factors $h_{i}$, belonging to the respective conjugacy classes $\operatorname{cyc}(\mu^{(i)})$

[TABLE]

The equivalence of the two follows from the monodromy homomorphism from the fundamental group of $\mathbf{P}^{1}$ punctured at the branch points into $S_{N}$ obtained by lifting closed loops from the base to the covering.

Denoting by

[TABLE]

the colength of the partition $\mu$ (the difference between the weight and the length), the Riemann Hurwitz theorem relates the Euler characteristic $\chi$ of the covering curve $\mathcal{C}$ to the sum of the colengths $\ell^{*}(\mu^{(i)})$ of the ramification profiles at the branch points

[TABLE]

where $g$ is the genus of $\mathcal{C}$.

A weight generating function $G(z)$ and its dual $\tilde{G}(z):=1/G(-z)$, may be represented in terms of an infinite set of constants (or indeterminates)

[TABLE]

either as an infinite product, or an infinite sum:

[TABLE]

where the coefficients $\{g_{i}\}_{i=1,\dots,\infty}$, $\{\tilde{g}_{i}\}_{i=1,\dots,\infty}$ in the (formal) Taylor series are, respectively, the elementary and complete symmetric functions

[TABLE]

We define the (associative, commutative) algebra

[TABLE]

of polynomials in the $g_{i}$’s over the field $\mathbb{Q}$ of rationals, which is the same (when the number of $c_{i}$’s is finite) as the algebra $\Lambda$ of symmetric functions of the $c_{i}$’s (over the coefficient field $\mathbb{Q}$).

Given a pair of partitions $(\mu,\nu)$ of $N$, the weighted double and single Hurwitz numbers are defined by the sums [20, 22]

[TABLE]

where $\sideset{}{{}^{\prime}}{\sum}$ denotes the sum over all $k$-tuples of partitions $\{\mu^{(1)},\dots,\mu^{(k)}\}$ other than the cycle type of the identity element $(1^{N})$, the weights

[TABLE]

and

[TABLE]

are, respectively, the monomial sum and “forgotten” symmetric function bases [31] of the algebra $\Lambda$ of symmetric functions in the variables $(c_{1},c_{2},\dots)$ and the partition $\lambda$ labelling the coefficients $m_{\lambda}({\bf c})$ and $f_{\lambda}({\bf c})$ in the sums have weight $|\lambda|=d$ and length $\ell(\lambda)=k$, with parts $\{\lambda_{i}\}$ equal to the colengths $\{\ell^{*}(\mu^{(i)})\}$, expressed in weakly decreasing order. The normalization factor $|\operatorname{aut}(\lambda)|$ is the order of the automorphism group in $S_{d}$ of the elements in the conjugacy class of cycle type $\lambda$, given by

[TABLE]

where $m_{i}(\lambda)$ is the number of parts of $\lambda$ (i.e. the number of colengths $\ell^{*}(\mu^{(j)})$) equal to $i$.

Remark 1.1**.**

By convention, if the partitions are not all of the same weight, we set

[TABLE]

and thus

[TABLE]

An equivalent definition of $H^{d}_{G}(\mu,\nu)$ and $H^{d}_{\tilde{G}}(\mu,\nu)$ (see [20] for the proof) is:

[TABLE]

where $\{e_{\lambda}({\bf c})\}$, $\{h_{\lambda}({\bf c})\}$ are the elementary and complete symmetric function bases for $\Lambda$ and $m^{\lambda}_{\mu\nu}$ is the number of monotonic $d=|\lambda|$ step paths in the Cayley graph of $S_{N}$ generated by the transpositions, starting from an element $h$ in the conjugacy class $\operatorname{cyc}(\nu)$ and ending in $\operatorname{cyc}(\mu)$, with signature $\lambda$ [20]; i.e., such that the parts of $\lambda$ are equal to the number of times any given second element $b_{i}$ in the steps $(a_{i},b_{i}),\ a_{i}<b_{i}$ of the path is repeated. These belong to the algebra $\mathbb{K}$:

[TABLE]

Letting $\tilde{H}(\mu^{(1)},\dots,\mu^{(k)})$ denote the number of connected coverings with given ramification profiles, the corresponding weighted Hurwitz numbers for connected coverings are denoted

[TABLE]

1.2 KP and $2D$-Toda $\tau$-functions of hypergeometric type as generating functions

Choosing a nonvanishing (small) parameter $\beta$, we define the *content product * coefficients

[TABLE]

where

[TABLE]

Introducing a further non vanishing parameter $\gamma$, we define the following parametric family of $2D$-Toda $\tau$-functions of hypergeometric type (at lattice site $N=0$), which are shown in [19, 20, 21] to be generating functions for the corresponding weighted Hurwitz numbers

[TABLE]

Here $s_{\lambda}({\bf t}),s_{\lambda}({\bf s})$ are Schur functions, viewed as functions of two infinite sequences of 2D-Toda flow variables ${\bf t}=(t_{1},t_{2},\dots)$, ${\bf s}=(s_{1},s_{2},\dots)$ with

[TABLE]

interpreted as power sum symmetric functions of a pair of auxiliary sets of variables.

Remark 1.2**.**

Note that all six standard bases $\{s_{\lambda},p_{\lambda},m_{\lambda},f_{\lambda},e_{\lambda},h_{\lambda}\}$ for $\Lambda$ introduced in [31] appear in the above definitions.

The corresponding generating functions for connected weighted Hurwitz numbers are given, as usual, by the logarithms

[TABLE]

We may view all these generating functions as elements of ${\mathbb{K}}[[{\bf t},{\bf s},\beta,\gamma]]$.

Below we consider only the case $G$; all formulae for $\tilde{G}$ are analogous.

1.3 Summary of results

In Section 2 we recall the infinite Grassmannian interpretation of KP and $2D$-Toda $\tau$-functions [40, 41, 44, 45, 48, 47] and the Baker function. For arbitrary elements $W=g(\mathcal{H}_{+})$ of the big cell of the infinite Grassmannian $Gr_{\mathcal{H}_{+}}(\mathcal{H})$, $g\in\operatorname{GL}(\mathcal{H})$, we introduce a dual pair of adapted bases $\{w_{k}^{g}(z)\}_{k\in\mathbf{Z}}$, $\{w_{k}^{g*}(z)\}_{k\in\mathbf{Z}}$, for which $W=\mathrm{span}\{w_{k}^{g}\}_{k\in\mathbf{N}^{+}}$ and the annihilator $W^{\perp}$ under the Hirota pairing is $W^{\perp}=\mathrm{span}\{w_{k}^{g*}\}_{k\in\mathbf{N}^{+}}$.

Section 3 provides, for general lower triangular group elements $g$, the fermionic vacuum state expectation value (VEV) representations of all the relevant quantities: the $\tau$-functions $\tau_{g}({\bf t})$, $\tau_{g}(N,{\bf t},{\bf s})$, the Baker function $\Psi^{-}_{g}(z,N,{\bf t})$ and its dual $\Psi^{+}_{g}(z,N,{\bf t})$, the adapted bases, the pair and multipair correlators $K^{g}_{2n}(z_{1},\dots,z_{n};w_{1},\dots,w_{n})$, and the multicurrent correlator $\mathcal{J}_{n}(x_{1},\dots,x_{n})$. Proposition 3.1 gives the VEV representation of the adapted bases and Proposition 3.2 gives the expansions of $\Psi^{-}_{g}(z,{\bf t})$ and $\Psi^{+}_{g}(z,{\bf t})$ in these bases. Proposition 3.3 provides the multiplicative recursion relation satisfied by these bases and Propositions 3.5, 3.6 the derivative relations obtained upon application of the Euler operator. Propositions 3.7, 3.10 express the $n$-pair correlator $\tilde{K}^{g}_{2n}(z_{1},\dots,z_{n};w_{1},\dots,w_{n})$ in terms of the $\tau$-function and Proposition 3.12 gives an expansion in binear products of the adapted basis elements. A Christoffel-Darboux type representation for the pair correlator $K_{2}^{g}(z,w)$ is derived in Proposition 3.13.

All this is specialized in Section 4 to the case of $\tau$-functions of hypergeometric type, which serve as generating functions for weighted Hurwitz numbers. The recursion matrices for the adapted bases are explicitly given in Propositions 4.1, 4.2. The classical and quantum spectral curves that enter in the topological recursion approach [12, 13, 11, 10] are deduced, both directly, using the series expansions of the adapted bases, and using the Kac-Schwarz operators that generate them from the Baker function at ${\bf t}={\bf 0}$. The Christoffel-Darboux representation of the pair correlator is given in Proposition 4.4 and the pairing matrix is expressed via a generating function in Proposition 4.5. It is shown in Proposition 4.7 how the multicurrent generators may be used as alternative generating functions for the weighted Hurwitz numbers.

In Section 5, a fermionic derivation is given for the generalized “cut-and-join” representation of the $\tau$ function, and the differential equations it satisfies with respect to the weighting parameters.

Section 6 illustrates the results for four specific examples: 1) the original case of simple (single and double) Hurwitz number studied by Okounkov [35] and Pandharipande [39]; 2) the case of strongly monotonic Hurwitz numbers or, equivalently, Belyi curves (with three branch points) studied in [9], [29]; 3) the case of weakly monotonic Hurwitz numbers, or signed geometrical enumerative Hurwitz numbers of refs. [16, 17, 20, 24, 21] and 4) the quantum Hurwitz numbers introduced in refs. [20, 21].

This paper is part of a series [1], [2] on embedding the $\tau$-function generating function approach to weighted Hurwitz numbers into the topological recursion scheme.

2 Infinite Grassmannian, Baker functions, adapted bases

2.1 The formal infinite Grassmannian, $\tau$-functions and Baker functions

In the related papers [1, 2], we use an extension of the Sato [40] and Segal-Wilson [44] formulations of the infinite dimensional Grassmannian on which the KP flows are understood to act, replacing the analytic or algebraic definitions of the underlying spaces and operators by formal series, including extensions by parameters $\{\beta,\gamma,{\bf c}\}$ etc. This will be necessary for the proper interpretation of the specialization to $\tau$-functions of hypergeometric type serving as generators for weighted Hurwitz numbers that appear in Sections 4 and 5. Our notational conventions derive from reinterpreting the elements $W\in\mathrm{Gr}_{\mathcal{H}_{+}}(\mathcal{H})$ of the Grassmannian formally, as subspaces $W\subset\mathbf{C}[[z,z^{-1}]]$ of the space of formal Laurent series in an indeterminate $z$, with complex coefficients. These extend what in [44] are viewed as closed subspaces of the Hilbert space

[TABLE]

of square integrable functions on the unit circle, with orthonormal basis $\{e_{i}\}_{i\in\mathbf{Z}}$ with $\mathcal{H}_{\pm}$ viewed as the spaces of complex Fourier series with negative ($-$) and non-negative ($+$) exponentials. But for the current section and the next, we construct the $\tau$-function, Baker function, adapted bases and associated correlators in the general setting, without reference to the formal series interpretation. In [40, 41] $\mathcal{H}$ is viewed as a completion of the space of positive and negative Laurent polynomials

[TABLE]

We use the labelling convention

[TABLE]

for the orthonormal basis, motivated by the identification of the vacuum element $|0\rangle$ in the associated Fermonic Fock space as the Dirac sea, in which the negative integers denote the filled negative energy states.

In the Segal-Wilson formulation, such subspaces $W\subset\mathcal{H}$ are required to be commensurable with $\mathcal{H}_{+}\subset\mathcal{H}$ in the sense that the orthogonal projection $W\rightarrow\mathcal{H}$ to $\mathcal{H}_{+}$ along $\mathcal{H}_{-}$ is required to be a Fredholm operator, whose index is called the virtual dimension of $W$, while the image under orthogonal projection $W\rightarrow\mathcal{H}_{-}$ is required, in some sense, to be “small” (either Hilbert-Schmidt, as in [44], or compact). Sato [40] uses a different notion, but for our purposes both the functional analytic [44] and differential algebraic [41, 40] versions are subsumed by replacing the Hilbert space $\mathcal{H}$ by the space of formal Laurent series $\mathbf{C}[[z,z^{-1}]]$, extended by the constants $\beta,\gamma$ and ${\bf c}$, viewed as indeterminates. We nevertheless retain the notation $\mathcal{H}=\mathcal{H}_{+}+\mathcal{H}_{-}$ for the space, and let

[TABLE]

(which might just be a formal residue) denote the complex scalar product in which the basis $\{e_{i}\}_{i\in\mathbf{Z}}$ is orthonormal, with the second equality valid on the smaller space $L^{2}(S^{1})$.

The precise definition of the infinite Lie algebra $\mathfrak{g}\mathfrak{l}(\mathcal{H})$ of endomorphisms of $\mathcal{H}$ and its connected Lie group $\operatorname{GL}(\mathcal{H})$ of admissible invertible endomorphisms will be omitted; we refer to [40], [44] for (different) particular choices, with the differences replaced, for our purposes, by requiring that their matrix representations within the orthonormal basis $\{e_{i}\}$ and elements in the associative algebra $\mathbf{C}[\beta,\gamma,\{g_{i}\}]$ of formal series in the auxiliary parameters $(\beta$, $\gamma)$ and the weighting constants $\{g_{i}\}_{i\in\mathbf{N}^{+}}$ (or ${\bf c}$), when appropriate, should have well defined products and commutators.

The two infinite abelian subgroups $\Gamma_{+}$, $\Gamma_{-}$ whose elements consist of multiplication by the exponential series

[TABLE]

where

[TABLE]

are central to the theory since, together with the discrete group of lattice translations, they generate the mKP and $2D$-Toda flows.

The KP $\tau$-function associated with an element $W\in\mathrm{Gr}_{\mathcal{H}_{+}}(\mathcal{H})$ of virtual dimension $N=0$ is, within a suitable choice of admissible basis that establishes an isomorphism between the two commensurable spaces $W$ and $\mathcal{H}_{+}$, the determinant of the orthogonal projection operator to $\mathcal{H}_{+}$ from the space

[TABLE]

obtained by applying the abelian group of $\Gamma_{+}$ flows to $W$

[TABLE]

It follows [40, 41, 27, 44] that this satisfies the Hirota bilinear relations

[TABLE]

where

[TABLE]

The Baker function $\Psi^{-}_{W}(x,{\bf t})$ which, for all values of the KP flow parameters ${\bf t}$, takes its values in the annihilator $W^{\perp}$ of $W$ under the Hirota bilinear pairing

[TABLE]

is then given by the Sato formula

[TABLE]

The dual Baker function, which takes its values in $W$ for all values of ${\bf t}$, is

[TABLE]

More generally, for $W_{N}$ of virtual dimension $N$, we similarly define $\tau_{W_{N}}({\bf t})$ as the determinant of the projections to the subspace

[TABLE]

namely

[TABLE]

In the fermionic Fock space, their image under the Plücker map (see below) is in the charge $N$ sector.

An equivalent view of the Grassmannian $\mathrm{Gr}_{\mathcal{H}_{+}}(\mathcal{H})$ is as a quotient $\operatorname{GL}(\mathcal{H})/\operatorname{GL}_{\mathcal{H}_{+}}(\mathcal{H})$ of the group $\operatorname{GL}(\mathcal{H})$ of admissible invertible endomorphisms of $\mathcal{H}$ by the stabilizer $\operatorname{GL}_{\mathcal{H}_{+}}(\mathcal{H})$ of the subspace $\mathcal{H}_{+}$, which is an infinite dimensional analog of the maximal parabolic subgroup of $\operatorname{GL}(N)$ leaving a fixed $k$-dimensional subspace invariant. Thus, choosing an element $W\in\mathrm{Gr}_{\mathcal{H}_{+}}(\mathcal{H})$ within the $\operatorname{GL}(\mathcal{H})$ orbit of $\mathcal{H}_{+}$

[TABLE]

is equivalent to choosing the left coset

[TABLE]

If a sequence of elements of the various virtual dimension components is chosen as

[TABLE]

for a fixed group element $g$, the resulting analogously defined sequence of KP $\tau$-functions $\tau_{W_{N}}({\bf t})$ satisfy the bilinear equations of the mKP hierarchy

[TABLE]

for $N^{\prime}\geq N$.

In this case, the group element $g$ is defined within right multiplication by an element of the simultaneous stabilizer of all the subspaces $\mathcal{H}_{+}^{N}$, $N\in\mathbf{Z}$, which is the infinite dimensional analog of the minimal parabolic (Borel) subgroups of $GL(N)$ consisting of invertible upper triangular matrices. That is, the mKP flows are interpreted as acting on the codimension $1$ nested sequence of subspaces

[TABLE]

i.e., infinite complete flags belonging to the $GL(\mathcal{H})$ orbit of the standard complete flag

[TABLE]

The dual infinite flag is given by the orthogonal annihilators

[TABLE]

2.2 Adapted bases

By applying the group element $g$ to the standard (monomial) basis $\{z^{k}\}_{k\in\mathbf{Z}}$ for $\mathcal{H}$, we obtain an adapted basis $\{w_{k}^{g}(z)\}_{k\in\mathbf{Z}}$ for $\mathcal{H}$ such that the positively labeled elements $\{w_{k}^{g}(z)\}_{k\in\mathbf{N}^{+}}$ span the subspace $W=g(\mathcal{H}_{+})$. We also introduce the dual basis $\{w_{k}^{g*}(z)\}_{k\in\mathbf{Z}}$ such that $\{w_{k}^{g*}(z)\}_{k\in\mathbf{N}^{+}}$ spans the subspace $W^{\perp}:=\iota\circ(g^{t})^{-1}\circ\iota(\mathcal{H}_{+})$ that annihilates $W$ under the Hirota bilinear pairing (2.11). Here $\iota$ is the involutive automorphism of $\mathcal{H}$ that relates the Hirota bilinear pairing to the scalar product (2.4)

[TABLE]

Define $\tilde{g}$ to be the conjugation of $g$ under this involution

[TABLE]

The corresponding matrix representation of $\tilde{g}$ in the standard basis $e_{i}$ is then

[TABLE]

Applying the group elements $g$ and $(\tilde{g}^{t})^{-1}$ to the standard monomial basis, we obtain the two bases,

[TABLE]

with inverse relations

[TABLE]

Then

[TABLE]

The bases $\{w_{k}^{g}\}_{k\in\mathbf{Z}}$, $\{w_{k}^{g*}\}_{k\in\mathbf{Z}}$ are dual under the Hirota bilinear pairing (2.11) in the sense that

[TABLE]

from which it follows that $W^{\perp}$ is, indeed, the annihilator of $W$.

We can make the further assumption that the group element $g$ is in fact represented by lower triangular matrices

[TABLE]

This means that $W\in\mathrm{Gr}_{\mathcal{H}_{+}}(\mathcal{H})$ must be in the “big cell”, but imposes no further restriction, since a representative of the left coset under the stabilizer of $\mathcal{H}_{+}$ may always be chosen in this lower triangular form.

It also implies that the sums (2.27) - (2.30) are semi-infinite, with the highest power of $z$ in the Fourier series for $w_{k}^{g}(z)$ and $w_{k}^{g*}(z)$ being $k-1$

[TABLE]

and

[TABLE]

3 Fermionic representations

3.1 Fermionic Fock space

We recall the conventions and notation for the fermionic operator approach to $\tau$-functions developed by Sato and his school [40, 41, 27]. Using the notation of Section 2.1, we denote the semi-infinite wedge product space (Fermonic Fock space)

[TABLE]

where $\mathcal{F}_{N}$ denotes the charge-$N$ sector, $N\in\mathbf{Z}$. The pairs $(\lambda,N)$ consisting of a partition $\lambda$ of any weight (with the convention that $\lambda_{i}=0$ for $i>\ell(\lambda)$) and an integer $N$ label the basis element in $\mathcal{F}$. These are denoted

[TABLE]

where

[TABLE]

is a strictly decreasing sequence of integers saturating, after a finite number of terms, in consecutively decreasing integers. These are viewed as the integer lattice points of occupied sites, and the vacuum element in each sector is given by the trivial partition

[TABLE]

The Fermi creation and annihilation operators $\{\psi_{i},\psi^{*}_{i}\}_{i\in\mathbf{Z}}$ are the generators of the Fermi Fock space representation of the Clifford algebra on $\mathcal{H}\oplus\mathcal{H}^{*}$ with quadratic form

[TABLE]

In the standard irreducible infinite Clifford module $\mathcal{F}$, these are given by the linear action of exterior and interior multiplication by the basis elements $\{e_{i}\}_{i\in\mathbf{Z}}$ for $\mathcal{H}$ and their duals $\{\tilde{e}^{i}\}_{i\in\mathbf{Z}}$ for $\mathcal{H}^{*}$

[TABLE]

This implies the generating relations of the Clifford algebra

[TABLE]

Normal ordering $:\mathcal{O}:$ of any finite product of such creation and annihilation operators is defined by placing all the annihilation factors $\psi^{*}_{i}$ to the right for $i\geq 0$ and all the creation factors $\psi_{i}$ to the left for $i<0$ (taking the signs following from anticommutativity into account). This implies the vanishing of vacuum expectation values (VEV’s) of all normally ordered operators (other than scalars)

[TABLE]

If the partition $\lambda$ is expressed in Frobenius notation as $(a_{1},\dots a_{r}|b_{1},\dots,b_{r})$, where the $a_{i}$’s and $b_{i}$’s are strictly decreasing sequences of of non-negative integers, denoting the number of boxes to the right of and below the $i$th diagonal box of the corresponding Young diagram (the “arm” and “leg” lengths), with the Frobenius index $r$ equal to the number of terms on the principal diagonal, we may express the basis element $|\lambda;N\rangle$ as

[TABLE]

The Fermi fields $\psi(z)$ and $\psi^{*}(z)$ are defined as generating series for these creation and annihilation operators

[TABLE]

Elements $g\in\operatorname{GL}(\mathcal{H})$ of the connected group of invertible endomorphisms of $\mathcal{H}$ may be viewed as elements of the larger group $\mathrm{Spin}(\mathcal{H}+\mathcal{H}^{*},\mathbf{Q})$ of endomorphisms of $\mathcal{H}+\mathcal{H}^{*}$ preserving the quadratic form $\mathbf{Q}$. If they are represented within the basis $\{e_{i}\}_{i\in\mathbf{Z}}$ by the doubly infinite matrix exponential

[TABLE]

with elements $\{A_{ij}\}_{i,j\in\mathbf{Z}}$, they have the Clifford representation

[TABLE]

The Plücker map into the projectivization $\mathbf{P}\mathcal{F}$ of the Fock space

[TABLE]

(where $[\dots]$ denotes the projective class) embeds the infinite Grassmannian $\mathrm{Gr}_{\mathcal{H}_{+}}(\mathcal{H})$ into $\mathbf{P}\mathcal{F}$ as the union of the orbits of the subspaces $\mathcal{H}_{+}^{N}$ under the Clifford action of $\operatorname{GL}(\mathcal{H})$ if $W$ has virtual dimension $N$. Thus the Plücker map intertwines the action of $\operatorname{GL}(\mathcal{H})$ on $Gr_{\mathcal{H}_{+}}(\mathcal{H})$ and the (projectivized) Clifford representation on $\mathcal{F}$

[TABLE]

In particular, the Clifford representation of the KP and 2D Toda flow groups $\Gamma_{+}$ and $\Gamma_{-}$ is given by

[TABLE]

where the operators $J_{i}$ are bilinears in the creation and annihilation operators

[TABLE]

that satisfy the commutation relations of the infinite Heisenberg algebra

[TABLE]

The projective representation may be viewed as a representation of this central extension of the abelian group generated by the elements $\gamma_{+}({\bf t})$, $\gamma_{-}({\bf s)}$, with

[TABLE]

The $J_{i}$’s may be viewed as Fourier components of the current operator

[TABLE]

The effect of conjugation of the creation and annihilation operators by a group element is

[TABLE]

if

[TABLE]

In particular,

[TABLE]

3.2 Fermionic construction of $\tau$-functions, Baker function

and adapted bases

In Sato’s approach to KP $\tau$-functions [40, 41], given an element $\hat{g}$ of the Fermionic representation of the Clifford algebra that satisfies the bilinear relation.

[TABLE]

where

[TABLE]

the following vacuum expectation value

[TABLE]

satisfies the Hirota bilinear relation (2.9) and hence is a KP $\tau$-function. In the following, we call such elements group-like elements [8]. In particular, if

[TABLE]

is the fermionic representation of the element $g\in\operatorname{GL}(\mathcal{H})$ with matrix representation $e^{A}$ in the standard $\{e_{i}\}_{i\in\mathbf{Z}}$ basis, $\hat{g}$ satisfies (3.26) and $\tau_{g}({\bf t})$ is the same as the $\tau$-function $\tau_{W}({\bf t})$ defined in (2.8) with $W=g\mathcal{H}_{+}$. The Baker function defined in (2.12) can then be expressed as the following VEV [40, 41]

[TABLE]

and the dual Baker function (2.13) is

[TABLE]

Since

[TABLE]

it follows that the $\tau$-function may, at least formally, be expressed as a sum over the Schur function basis

[TABLE]

where

[TABLE]

is the $\lambda$ Plücker coordinate of the Grassmannian element $W$ and the Schur functions $s_{\lambda}$ are viewed as functions of the KP flow variables ${\bf t}=(t_{1},t_{2},\dots)$.

More generally, for any such $\hat{g}$, we may define a lattice of KP $\tau$-functions generating solutions to the modified KP (mKP) hierarchy,

[TABLE]

which satisfy the extended system of Hirota equations (2.19) of the mKP hierarchy.

Introducing a second set of flow variables ${\bf s}=(s_{1},s_{2},\dots)$ we may define the $2D$ Toda lattice of $\tau$-functions

[TABLE]

which satisfy

[TABLE]

understood to hold identically in $\delta{\bf t}=(\delta t_{1},\delta t_{2},\dots),\ \delta{\bf s}:=(\delta s_{1},\delta s_{2},\dots)$ and in $N$, $N^{\prime}$.

In the following, we make the further assumption that the group element $g$ is represented by lower triangular matrices

[TABLE]

This means that

[TABLE]

Applying the group element $\hat{g}$, we obtain the fermionic VEV respresentation of the bases $\{w^{g}_{k\in\mathbf{Z}}\}$ , $\{w^{g*}_{k\in\mathbf{Z}}\}$ defined in (2.27), (2.28).

Proposition 3.1**.**

The bases defined in eqs. (2.27) , (2.28) have the following representation as fermionic VEV’s

[TABLE]

Proof.

Substitute the fourier series representations (3.10) of the fermionic field operators $\psi(z)$, $\psi^{*}(z)$ in the RHS of (3.41) and (3.42) and apply the conjugation relations (3.21) term by term. This results in the series representations (2.27), (2.28) for $\{w^{g}_{k\in\mathbf{Z}}\}$ , $\{w^{g*}_{k\in\mathbf{Z}}\}$, both for $k\geq 1$ and $k\leq 0$. ∎

Now define the sequence of subspaces $W_{N}^{g},W_{N}^{g*}$ , $N\in\mathbf{Z}$

[TABLE]

which coincide with (2.20), (2.22).

The charge $N$ sector Baker function and its dual

[TABLE]

can be expanded in these bases. For the case $N=0$, we have

Proposition 3.2**.**

[TABLE]

where

[TABLE]

Proof.

In eq. (3.30), insert $\mathbf{I}=\hat{g}\hat{g}^{-1}$ and the projection operator to the $N=1$ sector $\mathcal{F}_{1}\subset\mathcal{F}$ of the fermionic Fock space

[TABLE]

(where the sum is over all partitions $\lambda$). This gives

[TABLE]

In this sum, the only nonvanishing terms are the trivial partition $\lambda=\emptyset$ and the horizontal partitions $\lambda=(j),\ j=1,2,\dots$. To see this, use (3.21) to write the second factor in the sum (3.51) as

[TABLE]

Expressing the partition $\lambda$ in Frobenius notation as $(a_{1},\dots,a_{r}|b_{1},\dots,b_{r})$, the state $|\lambda;1\rangle$ is, by (3.9)

[TABLE]

It follows from the fermionic Wick theorem that

[TABLE]

vanishes unless either $\lambda=\emptyset$ or $r=1$ and $b_{1}=0$. For the first case, we have

[TABLE]

for the other case $\lambda=(j)$, we have

[TABLE]

by eq. (2.28), and

[TABLE]

proving eq.(3.47). Eq. (3.48) is proved similarly. ∎

3.3 Multiplicative recursion relations for general $\hat{g}$

Multiplying the adapted basis elements by $z$, it follows that they satisfy the recursion relations

Proposition 3.3**.**

[TABLE]

where

[TABLE]

Proof.

This can either be proved directly, by inverting the Fourier series expansions for $w_{k}^{g}(z)$ and $w_{k}^{g*}(z)$, or by using the fermionic identities

[TABLE]

and evaluating the RHS of the fermionic representations (3.41) and (3.42) for both $w_{k}^{g}(z)$ and $w_{k}^{g*}(z)$ by introducing a sum over a complete set of intermediate states in the $N=0$ or $N=-1$ fermionic charge sectors; i.e., by inserting the projection operator

[TABLE]

onto these sectors. ∎

The recursion matrix $\tilde{Q}^{+}$ may be split into block form

[TABLE]

where the range of indices are:

[TABLE]

It follows from the above computation that the fermionic representation of the recursion matrix elements is

Corollary 3.4**.**

[TABLE]

where

[TABLE]

Note that $\tilde{Q}^{-+}_{kl}$ has just one nonzero entry, in the lower triangular corner $(01)$, and both $\tilde{Q}^{--}_{kl}$ and $\tilde{Q}^{++}_{kl}$ are nearly lower triangular, with only a single nonzero diagonal immediately above the principal one.

Remark 3.1**.**

Proposition 3.3 provides a system of recursion relations satisfied by the adapted basis elements $\{w_{k}^{g}(z)\}$, $\{w_{k}^{g*}(z)\}$ in the sense that the recursion matrices $\tilde{Q}^{\pm}$ are almost lower triangular, with a single nonvanishing diagonal above the principle one. This means that, given all the preceding $\{w^{g}_{j}(z)\}$’s, for $j\leq k$, we can uniquely determine $w^{g}_{k+1}(z)$ (and similarly for $w^{g*}_{k+1}(z)$). This is only a finite recursion system in the case when the matrices $\tilde{Q}^{\pm}$ are of finite band width below the principle diagonal. In Section 4 it will be shown that this is indeed the case when we specialize to group elements of the convolution type, with a generating function $G(z)$ that is polynomial, and the number of nonvanishing parameters $s_{i}$ in the abelian group element $\hat{\gamma}_{-}({\bf s})$ finite.

3.4 Euler derivative relations for general $\hat{g}$

Applying the Euler operator

[TABLE]

to the adapted bases $\{w_{k}^{g}(z)\}$, $\{w_{k}^{g*}(z)\}$ gives the following linear relations

Proposition 3.5**.**

[TABLE]

where the lower triangular matrices $\tilde{P}^{\pm}$ have as entries

[TABLE]

Proof.

This can again be proved either directly, by inverting the Fourier series expansions for $w_{k}^{g}(z)$ and $w_{k}^{g*}(z)$, or by using the fermionic identities

[TABLE]

where $\hat{H}$ is the fermionic energy operator

[TABLE]

and again evaluating the RHS for the fermionic expressions by introducing the projector $\Pi_{N}$ to the $N=0$ or $N=-1$ fermionic charge sectors. ∎

The differential recursion matrix $\tilde{P}^{+}$ may again be split into block form

[TABLE]

where the range of indices are again:

[TABLE]

Proposition 3.6**.**

The fermionic representation of the Euler differential matrix elements is

[TABLE]

Note that both $\tilde{P}^{--}_{kl}$ and $\tilde{P}^{++}_{kl}$ are lower triangular.

Proof.

A direct evaluation of the matrix elements. ∎

Remark 3.2**.**

Proposition 3.6 provides a recursive system of linear differential relations satisfied by the adapted basis elements $\{w_{k}^{g}(z)\}$, $\{w_{k}^{g*}(z)\}$ in the sense that the matrices $\tilde{P}^{\pm}$ are lower triangular. This means that, given all the $\{w^{g}_{j}(z)\}$’s, for $j\leq k$, we can uniquely determine $\mathcal{D}w^{g}_{k}(z)$ (and similarly for $\mathcal{D}w^{g*}_{k}(z)$). This is again only a finite system in the case when the matrices $\tilde{P}^{\pm}$ are of finite band width below the principal diagonal. In Section 4 it will be shown that this is indeed the case when we require that the number of nonvanishing parameters $s_{i}$ in the abelian group element $\hat{\gamma}_{-}({\bf s})$ be finite.

3.5 The $n$-pair correlation function and Christoffel-Darboux type kernel

The $n$-pair correlation function is defined by the following VEV

[TABLE]

provided $|z_{i}|>|w_{i}|\ \forall i$. Adding the KP multitime flow variables gives the time dependent correlation function

[TABLE]

In particular, for $n=1$, ${\bf t}={\bf 0}$,

[TABLE]

This is equivalent to the following expression in terms of the $\tau$-function

Proposition 3.7**.**

[TABLE]

Proof.

See Appendix A.1 ∎

By Wick’s theorem, $\tilde{K}^{g}_{2n}$ may be expressed as a determinant in terms of the $2$-point function

Lemma 3.8**.**

[TABLE]

To express the multipair correlator $\tilde{K}^{g}_{2n}$ in terms of the $\tau$-function it is helpful to introduce the following notation.

Definition 3.1**.**

For a set of $n$ nonzero complex numbers ${\bf z}=(z_{1},z_{2},\dots z_{n})$, we use the notation $[{\bf z}^{-1}]$ to denote the infinite sequence $\{[{\bf z}]_{i}\}_{i=1,2,\cdots}$, of normalized power sums

[TABLE]

Lemma 3.9**.**

For $2n$ distinct nonzero complex parameters $\{z_{i},w_{i}\}_{i=1,\dots n}$ provided $|z_{i}|>|w_{i}|\ \forall i$, we have

[TABLE]

Proof.

See Appendix A.2. ∎

It follows that, for an arbitrary group element $g\in\operatorname{GL}(\mathcal{H})$, we can express the $2n$-point time dependent pair correlation function (3.88) in terms of the $\tau$-function.

Corollary 3.10**.**

[TABLE]

Proof.

(See Appendix A.2.) It is first proved for $n=1$, by applying Lemma 3.9 for $n=1$ to deduce

[TABLE]

where we have used eq. (3.23) and then extended to arbitrary $n$, either by induction or, equivalently, by the fermionic Wick theorem. ∎

Using Wick’s theorem, we have the following determinantal expression in terms of single pair correlators and in terms of the $\tau$-function.

Corollary 3.11**.**

[TABLE]

From the fermionic representation, it is also easy to see that the 2-point function at ${\bf t}={\bf 0}$ may be expressed as a series

Proposition 3.12**.**

[TABLE]

Proof.

A fermionic proof is given in Appendix A.3. Alternatively, eq. (3.97) may be proved using the $n=1$ case of (3.96) and the Schur function expansion (3.33) of the $\tau$-function. ∎

We also have the following Christoffel-Darboux type formula.

Proposition 3.13**.**

[TABLE]

where

[TABLE]

and $\tilde{Q}^{+}_{ij}$ is the recursion matrix appearing in eq. (3.60).

Proof.

Multiply the RHS of eq. (3.97) by $z-w$. Then the result follows from using the recursion relations (3.58) directly, together with the telescopic cancellation between the two types of terms coming from the $z$ and $w$ recursion relations. Alternatively, it may be derived directly from the VEV representation eq. (3.87) by using the fermionic identity

[TABLE]

and inserting sums over complete sets of intermediate states in both terms arising from the commutator. ∎

3.6 Multicurrent correlator for general $\hat{g}$

In the following, it will be convenient to use the inverse variables $x_{i}:=1/z_{i}$ in expressing the correlators.

Definition 3.2**.**

We denote the positive frequency part of the current operator as

[TABLE]

and define the multicurrent correlator $\mathcal{J}_{n}(x_{1},\dots,x_{n})$ as

[TABLE]

We also introduce the correlators $W_{n}(x_{1},\dots,x_{n})$, defined as multiple derivatives of the $\tau$-function

[TABLE]

where the commuting parametric family of first order gradient operators $\nabla(x_{i})$ in the ${\bf t}$-variables is defined by

[TABLE]

For later purposes, we also introduce a “connected” version of these quantities, defined by replacing $\tau_{g}$ by $\mathrm{ln}(\tau_{g})$

[TABLE]

Assuming $\tau_{g}({\bf t})$ to be normalized such the $\tau_{g}({\bf 0})=1$, we have, in particular,

[TABLE]

From the fermionic formula (3.28) for $\tau_{g}({\bf t})$ it follows that $W_{n}$ and the multicurrent correlator $\mathcal{J}_{n}$ are related by

Lemma 3.14**.**

[TABLE]

Proof.

This is immediate from applying (3.103) to the fermionic formula (3.28) for $\tau_{g}({\bf t})$. ∎

4 Specialization to the case of hypergeometric $\tau$-functions

4.1 Convolution products

An abelian group that is central to hypergeometric $\tau$-functions is obtained by extending the semigroup $C:=\{C_{\rho}\}$ consisting of formal convolution products with functions or distributions on the circle $S^{1}$ admitting a Fourier series representation

[TABLE]

i.e. the diagonal action on an element $w\in\mathcal{H}$ with Fourier representation

[TABLE]

consisting of multiplication of their Fourier coefficients

[TABLE]

to include the diagonal action of commuting differential operators of the form

[TABLE]

where $\mathcal{D}$ is the Euler operator (3.72) (or simply the vector field of infinitesimal rotations on the circle). (These may also formally be represented as a convolution with the series $\rho(z)$ in which the coefficients in (4.1) are $\rho_{i}=\exp\left(\sum_{j}\alpha_{j}(-i-1)^{j}\right)$.)

In the $L^{2}(S^{1})$ sense, the convolution product may be viewed as the integral

[TABLE]

Alternatively, in the formal Laurent series sense, the integral ${1\over 2\pi i}\oint_{S^{1}}f(\zeta){d\zeta\over\zeta}$ may be reinterpreted as a formal residue at $z=0$.

These are essential to the definition of KP or $2D$-Toda $\tau$-functions of hypergeometric type [37, 38], since they correspond to Grassmannian elements of the form

[TABLE]

for the zero virtual dimension component or, more generally

[TABLE]

in the virtual dimension $N$ sector.

In the following, the coefficients

[TABLE]

in the distributional (or formal) series $\rho(z)$ will always be understood as determined by the weight generating function $G$ and the constants $(\beta,\gamma)$, as in [22], such that

[TABLE]

Since the $\tau$-function is only projectively defined, assuming $\rho_{0}\neq 0$, we may choose it as $\rho_{0}=1$ and express the coefficients $\rho_{j}$ as finite products of the $\gamma G(i\beta)$’s and their inverses [22]

[TABLE]

Thus

[TABLE]

Although vanishing values $\rho_{j}=0$ for some negative coefficient $j=-n-1$ are allowed, the relations (4.10) then imply that

[TABLE]

and hence all the coefficients $\gamma^{|\lambda|}r_{\lambda}^{(G,\beta)}$ appearing in the $\tau$-functions (1.26), (1.27) vanish for partitions of length $\ell(\lambda)>n$.

4.2 Adapted basis, $\tau$-function and Baker function for the hypergeometric case

Since $\gamma_{-}({\bf s})$ is the generating function for the complete symmetric functions,

[TABLE]

specializing to the case

[TABLE]

we have

[TABLE]

The corresponding adapted bases $\{w^{(G,\beta,\gamma,{\bf s})}_{k}(z)\}_{k\in\mathbf{Z}}$ defined by applying the element $C_{\rho}\gamma_{-}(\beta^{-1}{\bf s})$ and its dual to the basis elements $\{e_{m}:=z^{-m-1}\}_{m\in\mathbf{Z}}$ are

[TABLE]

where the coefficients $\rho_{j}$ are defined in eq. (4.10). The dual basis, under the Hirota bilinear pairing (2.11), is defined by

[TABLE]

where $\iota$ is the involution (2.24). Simplifying the notation for the adapted bases in this case to

[TABLE]

these have the fermionic representation (cf. (3.41))

[TABLE]

which coincides with the basis defined in (4.17), and the dual basis, defined by (3.42), becomes

[TABLE]

which coincides with (4.18).

For $k\in\mathbf{N}^{+}$ they define corresponding elements of the Grassmanian:

[TABLE]

The fermionic representation of the group element associated to the weight generating function $\gamma G$ defined in (4.10) is of the form

[TABLE]

where

[TABLE]

with the algebra element $\hat{A}$ is

[TABLE]

We therefore have the following fermionic representation of the 2D Toda $\tau$-function $\tau^{(G,\beta,\gamma)}({\bf t},{\bf s})$ at $N=0$ defined by (1.26)

[TABLE]

.

Remark 4.1**.**

Such 2D Toda $\tau$-functions, referred to in [37, 38] as $\tau$-functions of hypergeometric type, were considered, for other purposes, in [30], but without identification of these as Hurwitz generating functions. Fermionic representations of particular families of the Hurwitz numbers were considered in [37, 38, 35, 20, 21, 22, 24, 46, 7, 5, 34, 36, 33]

Viewed as a parametric family of KP $\tau$-functions, after the substitution ${\bf s}\rightarrow\beta^{-1}{\bf s}$, the corresponding Baker function and its dual have the fermionic representation

[TABLE]

As particular cases of the expansions (3.47), (3.48), the Baker function and its dual for this class of hypergeometric $\tau$-functions can be expressed relative to the dual bases $\{w_{k}^{*}(z)\}$, $\{w_{k}(z)\}$ as

[TABLE]

where

[TABLE]

4.3 An alternative realization of convolution actions

It is easy to see that for (4.10) the convolution operator can formally be equivalently realized by the exponential of a differential operator, namely

[TABLE]

where the $T(x)$ is viewed as a formal Taylor series satisfying

[TABLE]

and $T(0)=0$ with

[TABLE]

This series, though nonunique, may be given an explicit expression as

[TABLE]

where

[TABLE]

the polynomials $\{p_{k}(x)\in x\,\mathbb{C}[x]\}$ are determined by

[TABLE]

and are closely related to the Bernoulli polynomials. In particular,

[TABLE]

Their generating function is

[TABLE]

Remark 4.2**.**

There are, of course, an infinity of other series $T(x)$ satisfying eqs. (4.35) and (4.36), obtained, e.g. by adding any periodic function $P(x)$ of unit period satisfying $P(0)=0$.

The fermionic operator $\hat{A}$ appearing in eqs. (4.26), may equivalently be expressed as

[TABLE]

which satisfies the commutation relations

[TABLE]

The adapted basis elements can equivalently be expressed as

[TABLE]

The linearly independent elements $\{w_{N+k}\}_{k\in\mathbf{N}^{+}}$ span an infinite nested sequence of subspaces

[TABLE]

of virtual dimension $N$

[TABLE]

This defines an infinite complete flag which, for fixed parameters ${\bf s}$ and convolution elements $\rho$, determines a chain of $\tau$-functions of the mKP hierarchy, defined as determinants of the corresponding orthogonal projection maps

[TABLE]

where the content product coefficient $r^{(G,\beta)}_{\lambda}$, in eq. (1.26) is replaced by the shifted version

[TABLE]

Alternatively, viewing the parameters ${\bf s}=(s_{1},s_{2}\dots)$ as a further infinite set of flow variables, this defines a hypergeometric $\tau$-function of the $2D$ Toda hierarchy. Viewing all parameters $\{N,{\bf t},{\bf s}\}$ on the same footing, as discrete or continuous flow parameters, we obtain a lattice of $\tau$-functions of the $2D$-Toda hierarchy.

4.4 Recursion relations in the hypergeometric case

We also introduce an alternative notation for these bases which is used in refs. [1, 2]

[TABLE]

where

[TABLE]

The basis elements $\{\Psi^{\pm}_{k}(x)\}$ defined in (4.59) are viewed as elements of the subspaces

[TABLE]

in the formal series sense. They may be understood as elements of formal power series analogs of the Sato-Segal-Wilson Grassmannian, consisting of subspaces of the Hilbert space spanned by powers $\{z^{k}\}_{k\in\mathbf{Z}}$ of the spectral parameter $z=1/x$, commensurable with the subspace spanned by the monomials $\{z^{k}\}_{k=0,1,\dots}$.

The Baker function $\Psi^{-}_{(G,\beta,\gamma)}(z=1/x,{\bf t},\beta^{-1}{\bf s})$ and its dual $\Psi^{+}_{(G,\beta,\gamma)}(z=1/x,{\bf t},\beta^{-1}{\bf s})$, are given by the Sato formula

[TABLE]

These may be viewed as elements of the subspaces $W^{\pm}_{(G,\beta,\gamma,{\bf s})}$ for all values of the KP flow parameters ${\bf t}$.

Specializing eqs. (3.58, 3.59) to this case, the basis elements $\{\Psi^{+}_{i}(x),\Psi^{-}_{i}(x)\}$ satisfy the recursion relations

Proposition 4.1**.**

[TABLE]

where the (nearly) upper triangular matrices $Q^{\pm}$ are the negatives of the transposes of $\tilde{Q}^{\mp}$

[TABLE]

In this case we can express them explicitly as

[TABLE]

Applying the Euler operator $\mathcal{D}$ to the basis elements $\{w_{k},w_{k}^{*}\}$, we obtain the following differential linear relations

Proposition 4.2**.**

[TABLE]

where the elements of the lower triangular matrices $\tilde{P}^{\pm}$ are

[TABLE]

Proof.

To see this, we may either use the general formulae (3.76) of Section 3.4, specialized to the case (4.16) or, more simply, proceed directly. Namely, from (4.52) we have

[TABLE]

where

[TABLE]

and similarly,

[TABLE]

This linear action is represented by the matrices $\tilde{P}^{\pm}$ defined in (4.70). ∎

In terms of the notation $\{\Psi_{k}^{+}(x)\}$, $\{\Psi_{k}^{-}(x)\}$, the differential linear relations are

[TABLE]

where $D$ is the Euler operator in the $x$-variable

[TABLE]

and the upper triangular matrices $P^{\pm}$ are the negatives of the transposes of $\tilde{P}^{\mp}$

[TABLE]

4.5 The spectral curve and Kac-Schwarz operators

4.5.1 The quantum and classical spectral curve

Now define the differential operator $R$ acting either on $\mathcal{H}$, or formally, on $\mathbf{C}[[z,z^{-1}]]$, as:

[TABLE]

which acts on the monomial $z^{i}$:

[TABLE]

Applying $R$ to the series for $\rho(z)$, we see this is an eigenvector with eigenvalue $1$

[TABLE]

Applying it to $w_{k}$ gives

[TABLE]

or iterating,

[TABLE]

Similarly, defining

[TABLE]

we have

[TABLE]

or iterating,

[TABLE]

It is clear that the operators $R$ and $R^{*}$ are invertible.

Equivalently, in terms of the $x=1/z$ variables, we have the operators

[TABLE]

and eqs.(4.85), (4.88) are equivalent to

[TABLE]

Using (4.85) and (4.88), we may re-express eqs. (4.73), (4.75) more compactly as

Proposition 4.3**.**

[TABLE]

or, equivalently,

[TABLE]

Remark 4.3** **(Quantum and classical spectral curves).

Since $\Psi^{-}_{0}(x)=w^{*}_{1}(z)$, $\Psi^{+}_{0}(x)=\gamma w_{1}(z)$ are the ${\bf t}={\bf 0}$ evaluation of the Baker function and its dual, respectively, the $k=0$ case of eqs. (4.94) and (4.95)

[TABLE]

is satisfied by the Baker function and its dual at ${\bf t}={\bf 0}$.

These are the quantum spectral curve appearing in refs. [1, 2], and its dual. Taking the classical limit amounts to replacing $\beta D$ by $xy$, where $y$ is the classical variable canonically conjugate to $x$. Thus, the classical spectral curve is

[TABLE]

and the dual is the same.

4.5.2 Kac-Schwarz operators

Definition 4.1**.**

If for a given $g$, an operator $a$ acting on ${\bf C}[[z,z^{-1}]]$ (e.g. a differential operator in $z$) stabilizes the element $W_{N}^{g}=g(\mathcal{H}^{N}_{+})$ of the Sato Grassmannian (resp. the dual Grassmannian element $W_{N}^{g\perp}$)

[TABLE]

or

[TABLE]

we call it a Kac-Schwarz operator (resp. dual Kac-Schwarz operator) for the corresponding KP $\tau$-function [28].

Remark 4.4**.**

Particular examples of Kac-Schwarz operators for weighted Hurwitz numbers were considered in [4, 5].

Definition 4.2**.**

Define the following operators:

[TABLE]

From (4.52) we see that

[TABLE]

so these are Kac-Schwarz (resp. dual Kac-Schwarz) operators, since they stabilize $W_{N}^{(G,\beta,\gamma,{\bf s})}$ (and $W_{N}^{(G,\beta,\gamma,{\bf s})\perp}$) for all $N$.

Similarly, define

Definition 4.3**.**

[TABLE]

Then (4.92) and (4.93) are equivalent to

[TABLE]

so these too are Kac-Schwarz operators for $W_{N}^{(G,\beta,\gamma{\bf s})}$ (and $W_{N}^{(G,\beta,\gamma{\bf s})\perp}$)

Moreover, for the operators

[TABLE]

we have

[TABLE]

Thus, these also are Kac-Schwarz (and dual Kac-Schwarz) operators for $W^{(G,\beta,\gamma,{\bf s})}$ and $W^{(G,\beta,\gamma,{\bf s})\perp}$. They satisfy the canonical commutation relations

[TABLE]

and completely specify the point of the Grassmanian.

More generally, for any $N\in\mathbf{Z}$, the operators

[TABLE]

give

[TABLE]

satisfy the canonical commutation relations

[TABLE]

and stabilize $W_{-N}^{(G,\beta,\gamma,{\bf s})}$ (and $W_{N}^{(G,\beta,\gamma,{\bf s})\perp}$).

4.6 The pair correlation function in the hypergeometric case

Specializing the Christoffel-Darboux expresssion (3.98) for the pair correlation function $\tilde{K}_{2}^{g}(z,w)$ to the case (4.24), it may be expressed as follows

Proposition 4.4**.**

[TABLE]

where

[TABLE]

Here we have used the fact that, for $g=C_{\rho}\gamma_{-}(\beta^{-1}{\bf s})$,$g_{00}g^{-1}_{-1,-1}=\gamma$.

Alternatively, in terms of the notation $\{\Psi^{\pm}_{j}\}_{j\in\mathbf{N}}$

[TABLE]

Equivalently, for

[TABLE]

we have

[TABLE]

where

[TABLE]

4.7 Finiteness and generating function for Christoffel-Darboux matrix

In this section we prove that, for polynomial $G(z)$ and $S(z)$, the numerator of the correlator (4.123), viewed as an integral operator kernel into $W$, is of finite rank, and that in fact the expression may be written as a finite sum.

Define the following power series in two variables $(r,t)$

[TABLE]

Proposition 4.5**.**

The generating function for the Christoffel-Darboux coefficients $A_{ij}$ defined by (4.128) is given by

[TABLE]

In particular, if $s_{i}=0,\ \forall\ i>L$ and $g_{i}=0,\ \forall\ i>M$, then

[TABLE]

To prove this, we begin with the following easily proved generalization of the orthogonality relations satisfied by the complete symmetric functions, valid in the case when only the first $L$ variables $(s_{1},\dots,s_{L},0,0,\cdots)$ are nonzero.

Lemma 4.6**.**

[TABLE]

Moreover, if $s_{i}=0,\ \forall\ i>L$, the following orthogonality relations are identically satisfied

[TABLE]

Proof.

For each $N\geq 0$, define the following generating series

[TABLE]

where the $S_{N}(x,y)$ are polynomials of degree $N$ in the variables $(x,y)$

[TABLE]

For all $k\in\mathbf{N}$, we also define two weighted versions of the polynomials $S_{N}(x,y)$

[TABLE]

Denoting the Euler operator in $x$

[TABLE]

and applying its $k$th power to $S(x,y)$, we obtain

[TABLE]

where $P(x)$ is the series

[TABLE]

For $x=y$ we have

[TABLE]

which proves the first equation in 4.133. The second equation follows from the substitution ${\bf s}\rightarrow-{\bf s}$.

If $s_{i}=0,\ \forall\ i>L$, then $P(x)$ and is a polynomial of degree $kL$. In this case (4.142) implies (4.134). ∎

We now turn to the proof of Proposition 4.5.

Proof.

It is obvious that it is sufficient to prove it for $\beta=1$. For (4.129) we have

[TABLE]

Then, from (4.141) and (4.142) we conclude

[TABLE]

where we introduced

[TABLE]

In the first term we can change the summation variable $m\mapsto m+1-N$ to get

[TABLE]

so that indeed for $i,j>0$

[TABLE]

If $M$ and $L$ are finite, then (4.129) is a polynomial of total degree $LM$, and (4.131) follows. ∎

For arbitrary $\beta$ we have

[TABLE]

which coincides with eq (4.128).

In terms of $\tilde{A}$ eq (4.131) is equivalent to

[TABLE]

and therefore

[TABLE]

Equivalently, we have the finite form of eq. (4.127)

[TABLE]

4.8 The multicurrent correlators as generating functions for weighted Hurwitz numbers

In this section, we specialize the multicurrent correlators (3.102) to the case of of hypergeometric $\tau$-functions, for which

[TABLE]

In this case

[TABLE]

The correlators $W_{n}$ are therefore expressed fermionically as

[TABLE]

We now introduce another set of generating functions for weighted Hurwitz numbers, both for the connected and nonconnected case.

Definition 4.4**.**

[TABLE]

where $m_{\mu}(x_{1},\dots,x_{n})$ denotes the monomial sum symmetric function corresponding to the partition $\mu$, and $(g,d)$ are related by the Riemann-Hurwitz formula

[TABLE]

Note that $F_{n}$, $\tilde{F}_{n}$ belong to $\mathbb{K}[x_{1},\dots,x_{n};{\bf s};\beta,\beta^{-1}][[\gamma]]$ and $\tilde{F}_{g,n}$ to $\mathbb{K}[x_{1},\dots,x_{n};{\bf s}][[\gamma]]$.

We may similarly define the multicurrent correlator $\tilde{W}_{g,n}({\bf s};x_{1},\dots,x_{n})$ for fixed genus $g$ and $n$ as the coefficients in the expansion

[TABLE]

The following result, relating the generating functions $F_{n},\tilde{F}_{n},\tilde{F}_{g,n}$ to the multicurrent correlators, is proved in [1].

Proposition 4.7**.**

[TABLE]

5 Bose-Fermi correspondence and “cut and join” operators

The cut-and-join description of the single and double Hurwitz numbers was introduced in [18, 49]. It is known that generating functions of general weighted Hurwitz numbers (or, equivalently, hypergeometric $\tau$-functions) can be described by several versions of cut-and-join equations, see e.g. [7, 5, 29, 32, 46, 43, 6]. Here we describe two different cut-and-join representations using the bosonization procedure to express these relations both bosonically and in terms of fermionic VEV’s.

5.1 Bose-Fermi correspondence and cut and join representation of the $\tau$-function

One of the cut-and-join type representations is a direct generalization of [18, 49] and is naturally parametrized by the coefficients of $\log(G)$

[TABLE]

introduced in Section 4.2. From (4.37) it follows that element (4.26) can be represented as

[TABLE]

where the operators

[TABLE]

commute with each other

[TABLE]

From (4.26) and (5.3), we may express the exponents $T_{i}$ in terms of the $A_{i}$’s as

[TABLE]

The bose-fermi correspondence is defined by the bosonization map

[TABLE]

from the (zero charge sector of the) Fermi Fock space $\mathcal{F}_{0}\subset\mathcal{F}$ to the corresponding sector $\mathcal{B}_{0}$ of the Bosonic Fock space $\mathcal{B}_{0}$. The latter is viewed as consisting of symmetric functions of an infinite number of auxiliary Bosonic variables $\{\xi_{a}\}_{a\in\mathbf{N}^{+}}$, in terms of which the KP flow parameters ${\bf t}=(t_{1},t_{2},\dots)$ are understood as normalized power sums

[TABLE]

This gives rise to the identity

[TABLE]

where ${\scriptstyle{{*}\atop{*}}}\dots\scriptstyle{{*}\atop{*}}$ denotes the bosonic normal ordering, which puts all derivatives with respect to $t_{k}$ to the right of all multiplications by $t_{k}$’s, and ${\phi}$ is the bosonic operator

[TABLE]

This allows us to construct a bosonic counterpart for any bilinear fermionic combination $\sum a_{ij}:\psi_{i}\psi^{*}_{j}:$. In particular, for the operators (5.5), we have the bosonic representation $Q_{k}$, which are polynomials in $t_{k}$ and $\frac{\partial}{\partial t_{k}}$, such that

[TABLE]

From (5.11) it immediately follows [43, 3] that a generating function of the bosonic counterparts of these operators $Q_{k}$ is

[TABLE]

The $Q_{k}$’s also generate a commutative algebra

[TABLE]

For example,

[TABLE]

From (5.13) we have

[TABLE]

This gives a cut-and-join type representation for the $\tau$-function (1.26):

[TABLE]

For the case of simple Hurwitz numbers, considered in Section 6.1 when $G(z)=\exp(z)$, if we put $\gamma=1$, (5.23) reduces to

[TABLE]

which coincides with the cut-and-join representation of [18, 49] for simple (double) Hurwitz numbers.

5.2 Cut and join equations

If the $A_{k}$’s are viewed as the independent parameters defining the weighting, the $\tau$-function $\tau^{(G,\beta,\gamma)}({\bf t},{\bf s})$ satisfies the generalized cut-and-join equation

[TABLE]

It follows from (5.13) and the commutation relations (5.15) that $\tau^{(G,\beta,\gamma)}({\bf t},{\bf s})$ also satisfies an infinite number of further cut-and-join equations corresponding to the variables $A_{k}$ and $\gamma$

[TABLE]

For the case where $G(z)$ is a polynomial, we have an alternative cut-and-join representation for the single Hurwitz numbers [50, 29]. Namely,

[TABLE]

Here

[TABLE]

where we have used (4.49). Thus

[TABLE]

where $V_{k}$ are the bosonic operators corresponding to $\mathop{\mathrm{res}}\limits_{z=0}\left(z^{-1}({\mathcal{D}}^{k}\psi(z))\psi^{*}(z)\right)$. For example

[TABLE]

6 Examples

To conclude, we focus upon four specific examples of weighted Hurwtz numbers: the simple (single and double) Hurwitz numbers of [39, 35]; the strongly monotonic Hurwitz numbers, corresponding to weights supported curves with theree branch points (Belyi curves) [9, 29, 22]; the weakly monotonic Hurwitz numbers, corresponding to signed weightings [16, 20, 22] and the quantum Hurwitz numbers introduced in [20, 22]. Explicit formulae are provided for: the content product coefficients entering in the $\tau$-function series, the parameters defining their fermionic respresentation, the weighting factors defining the Hurwitz numbers, and the classical and quantum spectral curves entering in the topological recursion approach.

6.1 Simple (single and double) Hurwitz numbers [39, 35]

This is the original case studied by Pandharipande and Okounkov. It corresponds to the exponential weight generating function

[TABLE]

The content product formula entering in the Schur function series (1.26), (1.27) for the $\tau$-function is

[TABLE]

The weight $W_{Exp}(\mu^{(1)},\dots,\mu^{(k)})$ is the Dirac measure (i.e. the characteristic function) supported on the partitions consisting only of $2$-cycles

[TABLE]

[TABLE]

and the exponents $T_{i}(\beta,\gamma)$ entering in (4.8), (4.10) and (4.24) are

[TABLE]

The quantum spectral curve is [5]:

[TABLE]

where $D=x{d\over dx}$ is the Euler operator, and the classical spectral curve is

[TABLE]

6.2 Three branch points (Belyi curves): strongly monotonic paths in the Cayley graph [9, 29, 22]

For this case, the weight generating function is linear

[TABLE]

The content product formula is

[TABLE]

where

[TABLE]

is the multiple Pochhammer symbol corresponding to the partition $\lambda$. The measure $W_{E}(\mu^{(1)},\dots,\mu^{(k)})$ is supported on the set of $k=1$ partitions, and hence corresponds to enumeration of coverings with just three branch points $(\mu^{(1)},\mu,\nu)$ (Belyi curves).

[TABLE]

The exponents $T_{i}(\beta)$ entering in (4.8), (4.10) and (4.24) are

[TABLE]

The quantum spectral curve is [5]:

[TABLE]

and the classical one is

[TABLE]

6.3 Signed Hurwitz numbers: weakly monotonic paths in the Cayley graph [16, 20, 22]

In this case the weight generating function is

[TABLE]

The content product coefficient is

[TABLE]

The measure $W_{H}(\mu^{(1)},\dots,\mu^{(k)})$ is thus given by the evaluation of the “forgotten” symmetric functions $f_{\lambda}({\bf c})$ for the partition with parts $\{\lambda_{i}\}=\{\ell^{*}(\mu^{(i)})\}$ at ${\bf c}=(1,0,\dots)$

[TABLE]

where $m_{i}(\lambda)$ is the number of colengths $\ell^{*}(\mu^{(j)})$ equal to $i$. The exponents $T_{i}(\beta)$ entering in (4.8), (4.10) and (4.24) are

[TABLE]

The quantum spectral curve is [5]:

[TABLE]

and the classical one is

[TABLE]

6.4 Simple quantum Hurwitz numbers [20, 22]

Choosing a real parameter $q$, $0\leq q\leq 1$, we select the parameters $c_{i}$ to be

[TABLE]

The weight generating function for this case is therefore

[TABLE]

where $\operatorname{Li}_{2}(q,z)$ is the quantum dilogarithm function

[TABLE]

This gives the “simple quantum” Hurwitz numbers introduced in [20], [22]. The content product coefficient for this case is

[TABLE]

The weight $W_{E^{\prime}(q)}(\mu^{(1)},\dots,\mu^{(k)})$ for this case is

[TABLE]

which may be viewed as the (unnormalized) probability measure of a Bosonic gas with energy levels

[TABLE]

associated to branch points with ramification profiles $\mu$, where

[TABLE]

The exponents $T_{i}(\beta)$ entering in (4.8), (4.10) and (4.24) are

[TABLE]

The quantum spectral curve is:

[TABLE]

and the classical one is

[TABLE]

Appendix A Fermionic representation of multipair correlators

In the following, we assume that $g$ is lower triangular. This implies, in particular, that the left vacuum is stabilized by both $\hat{g}$ and $\hat{g}^{-1}$

[TABLE]

A.1 Fermionic representation of $\tilde{K}_{2}^{g}(z,w)$

We provide here and in Appendix A.2 the details of the proofs of Proposition 3.7, Lemma 3.9 and Corollaries 3.10, 3.11. As defined in eq. (3.88), the pair correlator $\tilde{K}^{g}_{2}(z,w)$ may be expressed fermionically as

[TABLE]

We now prove Proposition 3.7, which states that this is equivalent to

[TABLE]

The general case will then follow from Wick’s theorem, as shown in the next subsection.

Proof.

We begin by proving the stronger relations, for all partitions $\lambda$,

[TABLE]

which, since these are valid for all $\lambda$, implies

[TABLE]

First, note that the LHS of (A.4) vanishes unless $\lambda$ is either the trivial partitition $\lambda=\emptyset$ , in which case

[TABLE]

or a hook partition $\lambda=(a|b)$, with arm length $a\geq 0$ and leg length $b\geq 0$, for which

[TABLE]

Turning to the RHS, by Wick’s theorem, $\langle 0|\psi(z)\psi^{*}(w)|\lambda\rangle$ and $\langle 0|\psi^{*}(w)\psi(z)|\lambda\rangle$ also vanish unless $\lambda=\emptyset$ or $\lambda=(a|b)$. For the first case, if $|z|>|w|$

[TABLE]

and similarly, if $|z|<|w|$

[TABLE]

For $\lambda=(a|b)$, and $|z|>|w|$

[TABLE]

by Wick’s theorem, while for $|z|<|w|$,

[TABLE]

Taking the scalar product of (A.5) with $\hat{g}|0\rangle$ then proves the equivalence of eq. (A.2) with (A.3).

∎

A.2 The $n$-pair correlator $\tilde{K}^{g}_{2n}({\bf z},{\bf w})$

For a set of $n$ nonzero complex numbers ${\bf z}=(z_{1},z_{2},\dots z_{n})$, we use the notations $[{\bf z}]$ and $[{\bf z}^{-1}]$ to denote the infinite sequence $\{[{\bf z}]_{i}\}_{i=1,2,\cdots}$, of normalized power sums

[TABLE]

We now proceed to the proof of Lemma 3.9, which is:

Lemma A.1**.**

For $2n$ distinct nonzero complex parameters $\{z_{i},w_{i}\}_{i=1,\dots n}$, we have

[TABLE]

and Corollaries 3.10, 3.11.

Proof.

We proceed by induction on $n$, recalling that the Cauchy determinant is given by

[TABLE]

where

[TABLE]

The case $n=1$ is what underlies Proposition 3.7, which was proved in Appendix A.1. Now assume it is valid for $n-1$

[TABLE]

and multiply both sides on the right by $\psi(z_{n})\psi^{*}(w_{n})$. The RHS becomes

[TABLE]

Using (3.23) and

[TABLE]

we have

[TABLE]

Substituting this into the RHS of (A.21) of the inductive hypothesis and using (A.18) for the Cauchy determinant, we obtain (A.17). ∎

Remark A.1**.**

The usual fermionic representation of the Baker function $\Psi_{g}(z,{\bf t})$ and its dual $\Psi_{g}^{*}(z,{\bf t})$ [41, 40] can be seen to follow as a corollary of the above.

Corollary A.2**.**

[TABLE]

Proof.

Taking the $n=1$ case of (A.17), evaluating the residue at $z=0$ and setting $w=z$, or the residue at $w=0$, gives the identities

[TABLE]

Substituting these into Sato’s formulae for $\Psi_{g}(z,{\bf t})$ and $\Psi_{g}^{*}(z,{\bf t})$ gives

[TABLE]

Substituting the relations (3.23) gives the result (A.25). ∎

A.3 Expansion of $\tilde{K}_{2}(z,w)$ in the adapted basis

We now prove Proposition 3.12, i.e. the expansions

[TABLE]

This can be done in two ways: the first is a direct calculation, using the expression (A.3) for $K_{g}(z,w)$ in terms of the $\tau$-function, to obtain a double power series expansion in $z$ and $w$ by specializing the Schur function expansion of $\tau_{g}({\bf t})$ at these values and the fact that the affine coordinates are just the hook Plücker coordinates. The second is a fermionic calculation, based on again introducing sums over a complete set of intermediate states. We here present the details of the latter.

Proof.

Proof of (A.29). For $|z|>|w|$, inserting a sum over a complete set of intermediate states, we have

[TABLE]

By Wick’s theorem, the only partitions that contribute to this sum are those of the form $\lambda=(1)^{j}$, $j=0,1,\dots$. Substituting

[TABLE]

into the sum therefore gives

[TABLE]

A similar calculation shows, for $|z|<|w|$

[TABLE]

∎

Acknowledgements. The work of A.A. is supported by IBS-R003-D1 and by RFBR grant 15-01-04217. B.E. wishes to thank the Centre de recherches mathématiques, Montréal, for the Aisenstadt Chair grant, and the FQRNT grant from the Québec government, that partially supported this joint project. G.C. acknowledges support from the European Research Council, grant ERC-2016-STG 716083 “CombiTop”, from the Agence Nationale de la Recherche, grant ANR 12-JS02-001-01 “Cartaplus” and from the City of Paris, grant “Émergences 2013, Combinatoire à Paris”. The work of J.H. was partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Fonds de recherche du Québec, Nature et technologies (FRQNT). He wishes to thank the Institut des Hautes Études Scientifiques, where much of this work was completed. It was also supported by the ERC Starting Grant no. 335739 “Quantum fields and knot homologies” funded by the European Research Council under the European Union’s Seventh Framework Programme. The authors would also like to thank the organizers of the January - March, 2017 thematic semester “Combinatorics and interactions” at the Institut Henri Poincaré, where they were participants during the completion of this work.