BKP and projective Hurwitz numbers

Abstract

We consider d-fold branched coverings of the projective plane $\mathbb {RP}^2$ and show that the hypergeometric tau function of the BKP hierarchy of Kac and van de Leur is the generating function for weighted sums of the related Hurwitz numbers. In particular, we get the $\mathbb {RP}^2$ analogues of the $\mathbb {CP}^1$ generating functions proposed by Okounkov and by Goulden and Jackson. Other examples are Hurwitz numbers weighted by the Hall–Littlewood and by the Macdonald polynomials. We also consider integrals of tau functions which generate Hurwitz numbers related to base surfaces with arbitrary Euler characteristics $\textsc {e}$, in particular projective Hurwitz numbers $\textsc {e}=1$.

Appendices

Appendix A: Hirota Equations for the BKP Tau Function with Two Discrete Time Variables

The BKP hierarchy we are interested in was introduced in [34]. In this paper, the BKP tau function $\tau ^\mathrm{B}(N,\mathbf {p})$ does not contain the discrete variable n. We need a slightly more general version of the BKP hierarchy which includes n as the higher time parameter [58, 59, 65]. The Hirota equations for the tau functions $\tau ^\mathrm{B}(N,n,\mathbf {p})$ of this modified BKP hierarchy read

$$\begin{aligned}&\oint \frac{dz}{2\pi i}z^{N'-N-1}e^{V(\mathbf {p}'-\mathbf {p},z)} \tau \left( N'-1,n,\mathbf {p}'-[z^{-1}]\right) \tau \left( N+1,n+1,\mathbf {p}+[z^{-1}]\right) \nonumber \\&\qquad + \oint \frac{dz}{2\pi i}z^{N-N'-3}e^{V(\mathbf {p}-\mathbf {p}',z)} \tau \left( N'+1,n+2,\mathbf {p}'+[z^{-1}]\right) \nonumber \\&\qquad \times \tau \left( N-1,n-1,\mathbf {p}-[z^{-1}]\right) \nonumber \\&\quad = \tau (N'+1,n+1,\mathbf {p}') \tau (N-1,n,\mathbf {p}) - \frac{1}{2}\left( 1-(-1)^{N'+N}\right) \nonumber \\&\qquad \times \tau \left( N',n+1,\mathbf {p}'|g\right) \tau (N,n,\mathbf {p}) \end{aligned}$$

(115)

and

$$\begin{aligned}&\oint \frac{dz}{2\pi i}z^{N'-N-2}e^{V(\mathbf {p}'-\mathbf {p},z)} \tau \left( N'-1,n-1,\mathbf {p}'-[z^{-1}]\right) \nonumber \\&\qquad \times \, \tau \left( N+1,n+1,\mathbf {p}+[z^{-1}]\right) \nonumber \\&\qquad + \oint \frac{dz}{2\pi i}z^{N-N'-2}e^{V(\mathbf {p}-\mathbf {p}',z)} \tau \left( N'+1,n+1,\mathbf {p}'+[z^{-1}]\right) \nonumber \\&\qquad \times \, \tau \left( N-1,n-1,\mathbf {p}'-[z^{-1}]\right) \nonumber \\&\quad = \frac{1}{2}\left( 1-(-1)^{N'+N}\right) \tau \left( N',n,\mathbf {p}'\right) \tau \left( N,n,\mathbf {p}\right) . \end{aligned}$$

(116)

Here $\mathbf {p}=(p_1,p_2,\dots )$, $\mathbf {p}'=(p'_1,p'_2,\dots )$. The notation $\mathbf {p}+[z^{-1}]$ denotes the set

$$\begin{aligned} \left( p_1+z^{-1},p_2+z^{-2}, p_3+z^{-3},\dots \right) \end{aligned}$$

and V is defined by (11). Equations (116) are the same as in [34], while (115) relate tau functions with different discrete time n and were given in [58, 59, 65]. Taking $N'=N+1$ and all $p_i=p_i',\,i\ne 2$ in (116), then picking up the terms linear in $p'_2-p_2$, we obtain (65). Taking $N'=N+1$ and all $p_i=p_i',\,i\ne 1$ in (115), then picking up the terms linear in $p'_1-p_1$, we obtain (66). The relation between the BKP hierarchy and the two- and three-component KP hierarchy was established in [65].

Appendix B: Hypergeometric BKP tau function—Fermionic formulae

Details may be found in [55, 58, 59]. Let $\{\psi _i$, $\psi _i^\dag$, $i \in \mathbb {Z}\}$ be Fermi creation and annihilation operators that satisfy the usual anticommutation relations and vacuum annihilation conditions

$$\begin{aligned}{}[\psi _i, \, \psi _j]_+ = \delta _{i,j}, \quad \psi _i | n\rangle = \psi _{-i-1} | n\rangle =0,\quad i< n. \end{aligned}$$

In contrast to the DKP hierarchy introduced in [32], for the BKP hierarchy introduced in [34], we need an additional Fermi mode $\phi$ which anticommutes with all the other Fermi operators except itself, for which $\phi ^2=1/2$, and $\phi |0\rangle =|0\rangle /\sqrt{2}$ [34]. Then the hypergeometric BKP tau function introduced in [58, 59] may be written as

$$\begin{aligned}&g(n)\tau ^\mathrm{B}_r(N,n,\mathbf {p}) \nonumber \\&\quad = \Big \langle n\big | e^{\sum _{m>0} \frac{1}{m} J_m p_m} e^{\sum _{i < 0} U_i \psi _i^\dag \psi _i -\sum _{i \ge 0} U_i \psi _i\psi _i^\dag } e^{\sum _{i>j} \psi _i\psi _j\,-\sqrt{2}\,\phi \sum _{i\in \mathbb {Z}} \psi _i}\big |n-N\Big \rangle \nonumber \\&\quad = \sum _{\begin{array}{c} \lambda \\ \ell (\lambda )\le N \end{array}} \,e^{-U_\lambda (n)} s_\lambda (\mathbf {p}) = g(n) \sum _{\begin{array}{c} \lambda \\ \ell (\lambda )\le N \end{array}} r_\lambda (n)s_\lambda (\mathbf {p}), \end{aligned}$$

(117)

where $J_m=\sum _{i\in \mathbb {Z}}\,\psi _i\psi ^\dag _{i+m}$, $m>0$, $U_\lambda (n)=\sum _{i} U_{h_i+n}$, $r(i)=e^{U_{i-1}-U_{i}}$, and

$$\begin{aligned} g(n):=\Big \langle n\big | e^{\sum _{i< 0} U_i \psi _i^\dag \psi _i -\sum _{i \ge 0} U_i \psi _i\psi _i^\dag } \big |n\Big \rangle =\left\{ \begin{array}{ll} e^{-U_0+\cdots -U_{n-1}}&{} \mathrm{if}\,\, n>0,\\ 1&{} \mathrm{if} \,\,n=0,\\ e^{U_{-1}+\cdots U_{n}}&{} \mathrm{if}\,\, n<0. \end{array}\right. \end{aligned}$$

(118)

In (117) the summation runs over all partitions whose lengths do not exceed N.

Remark 23

Note that, without the additional Fermi mode $\phi$, the summation range in (117) does include partitions with odd partition lengths. One can avoid this restriction by introducing a pair of DKP tau functions, which seems unnatural.

Apart from (117), the same series without the restriction $\ell (\lambda )\le N$ gives the BKP tau function. However, it is related to the single value $n=0$. The n-dependence destroys the simple form of this tau function [58, 59].