Spin Hurwitz Numbers Research Articles

Genus expansion of matrix models and hbar expansion of BKP hierarchy

We continue the investigation of the connection between the genus expansion of matrix models and the hbar expansion of integrable hierarchies. In this paper, we focus on the BKP hierarchy, which corresponds to the infinite-dimensional Lie algebra of type B. We consider the genus expansion of such important solutions as Brézin–Gross–Witten (BGW) model, Kontsevich model, and generating functions for spin Hurwitz numbers with completed cycles. We show that these partition functions with inserted parameter hbar , which controls the genus expansion, are solutions of the hbar -BKP hierarchy with good quasi-classical behavior. hbar -BKP language implies the algorithmic prescription for hbar -deformation of the mentioned models in terms of hypergeometric BKP tau -functions and gives insight into the similarities and differences between the models. Firstly, the insertion of hbar into the Kontsevich model is similar to the one in the BGW model, though the Kontsevich model seems to be a very specific example of hypergeometric tau -function. Secondly, generating functions for spin Hurwitz numbers appear to possess a different prescription for genus expansion. This property of spin Hurwitz numbers is not the unique feature of BKP: already in the KP hierarchy, one can observe that generating functions for ordinary Hurwitz numbers with completed cycles are deformed differently from the standard matrix model examples.

Kac-Schwarz operators of type B, quantum spectral curves, and spin Hurwitz numbers

Given a tau-function τ(t) of the BKP hierarchy satisfying τ(0)=1, we discuss the relation between its BKP-affine coordinates on the isotropic Sato Grassmannian and its BKP-wave function. Using this result, we formulate a type of Kac-Schwarz operators for τ(t) in terms of BKP-affine coordinates. As an example, we compute the affine coordinates of the BKP tau-function for spin single Hurwitz numbers with completed cycles, and find a pair of Kac-Schwarz operators (P,Q) satisfying [P,Q]=1. By doing this, we obtain the quantum spectral curve for spin single Hurwitz numbers.

Elements of spin Hurwitz theory: closed algebraic formulas, blobbed topological recursion, and a proof of the Giacchetto–Kramer–Lewański conjecture

In this paper, we discuss the properties of the generating functions of spin Hurwitz numbers. In particular, for spin Hurwitz numbers with arbitrary ramification profiles, we construct the weighed sums which are given by Orlov’s hypergeometric solutions of the 2-component BKP hierarchy. We derive the closed algebraic formulas for the correlation functions associated with these tau-functions, and under reasonable analytical assumptions we prove the loop equations (the blobbed topological recursion). Finally, we prove a version of topological recursion for the spin Hurwitz numbers with the spin completed cycles (a generalized version of the Giacchetto–Kramer–Lewański conjecture).

Generalized Brézin–Gross–Witten tau-function as a hypergeometric solution of the BKP hierarchy

In this paper, we prove that the generalized Brézin–Gross– Witten tau-function is a hypergeometric solution of the BKP hierarchy with simple weight generating function. We claim that it describes a spin version of the strictly monotone Hurwitz numbers. A family of the hypergeometric tau-functions of the BKP hierarchy, corresponding to the rational weight generating functions, is investigated. In particular, the cut-and-join operators are constructed, and the explicit description of the BKP Sato Grassmannian points is derived. Representatives of this family can be associated with interesting families of spin Hurwitz numbers including a spin version of the monotone Hurwitz numbers.

Natanzon-Orlov model and refined superintegrability

We reconsider the simple matrix model description of Hurwitz numbers proposed by S. Natanzon and A. Orlov, which uses the superintegrability property of the complex matrix model, and discuss a way of its possible supersymmetric extension to approach spin Hurwitz numbers.

Around spin Hurwitz numbers

We present a review of the spin Hurwitz numbers, which count the ramified coverings with spin structures. They are related to peculiar $Q$ Schur functions, which are actually related to characters of the Sergeev group. This allows one to put the whole story into the modern context of matrix models and integrable hierarchies. Hurwitz partition functions are actually broader than conventional $\tau$-functions, but reduce to them in particular circumstances. We explain, how a special $d$-soliton $\tau$-functions of KdV and Veselov-Novikov hierarchies generate the spin Hurwitz numbers $H^\pm\left( \Gamma^r_d \right)$ and $H^\pm\left( \Gamma^r_d,\Delta \right)$. The generating functions of the spin Hurwitz numbers are hypergeometric $\tau$-functions of the BKP integrable hierarchy, and we present their fermionic realization. We also explain how one can construct $\tau$-functions of this type entirely in terms of the $Q$ Schur functions. An important role in this approach is played by factorization formulas for the $Q$ Schur functions on special loci.

Superintegrability of Kontsevich matrix model

Many eigenvalue matrix models possess a peculiar basis of observables that have explicitly calculable averages. This explicit calculability is a stronger feature than ordinary integrability, just like the cases of quadratic and Coulomb potentials are distinguished among other central potentials, and we call it superintegrability. As a peculiarity of matrix models, the relevant basis is formed by the Schur polynomials (characters) and their generalizations, and superintegrability looks like a property langle characterrangle ,sim character. This is already known to happen in the most important cases of Hermitian, unitary, and complex matrix models. Here we add two more examples of principal importance, where the model depends on external fields: a special version of complex model and the cubic Kontsevich model. In the former case, straightforward is a generalization to the complex tensor model. In the latter case, the relevant characters are the celebrated Q Schur functions appearing in the description of spin Hurwitz numbers and other related contexts.

Cut-and-join structure and integrability for spin Hurwitz numbers

Spin Hurwitz numbers are related to characters of the Sergeev group, which are the expansion coefficients of the Q Schur functions, depending on odd times and on a subset of all Young diagrams. These characters involve two dual subsets: the odd partitions (OP) and the strict partitions (SP). The Q Schur functions Q_R with Rin hbox {SP} are common eigenfunctions of cut-and-join operators W_Delta with Delta in hbox {OP}. The eigenvalues of these operators are the generalized Sergeev characters, their algebra is isomorphic to the algebra of Q Schur functions. Similarly to the case of the ordinary Hurwitz numbers, the generating function of spin Hurwitz numbers is a tau -function of an integrable hierarchy, that is, of the BKP type. At last, we discuss relations of the Sergeev characters with matrix models.

A square root of Hurwitz numbers

We exhibit a generating function of spin Hurwitz numbers analogous to (disconnected) double Hurwitz numbers that is a tau function of the two-component BKP (2-BKP) hierarchy and is a square root of a tau function of the two-component KP (2-KP) hierarchy defined by related Hurwitz numbers.

A NOTE ON GUNNINGHAM’S FORMULA

Gunningham [‘Spin Hurwitz numbers and topological quantum field theory’, Geom. Topol.20(4) (2016), 1859–1907] constructed an extended topological quantum field theory (TQFT) to obtain a closed formula for all spin Hurwitz numbers. In this note, we use a gluing theorem for spin Hurwitz numbers to re-prove Gunningham’s formula. We also describe a TQFT formalism naturally induced by the gluing theorem.

Spin Hurwitz numbers and topological quantum field theory

Spin Hurwitz numbers count ramified covers of a spin surface, weighted by the size of their automorphism group (like ordinary Hurwitz numbers), but signed $\pm 1$ according to the parity of the covering surface. These numbers were first defined by Eskin-Okounkov-Pandharipande in order to study the moduli of holomorphic differentials on a Riemann surface. They have also been related to Gromov-Witten invariants of of complex 2-folds by work of Lee-Parker and Maulik-Pandharipande. In this paper, we construct a (spin) TQFT which computes these numbers, and deduce a formula for any genus in terms of the combinatorics of the Sergeev algebra, generalizing the formula of Eskin-Okounkov-Pandharipande. During the construction, we describe a procedure for averaging any TQFT over finite covering spaces based on the finite path integrals of Freed-Hopkins-Lurie-Teleman.

Equivalence of ELSV and Bouchard–Mariño conjectures for $$r$$ r -spin Hurwitz numbers

We propose two conjectures on Huwritz numbers with completed $(r+1)$-cycles, or, equivalently, on certain relative Gromov-Witten invariants of the projective line. The conjectures are analogs of the ELSV formula and of the Bouchard-Mari\~no conjecture for ordinary Hurwitz numbers. Our $r$-ELSV formula is an equality between a Hurwitz number and an integral over the space of $r$-spin structures, that is, the space of stable curves with an $r$th root of the canonical bundle. Our $r$-BM conjecture is the statement that $n$-point functions for Hurwitz numbers satisfy the topological recursion associated with the spectral curve $x = -y^r + \log y$ in the sense of Chekhov, Eynard, and Orantin. We show that the $r$-ELSV formula and the $r$-BM conjecture are equivalent to each other and provide some evidence for both.

Degree-three spin Hurwitz numbers

Recently, Gunningham \cite{G} calculated all spin Hurwitz numbers in terms of combinatorics of Sergeev algebra. In this paper, we use a spin curve degeneration to obtain a recursion formula for degree three spin Hurwitz numbers.

Spin Hurwitz numbers and the Gromov-Witten invariants of Kähler surfaces

The classical Hurwitz numbers which count coverings of a complex curve have an analog when the curve is endowed with a theta characteristic. These “spin Hurwitz numbers”, recently studied by Eskin, Okounkov and Pandharipande, are interesting in their own right. By the authors’ previous work, they are also related to the Gromov-Witten invariants of Kahler surfaces. We prove a recursive formula for spin Hurwitz numbers, which then gives the dimension zero GW invariants of Kahler surfaces with positive geometric genus. The proof uses a degeneration of spin curves, an invariant defined by the spectral flow of certain anti-linear deformations of ∂, and an interesting localization phenomenon for eigenfunctions that shows that maps with even ramification points cancel in pairs. The Hurwitz numbers of a complex curveD count covers with specified ramification type. Specifically, consider degree d (possibly disconnected) covering maps f : C → D with fixed ramification points q, . . . , q ∈ D and ramification given by m, . . . ,m where each m = (m1, · · · ,mili) is a partition of d. The Euler characteristic of C is related to the genus h of D and the partition lengths l(m) = li by the Riemann-Hurwitz formula χ(C) = 2d(1− h) + k ∑ i=1 ( l(m)− d). (1.1) In this context, there is an ordinary Hurwitz number ∑ 1 |Aut(f)| (1.2) that counts the covers f satisfying (1.1) mod automorphisms; the sum depends only on h and {mi}. Now fix a theta characteristic N on D, that is, a holomorphic line bundle with an isomorphism N = KD whereKD is the canonical bundle of D. The pair (D,N) is called a spin curve. By a well-known theorem of Mumford and Atiyah, the deformation class of the spin curve is completely characterized by the genus h of D and the parity p = (−1)h0(D,N) (1.3) Now consider degree d ramified covers f : C → D for which • each partition m is odd, i.e. each mj is an odd number. (1.4) In this case, the ramification divisor Rf of f is even and the twisted pullback bundle Nf = f ∗N ⊗O ( 1 2Rf ) (1.5) ∗partially supported by N.S.F. grants DMS-1206192 (J.L.) and DMS-1011793 (T.P.)

Stable pairs with descendents on local surfaces I: the vertical component

We study the full stable pair theory --- with descendents --- of the Calabi-Yau 3-fold $X=K_S$, where $S$ is a surface with a smooth canonical divisor $C$. By both $\mathbb C^*$-localisation and cosection localisation we reduce to stable pairs supported on thickenings of $C$ indexed by partitions. We show that only strict partitions contribute, and give a complete calculation for length-1 partitions. The result is a surprisingly simple closed product formula for these "vertical" thickenings. This gives all contributions for the curve classes $[C]$ and $2[C]$ (and those which are not an integer multiple of the canonical class). Here the result verifies, via the descendent-MNOP correspondence, a conjecture of Maulik-Pandharipande, as well as various results about the Gromov-Witten theory of $S$ and spin Hurwitz numbers.