Cut-and-join structure and integrability for spin Hurwitz numbers
- PDF / 462,062 Bytes
- 16 Pages / 595.276 x 790.866 pts Page_size
- 64 Downloads / 184 Views
Regular Article - Theoretical Physics
Cut-and-join structure and integrability for spin Hurwitz numbers A. Mironov1,2,3,a , A. Morozov2,3,4,b , S. Natanzon2,5,c 1
Lebedev Physics Institute, Moscow 119991, Russia ITEP, Moscow 117218, Russia 3 Institute for Information Transmission Problems, Moscow 127994, Russia 4 MIPT, Dolgoprudny 141701, Russia 5 HSE University, Moscow, Russia
2
Received: 8 August 2019 / Accepted: 13 January 2020 / Published online: 6 February 2020 © The Author(s) 2020
Abstract 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 R ∈ SP are common eigenfunctions of cut-and-join operators W with ∈ 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 τ -function of an integrable hierarchy, that is, of the BKP type. At last, we discuss relations of the Sergeev characters with matrix models.
1 Introduction This month it is exactly ten years from the publication of [1,2] which introduced the commutative ring of general cutand-join operators with linear group characters as common eigenfunctions and symmetric group characters as the corresponding eigenvalues. Since then, these operators have found a lot of applications in mathematical physics, from matrix models to knot theory, and led to a crucially important and still difficult notion of Hurwitz τ -functions. A variety of further generalizations was considered, from q, t-deformations [3,4] to the Ooguri-Vafa partition functions [5–11] and various non-commutative extensions [12,13]. One of the most important generalizations is a construction of open Hurwitz numbers [14–16]: an infinite-dimensional counterpart of the Hurwitz theory realization of algebraic open-closed string a e-mails:
[email protected]; [email protected]
b e-mail:
[email protected]
c e-mail:
[email protected]
model á la Moore and Lizaroiu equipped with the CardyFrobenius algebra, the closed and open sectors being represented by conjugation classes of permutations and the pairs of permutations, i.e. by the algebra of Young diagrams and bipartite graphs respectively. Note that the original construction essentially involves the characters of linear groups and symmetric groups (another manifestation of the Schur–Weyl duality) understood as embedded into the linear group G L(∞) and the symmetric group S∞ . However, an obvious direction of changing this group set-up remained poorly explored. In the present paper, we discuss this interesting subject with the hope that it would add essential new colors to the picture and give rise to many new applications. That is, instead of the Schur polynomials (characters of linear groups
Data Loading...
