Quantum Cohomology via Vicious and Osculating Walkers

Abstract

Abstract.We relate the counting of rational curves intersecting Schubert varieties of the Grassmannian to the counting of certain non-intersecting lattice paths on the cylinder, so-called vicious and osculating walkers. These lattice paths form exactly solvable statistical mechanics models and are obtained from solutions to the Yang-Baxter equation. The eigenvectors of the transfer matrices of these models yield the idempotents of the Verlinde algebra of the gaugedû(n) k -WZNW model. The latter is known to be closely related to the small quantum cohomology ring of the Grassmannian. We establish further that the partition functions of the vicious and osculating walker model are given in terms of Postnikov's toric Schur functions and can be interpreted as generating functions for GromovWitten invariants. We reveal an underlying quantum group structure in terms of YangBaxter algebras and use it to give a generating formula for toric Schur functions in terms of divided difference operators which appear in known representations of the nil-Hecke algebra.Mathematics Subject Classification (2010). 14N35, 05E05, 05A15, 05A19, 82B23.

Quantum integrability and generalised quantum Schubert calculus

“…In this article we shall instead investigate the above connection for the Grassmannians Gr n,N = Gr n (C N ) themselves rather than their cotangent spaces based on the earlier findings in [40], [38] and [26]; see also the work on non-quantum GL(N )-equivariant cohomology in [60]. In the present setting it is initially not clear which quantum group to expect.…”

“…, N using the theory of exactly solvable lattice models in statistical mechanics [4]. While the latter appear in theoretical physics, we shall use them here as abstract combinatorial objects -they define a weighted counting of non-intersecting lattice paths as described for β = 0 in [38] -which can be rigorously defined in purely mathematical terms using Yang-Baxter algebras. The weights or probabilities attached to the lattice models depend on • a variable β (the anisotropy parameter of the six-vertex model) entering the multiplicative formal group law [59], [15] and its inverse, (1.1) x ⊕ y = x + y + βxy and x ⊖ y = x − y 1 + βy ,…”

Izergin-Korepin Analysis on the Projected Wavefunctions of the Generalized Free-Fermion Model

“…Deriving algebraic combinatorial properties of symmetric functions using their integrable model realizations is an active line of research. See [36][37][38][39][40] for more examples on Cauchy-type identities and more recent studies on the Littlewood-Richardson coefficients by [33,41].…”

Dual wavefunction of the Felderhof model

“…The integrable five-vertex model which is the t = 0 limit of the L-operator (2.8), which gives the Schur polynomials, can be regarded as special limits of both the Felderhof model and the XXZ model. See [21,22,29,30,31,32,33] for examples on the recent investigations on the combinatorics of the symmetric polynomials from the viewpoint of partition functions, in which the combinatorial identities of various symmetric polynomials such as the Schur, Grothendieck, Hall-Littlewood and their noncommutative versions are derived.…”