Quantum cohomology via vicious and osculating walkers

Korff, C. (2014) Quantum cohomology via vicious and osculating walkers. Letters in Mathematical Physics, 104(7), pp. 771-810. (doi:10.1007/s11005-014-0685-2)

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Publisher's URL: http://dx.doi.org/10.1007/s11005-014-0685-2

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 gauged u^(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 Gromov–Witten invariants. We reveal an underlying quantum group structure in terms of Yang–Baxter 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.

Item Type:Articles
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Korff, Dr Christian
Authors: Korff, C.
Subjects:Q Science > QA Mathematics
Q Science > QC Physics
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Research Group:Integrable Systems and Mathematical Physics
Journal Name:Letters in Mathematical Physics
Publisher:Springer
ISSN:0377-9017
ISSN (Online):1573-0530
Published Online:06 March 2014
Copyright Holders:Copyright © 2014 The Author
First Published:First published in Letters in Mathematical Physics 104(7):771-810
Publisher Policy:Reproduced under a Creative Commons License
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Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
569021Integrable Systems and Tropical GeometryChristian KorffEngineering & Physical Sciences Research Council (EPSRC)EP/I037636/1M&S - MATHEMATICS