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, Professor 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 |
| Related URLs: |
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