Proceedings of Symposia in Pure Mathematics Volume 88, 2014 http://dx.doi.org/10.1090/pspum/088/01483 Integrable lattice models from four-dimensional field theories Kevin Costello Abstract. This paper gives a general construction of an integrable lattice model (and a solution of the Yang-Baxter equation with spectral parameter) from a four-dimensional field theory which is a mixture of topological and holo- morphic. Spin-chain models arise in this way from a twisted, deformed version of N = 1 gauge theory. This note is based on the longer paper arXiv:1303.2632. 1. Introduction Integrable lattice models have a long and fruitful history in physics, dating back to Heisenberg’s work on the XXX model. Integrability of the XXX and related models was proved by Baxter, Bethe, Yang and others in the 60’s and 70’s. A key insight of this work is that integrability follows from the fact that the vertex interaction of the model encoded by the R-matrix satisfies the Yang-Baxter equation. If we take an integrable model and perturb it a small amount, it will typically no longer be integrable. The physical properties of the perturbed model will be essentially identical, however. One can therefore ask the following question: where do integrable models come from? 1 In this note (which is a summary of the long paper [5]) I propose the follow- ing answer: integrable models arise from four-dimensional field theories which are topological in two real directions and holomorphic in one complex direction. I show that every such field theory, equipped with some line operators in the topological directions, leads to a two-dimensional integrable lattice model. The correspondence is as follows. (1) The partition function of the lattice model is equal to the expectation value of a configuration of line operators on a product of a Riemann surface Σ and a topological two-torus T 2. (2) The Hilbert space of the lattice model is the Hilbert space of the field theory on a Riemann surface Σ times a topological S1, in the presence of line operators which end at points on the circle. (3) The transfer matrix is the operator on the Hilbert space associated to Σ × S1 arising from a line operator parallel to the S1. Partially supported by NSF grant DMS 1007168, by a Sloan Fellowship, and by a Simons Fellowship in Mathematics. 1This paragraph paraphrases some comments made by Okounkov in a lecture in 2013. c 2014 American Mathematical Society 3
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