4 KEVIN COSTELLO (4) The spectral parameter is a meromorphic parameter on Σ. (5) The Boltzmann weights (or R-matrix) of the lattice model arises from the operator product expansion of line operators. Kapustin showed that any N = 2 field theory admits a twist of this form. I showed in [5] that N = 1 pure gauge theory can be deformed and twisted to yield a theory of this form. This deformed N = 1 gauge theory has a Wilson operator invariant under the supercharge we use to twist. The main result of [5], which I sketch here, states that the integrable lattice model associated to a twisted, deformed N = 1 gauge theory, with gauge group SU(n) and Wilson operator in a representation V of SU(n), is the higher spin-chain system associated to the SU(n) representation V . (Thus, in the case that n = 2 and V is the fundamental representation, we find the Heisenberg XXX model). The result generalises to other semi-simple gauge groups: however, for G = SU(n), the Wilson operator associated to a G-representation V may have a quan- tum anomaly. This anomaly occurs if V can not be lifted to a representation of the Yangian Y (g) of the Lie algebra g of G. Kapustin’s holomorphic/topological twist of N = 2 gauge theories admits both Wilson and t’Hooft operators. The construction of this paper, applied to Kapustin’s twist, will yield an integrable lattice model associated to any N = 2 theory, whose partition function is the expectation value of a configuration of Wilson and t’Hooft operators. There are several other known relationships between integrable lattice mod- els and four-dimensional field theories. One was introduced by Nekrasov and Shatashvili in [18], and developed mathematically by Maulik and Okounkov in the beautiful paper [16]. It seems that these two connections between field theories and integrable systems are completely different. Indeed, Nekrasov and Shatashvili show that the spin-chain system for an ADE group G is associated to the N = 2 quiver gauge theory with ADE quiver corresponding to G, whereas in this paper we find that the same spin-chain system arises from the N = 1 gauge theory with gauge group G. Another relationship between integrable systems and gauge theories was devel- oped by Yamazaki in [22]. Again, this appears to be different from the relationship developed here, in that in Yamazaki’s work the Yang-Baxter equation is derived from Seiberg duality, whereas here the Yang-Baxter equation is much easier to derive. Of course, there is also the well-known connection between N = 4 gauge theory and the Yangian (see e.g. [10]). This as also, as far as I know, unrelated to the results of this note. 1.1. Acknowledgements. I’d like to thank Kolya Reshitikhin, Nick Rozen- blyum, Josh Shadlen, and Edward Witten for helpful conversations. 2. Integrable lattice models In this section, I will define the concept of integrable lattice model from the vertex-model point of view (i.e. from the discrete version of the path-integral approach to quantum field theory). Let V, W be finite-dimensional vector spaces. Let ˇ R : V W W V
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