Surface defects and elliptic quantum groups

Published for SISSA by Springer Received: February 14, 2017 Accepted: May 22, 2017 Published: June 5, 2017 Junya Yagi Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland E-mail: Abstract: A brane construction of an integrable lattice model is proposed. The model is composed of Belavin’s R-matrix, Felder’s dynamical R-matrix, the Bazhanov-SergeevDerkachov-Spiridonov R-operator and some intertwining operators. This construction implies that a family of surface defects act on supersymmetric indices of four-dimensional N = 1 supersymmetric field theories as transfer matrices related to elliptic quantum groups. Keywords: Lattice Integrable Models, p-branes, Quantum Groups, Supersymmetric Gauge Theory ArXiv ePrint: 1701.05562 Open Access, c The Authors. Article funded by SCOAP3 . doi:10.1007/JHEP06(2017)013 JHEP06(2017)013 Surface defects and elliptic quantum groups Contents 1 2 Integrable lattice models of elliptic type 2.1 Belavin model 2.2 Felder’s dynamical R-matrix 2.3 Vertex-face correspondence and Jimbo-Miwa-Okado model 2.4 Bazhanov-Sergeev model 4 5 7 9 9 3 Unification 3.1 Intertwining operators 3.2 Yang-Baxter equations with one dashed line 3.3 Yang-Baxter equations with two dashed lines 3.4 L-operators and elliptic quantum groups 13 13 14 17 18 4 Surface defects and elliptic quantum groups 4.1 Brane construction 4.2 N = 1 supersymmetric quiver gauge theories and surface defects 4.3 Fusion procedure and exterior powers of the vector representation 20 21 23 25 1 Introduction The aim of this paper is twofold. It is an attempt to offer a fresh perspective on integrable lattice models of elliptic type, such as Baxter’s eight-vertex model [1, 2], by embedding them into string theory. At the same time, it seeks to provide a new approach to the study of four-dimensional supersymmetric field theories by connecting their surface operators to those models. More specifically, I propose a brane construction of an integrable lattice model that is composed of Belavin’s R-matrix [3], Felder’s dynamical R-matrix [4, 5], the BazhanovSergeev-Derkachov-Spiridonov (BSDS) R-operator [6–8] and intertwining operators between the first two R-matrices [9, 10]. This construction allows us to map a family of surface defects in N = 1 supersymmetric field theories to transfer matrices related to Felder’s elliptic quantum groups. The proposal builds on a recent development [11] in the correspondence between N = 1 supersymmetric field theories and integrable lattice models [12–15]. Therefore, let me first review the relevant results. One side of the correspondence is a class of theories realized by certain configurations of 5-branes in string theory, referred to as brane tilings [16, 17]. A theory in this class –1– JHEP06(2017)013 1 Introduction (a) (b) Figure 1. (a) A periodic lattice colored in a checkerboard-like pattern. (b) The quiver associated with the lattice. Each node is an SU(N ) gauge group and each arrow is a matter multiplet. –2– JHEP06(2017)013 has multiple SU(N ) gauge and flavor groups as well as matter fields transforming in bifundamental representations under these groups, where N is an integer fixed by the brane configuration. It is an example of a quiver gauge theory; its field content can be encapsulated in a planar quiver diagram. The quiver itself is specified by a square lattice whose faces are colored in a checkerboard-like pattern, which encodes the topology of the 5-branes interwoven in a ten-dimensional spacetime. The lattice has two kinds of “black” faces, either light shaded or dark shaded, and two shaded faces sharing a vertex is required to be of different kinds. Figure 1 shows an example of such a lattice and the associated quiver. The other side of the correspondence is the Bazhanov-Sergeev model [6, 7], defined on the same tricolor checkerboard lattice. To each unshaded face is assigned a continuous spin variable that takes values in a maximal torus of SU(N ). The Boltzmann weight, or R-operator, of the model is an integral operator involving elliptic gamma functions. It solves the Yang-Baxter equation, ensuring that the model is integrable. In [12, 13], it was discovered that the supersymmetric index of the gauge theory, formulated on the three-sphere S 3 , coincides with the partition function of the lattice model. Under this correspondence, the gauge and flavor groups are mapped to the spin sites, while the matters are interpreted as interactions between spins located at different sites. The Yang-Baxter equation translates on the gauge theory side to the invariance of the index under Seiberg duality [18], which relates two theories describing the same infrared physics. As elucidated in [14], what underlies this correspondence is the structure of a twodimensional topological quantum field theory (TQFT), equipped with line operators that are localized in an extra dimension. From the fact that the supersymmetric index is invariant under continuous deformations of the checkerboard lattice, one deduces that it is captured by a correlation function of line operators in a TQFT. A general argument in open/closed TQFTs [15] then shows that the correlation function equals the partition function of a lattice model, the Bazhanov-Sergeev model in this case. Finally, the TQFT has a hidden extra dimension that emerges via string dualities, and its existence implies the integrability of the model [19, 20]. Things get more interesting when we introduce D3-branes that end on the 5-branes. With their configurations chosen appropriately, these additional branes create in the gauge theory surface defects preserving half of the N = 1 supersymmetry. In the checkerboard lattice they are supported along curves, which we represent by dashed lines as in figure 2(a). The same reasoning as above, applied to this situation, leads to the conclusion [11] that (a) (b) (c) Figure 2. (a) A dashed line representing a D3-brane. It acts on the Bazhanov-Sergeev model as a transfer matrix. (b) The L-operator from which the transfer matrix is constructed. It is also an L-operator for the Belavin model. (c) An L-operator for Felder’s R-matrix. It is obtained from the L-operator for the Belavin model by an interchange of the left and right halves. (b) Figure 3. (a) A lattice formed by D3-branes in a shaded background supports the Belavin model. (b) The same lattice in an unshaded background gives rise to the Jimbo-Miwa-Okado model. they act on the lattice model as transfer matrices that consist of so-called L-operators. An example of an L-operator is depicted in figure 2(b). In [11], the relevant L-operator was identified for N = 2, based on the analysis carried out in [8] and independent gauge theory computations. It was found that this L-operator is essentially Sklyanin’s L-operator [21], and satisfies a defining “RLL relation” not only with the BSDS R-operator but also with Baxter’s R-matrix for the eight-vertex model—a property th (...truncated)