Title and Abstract
Junya Yagi
Cluster algebras and 3D integrable systems
Solutions of Zamolodchikov’s tetrahedron equation define integrable 3D lattice models in statistical mechanics, just as solutions of the Yang-Baxter equation define integrable 2D lattice models. I will explain how we can construct solutions of the tetrahedron equation using quantum cluster algebras. This is based on joint work with Xiao-yue Sun, Rei Inoue, Atsuo Kuniba and Yuji Terashima
Dan Xie
On duality of 4d N=1 non-abelian gauge theory
I will discuss the constraints on the meson spectrum formed by tensor fields so that a Seiberg-like duality would work. Our results cover known examples and provide a large class of new models with duality property.
Yiwen Pan
Schur index and modularity
Under the 4d/2d correspondence, the Schur index of a 4d N=2 SCFT is identified with the vacuum character of the associated chiral algebra. In the presence of non-local operators, the index is believed to be related to non-vacuum characters. We study the modular property of the Schur index with or without non-local operators. We show that the vortex defect index is inside the modular orbit of the original Schur index, and line operator index also appears in the modular orbit. We also study the high temperature behavior of the index which gives closed-form $S^3$ partition function of a class of 3d N= 4 theories.
Yang Lei
Modular factorization of superconformal index
Superconformal indices of four-dimensional N=1 gauge theories factorize into holomorphic blocks. We interpret this as a modular property resulting from the combined action of an SL(3,Z) and SL(2,Z)⋉Z2 transformation. The former corresponds to a gluing transformation and the latter to an overall large diffeomorphism, both associated with a Heegaard splitting of the underlying geometry. The extension to more general transformations leads us to argue that a given index can be factorized in terms of a family of holomorphic blocks parametrized by modular (congruence sub)groups. We find precise agreement between this proposal and new modular properties of the elliptic Γ function. This allows us to establish the “modular factorization” of superconformal lens indices of general N=1 gauge theories. Based on this result, we systematically prove that a normalized part of superconformal lens indices defines a non-trivial first cohomology class associated with SL(3,Z). Finally, we use this framework to propose a formula for the general lens space index.
Kaiwen Sun
2d CFTs, Borcherds products and hyperbolization of affine Lie algebras
In 1983, Feingold and Frenkel posed a question about possible relations between affine Lie algebras, hyperbolic Kac-Moody algebras and Siegel modular forms. We give an automorphic answer to this question and its generalization. We classify Borcherds-Kac-Moody algebras whose denominators define reflective automorphic products of singular weight. As a consequence, we prove that there are exactly 81 affine Lie algebras which have nice extensions to BKM algebras. We find that 69 of them appear in Schellekens’ list of holomorphic CFT of central charge 24, while 8 of them correspond to the N=1 structures of holomorphic SCFT of central charge 12 composed of 24 chiral fermions. The last 4 cases are related to exceptional modular invariants from nontrivial automorphisms of fusion algebras. This is based on a joint paper with Haowu Wang and Brandon Williams.