We conjecture a formula for the Schur index of four-dimensional $$ \mathcal{N}=2 $$
 theories in the presence of boundary conditions and/or line defects, in terms of the low-energy effective Seiberg-Witten description of the system together with massive BPS excitations. We test our proposal in a variety of examples for SU(2) gauge theories, either conformal or asymptotically free. We use the conjecture to compute these defect-enriched Schur indices for theories which lack a Lagrangian description, such as Argyres-Douglas theories. We demonstrate in various examples that line defect indices can be expressed as sums of characters of the associated two-dimensional chiral algebra and that for Argyres-Douglas theories the line defect OPE reduces in the index to the Verlinde algebra.

/pdf/infrared-computations-of-defect-schur-indices-1ykt0kfb7n.pdf

Infrared computations of defect Schur indices

Let V be a simple vertex operator algebra containing a rank n Heisenberg vertex algebra H and let C = Com(H;V) be the coset of H in V. Assuming that the module categories of interest are vertex tensor categories in the sense of Huang, Lepowsky and Zhang, a Schur-Weyl type duality for both simple and indecomposable but reducible modules is proven. Families of vertex algebra extensions of C are found and every simple C-module is shown to be contained in at least one V-module. A corollary of this is that if V is rational, C2-cofinite and CFT-type, and Com(C;V) is a rational lattice vertex operator algebra, then C is likewise rational. These results are illustrated with many examples and the C1-cofiniteness of certain interesting classes of modules is established.

Schur–weyl duality for heisenberg cosets

We conjecture a formula for the Schur index of N=2 four-dimensional theories in the presence of boundary conditions and/or line defects, in terms of the low-energy effective Seiberg-Witten description of the system together with massive BPS excitations. We test our proposal in a variety of examples for SU(2) gauge theories, either conformal or asymptotically free. We use the conjecture to compute these defect-enriched Schur indices for theories which lack a Lagrangian description, such as Argyres-Douglas theories. We demonstrate in various examples that line defect indices can be expressed as sums of characters of the associated two-dimensional chiral algebra and that for Argyres-Douglas theories the line defect OPE reduces in the index to the Verlinde algebra.

/pdf/infrared-computations-of-defect-schur-indices-ps3o2x2x5q.pdf

Infrared Computations of Defect Schur Indices

Singular Gelfand–Tsetlin modules of gl(n)

We define an action of the (double of) Cohomological Hall algebra of Kontsevich and Soibelman on the cohomology of the moduli space of spiked instantons of Nekrasov. We identify this action with the one of the affine Yangian of $\mathfrak{gl}(1)$. Based on that we derive the vertex algebra at the corner $\mathcal{W}_{r_1,r_2,r_3}$ of Gaiotto and Rapcak. We conjecture that our approach works for a big class of Calabi-Yau categories, including those associated with toric Calabi-Yau $3$-folds.

Cohomological Hall algebras, vertex algebras and instantons

We provide a classification and explicit bases of tableaux of all irreducible generic Gelfand{Tsetlin modules for the Lie algebra gl(n).

/pdf/irreducible-generic-gelfand-tsetlin-modules-of-gl-n-3iyn5ptdyp.pdf

Irreducible Generic Gelfand-Tsetlin Modules of gl(n)

For an admissible affine vertex algebra $${V_k{(\mathfrak{g})}}$$
 of type A, we describe a new family of relaxed highest weight representations of $${V_k{(\mathfrak{g})}}$$
 . They are simple quotients of representations of the affine Kac–Moody algebra $${\widehat{\mathfrak{g}}}$$
 induced from the following $${\mathfrak{g}}$$
 -modules: (1) generic Gelfand–Tsetlin modules in the principal nilpotent orbit, in particular all such modules induced from $${\mathfrak{sl}_2}$$
 ; (2) all Gelfand–Tsetlin modules in the principal nilpotent orbit that are induced from $${\mathfrak{sl}_3}$$
 ; (3) all simple Gelfand–Tsetlin modules over $${\mathfrak{sl}_3}$$
 . This in particular gives the classification of all simple positive energy weight representations of $${V_k{(\mathfrak{g})}}$$
 with finite dimensional weight spaces for $${\mathfrak{g}=\mathfrak{sl}_3}$$
 .

Weight Representations of Admissible Affine Vertex Algebras

For an admissible affine vertex algebra $V_k(\mathfrak{g})$ of type $A$, we describe a new family of relaxed highest weight representations of $V_k(\mathfrak{g})$. They are simple quotients of representations of the affine Kac-Moody algebra $\widehat{\mathfrak{g}}$ induced from the following $\mathfrak{g}$-modules: 1) generic Gelfand-Tsetlin modules in the principal nilpotent orbit, in particular all such modules induced from $\mathfrak{sl}_2$; 2) all Gelfand-Tsetlin modules in the principal nilpotent orbit which are induced from $\mathfrak{sl}_3$; 3) all simple Gelfand-Tsetlin modules over $\mathfrak{sl}_3$. This in particular gives the classification of all simple positive energy weight representations of $V_k(\mathfrak{g})$ with finite dimensional weight spaces for $\mathfrak{g}=\mathfrak{sl}_3$.

Luis Enrique Ramirez

Papers

Singular Gelfand–Tsetlin modules of gl(n)

Irreducible Generic Gelfand-Tsetlin Modules of gl(n)

Weight Representations of Admissible Affine Vertex Algebras

Weight representations of admissible affine vertex algebras

Combinatorial construction of Gelfand–Tsetlin modules for gln