Abelian mirror symmetry of $$ \mathcal{N} $$ = (2, 2) boundary conditions

Abstract

We evaluate half-indices of $$ \mathcal{N} $$ N = (2, 2) half-BPS boundary conditions in 3d $$ \mathcal{N} $$ N = 4 supersymmetric Abelian gauge theories. We confirm that the Neumann boundary condition is dual to the generic Dirichlet boundary condition for its mirror theory as the half-indices perfectly match with each other. We find that a naive mirror symmetry between the exceptional Dirichlet boundary conditions defining the Verma modules of the quantum Coulomb and Higgs branch algebras does not always hold. The triangular matrix obtained from the elliptic stable envelope describes the precise mirror transformation of a collection of half-indices for the exceptional Dirichlet boundary conditions.

Dualities and flavored indices of M2-brane SCFTs

“…This definition is also compatible with the half-and quarter-indices of 4d N = 4 SYM theory studied in[92] including the half-indices of 3d N = 4 gauge theories analyzed in[37,38].…”

“…Such BPS boundary conditions are studied in the ABJM theory [34][35][36]. The dualities of N = (0, 4) boundary conditions and N = (2, 2) boundary conditions in 3d N = 4 Abelian gauge theories are studied by evaluating the half-indices [37,38] and by engineering them in brane setup [39]. They will generalize the dualities of the M2-brane SCFTs associated with the 3d N = 4 ADHM theories.…”

Interfaces and Quantum Algebras, I: Stable Envelopes

“…CS on the background field configuration that consists of: flat connections z a , , x k =i on E τ , and a connection for the i-th flavor symmetry that is flat along E τ but changes from x i to qx i as we go around S 1 Such boundary conditions are sometimes called thimble boundary conditions [85,111] and are realized via the exceptional Dirichlet boundary conditions in gauge theory [85] (they have also played role in the recent papers [127,128]). Since the state |Ψ p is an "in" state, the corresponding boundary conditions have to be imposed on the past boundary at some y = y − < 0.…”

Seiberg-like dualities for orthogonal and symplectic 3d $$ \mathcal{N} $$ = 2 gauge theories with boundaries

“…See[47,48] for the study of dualities of N = (0, 4) boundary conditions and[49,50] for N = (2, 2) boundary conditions in 3d N = 4 supersymmetric gauge theories.2 In fact, Chern-Simons theories on compact manifolds without boundary are also related to CFTs[51,52].…”