Supersymmetric boundary conditions in three-dimensionalN=2theories

Abstract

We study supersymmetric boundary conditions in three-dimensional N ¼ 2 Landau-Ginzburg models and Abelian gauge theories. In the Landau-Ginzburg model the boundary conditions that preserve (1,1) supersymmetry (A-type) and (2,0) supersymmetry (B-type) on the boundary are classified in terms of subspaces of the target space (''brane''). An A-type brane is a Lagrangian submanifold on which the imaginary part of the superpotential is constant, while a B-type brane is a holomorphic submanifold on which the superpotential is constant. We also consider the N ¼ 2 Maxwell theory with boundary and the Abelian duality. Finally we make some comments on N ¼ 2 SQED with boundary condition and the mirror symmetry.1 Here we use the term ''membrane'' in a broad meaning. We do not claim that our membrane is the same as that considered in M theory.

On supersymmetry, boundary actions and brane charges

“…With fewer supersymmetries, the general BC and their interplay with dualities are still largely unexplored. (See [21][22][23] for the 3d case. )…”

Localization of 4d $$ \mathcal{N} $$ = 1 theories on 𝔻2× 𝕋2

“…The formulation of supersymmetric theories on manifolds with boundaries is also necessary in order to study interesting aspects which would be lost otherwise, such as the physics of boundary conditions, interfaces and bulk/boundary coupled systems. Once again, 3d N = 2 theories have provided a very useful laboratory so far [67][68][69][70], and the lift to 4d N = 1 theories provides another strong motivation for this paper (for a recent general analysis we refer to [71]). An interesting and localization-friendly approach has been recently put forward in [72] for 3d N = 2 theories and a class of dual boundary conditions preserving 2d N = (0, 2) supersymmetry on the boundary, including the familiar Dirichlet and Neumann conditions.…”

(0,4) brane box models

“…It is important to note that as 3d N = 2 supersymmetric theories with a superpotential W 3d (Φ) admits N = (0, 2) supersymmetric boundary conditions with W (Φ) being constant [22], the condition (2.22) can be relaxed so that E • J = W 3d (Φ) (2.23) if N = (0, 2) theories live on a boundary of 3d N = 2 theories [23,24]. This is a 3d analogue of the Warner problem [25] so that (2.23) exhibits holomorphic factorization of 3d bulk superpotential.…”

“…gram for adding N f D5-and N ′ f D5 ′ -branes to D1-branes on Z k × Z k ′ in e square boxes which are horizontally aligned represent the SU (N f ) flavor green square boxes which are vertically aligned represent the SU (N ′ f ) flavor ion of D0-D8 brane system Banks:1997zs, Bachas:1997kn [22,23].…”

“…We will see that the N = (0, 4) theories which arise from D1-D5-KK system discussed in Behrndt:1998nt, Kutasov:1998zh, Berenstein:1998rr, Sugawara:1999qp, Bena:2005ay [23,24,25,26,27] is a special case of our model. 4 To see this one can take the T-dual configuration of D0-D8 brane system Banks:1997zs, Bachas:1997kn [21,22].…”