Junctions of surface operators and categorification of quantum groups

Abelian dualities of $$ \mathcal{N} $$ = (0, 4) boundary conditions

“…While the N = (1, 1) boundary conditions flow to the special Lagrangian branes, the N = (0, 2) boundary conditions flow to the holomorphic branes in the IR sigma-models [14]. The N = (0, 2) boundary conditions in 3d N = 2 gauge theories are also important in the study of 4d-2d relation [13,15,16,17,18,19,20], which is involved with the compactification of M5-branes on 4-manifold. The half-BPS boundary conditions in three-dimensional N = 4 supersymmetric gauge theories are studied in [21,22].…”

Four-dimensional quantum gravity with a cosmological constant from three-dimensional holomorphic blocks

“…In particular, the holomorphic blocks that play a key role in defining the present model have an M-theory interpretation, and closely relate to supersymmetric gauge theory, as well as surface operators with junctions [14][15][16][17][18]. While not holographic in any standard sense our model does exhibit a rich interplay between three-and four-dimensional topological and gravitational theories (for recent work on this interplay in lower dimensions, see e.g.…”

On Supersymmetric Interface Defects, Brane Parallel Transport, Order-Disorder Transition and Homological Mirror Symmetry

“…• Appearance of Young diagram crystals in the problem about categorified braid group action colored with m−1 colors is not surprising and accounts to the skew Howe duality (see e.g. [45,95]) where braid functors Φ i,i+1 are associated with a categorified action of raising/lowering operators in corresponding sl(m). A similarity of the braid action with the crystal melting problem suggests that the action of a BPS algebra on molten crystals of Calabi-Yau 3-folds (see e.g.…”

“…A concept simpler than an Abelian category C capturing its partial behavior is its Grothendieck group K 0 (C). A transition from C to K 0 (C) may be called a decategorification process, and the inverse process is usually referred to as a categorification process [45]. Physically, a relation between a category and its Grothendieck group is similar to a relation between the graded BPS Hilbert space and its supersymmetric index:…”

“…We will work with expansions in both s and t. The complexified gauge group acts on quantum degrees of freedom by corresponding algebraic shift actions. So that a quantum field δφ a→b associated with arrow a → b in the quiver is shifted by an algebraic element g as {g a } m−1 a=1 ∈ g : t δφ a→b → t δφ a→b + t g b • φ a→b − t φ a→b • g a , (F. 45) where φ a→b is a classical vacuum average for the field associated with arrow a → b. First we find independent deformation fields describing the tangent vector space to F-term constraint then use the gauge transformations to eliminate gauge-dependent degrees of freedom.…”