Junctions of surface operators and
                    categorification of quantum groups

Chun, Sungbong; Gukov, Sergei; Roggenkamp, Daniel

doi:10.1090/conm/684/13749

Cited by 20 publications

(39 citation statements)

References 142 publications

(247 reference statements)

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“…The resulting quotient of the (perhaps) more familiar quantum group U q (sl 2 ) is, in fact, finite-dimensional, namely 2p 3 -dimensional. 7 It has 2p irreducible representa-6 See [29] for a friedly introduction and a physical realization of the Lusztig quantum groups in the setup of [4,6] that leads to Za(M3). In a two-dimensional description of this setup, E and F generators of the quantum group correspond to half-BPS interfaces of a 2d N = (2, 2) CFT (Kazama-Suzuki model), so that the quantum group emerges as an algebra of interfaces.…”

Section: -Manifoldsmentioning

confidence: 99%

3d modularity

Cheng

Chun

Ferrari

et al. 2019

J. High Energ. Phys.

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159

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Section: -Manifoldsmentioning

confidence: 99%

3d modularity

Cheng

Chun

Ferrari

et al. 2019

J. High Energ. Phys.

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159

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“…[21,96,97,134,135], cf. the basic idea in [136], all ultimately tracing back to the physics of M2-M5 and related M5-M5 brane systems from [94,95].)…”

Section: Jhep10(2016)108mentioning

confidence: 99%

Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 $$ \mathcal{N}=4 $$ gauge theory

Bullimore

Dimofte

Gaiotto

et al. 2016

J. High Energ. Phys.

106

229

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Abstract:We introduce several families of N = (2, 2) UV boundary conditions in 3d N = 4 gauge theories and study their IR images in sigma-models to the Higgs and Coulomb branches. In the presence of Omega deformations, a UV boundary condition defines a pair of modules for quantized algebras of chiral Higgs-and Coulomb-branch operators, respectively, whose structure we derive. In the case of abelian theories, we use the formalism of hyperplane arrangements to make our constructions very explicit, and construct a half-BPS interface that implements the action of 3d mirror symmetry on gauge theories and boundary conditions. Finally, by studying two-dimensional compactifications of 3d N = 4 gauge theories and their boundary conditions, we propose a physical origin for symplectic duality -an equivalence of categories of modules associated to families of Higgs and Coulomb branches that has recently appeared in the mathematics literature, and generalizes classic results on Koszul duality in geometric representation theory. We make several predictions about the structure of symplectic duality, and identify Koszul duality as a special case of wall crossing.

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“…The first major piece of the desired structure came with the construction of Khovanov-Rozansky homology [5][6][7] that belongs to the lower left corner in Figure 1. This corner is by far the most developed element of the sought after 2d-4d TQFT on D ⊂ M 4 , and even that only for M 4 = R 4 and D = R × K. Its physical interpretation, proposed in [8], led to many new predictions and connections between various areas, which include knot contact homology [9], gauge theory [10,11], and algebras of interfaces [12,13], just to name a few. (A more complete account of these connections can be found, e.g., in [13,14].…”

Section: Numbersmentioning

confidence: 99%

“…Since the R-symmetry U (1) R of the 4d N = 2 theory on M 4 is precisely the Rsymmetry (under the same name) of 2d N = (0, 2) theory T [M 4 ] on Σ, we can determine U (1) R anomaly of the latter by integrating (3.47) over M 4 , with G = U (1) and E = Λ 2,+ (M 4 ). In particular, if we choose Σ such that 4d N = 2 theory is a theory of a single 13 In terms of the conformal anomaly coefficients, c = 1 6 nv + 1 12 n h and a = 5 24 nv + 1 24 n h , the same expression reads:…”

Section: Anomalies: Intersecting M5-branes and Basic Classesmentioning

confidence: 99%

Vertex Algebras and 4-Manifold Invariants

Dedushenko¹,

Gukov²,

Putrov³

2018

Geometry and Physics: Volume I

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112

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We propose a way of computing 4-manifold invariants, old and new, as chiral correlation functions in half-twisted 2d N = (0, 2) theories that arise from compactification of fivebranes. Such formulation gives a new interpretation of some known statements about Seiberg-Witten invariants, such as the basic class condition, and gives a prediction for structural properties of the multi-monopole invariants and their non-abelian generalizations. 5 Non-abelian generalizations 51 5.1 Coulomb-branch index and new 4-manifold invariants 54 A Curvature of the canonical connection on the universal bundle 57 A.1 SU (N f ) invariance and the moment map 59 B Topological twist of SQED 60 1.1 Unorthodox invariants of smooth 4-manifolds Searching for new invariants of smooth 4-manifolds, that potentially could go beyond the Seiberg-Witten and Donaldson invariants, it was proposed in [1] to consider a twodimensional quantum field theory T [M 4 , G] as a rather unusual invariant of smooth structures on a 4-manifold M 4 . Specifically, T [M 4, G] is a 2d N = (0, 2) superconformal theory that, apart from M 4 , also depends on a choice of a root system G and is invariant under the Kirby moves. Luckily, conformal field theories in two dimensions exhibit rich mathematical structure which, on the one hand, is rich enough to (potentially) describe the wild world of smooth 4-manifolds and, on the other hand, is rigorous enough to hope for a precise mathematical definition of the invariant T [M 4 , G]. In fact, for many practical purposes and applications in this paper, a mathematically inclined reader can think of T [M 4 , G] as a functor that assigns a vertex operator algebra (VOA) to a smooth 4-manifold M 4 (and a "gauge" group G).Composing it with other functors that assign various quantities to 2d conformal theories, one can obtain more conventional invariants of smooth 4-manifolds:Here, Z can be any invariant of a 2d conformal theory with N = (0, 2) supersymmetry, e.g., its elliptic genus, chiral ring, moduli space of marginal couplings, or central charge. Since 2d theory T [M 4 ; G] is systematically determined by M 4 and invariant under the Kirby moves, all such invariants lead to various 4-manifold invariants; some are simple and some are quite powerful. In particular, it was conjectured in [1] that chiral ring of the theory T [M 4; G] for G = SU (2) or, equivalently, its Q + -cohomology knows about Donaldson invariants of M 4 . One of the main goals in this paper is to present some evidence to this conjecture and to build a bridge between VOA G [M 4 ] and more traditional 4-manifold invariants. Conjecturally, at least for manifolds with b + 2 > 1, one can trade SU (2) Donaldson invariants for Seiberg-Witten invariants defined in a simpler gauge theory, with gauge group G = U (1), which would be too trivial if not for an extra ingredient, the additional spinor fields. We wish to study how these invariants are realized in 2d theory T [M 4 ; G] with G = SU (2) and G = U (1), respectively. In particular, we shall see that in 2d realizati...

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Junctions of surface operators and categorification of quantum groups

Cited by 20 publications

References 142 publications

3d modularity

3d modularity

Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 $$ \mathcal{N}=4 $$ gauge theory

Vertex Algebras and 4-Manifold Invariants

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