Ω-deformation and quantization

Abstract

We formulate a deformation of Rozansky-Witten theory analogous to the Ω-deformation. It is applicable when the target space X is hyperkähler and the spacetime is of the form R×Σ, with Σ being a Riemann surface. In the case that Σ is a disk, the Ω-deformed Rozansky-Witten theory quantizes a symplectic submanifold of X, thereby providing a new perspective on quantization. As applications, we elucidate two phenomena in fourdimensional gauge theory from this point of view. One is a correspondence between the Ω-deformation and quantization of integrable systems. The other concerns supersymmetric loop operators and quantization of the algebra of holomorphic functions on a hyperkähler manifold.

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

“…3 It is thus generally expected that symplectic duality should encode mathematical aspects of three-dimensional mirror symmetry, which exchanges the Higgs and Coulomb branches of N = 4 SCFT's. 4 The most rudimentary aspects of symplectic duality can readily be given a direct physical interpretation. Consider a gauge theory that satisfies the two properties above.…”

“…This is a mirror of the standard Ω-deformation. The Ω-deformation is known to localize a non-linear sigma model with hyperkähler target space M to a supersymmetric quantum mechanics whose operator algebraĈ[M] quantizes the Poisson algebra C[M] of holomorphic functions on M [4]. We similarly expect the Ω-deformation to localize a gauge theory to a gauged supersymmetric quantum mechanics, in which a quantization of the chiral ringĈ[M H ] appears as the gauge-invariant part of the operator algebra associated to a quantization of the matter fields [5].…”

“…There are two bulk vacua, or fixed points of the rotation: the North pole of CP 1 , where µ H,R = t R 2 and σ = − m R 2 ; and the South pole, where µ H,R = − t R 2 and σ = m R 2 . Gradient flows for the real moment map h m = m R µ H,R preserve the slice X i Y i = 0 depicted in figure (4). Depending on the sign of m R , one may have either gradient flows from the North to the South pole, or vice versa, corresponding to the existence of a single dynamical domain wall between the two vacua.…”

“…The holomorphic functions on the Higgs and There are two interesting deformations of 3d N = 4 theories that turn the chiral rings into non-commutative algebras: standard and twisted Omega-backgrounds. In the 3d N = 4 context, this was studied in [4,5] (see below for other connections). The Omega backgrounds mix supersymmetry transformations with rotations of some R 2 ⊂ R 3 , and effectively reduce the 3d theory to one-dimensional supersymmetric quantum mechanics supported on the fixed axis of rotations.…”

Non-Abelian mirror symmetry beyond the chiral ring

“…Along the way, we extract various operator product expansion (OPE) coefficients for the quantized Higgs and Coulomb branches. We conclude by offering some perspectives on how the topological subsectors of the E-type quivers might shed light on their non-Lagrangian duals.2 A variation of the Ω-background [25][26][27] leads to related deformation quantizations of the Higgs and Coulomb branches [28][29][30]. It would be interesting to spell out the explicit relation to the TQM sector.3 Specifically, the abelian A-type mirror symmetries were analyzed in [32], and a simple N = 8 nonabelian A-type mirror symmetry was analyzed in [33].…”