N = 2 $$ \mathcal{N}=2 $$ supersymmetric gauge theories on S2 × S2 and Liouville Gravity

159

Abstract:We determine the spectrum and correlation functions of Liouville theory with a central charge less than (or equal) one. This completes the definition of Liouville theory for all complex values of the central charge. The spectrum is always spacelike, and there is no consistent timelike Liouville theory. We also study the non-analytic conformal field theories that exist at rational values of the central charge. Our claims are supported by numerical checks of crossing symmetry. We provide Python code for computing Virasoro conformal blocks, and correlation functions in Liouville theory and (generalized) minimal models.

Section: Resultsmentioning

confidence: 82%

Liouville theory with a central charge less than one

Ribault

Santachiara

2015

159

“…As discussed in [22] (see also [23,63]), four-dimensional N = 2 supersymmetric theories can be placed on compact Euclidean manifolds with metric g µν if one can find symplectic-Majorana spinors ξ I andξ I solving the generalized Killing spinor equations…”

Section: A2 σ-Matrices and Fierz Identitiesmentioning

confidence: 99%

Ellipsoid partition function from Seiberg-Witten monopoles

Pan

Peelaers

2015

Abstract:We study Higgs branch localization of N = 2 supersymmetric theories placed on compact Euclidean manifolds. We analyze the resulting localization equations in detail on the four-sphere and find that in this case the path integral is dominated by vortex-like configurations as well as singular Seiberg-Witten monopoles located at the north and south pole. The partition function is written accordingly.

“…In this paper, we would like to derive a 4d-2d duality that has not yet been listed in [3,Introduction]. To this end, note that in [4], a supersymmetric theory was constructed on S 2 × S 2 , and its partition function was computed using the technique of localization. The authors observed that the building blocks of the partition function contained three-point functions and conformal blocks of Liouville gravity [5,6], and so they conjectured a novel AGT-like correspondence between the 4d supersymmetric theory on S 2 × S 2 and 2d Liouville gravity.…”

Section: Jhep05(2017)121mentioning

confidence: 99%

“…4 This amounts to inserting concrete values for the supergravity fields G and α, which emerged from reduction of the frame field in six dimensions. Then we solve this equation.…”

Section: Jhep05(2017)121mentioning

confidence: 99%

See 1 more Smart Citation

Four-dimensional N $$ \mathcal{N} $$ = 2 supersymmetric theory with boundary as a two-dimensional complex Toda theory

Luo

Tan

Vasko

et al. 2017

We perform a series of dimensional reductions of the 6d, N = (2, 0) SCFT on S 2 × Σ × I × S 1 down to 2d on Σ. The reductions are performed in three steps: (i) a reduction on S 1 (accompanied by a topological twist along Σ) leading to a supersymmetric Yang-Mills theory on S 2 × Σ × I, (ii) a further reduction on S 2 resulting in a complex Chern-Simons theory defined on Σ × I, with the real part of the complex Chern-Simons level being zero, and the imaginary part being proportional to the ratio of the radii of S 2 and S 1 , and (iii) a final reduction to the boundary modes of complex Chern-Simons theory with the Nahm pole boundary condition at both ends of the interval I, which gives rise to a complex Toda CFT on the Riemann surface Σ. As the reduction of the 6d theory on Σ would give rise to an N = 2 supersymmetric theory on S 2 × I × S 1 , our results imply a 4d-2d duality between four-dimensional N = 2 supersymmetric theory with boundary and two-dimensional complex Toda theory.