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#### Elliptic hypergeometry of supersymmetric dualities II. Orthogonal groups, knots, and vortices

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1107.5788

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art%3A10.1007%2Fs00220-013-1861-4.pdf

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##### Citation

Spiridonov, V. P., & Vartanov, G. (2014). Elliptic hypergeometry of supersymmetric
dualities II. Orthogonal groups, knots, and vortices.* Communications in Mathematical Physics,*
*325*(2), 421-486. doi:10.1007/s00220-013-1861-4.

Cite as: https://hdl.handle.net/11858/00-001M-0000-0012-14C9-F

##### Abstract

We consider Seiberg electric-magnetic dualities for four-dimensional

$\mathcal{N}=1$ SYM theories with $SO({N})$ gauge group. For all such theories

we construct superconformal indices (SCIs) in terms of the elliptic

hypergeometric integrals. Equalities of these indices for dual theories lead

both to known special function identities and new highly nontrivial conjectural

relations for integrals. In particular, we describe a number of new elliptic

beta integrals associated with the $s$-confining theories with the spinor

matter. Reductions of some dualities from $SP(2{N})$ to $SO(2{N})$ or

$SO(2{N}+1)$ gauge groups are described. Interrelation of SCIs and the Witten

anomaly is briefly discussed. Possible applications of the elliptic

hypergeometric integrals to a two-parameter deformation of the two-dimensional

conformal field theory and related matrix models are indicated. Connections of

the reduced SCIs with the state integrals of the knot theory, generalized AGT

duality for $(3+3)$-dimensional theories, and the 2d vortex partition function

are described.

$\mathcal{N}=1$ SYM theories with $SO({N})$ gauge group. For all such theories

we construct superconformal indices (SCIs) in terms of the elliptic

hypergeometric integrals. Equalities of these indices for dual theories lead

both to known special function identities and new highly nontrivial conjectural

relations for integrals. In particular, we describe a number of new elliptic

beta integrals associated with the $s$-confining theories with the spinor

matter. Reductions of some dualities from $SP(2{N})$ to $SO(2{N})$ or

$SO(2{N}+1)$ gauge groups are described. Interrelation of SCIs and the Witten

anomaly is briefly discussed. Possible applications of the elliptic

hypergeometric integrals to a two-parameter deformation of the two-dimensional

conformal field theory and related matrix models are indicated. Connections of

the reduced SCIs with the state integrals of the knot theory, generalized AGT

duality for $(3+3)$-dimensional theories, and the 2d vortex partition function

are described.