Abstract
We investigate the relation between 3d mathcal{N} = 2 theories and 2d free field correlators or Dotsenko-Fateev (DF) integrals for Liouville CFT. We show that the S2× S1 partition functions of some known 3d Seiberg-like dualities reduce, in a suitable 2d limit, to known basic duality identities for DF correlators. These identities are applied in a variety of contexts in CFT, as for example in the derivation of the DOZZ 3-point function. Reversing the logic, we can try to guess new 3d IR dualities which reduce to more intricate duality relations for the DF correlators. For example, we show that a recently proposed duality relating the U(N) theory with one flavor and one adjoint to a WZ model can be regarded as the 3d ancestor of the evaluation formula for the DF integral representation of the 3-point correlator. We are also able to interpret the analytic continuation in the number of screening charges, which is performed on the CFT side to reconstruct the DOZZ 3-point function, as the geometric transition relating the 3d U(N) theory to the 5d T2 theory.
Highlights
We are able to interpret the analytic continuation in the number of screening charges, which is performed on the CFT side to reconstruct the DOZZ 3-point function, as the geometric transition relating the 3d U(N ) theory to the 5d T2 theory
One possibility consists of the Higgs limit, where the matter fields remain light while the Coulomb branch is lifted
The Higgs limit typically results in the partition function on D2 or S2 of a GLSM, where the contribution of a 2d chiral is written in terms of gamma functions
Summary
As we mentioned in the Introduction, the evaluation formula of the IN integral (1.8) can be uplifted to the genuine 3d duality recently proposed in [19], which can be considered as a non-abelian generalization of the duality between SQED with one flavor and the XYZ model [20]. Their charges under the global symmetries are listed in table 1 These operators directly map under the duality into the 3N singlets of the WZ theory. Notice that by looking at the argument of the double-sine functions we can read out the R-charges of the various fields in (2.6) and see that they are consistent with our parametrization in table 1. This integral identity already appeared in the mathematical literature in [28] and connected to this 3d duality in [29]
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