From 3d dualities to 2d free field correlators and back

Abstract

We investigate the relation between 3d N = 2 theories and 2d free field correlators or Dotsenko-Fateev (DF) integrals for Liouville CFT. We show that the S 2 × S 1 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 T 2 theory.

Supersymmetric indices of 3d S-fold SCFTs

“…There are also another class of theories that have been recently studied in [63,64]. It would be interesting to study whether such theories still have similar properties to quiver theories that contain T -link associated with T (U (N )).…”

3d dualities from 2d free field correlators: recombination and rank stabilization

“…Recently in [1] it was observed that the 3d N = 2 duality relating the U(N ) theory with one adjoint and one flavor to a Wess-Zumino (WZ) model with 3N chiral fields proposed in [2] enjoys an intriguing relation to the 3-point correlator in Liouville CFT in the free field realization. More precisely, the observation is that the integral identity encoding the equality of the S 2 ×S 1 partition functions of the U(N ) theory and of the WZ model, reduces in a suitable Coulomb limit, to a complex integral identity providing the evaluation formula for the Liouville 3-point correlator in the free field representation.…”

“…where Z int is the contribution of the classical action evaluated on the BPS vacua and of the quadratic fluctations. In [1] it was shown how, in the Coulomb limit, where we shrink the ratio β between the radius of the S 1 of the S 2 and suitably rescale the mass parameters, the sum over magnetic fluxes can be approximated by an integration and (1.1) reduces to a complex integral:…”

“…In [1] it has been shown that also the fundamental duality identity of [5] can be obtained by taking the Coulomb limit of an identity expressing the equality of the S 2 × S 1 partition functions of pair of dual theories related by a 3d Seiberg-like duality with monopole superpotential derived in [6]. The two specializations relevant for the recursion in particular correspond to the Aharony duality [7] for the U(N c ) SQCD theory with N f = N c flavors and to the one-monopole duality for the U(N c ) SQCD with N f = N c + 1 flavors and the negative fundamental monopole operator turned on in the superpotential [6].…”

Duality walls in the 4d $$ \mathcal{N} $$ = 2 SU(N) gauge theory with 2N flavours

“…Although we shall not provide an analytic proof 3 of the relation (2.6) in this paper, it can be checked perturbatively by expanding both sides as power series in x and matching each order of the power expansion. Moreover, as a further support of (2.6), one may take an appropriate 2d limit of the index of each side in (2.6) to obtain certain complex integrals [38], which are related to CFT free field correlators; the equality of such integrals was proposed in [39,40].…”