q-Virasoro Modular Triple

Abstract

Inspired by 5d supersymmetric Yang-Mills theories placed on the compact space S 5 , we propose an intriguing algebraic construction for the q-Virasoro algebra. We show that, when multiple q-Virasoro "chiral" sectors have to be fused together, a natural SL(3, Z) structure arises. This construction, which we call the modular triple, is consistent with the observed triple factorization properties of supersymmetric partition functions derived from localization arguments. We also give a 2d CFT-like construction of the modular triple, and conjecture for the first time a (non-local) Lagrangian formulation for a q-Virasoro model, resembling ordinary Liouville theory.

Bootstrapping the S5 partition function

“…While our Higgs branch result formally follows from the residue calculation of the matrix integral, we do not have a rigorous proof of its equivalence with the Coulomb branch expression due to some assumption on the contributing poles. However, we support our proposal by showing that it has all the expected properties, both from the the gauge theory and 5d AGT perspectives: i) the partition function has poles at special values of the mass parameters due to the pinching mechanism of the integration contour, and taking the residues can be interpreted as RG flows to defect theories; our proposal indeed assumes the form of a discrete sum whose summands capture the partition functions of codimension 2 and 4 defects which are consistent with the Higgs phase from the brane picture; ii) our proposal looks like the compact space version of a similar result in the 5d Ω-background [58], which can be proven combinatorially; iii) the defect partition functions can manifestly be constructed through the q-Virasoro modular triple [59], in agreement with [35,36]; quite interestingly, it turns out that the free boson correlators form a natural basis for solutions to q-deformed Ward identities or q-conformal blocks, and a similar approach has been recently used to study W 3 conformal blocks of Toda 2d CFT [60].…”

“…K n (1) ,n (2) ,n (3) = 1) nor of the weight factor without explicitly referring to some gauge theory argument. However, from the algebraic viewpoint, besides the construction of the proposal being very simple and natural, we may observe that: i) thanks to the exponentiation {n (i) } Z n (1) ,n (2) ,n (3) ∼ exp( i S (i) ), when this operator is sandwiched between external Fock states with finite momentum (as necessary in order to have a pure defect partition function/Dotsenko-Fateev q-conformal block) the right number of screening charges is automatically picked up; also, the exponentiation hints to a worldsheet model similar to Liouville 2d CFT proposed in [59]; ii) the simple and specific weight factor may be needed once analytic, modular or crossing symmetry properties are imposed [35]; iii) the open questions may be solved by studying the more elementary Ding-Iohora-Miki algebra [81,82], the fundamental symmetry of matrix models arising in the present gauge theory context [83][84][85][86], which in our setup would give rise to the q-Virasoro algebra together with and additional Heisenberg algebra [87] which may play an important role here (similarly to the undeformed 4d case [88]). What we have discussed in this short note deserve further studies, also in view of a possible parallel story on S 4 which we leave for future work.…”

“…This construction can obviously be extended to include an arbitrary number of commuting copies of the algebra. However, when considering three copies with deformation parameters entering as in Table 1, a peculiar construction, the q-Virasoro modular triple, arises [59]. In this section, we show how this algebraic setting naturally leads to (2.8).…”

3D mirror symmetry fromS-duality

“…Finally, it is worth noting that the type of 3d/1d defects that we have considered in this paper appear in the Higgs branch localization approach to SQCD on S 5 [43], whose partition functions are identified with correlators in the q-Virasoro modular triple [80]. Therefore, another interesting route of investigation would be the study of type IIB SL(2, Z) symmetry from the viewpoint of the BPS/CFT and 5d AGT correspondences [81][82][83][84][85][86][87][88][89][90][91][92][93][94][95] and the DIM algebra [96,97], whose representation theory is known to govern the topological amplitudes associated to toric CY 3-folds or (p, q)-webs [98][99][100][101].…”

3d expansions of 5d instanton partition functions

“…Therefore, we expect the expansions discussed in this note will have natural extensions to compact spaces. The S 5 case is currently under investigation [49], and the relevant algebraic setting provided by the q-Virasoro modular triple has recently been constructed in [50] (see also [14] for earlier work in the context of 5d AGT).…”