3d and 5d Gauge Theory Partition Functions as q-deformed CFT Correlators

Abstract

© 2014, Springer Science+Business Media Dordrecht. 3d (Formula Presented.) partition functions on the squashed three-sphere (Formula Presented.) and on the twisted product (Formula Presented.) have been shown to factorize into sums of squares of solid tori partition functions, the so-called holomorphic blocks. The same set of holomorphic blocks realizes (Formula Presented.) and (Formula Presented.) partition functions but the two cases involve different inner products, the S-pairing and the id-pairing, respectively. We define a class of q-deformed CFT correlators where chiral blocks are controlled by a deformation of Virasoro symmetry and are paired by S-pairing and id-pairing, respectively. Applying the bootstrap approach to a class of degenerate correlators we are able to derive three-point functions. We show that degenerate correlators can be mapped to 3d partition functions while the crossing symmetry of correlators corresponds to the flop symmetry of 3d gauge theories. We explore how non-degenerate q-deformed correlators are related to 5d partition functions. We argue that id-pairing correlators are associated with the superconformal index on (Formula Presented.) while S-pairing three-point function factors capture the one-loop part of S5 partition functions. This is consistent with the interpretation of (Formula Presented.) and (Formula Presented.) gauge theories as codimension two defect theories inside (Formula Presented.) and S5, respectively

q-Virasoro Modular Double and 3d Partition Functions

“…leading to a dual 2d CFT-like description in the spirit of the AGT correspondence [49][50][51] and similar to the proposal of [52,53] for the SQED. The origin of the q-deformation has been proposed to lie in the little string deformation of the 6d (2, 0) theory [54].…”

“…As recently stressed in [84], these type of deformed Ward identities may be regarded as a further step towards a proper definition of q-CFT theories, a deformation of ordinary 2d CFTs. While most of the studies have so far focused on the chiral description, our work shows that it is possible to consistently couple different chiral sectors into well-defined modular invariant objects [52,53]. This is familiar in 2d CFTs, where the invariance w.r.t the Moore-Seiberg groupoid puts severe constraints on the physical theories [126].…”

“…While in the chiral case (i.e., 5d theories on R 4 × S 1 ) the answer is clearly affirmative according to the results of [54,68,73,96], in the non-chiral case (i.e., 5d theories on compact spaces) the answer is not straightforward (the rank 1 case was studied in [52,53]). First of all, the free boson realization of W q,t explicitly breaks the q, t symmetry of the 5d theory, which is however restored in the large N limit (i.e., sending the number of screening currents to infinity or going to the affine case).…”

Localisation on Sasaki-Einstein manifolds from holomorphic functions on the cone

“…Subsequently, the perturbative partition functions of Y p,q and L a,b,c were calculated in [23,24]. The recent work [24] conjectures the full partition functions using factorization [30][31][32].…”

Toric Calabi-Yau threefolds as quantum integrable systems. ℛ $$ \mathrm{\mathcal{R}} $$ -matrix and ℛ T T $$ \mathrm{\mathcal{R}}\mathcal{T}\mathcal{T} $$ relations

“…The spectral dual interpretation of the resulting amplitude is the partition function of a 6d linear quiver gauge theory compactified on a two-dimensional torus. The AGT relations in this case [121][122][123][124][125][126][127][128][129][130][131][132][133][134] give the conformal block of the q-deformed W -algebras on torus, or the spherical conformal block of the affine W -algebra [98,99].…”