Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

3d & 5d gauge theory partition functions as q-deformed CFT correlators

Fabrizio Nieri, Sara Pasquetti, Filippo Passerini

TL;DR

The paper develops a unified framework linking 3d and 5d ${\cal N}=2$/$\mathcal N=1$ gauge theory partition functions to $q$-deformed CFT correlators controlled by ${\cal V}ir_{q,t}$. It shows that 3d partition functions factorize into holomorphic blocks and that flop symmetry is realized as crossing symmetry in suitably paired $q$-CFT correlators, with degenerate insertions mapping to 3d defect partition functions. Through a bootstrap construction for both ${S}$-pairing and ${id}$-pairing, it derives explicit three-point functions and demonstrates that degenerate correlators reproduce 3d $Z_S$ and $Z_{\rm id}$, while non-degenerate correlators map to 5d partition functions on $S^4\times S^1$ and $S^5$. This extends the AGT paradigm to higher dimensions, providing a robust dictionary between holomorphic blocks, defect theories, and Nekrasov functions, and suggesting a deep CFT underpinning for higher-dimensional gauge theory partition functions with potential computational leverage for exact results.

Abstract

3d N=2 partition functions on the squashed three-sphere and on the twisted product S2xS1 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 squashed three-sphere and S2xS1 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 conformal 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 CFT 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 to the superconformal index on S4xS1 while S-pairing three-point function factors capture the one-loop part of S5 partition functions. This is consistent with the interpretation of S2xS1 and squashed three-sphere gauge theories as codimension two defect theories inside S4xS1 and S5 respectively.

3d & 5d gauge theory partition functions as q-deformed CFT correlators

TL;DR

The paper develops a unified framework linking 3d and 5d / gauge theory partition functions to -deformed CFT correlators controlled by . It shows that 3d partition functions factorize into holomorphic blocks and that flop symmetry is realized as crossing symmetry in suitably paired -CFT correlators, with degenerate insertions mapping to 3d defect partition functions. Through a bootstrap construction for both -pairing and -pairing, it derives explicit three-point functions and demonstrates that degenerate correlators reproduce 3d and , while non-degenerate correlators map to 5d partition functions on and . This extends the AGT paradigm to higher dimensions, providing a robust dictionary between holomorphic blocks, defect theories, and Nekrasov functions, and suggesting a deep CFT underpinning for higher-dimensional gauge theory partition functions with potential computational leverage for exact results.

Abstract

3d N=2 partition functions on the squashed three-sphere and on the twisted product S2xS1 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 squashed three-sphere and S2xS1 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 conformal 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 CFT 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 to the superconformal index on S4xS1 while S-pairing three-point function factors capture the one-loop part of S5 partition functions. This is consistent with the interpretation of S2xS1 and squashed three-sphere gauge theories as codimension two defect theories inside S4xS1 and S5 respectively.

Paper Structure

This paper contains 19 sections, 154 equations, 2 figures.

Table of Contents

  1. Introduction
  2. 3d partition functions
  3. The superconformal index factorization
  4. Ellipsoid partition function factorization
  5. Flop invariance
  6. $q-$deformed CFT correlation functions
  7. The bootstrap approach to Liouville theory
  8. $q-$deforming the bootstrap
  9. $id$-pairing three-point functions
  10. $S$-pairing three-point functions
  11. 3d partition functions as $q$-deformed CFT correlators
  12. 5d partition functions as $q$-deformed CFT correlators
  13. Partition function on $S^4\times S^1$ and $q$-deformed CFT
  14. Partition function on squashed $S^5$ and $q$-deformed CFT
  15. Index factorization
  16. ...and 4 more sections

Figures (2)

  • Figure 1: The $s$-channel decomposition of the correlator given in formula (\ref{['scorre']}). The dashed line is associated to the degenerate state $\alpha_2=-\frac{b_0}{2}$.
  • Figure 2: The $u$-channel decomposition of the correlator given in formula (\ref{['ucorre']}). The dashed line is associated to the degenerate state $\alpha_2=-\frac{b_0}{2}$.