We relate Liouville/Toda CFT correlators on Riemann surfaces with boundaries and cross-cap states to supersymmetric observables in four-dimensional gauge theories. Our construction naturally involves four-dimensional theories with fields defined on different  quotients of the sphere (hemisphere and projective space) but nevertheless interacting with each other. The six-dimensional origin is a  quotient of the setup giving rise to the usual AGT correspondence. To test the correspondence, we work out the partition function of four-dimensional theories by combining a 3d lens space and a 4d hemisphere partition functions. The same technique reproduces known partition functions in a form that lets us easily check two-dimensional Seiberg-like dualities on this nonorientable space. As a bonus we work out boundary and cross-cap wavefunctions in Toda CFT.



1 Introduction

The AGT correspondence relates correlation functions in Liouville/Toda 2d CFT on a Riemann surface and 4d gauge theories on  AGT. The interplay between these two completely different setups has provided new ways to obtain and motivate new results on both sides, mainly for four dimensional gauge theories. From the two dimensional perspective this correspondence has been studied on Riemann surfaces with punctures and arbitrary genus. Nevertheless, a question originally posed in AGT has not been addressed in the literature:

What does CFT on surfaces with boundaries correspond to on the gauge theory side?

In 2d CFT, adding boundaries to the surface in which the theory lives has proven to be very fruitful towards understanding their structure. Important progress in this direction was initiated by Cardy in a seminal paper Cardy:1989ir. Given the importance of his construction in CFT one is led to wonder what it teaches us about four dimensional gauge theories. This is the motivation for this work, and we take a first step by answering the question raised above. A second motivation is to learn more about the 6d theory that gives rise to the AGT correspondence.

The 6d theory admits no supersymmetric boundary conditions, because of chirality. Instead, 2d boundaries (and cross-caps, as we will see) arise from a  quotient of the usual AGT setup.1 The 2d surface is a  quotient of a closed Riemann surface  and we consider the 6d theory on several quotients . Since the AGT correspondence relates (anti)holomorphic parts of 2d CFT correlators to (anti)instantons the two poles of the ellipsoid , and since the  reverses the orientation on  it must concurrently exchange the poles of .

Discrete quotients of the AGT setup considered in previous works fixed the poles of the ellipsoid, where instantons and anti-instantons are located. For example Belavin:2011pp; Belavin:2011tb; Belavin:2011sw; Bonelli:2012ny considered  subgroups of and rotations; instanton contributions matching Virasoro conformal blocks are then changed to those of a different chiral algebra. In contrast, our  exchanges poles of the ellipsoid, hence identifies the instanton and anti-instanton sums. Correspondingly, the  action identifies left-moving and right-moving modes of the CFT. An interesting extension would be to combine this  with our  quotients to learn about boundary states of the parafermionic CFTs corresponding to , including super Liouville for , and super Liouville obtained by adding a surface operator on the gauge theory side Belavin:2012uf.

Our main tool to probe the proposed correspondence is supersymmetric localization. The  action must thus leave the localizing supercharge invariant, and in particular its square. In terms of embedding coordinates the ellipsoid is defined by


with poles at , and the supercharge squares to rotations in the and planes. We thus restrict our attention to the reflection around the equator of and to the antipodal map:


The quotient is the (squashed) hemisphere and projective space , respectively.

In LABEL:fig:6dorbifold we show a simple example of our setup. We begin with a Riemann surface  composed of a torus with a single circular boundary. In the same figure we show the Schottky double  which has no boundary but has genus . In the right panel we show how this construction naturally lifts to 6d, with the involution acting simultaneously on and .

The 4d theory corresponding to the surface with boundaries is then obtained by a  identification of fields in the 4d theory


  1. A toy example to keep in mind is that the quotient gives rise to a half-line upon reduction on . This is true for both actions and on , analogous to the two actions in (1.2).
Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
Add comment
Loading ...
This is a comment super asjknd jkasnjk adsnkj
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test description