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.
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.
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
- 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).