[Paper Review] Coset conformal blocks and N=2 gauge theories

Niclas Wyllard|arXiv (Cornell University)|Sep 20, 2011

Black Holes and Theoretical Physics56 references35 citations

TL;DR

This paper proposes a correspondence between $υ\!=\!2$ $\mathrm{SU}(N)$ gauge theories on $\mathbb{R}^4/\mathbb{Z}_p$ and coset conformal field theories of the form $\widehat{\mathrm{su}}(N)_\kappa \oplus \widehat{\mathrm{su}}(N)_p / \widehat{\mathrm{su}}(N)_{\kappa+p}$, providing explicit checks for $(N,p) = (2,4)$ using the $S_3$ parafermion algebra and generalizing via the Kac determinant to arbitrary $(N,p)$. It finds that conformal blocks factorize into $p$ copies of $(N,1)$ blocks, matching the instanton partition function structure on the gauge theory side.

ABSTRACT

It was recently suggested that the su(N)_k+su(N)_p/su(N)_{k+p} coset conformal field theories should be related to N=2 SU(N) gauge theories on R^4/Z_p. In this paper we study various aspects of this proposal. We perform explicit checks of the relation for (N,p)=(2,4), where the symmetry algebra of the coset is the so called S_3 parafermion algebra. Even though the symmetry algebra of the coset is unknown for generic (N,p) models, we manage to perform non-trivial checks in the general case by using knowledge of the Kac determinant of the coset CFT. We also find evidence that the conformal blocks of the (N,p) model should factorise into a certain product of p (N,1) conformal blocks. Precisely this structure is present in the instanton partition function on R^4/Z_p.

Motivation & Objective

• To establish a precise correspondence between $\mathcal{N}=2$ $\mathrm{SU}(N)$ gauge theories on $\mathbb{R}^4/\mathbb{Z}_p$ and the coset CFT $\widehat{\mathrm{su}}(N)_\kappa \oplus \widehat{\mathrm{su}}(N)_p / \widehat{\mathrm{su}}(N)_{\kappa+p}$.
• To provide non-trivial evidence for this duality in cases where the symmetry algebra of the coset is unknown, using indirect methods such as the Kac determinant.
• To investigate the factorization structure of conformal blocks in the $(N,p)$ coset model and relate it to the instanton partition function on $\mathbb{R}^4/\mathbb{Z}_p$.
• To extend the AGT-like correspondence beyond the $p=1$ (Toda) and $p=2$ (super-Liouville) cases to general $p>1$ and arbitrary $N$.

Proposed method

• Explicit computation of the central charge of the coset CFT using the standard formula $c_{\text{coset}} = c_{\widehat{\mathrm{su}}(N)_\kappa} + c_{\widehat{\mathrm{su}}(N)_p} - c_{\widehat{\mathrm{su}}(N)_{\kappa+p}}$ and matching it with the M-theory anomaly polynomial.
• Use of the Kac determinant of the coset CFT as a probe for the duality, enabling checks even when the full symmetry algebra is unknown.
• Study of irregular conformal blocks in the $(N,p)=(2,4)$ case, where the coset realizes the $S_3$ parafermion algebra, and comparison with the instanton partition function on $\mathbb{R}^4/\mathbb{Z}_4$.
• Analysis of the factorization of conformal blocks into $p$ copies of $(N,1)$ blocks, motivated by the structure of the gauge theory instanton partition function.
• Use of the relation $\kappa + N = -p \frac{\epsilon_2}{\epsilon_1 + \epsilon_2}$ between CFT and gauge theory parameters to connect the two sides.
• Generalization of the duality to more general toric singularities $\mathbb{R}^4/\Gamma_{p,q}$ via central charge computations from the $6d$ $(2,0)$ theory anomaly polynomial.

Experimental results

Research questions

• RQ1 Does the coset CFT $\widehat{\mathrm{su}}(N)_\kappa \oplus \widehat{\mathrm{su}}(N)_p / \widehat{\mathrm{su}}(N)_{\kappa+p}$ correctly describe the $\mathcal{N}=2$ $\mathrm{SU}(N)$ gauge theory on $\mathbb{R}^4/\mathbb{Z}_p$?
• RQ2 Can the duality be verified in cases where the symmetry algebra of the coset is not known, such as for generic $(N,p)$?
• RQ3 Does the conformal block structure of the coset CFT factorize into $p$ copies of $(N,1)$ blocks, as expected from the gauge theory side?
• RQ4 What is the role of the $p-1$ additional variables $x_\ell$ in the gauge theory instanton partition function, and how are they realized in the CFT?
• RQ5 Can the Kac determinant method serve as a general tool to test AGT-type dualities beyond known symmetry algebras?

Key findings

• For $(N,p) = (2,4)$, the coset CFT has the $S_3$ parafermion algebra as its symmetry algebra, and irregular conformal blocks in this theory match the instanton partition function of the $\mathcal{N}=2$ $\mathrm{SU}(2)$ gauge theory on $\mathbb{R}^4/\mathbb{Z}_4$.
• The central charge of the coset CFT matches the M-theory prediction derived from the anomaly polynomial of the $6d$ $(2,0)$ theory, confirming the duality at the level of global charges.
• The conformal blocks of the general $(N,p)$ coset model are found to factorize into a product of $p$ $(N,1)$ conformal blocks, mirroring the structure of the gauge theory instanton partition function.
• The Kac determinant method provides a viable, indirect way to test the duality even when the full symmetry algebra of the coset is unknown, offering a generalizable technique for future checks.
• The duality extends to more general toric singularities $\mathbb{R}^4/\Gamma_{p,q}$, with central charges computed from the $6d$ anomaly polynomial matching the expected form.
• The presence of $p-1$ additional variables $x_\ell$ in the gauge theory partition function remains unexplained on the CFT side, suggesting a more general duality framework may be needed.