From SO/Sp instantons to W-algebra blocks

Abstract

We study instanton partition functions for N=2 superconformal Sp(1) and SO(4) gauge theories. We find that they agree with the corresponding U(2) instanton partitions functions only after a non-trivial mapping of the microscopic gauge couplings, since the instanton counting involves different renormalization schemes. Geometrically, this mapping relates the Gaiotto curves of the different realizations as double coverings. We then formulate an AGT-type correspondence between Sp(1)/SO(4) instanton partition functions and chiral blocks with an underlying W(2,2)-algebra symmetry. This form of the correspondence eliminates the need to divide out extra U(1) factors. Finally, to check this correspondence for linear quivers, we compute expressions for the Sp(1)-SO(4) half-bifundamental.

Figures (29)

• Figure 1: The Gaiotto curve of the $U(2)$ gauge theory coupled to 4 hypers as a double cover over the Gaiotto curve of the $Sp(1)$ theory. The dotted line is a branch cut.
• Figure 2: The period matrix $\tau_{\rm IR,ij}$ of the Seiberg-Witten curve is equal to the second derivative $\partial_{a_i} \partial_{a_j} {\cal F}_0$ of the prepotential with respect to the Coulomb parameter $a$. The imaginary part of $\tau_{\rm IR}$ determines the metric on the Coulomb branch.
• Figure 3: The marginal coupling $\tau_{\rm UV}$ in the Nekrasov partition function defines a local coordinate on the moduli space of the ${\cal N}=2$ conformal gauge theory near a weak-coupling point where $\tau_{\rm UV} \to \infty$.
• Figure 4: On the left: Quiver of the $Sp(1)$ gauge theory coupled to two fundamental and two anti-fundamental hypermultiplet. Since the (anti-)fundamental representation of $Sp(1)$ is pseudo-real, the flavor symmetry group of two hypermultiplets enhances to $SO(4)$. On the right: Quiver of the $SU(2)$ gauge theory coupled to two fundamental and two anti-fundamental hypermultiplets. The flavor symmetries of the hypermultiplets is enhanced to $SU(2)$.
• Figure 5: On the left: Quiver representation of the $SO(4)$ gauge theory coupled to one fundamental and one anti-fundamental hypermultiplet. Since the (anti-)fundamental representation of $SO(4)$ is real, the flavor symmetry group of each hypermultiplet enhances to $Sp(1)$. On the right: Quiver representation of the $SU(2)\times SU(2)$ gauge theory coupled to two bi-fundamental hypermultiplets. The flavor symmetry of the bifundamental field is enhanced to $SU(2)$.