Abstract
We study N = 2 supersymmetric gauge theories with gauge group SU(2) coupled to fundamental flavours, covering all asymptotically free and conformal cases. We re-derive, from the conformal field theory perspective, the differential equations satisfied by omega deformed instanton partition functions. We confirm their validity at leading order in one of the deformation parameters via a saddle-point analysis of the partition function. In the semi-classical limit we show that these differential equations take a form amenable to exact WKB analysis. We compute the monodromy group associated to the differential equations in terms of deformed and Borel resummed Seiberg-Witten data. For each case, we study pairs of Stokes graphs that are related by flips and pops, and show that the monodromy groups allow one to confirm the Stokes automorphisms that arise as the phase of the deformation parameter is varied. Finally, we relate the Borel resummed monodromies with the traditional Seiberg-Witten variables in the semi-classical limit.
Highlights
For some time we have been able to compute the low-energy effective action of N = 2 supersymmetric gauge theories in four dimensions
We study pairs of Stokes graphs that are related by flips and pops, and show that the monodromy groups allow one to confirm the Stokes automorphisms that arise as the phase of 1 is varied
Using the exact WKB analysis, we study the resulting differential equations satisfied by theregular conformal blocks
Summary
We study the differential equation that the instanton partition function with surface operator insertion satisfies This corresponds to an analysis of null vector decoupling equations in the presence of irregular blocks. Using the exact WKB analysis, we study the resulting differential equations satisfied by the (ir)regular conformal blocks (equivalently, the 1-deformed surface operator partition function). This allows us to compute the monodromy group of each of the differential equations as a function of (i) the parameters of the differential equations, and (ii) the Borel resummed monodromies that are properties of individual solutions. Our broader goal is to communicate the extreme generality of the correspondence between 1-deformed N = 2 gauge theories — their instanton partition functions with surface operator insertions — and certain Schrodinger equations amenable to exact WKB analysis. Check of the semi-classical differential equations via the saddle-point analyses of Nekrasov partition functions [35]
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