On the convergence of Nekrasov functions. (English) Zbl 07826569
Summary: In this note, we present some results on the convergence of Nekrasov partition functions as power series in the instanton counting parameter. We focus on \(U(N) \; \mathcal{N}=2\) gauge theories in four dimensions with matter in the adjoint and in the fundamental representations of the gauge group, respectively, and find rigorous lower bounds for the convergence radius in the two cases: if the theory is conformal, then the series has at least a finite radius of convergence, while if it is asymptotically free it has infinite radius of convergence. Via AGT correspondence, this implies that the related irregular conformal blocks of \(W_N\) algebrae admit a power expansion in the modulus converging in the whole plane. By specifying to the \(SU(2)\) case, we apply our results to analyze the convergence properties of the corresponding Painlevé \(\tau\)-functions.
MSC:
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
| 81R05 | Finite-dimensional groups and algebras motivated by physics and their representations |
| 81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |
| 34M55 | Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies |
Keywords:
Nekrasov partition functions; instanton counting parameter; Painlevé \(\tau\)-functions; AGT correspondenceReferences:
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