<p>We study BPS surface observables of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {N}=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">N</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> four dimensional <i>SU</i>(2) gauge theory in gravitational <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>-background at perturbative and at Argyres–Douglas superconformal fixed points. This is done by formulating the equivariant gauge theory on the blow-up of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\mathbb {C}}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> and considering the decoupling Nekrasov–Shatashvili limit. We show that in this limit the blow-up equations are solved by corresponding Painlevé <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {T}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">T</mi> </math></EquationSource> </InlineEquation>-functions and exploit operator/state correspondence to compute their expansion in an integer basis, given in terms of the moduli of the quantum Seiberg–Witten curve. We study the modular properties of these solutions and show that they do directly lead to BCOV holomorphic anomaly equations for the corresponding topological string partition function. The resulting <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {T}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">T</mi> </math></EquationSource> </InlineEquation>-functions are holomorphic and modular and as such they provide a natural non-perturbative completion of topological strings partition functions.</p>

Surface Observables in Gauge Theories, Modular Painlevé Tau Functions and Non-perturbative Topological Strings

期刊论文
发表日期：

2026年4月4日

查看全文

Giulio Bonelli,
Pavlo Gavrylenko,
Ideal Majtara,
Alessandro Tanzini

摘要

We study BPS surface observables of $\mathcal {N}=2$ $N = 2$ four dimensional SU(2) gauge theory in gravitational $\Omega $ $Ω$ -background at perturbative and at Argyres–Douglas superconformal fixed points. This is done by formulating the equivariant gauge theory on the blow-up of ${\mathbb {C}}^2$ $C^{2}$ and considering the decoupling Nekrasov–Shatashvili limit. We show that in this limit the blow-up equations are solved by corresponding Painlevé $\mathcal {T}$ $T$ -functions and exploit operator/state correspondence to compute their expansion in an integer basis, given in terms of the moduli of the quantum Seiberg–Witten curve. We study the modular properties of these solutions and show that they do directly lead to BCOV holomorphic anomaly equations for the corresponding topological string partition function. The resulting $\mathcal {T}$ $T$ -functions are holomorphic and modular and as such they provide a natural non-perturbative completion of topological strings partition functions.