Magnetic expansion of Nekrasov theory: the SU(2) pure gauge theory

Wei He; Yan-Gang Miao

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Magnetic expansion of Nekrasov theory: the SU(2) pure gauge theory

Wei He, Yan-Gang Miao

High Energy Physics - Theory · Submitted 7 Jun 2010

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We obtain the magnetic and dyonic expansions of the Nekrasov theory by studying the property of hypergeometric functions in the magnetic and dyonic regions on the moduli space.

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It is recently claimed by Nekrasov and Shatashvili that the $\mathcal {N}=2$ gauge theories in the $\Omega$ background with $\epsilon_1=\hbar, \epsilon_2=0$ are related to the quantization of certain algebraic integrable systems. We study the special case of SU(2) pure gauge theory, the corresponding integrable model is the A$_1$ Toda model, which reduces to the sine-Gordon quantum mechanics problem. The quantum effects can be expressed as the WKB series written analytically in terms of hypergeometric functions. We obtain the magnetic and dyonic expansions of the Nekrasov theory by studying the property of hypergeometric functions in the magnetic and dyonic regions on the moduli space. We also discuss the relation between the electric-magnetic duality of gauge theory and the action-action duality of the integrable system.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	It is recently claimed by Nekrasov and Shatashvili that the $\mathcal {N}=2$ gauge theories in the $\Omega$ background with $\epsilon_1=\hbar, \epsilon_2=0$ are related to the quantization of certain algebraic integrable systems.	background 1.00	0.03	0.08	asserted 0.88
s1	We study the special case of SU(2) pure gauge theory, the corresponding integrable model is the A$_1$ Toda model, which reduces to the sine-Gordon quantum mechanics problem.	contribution 0.96	0.61	0.10	asserted 1.00
s2	The quantum effects can be expressed as the WKB series written analytically in terms of hypergeometric functions.	result 0.88	0.79	0.14	asserted 1.00
s3	We obtain the magnetic and dyonic expansions of the Nekrasov theory by studying the property of hypergeometric functions in the magnetic and dyonic regions on the moduli space.	contribution 0.73	0.95	0.42	asserted 1.00
s4	We also discuss the relation between the electric-magnetic duality of gauge theory and the action-action duality of the integrable system.	interpretation 0.52	0.76	0.18	asserted 0.72

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Original abstract

It is recently claimed by Nekrasov and Shatashvili that the $\mathcal {N}=2$ gauge theories in the $\Omega$ background with $\epsilon_1=\hbar, \epsilon_2=0$ are related to the quantization of certain algebraic integrable systems. We study the special case of SU(2) pure gauge theory, the corresponding integrable model is the A$_1$ Toda model, which reduces to the sine-Gordon quantum mechanics problem. The quantum effects can be expressed as the WKB series written analytically in terms of hypergeometric functions. We obtain the magnetic and dyonic expansions of the Nekrasov theory by studying the property of hypergeometric functions in the magnetic and dyonic regions on the moduli space. We also discuss the relation between the electric-magnetic duality of gauge theory and the action-action duality of the integrable system.

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References

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Quantum Geometry of Resurgent Perturbative/Nonperturbative Relations
hep-th 2017-01 conditional novelty 6.0 of 10

For quantum potentials with a genus-one classical elliptic curve, the all-orders WKB action determines the all-orders dual (tunneling) action, and for four special arithmetic potentials the relation takes a universal ...
Resurgence and the Nekrasov-Shatashvili Limit: Connecting Weak and Strong Coupling in the Mathieu and Lam'e Systems
hep-th 2015-01 conditional novelty 6.0 of 10

Exact WKB analysis unifies the three spectral regimes of the Mathieu and Lamé equations and proves that non-perturbative effects are encoded in perturbation theory through the Picard-Fuchs equation for the quantized e...
Quantum Hitchin Systems via beta-deformed Matrix Models
hep-th 2011-04 conditional novelty 6.0 of 10

In the Nekrasov-Shatashvili limit, the Schwinger-Dyson loop equations of the beta-deformed Dotsenko-Fateev matrix model yield the Schroedinger-type equations whose potentials are the quantum Hitchin Hamiltonians on th...
Deformed Prepotential, Quantum Integrable System and Liouville Field Theory
hep-th 2010-06 conditional novelty 6.0 of 10

The null state condition for a degenerate field in Liouville theory yields the Schrödinger equations of the integrable systems tied to N=2 gauge theories, and the monodromies of the conformal block reproduce the defor...

How the literature uses it

18 arXiv papers cite this in 46 sentences

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Papers citing it · 38 sentences without a confident label

Uses its method or tool 1 paper

We then find the expansion of the SW periods around $u=Λ_0^2$, which are given by [ref] [equation] where $ũ :=u-Λ_0^2$.

Quantum periods and prepotential in ${\cal N}=2$ SU(2) SQCD · Quantum SW curve for $N=2$ SU(2) SQCD · 0.90

Compares with it 1 paper

They indeed match [ref] .

Mathieu equation and Elliptic curve · 5th order proof · 0.99

Cites it as background 5 papers

The results in this section have already been given in earlier works, see [ref] .

Combinatorial approach to Mathieu and Lam\'e equations · Review results of Mathieu equation · 0.84
In this paper we try to present the problem as a differential equation problem, for relevant physics background, see [ref] and [ref] , and references therein.

Mathieu equation and Elliptic curve · Introduction · 0.98
For further results on the WKB analysis of this equation, see [ref] .

Pure N=2 Super Yang-Mills and Exact WKB · WKB analysis of the Mathieu equation · 0.98

What it cites · 13 sentences without a confident label

Uses the method or tool of 4 papers

The problem is solved through the localization technique, this can be achieved only after embedding the $𝒩=2$ gauge theory into the so called $Ω$-background [ref] .

Seiberg-Witten Theory and Random Partitions · Introduction · 0.85
The problem is solved through the localization technique, this can be achieved only after embedding the $𝒩=2$ gauge theory into the so called $Ω$-background [ref] .

Issues in Topological Gauge Theory · Introduction · 0.85
The problem is solved through the localization technique, this can be achieved only after embedding the $𝒩=2$ gauge theory into the so called $Ω$-background [ref] .

Integrating Over Higgs Branches · Introduction · 0.85

Compares with 2 papers

Expanding $a(u)$ and $a_D(u)$ near $∞$, up to the $ħ^4$ order we have [equation] This result has been obtained in [ref] and [ref] , through different methods.

Nekrasov Functions and Exact Bohr-Sommerfeld Integrals · Electric expansion · 0.83
Comparing our results with that in [ref] , we know that the Nekrasov theory with $ϵ_1=-ϵ_20$ and $ϵ_10,ϵ_2=0$ will give the same $ℱ^(0)(a)$ and $ℱ_D^(0)(a_D)$ which is the Seiberg-Witten theory, but for higher order corrections, $ℱ^(g)(a)$ and $ℱ_D^(g)(a_D)$ for $g≥1$, they two cases give different results. $∙$ The case $ϵ_1=ϵ_2$ is explored in [ref] .

Localization of gauge theory on a four-sphere and supersymmetric Wilson loops · Conclusion · 0.99

Cites as background 7 papers

In [ref] the authors study the SU(2) theory, using the $Γ(2)⊂SL(2,ℤ)$ (quasi)modular property of $F^(g)(τ,τ̅)$, the magnetic expansion of $ℱ_D^(g)(a_D)$ was obtained by the limit $τ̅_D=-1τ̅→∞$. $∙$ The case of $ϵ_1=ħ, ϵ_2=0$ is related to the quantization of integrable systems, initiated in [ref] .

Quantization of Integrable Systems and Four Dimensional Gauge Theories · Conclusion · 0.83
A rather interesting discovery is recently presented in [ref] , where the relation between the Nekrasov theory and the quantization of algebraic integrable system is established.

Supersymmetric vacua and Bethe ansatz · Introduction · 0.88
It was noted in [ref] that the Seiberg-Witten solution of gauge theory is related to the classical integrable system, more precisely, the Seiberg-Witten curve of the gauge theory is identical to the spectral curve of the classical integrable system if we suitably identify physical quantities on the two sides, see review [ref] .

Lectures on Supersymmetric Yang-Mills Theory and Integrable Systems · Introduction · 0.97

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