4D N=2 SYM理论的瞬时子分区函数的有限ε2校正

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"这篇论文是关于4维N = 2超杨-米尔斯（SYM）理论中的瞬时子分区函数的有限γ2校正，特别是在Nekrasov-Shatashvili极限ε2→0的情况下的研究。作者Jean-Emile Bourgine和Davide Fioravanti来自意大利博洛尼亚的INFN Sezione di Bologna和Dipartimento di Fisica e Astronomia。他们探讨了热力学Bethe Ansatz方程的出现，该方程在Mayer展开技术中被发现，并在此基础上进行场论分析。" 在4D N = 2 Super Yang-Mills理论中，瞬时子分区函数是一个关键概念，它涉及量子场论中的非微扰效应。该理论具有对称性N = 2超对称性，这是一种特殊的对称性，允许物理系统同时拥有 bosonic（玻色子）和fermionic（费米子）自由度，并在量子层面上保持平衡。文章的核心是研究当ε2参数趋于零时的Nekrasov-Shatashvili极限，这个极限通常与量子可积系统和Bethe Ansatz方法相关联。热力学Bethe Ansatz是一种用于求解多体量子系统能量谱的有效方法，尤其在统计力学和量子链中。在Mayer展开技术中，复杂的相互作用被转化为易于处理的图论问题，从而揭示了Bethe Ansatz方程。在本研究中，作者不仅回顾了这一方程的出现，还试图将这种解析技巧与场论的其他方面结合起来。作者引入了一个新的运算符∇，这个运算符能够区分分区函数积分轮廓内的奇点与轮廓外的奇点。这有助于解决和解析处理这些奇点，从而使计算更加精确。在场论中，积分轮廓的选择可以极大地影响结果，尤其是在处理具有非平凡奇异点的积分时。因此，这个新运算符的引入被认为是自然且有解决力的。文章的最终发表是在《Physics Letters B》上，这是一个开放获取的期刊，意味着所有读者都可以免费阅读和使用文章内容。此外，根据CC BY许可，这篇文章允许作者保留版权，同时允许他人在遵守一定条件的情况下使用和分发作品。这篇研究论文为4D N = 2 SYM理论的瞬时子分区函数提供了深入的理解，特别是在Nekrasov-Shatashvili极限下的行为。通过结合热力学Bethe Ansatz和新的运算符，作者们为理解和计算这类复杂物理系统提供了新的工具和洞察。

Physics Letters B 750 (2015) 139–146

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Finite ε

-corrections to the N = 2 SYM prepotential

Jean-Emile Bourgine, Davide Fioravanti

∗

Sezione INFN di Bologna, Dipartimento di Fisica e Astronomia, Università di Bologna, Via Irnerio 46, Bologna, Italy

a r t i c l e i n f o a b s t r a c t

Article history:

Received

22 July 2015

Accepted

1 September 2015

Available

online 5 September 2015

Editor: N.

Lambert

We derive the ﬁrst ε

-correction of the instanton partition functions in 4D N = 2Super Yang–Mills

(SYM) to the Nekrasov–Shatashvili limit ε

→ 0. In the latter we recall the emergence of the famous

Thermodynamic Bethe Ansatz-like equation which has been found by Mayer expansion techniques. Here

we combine eﬃciently these to ﬁeld theory arguments. In a nutshell, we ﬁnd natural and resolutive the

introduction of a new operator ∇ that distinguishes the singularities within and outside the integration

contour of the partition function.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

The Omega background was ﬁrst introduced in N = 2SUSY

gauge theories to regularise the inﬁnite volume of R

in the com-

putation

of instanton contributions to the partition function by lo-

calisation

[1]. More recently, it has also proven to be a formidable

way to preserve integrability of these theories upon deformation.

In fact, this background is characterised by two equivariant defor-

mation

parameters ε

and ε

associated to the breaking of the

Lorentz invariant four dimensional space into C × C, but still the

Nekrasov instanton partition function exhibits an integrable struc-

ture

in the form of covariance under the Spherical Hecke central

(SHc) algebra [2,3] (which is formally equivalent to a W

∞

algebra).

The presence of this algebra shed light on the conjecture by [4,5]

a duality between these four dimensional theories and the fam-

ily

of Toda conformal ﬁeld theories in two dimensions.

Moreover,

the SHc algebra is closely related to a tensorial version of the in-

tegrable

Calogero–Moser Hamiltonian and led to the construction

of one of the most basic objects of quantum integrable theories,

namely an (instanton) R-matrix [9].

the Nekrasov–Shatashvili (NS) limit ε

→ 0, the theory pos-

sesses

only one (non-zero) equivariant parameter ε

and, besides,

its 2D underlying integrable structure is better understood [10].

The latter deﬁnes, indeed, a quantisation of a Hitchin system as-

sociated

to the Seiberg–Witten curve (see [11] and the references

Corresponding author.

E-mail

addresses: bourgine@bo.infn.it (J.-E. Bourgine), ﬁoravanti@bo.infn.it

(D. Fioravanti).

The conjecture has been partially proved either using the basis of AFLT states

[6], or a set of generalised Jack polynomials [7,8].

therein) which can now be characterised by a quantum curve [12].

And this curve is, actually, equivalent to a TQ-relation upon a

(quantum) change of variables [13], while the new integrable sys-

tem

is the bi-spectral dual of the previous one [14]. Of course, of

great interest for integrable model theory is the TQ-relation, which

can also be obtained directly from the Nekrasov partition function

by extremising the sum over Young diagrams [15], a technique that

goes back to [16]. Moreover, another key feature of integrability

has come out of the blue since the original paper [10] in the form

of a non-linear integral equation (NLIE) in the complex plane [17]

resembling

a Thermodynamical Bethe Ansatz (TBA) equation [18].

Subsequently, this equation was derived in full detail by [19,20]

upon

using Mayer cluster expansion techniques [21,22].

the present letter, we go beyond this limit and present an

explicit formula for the ﬁrst ε

-correction to the prepotential. This

result should pave the way to a better understanding of the full

-deformation and its meaning as quantum integrable system

characterised by a TBA/NLIE. This should ultimately lead to a richer

algebraic structure similar to the SHc Hopf algebra as we know

how, along the other way around, the NS limit of the SHc algebra

can be obtained [23]. For convenience, we focus on N = 2SYM

with a single SU(N

) gauge group and a number of fundamental

ﬂavours. However, we believe that our mathematical construction

and results will be easily generalisable to arbitrary quivers (along

the lines of [23]).

fact, with respect to the latter reference (exploiting the

methods of Mayer cluster expansion) we shall add an eﬃcient ﬁeld

theory argument. The latter is ﬁrstly used to disentangle the long-

and

short-range interactions between instantons: this is performed

in Section 2. Then, we treat the short-range interactions by the

Mayer cluster expansion technique in Section 3 and achieve for-

http://dx.doi.org/10.1016/j.physletb.2015.09.002

0370-2693/

SCOAP

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