We show that the dual partition function of the pure $mathcal N=2$ $SU(2)$ gauge theory in the self-dual $Omega$-background (a) is given by Fredholm determinant of a generalized Bessel kernel and (b) coincides with the tau function associated to the general solution of the Painleve III equation of type $D_8$ (radial sine-Gordon equation). In particular, the principal minor expansion of the Fredholm determinant yields Nekrasov combinatorial sums over pairs of Young diagrams.
{"title":"Pure 𝑆𝑈(2) gauge theory partition function\u0000 and generalized Bessel kernel","authors":"P. Gavrylenko, O. Lisovyy","doi":"10.1090/PSPUM/098/01727","DOIUrl":"https://doi.org/10.1090/PSPUM/098/01727","url":null,"abstract":"We show that the dual partition function of the pure $mathcal N=2$ $SU(2)$ gauge theory in the self-dual $Omega$-background (a) is given by Fredholm determinant of a generalized Bessel kernel and (b) coincides with the tau function associated to the general solution of the Painleve III equation of type $D_8$ (radial sine-Gordon equation). In particular, the principal minor expansion of the Fredholm determinant yields Nekrasov combinatorial sums over pairs of Young diagrams.","PeriodicalId":384712,"journal":{"name":"Proceedings of Symposia in Pure\n Mathematics","volume":"80 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124464288","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present several recent developments on ELSV-type formulae and topological recursion concerning Chiodo classes and several kind of Hurwitz numbers. The main results appeared in D. Lewanski, A. Popolitov, S. Shadrin, D. Zvonkine, "Chiodo formulas for the r-th roots and topological recursion", Lett. Math. Phys. (2016).
本文介绍了关于Chiodo类和几种Hurwitz数的elv型公式和拓扑递推的最新进展。主要结果出现在D. Lewanski, A. Popolitov, S. Shadrin, D. Zvonkine，“关于r- n根和拓扑递归的Chiodo公式”，Lett。数学。理论物理。(2016)。
{"title":"On ELSV-type formulae, Hurwitz numbers and\u0000 topological recursion","authors":"D. Lewanski","doi":"10.1090/PSPUM/100/01764","DOIUrl":"https://doi.org/10.1090/PSPUM/100/01764","url":null,"abstract":"We present several recent developments on ELSV-type formulae and topological recursion concerning Chiodo classes and several kind of Hurwitz numbers. The main results appeared in D. Lewanski, A. Popolitov, S. Shadrin, D. Zvonkine, \"Chiodo formulas for the r-th roots and topological recursion\", Lett. Math. Phys. (2016).","PeriodicalId":384712,"journal":{"name":"Proceedings of Symposia in Pure\n Mathematics","volume":"74 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124602349","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
When the action of a reductive group on a projective variety has a suitable linearisation, Mumford’s geometric invariant theory (GIT) can be used to construct and study an associated quotient variety. In this article we describe how Mumford’s GIT can be extended effectively to suitable actions of linear algebraic groups which are not necessarily reductive, with the extra data of a graded linearisation for the action. Any linearisation in the traditional sense for a reductive group action induces a graded linearisation in a natural way. The classical examples of moduli spaces which can be constructed using Mumford’s GIT are moduli spaces of stable curves and of (semi)stable bundles over a fixed nonsingular curve. This more general construction can be used to construct moduli spaces of unstable objects, such as unstable curves or unstable bundles (with suitable fixed discrete invariants in each case, related to their singularities or Harder–Narasimhan type). In algebraic geometry it is often useful to be able to construct quotients of algebraic varieties by linear algebraic group actions; in particular moduli spaces (or stacks) can be constructed in this way. When the linear algebraic group is reductive, and we have a suitable linearisation for its action on a projective variety, we can use Mumford’s geometric invariant theory (GIT) to construct and study such quotient varieties [32]. The aim of this article is to describe how Mumford’s GIT can be extended effectively to actions of a large family of linear algebraic groups which are not necessarily reductive, with the extra data of a graded linearisation for the action. Any linearisation in the traditional sense for a reductive group action can be regarded as a graded linearisation in a natural way. When a linear algebraic group over an algebraically closed field k of characteristic 0 is a semidirect productH = U ⋊R of its unipotent radical U and a reductive subgroupR ∼= H/U which contains a central one-parameter subgroup λ : Gm → Rwhose adjoint action on the Lie algebra of U has only strictly positive weights, we will see that any linearisation for an action of H on a projective variety X becomes graded if it is twisted by an appropriate (rational) character, and then many of the good properties of Mumford’s GIT hold. Many non-reductive linear algebraic group actions arising in algebraic geometry are actions of groups of this form: for example, any parabolic subgroup of a reductive group has this form, as does the automorphism group of any complete simplicial toric variety [11], and the group of k-jets of germs of biholomorphisms of (C, 0) for any positive integers k and p [6]. Example 0.1. The automorphism group of the weighted projective plane P(1, 1, 2) with weights 1,1 and 2 is Aut(P(1, 1, 2)) ∼= R⋉ U where R ∼= (GL(2)×Gm)/Gm ∼= GL(2) is reductive and U ∼= (k+)3 is unipotent with elements given by (x, y, z) 7→ (x, y, z + λx2 + μxy + νy2) for (λ, μ, ν) ∈ k3. Early work on this project was suppo
当约化群对一个投影变项的作用具有适当的线性化时，Mumford的几何不变理论(GIT)可以用来构造和研究一个相关的商变项。在本文中，我们描述了Mumford的GIT如何有效地扩展到线性代数群的适当动作，这些动作不一定是约化的，并且具有动作的分级线性化的额外数据。任何传统意义上的线性化对于还原性群作用都会以自然的方式引起梯度线性化。可以用Mumford的GIT构造模空间的经典例子是稳定曲线和固定非奇异曲线上的(半)稳定束的模空间。这种更一般的构造可以用来构造不稳定对象的模空间，例如不稳定曲线或不稳定束(每种情况下都有合适的固定离散不变量，与它们的奇点或hard - narasimhan型相关)。在代数几何中，利用线性代数群作用构造代数变量的商是很有用的;特别是模空间(或堆栈)可以用这种方式构造。当线性代数群是约化的，并且我们有一个合适的线性化它对一个射影变量的作用，我们可以使用Mumford的几何不变理论(GIT)来构造和研究这样的商变量[32]。本文的目的是描述Mumford的GIT如何有效地扩展到一大族线性代数群的动作，这些动作不一定是约化的，并且具有动作的分级线性化的额外数据。对于约化群作用，传统意义上的任何线性化都可以看作是自然的梯度线性化。当特征为0的代数闭域k上的线性代数群是其单幂根群U和包含中心单参数子群λ的约化子群pr ~ = H/U的半直积th = U * R时:Gm→r，其在U的李代数上的伴随作用只有严格的正权，我们将看到H在一个投影变量X上的作用的任何线性化，如果它被一个适当的(有理)特征扭曲，那么它就变成了分级，然后Mumford的GIT的许多好的性质成立。代数几何中出现的许多非约化线性代数群作用都是这种形式的群的作用:例如，约化群的任何抛物子群都具有这种形式，任何完全简单环型簇[11]的自同构群也具有这种形式，对于任何正整数k和p[6]的(C, 0)生物纯态的胚的k-射流群也具有这种形式。例0.1。权为1,1和2的加权射影平面P(1,1,2)的自同态群为Aut(P(1,1,2)) ~ = R × U，其中R ~ = (GL(2)×Gm)/Gm ~ = GL(2)是约化的，U ~ = (k+)3是单幂的，对于(λ， μ， ν)∈k3，元素为(x, y, z) 7→(x, y, z + λx2 + μxy + νy2)。该项目的早期工作得到了工程与物理科学研究委员会的支持[批准号GR/T016170/1,EP/G000174/1]。Brent Doran获得瑞士国家科学基金奖200021-138071的部分资助。
{"title":"Graded linearisations","authors":"Gergely B'erczi, B. Doran, F. Kirwan","doi":"10.1090/pspum/099/01","DOIUrl":"https://doi.org/10.1090/pspum/099/01","url":null,"abstract":"When the action of a reductive group on a projective variety has a suitable linearisation, Mumford’s geometric invariant theory (GIT) can be used to construct and study an associated quotient variety. In this article we describe how Mumford’s GIT can be extended effectively to suitable actions of linear algebraic groups which are not necessarily reductive, with the extra data of a graded linearisation for the action. Any linearisation in the traditional sense for a reductive group action induces a graded linearisation in a natural way. The classical examples of moduli spaces which can be constructed using Mumford’s GIT are moduli spaces of stable curves and of (semi)stable bundles over a fixed nonsingular curve. This more general construction can be used to construct moduli spaces of unstable objects, such as unstable curves or unstable bundles (with suitable fixed discrete invariants in each case, related to their singularities or Harder–Narasimhan type). In algebraic geometry it is often useful to be able to construct quotients of algebraic varieties by linear algebraic group actions; in particular moduli spaces (or stacks) can be constructed in this way. When the linear algebraic group is reductive, and we have a suitable linearisation for its action on a projective variety, we can use Mumford’s geometric invariant theory (GIT) to construct and study such quotient varieties [32]. The aim of this article is to describe how Mumford’s GIT can be extended effectively to actions of a large family of linear algebraic groups which are not necessarily reductive, with the extra data of a graded linearisation for the action. Any linearisation in the traditional sense for a reductive group action can be regarded as a graded linearisation in a natural way. When a linear algebraic group over an algebraically closed field k of characteristic 0 is a semidirect productH = U ⋊R of its unipotent radical U and a reductive subgroupR ∼= H/U which contains a central one-parameter subgroup λ : Gm → Rwhose adjoint action on the Lie algebra of U has only strictly positive weights, we will see that any linearisation for an action of H on a projective variety X becomes graded if it is twisted by an appropriate (rational) character, and then many of the good properties of Mumford’s GIT hold. Many non-reductive linear algebraic group actions arising in algebraic geometry are actions of groups of this form: for example, any parabolic subgroup of a reductive group has this form, as does the automorphism group of any complete simplicial toric variety [11], and the group of k-jets of germs of biholomorphisms of (C, 0) for any positive integers k and p [6]. Example 0.1. The automorphism group of the weighted projective plane P(1, 1, 2) with weights 1,1 and 2 is Aut(P(1, 1, 2)) ∼= R⋉ U where R ∼= (GL(2)×Gm)/Gm ∼= GL(2) is reductive and U ∼= (k+)3 is unipotent with elements given by (x, y, z) 7→ (x, y, z + λx2 + μxy + νy2) for (λ, μ, ν) ∈ k3. Early work on this project was suppo","PeriodicalId":384712,"journal":{"name":"Proceedings of Symposia in Pure\n Mathematics","volume":"265 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127547421","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We survey here the construction and the basic properties of descendent invariants in the theory of stable pairs on nonsingular projective 3-folds. The main topics covered are the rationality of the generating series, the functional equation, the Gromov-Witten/Pairs correspondence for descendents, the Virasoro constraints, and the connection to the virtual fundamental class of the stable pairs moduli space in algebraic cobordism. In all of these directions, the proven results constitute only a small part of the conjectural framework. A central goal of the article is to introduce the open questions as simply and directly as possible.
{"title":"Descendents for stable pairs on\u0000 3-folds","authors":"R. Pandharipande","doi":"10.1090/PSPUM/099/01743","DOIUrl":"https://doi.org/10.1090/PSPUM/099/01743","url":null,"abstract":"We survey here the construction and the basic properties of descendent invariants in the theory of stable pairs on nonsingular projective 3-folds. The main topics covered are the rationality of the generating series, the functional equation, the Gromov-Witten/Pairs correspondence for descendents, the Virasoro constraints, and the connection to the virtual fundamental class of the stable pairs moduli space in algebraic cobordism. In all of these directions, the proven results constitute only a small part of the conjectural framework. A central goal of the article is to introduce the open questions as simply and directly as possible.","PeriodicalId":384712,"journal":{"name":"Proceedings of Symposia in Pure\n Mathematics","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121227268","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This note announces results on the relations between the approach of Beilinson and Drinfeld to the geometric Langlands correspondence based on conformal field theory, the approach of Kapustin and Witten based on $N=4$ SYM, and the AGT-correspondence. The geometric Langlands correspondence is described as the Nekrasov-Shatashvili limit of a generalisation of the AGT-correspondence in the presence of surface operators. Following the approaches of Kapustin - Witten and Nekrasov - Witten we interpret some aspects of the resulting picture using an effective description in terms of two-dimensional sigma models having Hitchin's moduli spaces as target-manifold.
{"title":"Supersymmetric field theories and geometric\u0000 Langlands: The other side of the coin","authors":"A. Balasubramanian, J. Teschner","doi":"10.1090/PSPUM/098/01723","DOIUrl":"https://doi.org/10.1090/PSPUM/098/01723","url":null,"abstract":"This note announces results on the relations between the approach of Beilinson and Drinfeld to the geometric Langlands correspondence based on conformal field theory, the approach of Kapustin and Witten based on $N=4$ SYM, and the AGT-correspondence. The geometric Langlands correspondence is described as the Nekrasov-Shatashvili limit of a generalisation of the AGT-correspondence in the presence of surface operators. Following the approaches of Kapustin - Witten and Nekrasov - Witten we interpret some aspects of the resulting picture using an effective description in terms of two-dimensional sigma models having Hitchin's moduli spaces as target-manifold.","PeriodicalId":384712,"journal":{"name":"Proceedings of Symposia in Pure\n Mathematics","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116664955","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study symplectic properties of monodromy map for second order linear equation with meromorphic potential having only simple poles on a Riemann surface. We show that the canonical symplectic structure on the cotangent bundle $T^*M_{g,n}$ implies the Goldman bracket on the corresponding character variety under the monodromy map, thereby extending the recent results of the paper of M.Bertola, C.Norton and the author from the case of holomorphic to meromorphic potentials with simple poles.
{"title":"Periods of meromorphic quadratic\u0000 differentials and Goldman bracket","authors":"D. Korotkin","doi":"10.1090/PSPUM/100/01763","DOIUrl":"https://doi.org/10.1090/PSPUM/100/01763","url":null,"abstract":"We study symplectic properties of monodromy map for second order linear equation with meromorphic potential having only simple poles on a Riemann surface. We show that the canonical symplectic structure on the cotangent bundle $T^*M_{g,n}$ implies the Goldman bracket on the corresponding character variety under the monodromy map, thereby extending the recent results of the paper of M.Bertola, C.Norton and the author from the case of holomorphic to meromorphic potentials with simple poles.","PeriodicalId":384712,"journal":{"name":"Proceedings of Symposia in Pure\n Mathematics","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134226983","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose a new approach to the topological recursion of Eynard-Orantin based on the notion of Airy structure, which we introduce in the paper. We explain why Airy structure is a more fundamental object than the one of the spectral curve. We explain how the concept of quantization of Airy structure leads naturally to the formulas of topological recursion as well as their generalizations. The notion of spectral curve is also considered in a more general framework of Poisson surfaces endowed with foliation. We explain how the deformation theory of spectral curves is related to Airy structures. Few other topics (e.g. the Holomorphic Anomaly Equation) are also discussed from the general point of view of Airy structures.
{"title":"Airy structures and symplectic geometry of\u0000 topological recursion","authors":"M. Kontsevich, Y. Soibelman","doi":"10.1090/PSPUM/100/01765","DOIUrl":"https://doi.org/10.1090/PSPUM/100/01765","url":null,"abstract":"We propose a new approach to the topological recursion of Eynard-Orantin based on the notion of Airy structure, which we introduce in the paper. We explain why Airy structure is a more fundamental object than the one of the spectral curve. We explain how the concept of quantization of Airy structure leads naturally to the formulas of topological recursion as well as their generalizations. The notion of spectral curve is also considered in a more general framework of Poisson surfaces endowed with foliation. We explain how the deformation theory of spectral curves is related to Airy structures. Few other topics (e.g. the Holomorphic Anomaly Equation) are also discussed from the general point of view of Airy structures.","PeriodicalId":384712,"journal":{"name":"Proceedings of Symposia in Pure\n Mathematics","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125503842","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We offer a new proof that two closed oriented 4–manifolds are cobordant if their signatures agree, in the spirit of Lickorish’s proof [6] that all closed oriented 3–manifolds bound 4–manifolds. Where Lickorish uses Heegaard splittings we use trisections. In fact we begin with a subtle recasting of Lickorish’s argument: Instead of factoring the gluing map for a Heegaard splitting as a product of Dehn twists, we encode each handlebody in a Heegaard splitting in terms of a Morse function on the surface and build the 4–manifold from a generic homotopy between the two functions. This extends up a dimension by encoding a trisection of a closed 4–manifold as a triangle (circle) of functions and constructing an associated 5– manifold from an extension to a 2–simplex (disk) of functions. This borrows ideas from Hatcher and Thurston’s proof [3] that the mapping class group of a surface is finitely presented.
{"title":"Functions on surfaces and constructions of\u0000 manifolds in dimensions three, four and five","authors":"David T. Gay","doi":"10.1090/pspum/102/06","DOIUrl":"https://doi.org/10.1090/pspum/102/06","url":null,"abstract":"We offer a new proof that two closed oriented 4–manifolds are cobordant if their signatures agree, in the spirit of Lickorish’s proof [6] that all closed oriented 3–manifolds bound 4–manifolds. Where Lickorish uses Heegaard splittings we use trisections. In fact we begin with a subtle recasting of Lickorish’s argument: Instead of factoring the gluing map for a Heegaard splitting as a product of Dehn twists, we encode each handlebody in a Heegaard splitting in terms of a Morse function on the surface and build the 4–manifold from a generic homotopy between the two functions. This extends up a dimension by encoding a trisection of a closed 4–manifold as a triangle (circle) of functions and constructing an associated 5– manifold from an extension to a 2–simplex (disk) of functions. This borrows ideas from Hatcher and Thurston’s proof [3] that the mapping class group of a surface is finitely presented.","PeriodicalId":384712,"journal":{"name":"Proceedings of Symposia in Pure\n Mathematics","volume":"60 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127998628","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Hilbert series is a generating function that enumerates gauge invariant chiral operators of a supersymmetric field theory with four supercharges and an R-symmetry. In this article I review how counting dressed 't Hooft monopole operators leads to a formula for the Hilbert series of a 3d $mathcal{N}geq 2$ gauge theory, which captures precious information about the chiral ring and the moduli space of supersymmetric vacua of the theory.
{"title":"3d supersymmetric gauge theories and Hilbert\u0000 series","authors":"S. Cremonesi","doi":"10.1090/PSPUM/098/01728","DOIUrl":"https://doi.org/10.1090/PSPUM/098/01728","url":null,"abstract":"The Hilbert series is a generating function that enumerates gauge invariant chiral operators of a supersymmetric field theory with four supercharges and an R-symmetry. In this article I review how counting dressed 't Hooft monopole operators leads to a formula for the Hilbert series of a 3d $mathcal{N}geq 2$ gauge theory, which captures precious information about the chiral ring and the moduli space of supersymmetric vacua of the theory.","PeriodicalId":384712,"journal":{"name":"Proceedings of Symposia in Pure\n Mathematics","volume":"156 ","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120875412","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper provides an introduction to the mathematical notion of emph{quantum curves}. We start with a concrete example arising from a graph enumeration problem. We then develop a theory of quantum curves associated with Hitchin spectral curves. A conjecture of Gaiotto, which predicts a new construction of opers from a Hitchin spectral curve, is explained. We give a step-by-step detailed description of the proof of the conjecture for the case of rank $2$ Higgs bundles. Finally, we identify the two concepts of textit{quantum curve} arising from the topological recursion formalism with the limit oper of Gaiotto's conjecture.
{"title":"A journey from the Hitchin section to the\u0000 oper moduli","authors":"Olivia Dumitrescu","doi":"10.1090/PSPUM/098/01726","DOIUrl":"https://doi.org/10.1090/PSPUM/098/01726","url":null,"abstract":"This paper provides an introduction to the mathematical notion of emph{quantum curves}. We start with a concrete example arising from a graph enumeration problem. We then develop a theory of quantum curves associated with Hitchin spectral curves. A conjecture of Gaiotto, which predicts a new construction of opers from a Hitchin spectral curve, is explained. We give a step-by-step detailed description of the proof of the conjecture for the case of rank $2$ Higgs bundles. Finally, we identify the two concepts of textit{quantum curve} arising from the topological recursion formalism with the limit oper of Gaiotto's conjecture.","PeriodicalId":384712,"journal":{"name":"Proceedings of Symposia in Pure\n Mathematics","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130748752","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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