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}
This is an introduction to a provisional mathematical definition of Coulomb branches of $3$-dimensional $mathcal N=4$ supersymmetric gauge theories, studied in arXiv:1503.03676, arXiv:1601.03586. This is an expanded version of an article arXiv:1612.09014 appeared in the 61st DAISUUGAKU symposium proceeding (2016), written originally in Japanese.
{"title":"Introduction to a provisional mathematical\u0000 definition of Coulomb branches of 3-dimensional 𝒩=4\u0000 gauge theories","authors":"H. Nakajima","doi":"10.1090/PSPUM/099/01741","DOIUrl":"https://doi.org/10.1090/PSPUM/099/01741","url":null,"abstract":"This is an introduction to a provisional mathematical definition of Coulomb branches of $3$-dimensional $mathcal N=4$ supersymmetric gauge theories, studied in arXiv:1503.03676, arXiv:1601.03586. This is an expanded version of an article arXiv:1612.09014 appeared in the 61st DAISUUGAKU symposium proceeding (2016), written originally in Japanese.","PeriodicalId":384712,"journal":{"name":"Proceedings of Symposia in Pure\n Mathematics","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114887726","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 show that the double quantization of Seiberg-Witten spectral curve for $Gamma$-quiver gauge theory defines the generating current of W$(Gamma)$-algebra in the free field realization. We also show that the partition function is given as a correlator of the corresponding W$(Gamma)$-algebra, which is equivalent to the AGT relation under the gauge/quiver (spectral) duality.
{"title":"Double quantization of Seiberg–Witten\u0000 geometry and W-algebras","authors":"Taro Kimura","doi":"10.1090/pspum/100/01762","DOIUrl":"https://doi.org/10.1090/pspum/100/01762","url":null,"abstract":"We show that the double quantization of Seiberg-Witten spectral curve for $Gamma$-quiver gauge theory defines the generating current of W$(Gamma)$-algebra in the free field realization. We also show that the partition function is given as a correlator of the corresponding W$(Gamma)$-algebra, which is equivalent to the AGT relation under the gauge/quiver (spectral) duality.","PeriodicalId":384712,"journal":{"name":"Proceedings of Symposia in Pure\n Mathematics","volume":"65 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126549996","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 prove that conformal nets of finite index are an instance of the notion of a factorization algebra. This result is an ingredient in our proof that, for $G=SU(n)$, the Drinfel'd center of the category of positive energy representations of the based loop group is equivalent to the category of positive energy representations of the free loop group.
{"title":"Conformal nets are factorization\u0000 algebras","authors":"A. Henriques","doi":"10.1090/pspum/098/01749","DOIUrl":"https://doi.org/10.1090/pspum/098/01749","url":null,"abstract":"We prove that conformal nets of finite index are an instance of the notion of a factorization algebra. This result is an ingredient in our proof that, for $G=SU(n)$, the Drinfel'd center of the category of positive energy representations of the based loop group is equivalent to the category of positive energy representations of the free loop group.","PeriodicalId":384712,"journal":{"name":"Proceedings of Symposia in Pure\n Mathematics","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121155780","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 construct an open substack $Usubsetmathcal{M}_{g,1}$ with the complement of codimension $ge 2$ and a morphism from $U$ to a weighted projective stack, which sends the Weierstrass locus $mathcal{W}cap U$ to a point, and maps $mathcal{M}_{g,1}setminusmathcal{W}$ isomorphically to its image. The proof uses alternative birational models of $mathcal{M}_{g,1}$ and $mathcal{M}_{g,2}$ from arXiv:1509.07241.
{"title":"Contracting the Weierstrass locus to a\u0000 point","authors":"A. Polishchuk","doi":"10.1090/PSPUM/098/01725","DOIUrl":"https://doi.org/10.1090/PSPUM/098/01725","url":null,"abstract":"We construct an open substack $Usubsetmathcal{M}_{g,1}$ with the complement of codimension $ge 2$ and a morphism from $U$ to a weighted projective stack, which sends the Weierstrass locus $mathcal{W}cap U$ to a point, and maps $mathcal{M}_{g,1}setminusmathcal{W}$ isomorphically to its image. The proof uses alternative birational models of $mathcal{M}_{g,1}$ and $mathcal{M}_{g,2}$ from arXiv:1509.07241.","PeriodicalId":384712,"journal":{"name":"Proceedings of Symposia in Pure\n Mathematics","volume":"263 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116417616","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}
In a series of papers the authors associated to an $L^2$-acyclic group $Gamma$ an invariant $mathcal{P}(Gamma)$ that is a formal difference of polytopes in the vector space $H_1(Gamma;Bbb{R})$. This invariant is in particular defined for most 3-manifold groups, for most 2-generator 1-relator groups and for all free-by-cyclic groups. In most of the above cases the invariant can be viewed as an actual polytope. In this survey paper we will recall the definition of the polytope invariant $mathcal{P}(Gamma)$ and we state some of the main properties. We conclude with a list of open problems.
{"title":"Groups and polytopes","authors":"Stefan Friedl, W. Luck, Stephan Tillmann","doi":"10.1090/pspum/102/05","DOIUrl":"https://doi.org/10.1090/pspum/102/05","url":null,"abstract":"In a series of papers the authors associated to an $L^2$-acyclic group $Gamma$ an invariant $mathcal{P}(Gamma)$ that is a formal difference of polytopes in the vector space $H_1(Gamma;Bbb{R})$. This invariant is in particular defined for most 3-manifold groups, for most 2-generator 1-relator groups and for all free-by-cyclic groups. In most of the above cases the invariant can be viewed as an actual polytope. \u0000In this survey paper we will recall the definition of the polytope invariant $mathcal{P}(Gamma)$ and we state some of the main properties. We conclude with a list of open problems.","PeriodicalId":384712,"journal":{"name":"Proceedings of Symposia in Pure\n Mathematics","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123825097","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}
There is a strong analogy between compact, torsion-free $G_2$-manifolds $(X,varphi,*varphi)$ and Calabi-Yau 3-folds $(Y,J,g,omega)$. We can also generalize $(X,varphi,*varphi)$ to 'tamed almost $G_2$-manifolds' $(X,varphi,psi)$, where we compare $varphi$ with $omega$ and $psi$ with $J$. Associative 3-folds in $X$, a special kind of minimal submanifold, are analogous to $J$-holomorphic curves in $Y$. Several areas of Symplectic Geometry -- Gromov-Witten theory, Quantum Cohomology, Lagrangian Floer cohomology, Fukaya categories -- are built using 'counts' of moduli spaces of $J$-holomorphic curves in $Y$, but give an answer depending only on the symplectic manifold $(Y,omega)$, not on the (almost) complex structure $J$. We investigate whether it may be possible to define interesting invariants of tamed almost $G_2$-manifolds $(X,varphi,psi)$ by 'counting' compact associative 3-folds $Nsubset X$, such that the invariants depend only on $varphi$, and are independent of the 4-form $psi$ used to define associative 3-folds. We conjecture that one can define a superpotential $Phi_psi:{mathcal U}toLambda_{>0}$ 'counting' associative $mathbb Q$-homology 3-spheres $Nsubset X$ which is deformation-invariant in $psi$ for $varphi$ fixed, up to certain reparametrizations $Upsilon:{mathcal U}to{mathcal U}$ of the base ${mathcal U}=$Hom$(H_3(X;{mathbb Z}),1+Lambda_{>0})$, where $Lambda_{>0}$ is a Novikov ring. Using this we define a notion of '$G_2$ quantum cohomology'. These ideas may be relevant to String Theory or M-Theory on $G_2$-manifolds. We also discuss Donaldson and Segal's proposal in arXiv:0902.3239, section 6.2, to define invariants 'counting' $G_2$-instantons on tamed almost $G_2$-manifolds $(X,varphi,psi)$, with 'compensation terms' counting weighted pairs of a $G_2$-instanton and an associative 3-fold, and suggest some modifications to it.
{"title":"Conjectures on counting associative 3-folds\u0000 in 𝐺₂-manifolds","authors":"D. Joyce","doi":"10.1090/PSPUM/099/01739","DOIUrl":"https://doi.org/10.1090/PSPUM/099/01739","url":null,"abstract":"There is a strong analogy between compact, torsion-free $G_2$-manifolds $(X,varphi,*varphi)$ and Calabi-Yau 3-folds $(Y,J,g,omega)$. We can also generalize $(X,varphi,*varphi)$ to 'tamed almost $G_2$-manifolds' $(X,varphi,psi)$, where we compare $varphi$ with $omega$ and $psi$ with $J$. Associative 3-folds in $X$, a special kind of minimal submanifold, are analogous to $J$-holomorphic curves in $Y$. \u0000Several areas of Symplectic Geometry -- Gromov-Witten theory, Quantum Cohomology, Lagrangian Floer cohomology, Fukaya categories -- are built using 'counts' of moduli spaces of $J$-holomorphic curves in $Y$, but give an answer depending only on the symplectic manifold $(Y,omega)$, not on the (almost) complex structure $J$. \u0000We investigate whether it may be possible to define interesting invariants of tamed almost $G_2$-manifolds $(X,varphi,psi)$ by 'counting' compact associative 3-folds $Nsubset X$, such that the invariants depend only on $varphi$, and are independent of the 4-form $psi$ used to define associative 3-folds. \u0000We conjecture that one can define a superpotential $Phi_psi:{mathcal U}toLambda_{>0}$ 'counting' associative $mathbb Q$-homology 3-spheres $Nsubset X$ which is deformation-invariant in $psi$ for $varphi$ fixed, up to certain reparametrizations $Upsilon:{mathcal U}to{mathcal U}$ of the base ${mathcal U}=$Hom$(H_3(X;{mathbb Z}),1+Lambda_{>0})$, where $Lambda_{>0}$ is a Novikov ring. Using this we define a notion of '$G_2$ quantum cohomology'. These ideas may be relevant to String Theory or M-Theory on $G_2$-manifolds. \u0000We also discuss Donaldson and Segal's proposal in arXiv:0902.3239, section 6.2, to define invariants 'counting' $G_2$-instantons on tamed almost $G_2$-manifolds $(X,varphi,psi)$, with 'compensation terms' counting weighted pairs of a $G_2$-instanton and an associative 3-fold, and suggest some modifications to it.","PeriodicalId":384712,"journal":{"name":"Proceedings of Symposia in Pure\n Mathematics","volume":"147 10","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133051287","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}
Maximally supersymmetric gauge theory in four dimensions admits local boundary conditions which preserve half of the bulk supersymmetries. The S-duality of the bulk gauge theory can be extended in a natural fashion to act on such half-BPS boundary conditions. The purpose of this note is to explain the role these boundary conditions can play in the Geometric Langlands program. In particular, we describe how to obtain pairs of Geometric Langland dual objects from S-dual pairs of half-BPS boundary conditions.
{"title":"S-duality of boundary conditions and the\u0000 Geometric Langlands program","authors":"D. Gaiotto","doi":"10.1090/PSPUM/098/01721","DOIUrl":"https://doi.org/10.1090/PSPUM/098/01721","url":null,"abstract":"Maximally supersymmetric gauge theory in four dimensions admits local boundary conditions which preserve half of the bulk supersymmetries. The S-duality of the bulk gauge theory can be extended in a natural fashion to act on such half-BPS boundary conditions. The purpose of this note is to explain the role these boundary conditions can play in the Geometric Langlands program. In particular, we describe how to obtain pairs of Geometric Langland dual objects from S-dual pairs of half-BPS boundary conditions.","PeriodicalId":384712,"journal":{"name":"Proceedings of Symposia in Pure\n Mathematics","volume":"108 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131131850","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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