We describe an extension of Morse theory to smooth functions on compact Riemannian manifolds, without any non-degeneracy assumptions except that the critical locus must have only finitely many connected components.
{"title":"Morse Theory without Non-Degeneracy","authors":"Frances Kirwan;Geoffrey Penington","doi":"10.1093/qmath/haaa064","DOIUrl":"https://doi.org/10.1093/qmath/haaa064","url":null,"abstract":"We describe an extension of Morse theory to smooth functions on compact Riemannian manifolds, without any non-degeneracy assumptions except that the critical locus must have only finitely many connected components.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 1-2","pages":"455-514"},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmath/haaa064","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49950712","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We define and study the motive of the moduli stack of vector bundles of fixed rank and degree over a smooth projective curve in Voevodsky’s category of motives. We prove that this motive can be written as a homotopy colimit of motives of smooth projective Quot schemes of torsion quotients of sums of line bundles on the curve. When working with rational coefficients, we prove that the motive of the stack of bundles lies in the localizing tensor subcategory generated by the motive of the curve, using Białynicki-Birula decompositions of these Quot schemes. We conjecture a formula for the motive of this stack, inspired by the work of Atiyah and Bott on the topology of the classifying space of the gauge group, and we prove this conjecture modulo a conjecture on the intersection theory of the Quot schemes.
{"title":"On the Voevodsky Motive of the Moduli Stack of Vector Bundles on a Curve","authors":"Victoria Hoskins;Simon Pepin Lehalleur","doi":"10.1093/qmathj/haaa023","DOIUrl":"https://doi.org/10.1093/qmathj/haaa023","url":null,"abstract":"We define and study the motive of the moduli stack of vector bundles of fixed rank and degree over a smooth projective curve in Voevodsky’s category of motives. We prove that this motive can be written as a homotopy colimit of motives of smooth projective Quot schemes of torsion quotients of sums of line bundles on the curve. When working with rational coefficients, we prove that the motive of the stack of bundles lies in the localizing tensor subcategory generated by the motive of the curve, using Białynicki-Birula decompositions of these Quot schemes. We conjecture a formula for the motive of this stack, inspired by the work of Atiyah and Bott on the topology of the classifying space of the gauge group, and we prove this conjecture modulo a conjecture on the intersection theory of the Quot schemes.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 1-2","pages":"71-114"},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmathj/haaa023","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49950827","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The construction of Atiyah, Drinfeld, Hitchin and Manin provided complete description of all instantons on Euclidean four-space. It was extended by Kronheimer and Nakajima to instantons on ALE spaces, resolutions of orbifolds $mathbb{R}^4/Gamma$ by a finite subgroup Γ⊂SU(2). We consider a similar classification, in the holomorphic context, of instantons on some of the next spaces in the hierarchy, the ALF multi-Taub-NUT manifolds, showing how they tie in to the bow solutions to Nahm’s equations via the Nahm correspondence. Recently Nakajima and Takayama constructed the Coulomb branch of the moduli space of vacua of a quiver gauge theory, tying them to the same space of bow solutions. One can view our construction as describing the same manifold as the Higgs branch of the mirror gauge theory as described by Cherkis, O’Hara and Saemann. Our construction also yields the monad construction of holomorphic instanton bundles on the multi-Taub-NUT space for any classical compact Lie structure group.
{"title":"Instantons and Bows for the Classical Groups","authors":"Sergey A Cherkis;Jacques Hurtubise","doi":"10.1093/qmath/haaa034","DOIUrl":"https://doi.org/10.1093/qmath/haaa034","url":null,"abstract":"The construction of Atiyah, Drinfeld, Hitchin and Manin provided complete description of all instantons on Euclidean four-space. It was extended by Kronheimer and Nakajima to instantons on ALE spaces, resolutions of orbifolds \u0000<tex>$mathbb{R}^4/Gamma$</tex>\u0000 by a finite subgroup Γ⊂SU(2). We consider a similar classification, in the holomorphic context, of instantons on some of the next spaces in the hierarchy, the ALF multi-Taub-NUT manifolds, showing how they tie in to the bow solutions to Nahm’s equations via the Nahm correspondence. Recently Nakajima and Takayama constructed the Coulomb branch of the moduli space of vacua of a quiver gauge theory, tying them to the same space of bow solutions. One can view our construction as describing the same manifold as the Higgs branch of the mirror gauge theory as described by Cherkis, O’Hara and Saemann. Our construction also yields the monad construction of holomorphic instanton bundles on the multi-Taub-NUT space for any classical compact Lie structure group.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 1-2","pages":"339-386"},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmath/haaa034","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49950709","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose a new axiom system for unitary quantum field theories on curved space-time backgrounds, by postulating that the partition function and the correlators extend analytically to a certain domain of complex-valued metrics. Ordinary Riemannian metrics are contained in the allowable domain, while Lorentzian metrics lie on its boundary.
{"title":"Wick Rotation and the Positivity of Energy in Quantum Field Theory","authors":"Maxim Kontsevich;Graeme Segal","doi":"10.1093/qmath/haab027","DOIUrl":"https://doi.org/10.1093/qmath/haab027","url":null,"abstract":"We propose a new axiom system for unitary quantum field theories on curved space-time backgrounds, by postulating that the partition function and the correlators extend analytically to a certain domain of complex-valued metrics. Ordinary Riemannian metrics are contained in the allowable domain, while Lorentzian metrics lie on its boundary.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 1-2","pages":"673-699"},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49950718","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It is well known that there are no stable bundles of rank greater than 1 on the projective line. In this paper, our main purpose is to study the existence problem for stable coherent systems on the projective line when the number of sections is larger than the rank. We include a review of known results, mostly for a small number of sections.
{"title":"Coherent Systems on the Projective Line","authors":"Peter Newstead;Montserrat Teixidor i Bigas","doi":"10.1093/qmathj/haaa024","DOIUrl":"https://doi.org/10.1093/qmathj/haaa024","url":null,"abstract":"It is well known that there are no stable bundles of rank greater than 1 on the projective line. In this paper, our main purpose is to study the existence problem for stable coherent systems on the projective line when the number of sections is larger than the rank. We include a review of known results, mostly for a small number of sections.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 1-2","pages":"115-136"},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmathj/haaa024","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49950826","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Consider a smooth projective 3-fold $X$ satisfying the Bogomolov–Gieseker conjecture of Bayer–Macrì–Toda (such as ${mathbb{P}}^3$ , the quintic 3-fold or an abelian 3-fold). Let $L$ be a line bundle supported on a very positive surface in $X$ . If $c_1(L)$ is a primitive cohomology class, then we show it has very negative square.
{"title":"An Application of Wall-Crossing to Noether–Lefschetz Loci","authors":"S Feyzbakhsh;R P Thomas;C Voisin","doi":"10.1093/qmathj/haaa022","DOIUrl":"https://doi.org/10.1093/qmathj/haaa022","url":null,"abstract":"Consider a smooth projective 3-fold \u0000<tex>$X$</tex>\u0000 satisfying the Bogomolov–Gieseker conjecture of Bayer–Macrì–Toda (such as \u0000<tex>${mathbb{P}}^3$</tex>\u0000, the quintic 3-fold or an abelian 3-fold). Let \u0000<tex>$L$</tex>\u0000 be a line bundle supported on a very positive surface in \u0000<tex>$X$</tex>\u0000. If \u0000<tex>$c_1(L)$</tex>\u0000 is a primitive cohomology class, then we show it has very negative square.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 1-2","pages":"51-70"},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmathj/haaa022","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49950707","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper, completed in its present form by the second author after the first author passed away in 2019, describes an intended continuation of the previous joint work on anyons in geometric models of matter. This part outlines a construction of anyon tensor networks based on four-dimensional orbifold geometries and braid representations associated with surface-braids defined by multisections of the orbifold normal bundle of the surface of orbifold points.
{"title":"Anyon Networks from Geometric Models of Matter","authors":"Michael Atiyah;Matilde Marcolli","doi":"10.1093/qmath/haab004","DOIUrl":"https://doi.org/10.1093/qmath/haab004","url":null,"abstract":"This paper, completed in its present form by the second author after the first author passed away in 2019, describes an intended continuation of the previous joint work on anyons in geometric models of matter. This part outlines a construction of anyon tensor networks based on four-dimensional orbifold geometries and braid representations associated with surface-braids defined by multisections of the orbifold normal bundle of the surface of orbifold points.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 1-2","pages":"717-733"},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmath/haab004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49950715","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article is an introduction to newly discovered relations between volumes of moduli spaces of Riemann surfaces or super Riemann surfaces, simple models of gravity or supergravity in two dimensions, and random matrix ensembles. (The article is based on a lecture at the conference on the Mathematics of Gauge Theory and String Theory, University of Auckland, January 2020)
{"title":"Volumes And Random Matrices","authors":"Edward Witten","doi":"10.1093/qmath/haaa035","DOIUrl":"https://doi.org/10.1093/qmath/haaa035","url":null,"abstract":"This article is an introduction to newly discovered relations between volumes of moduli spaces of Riemann surfaces or super Riemann surfaces, simple models of gravity or supergravity in two dimensions, and random matrix ensembles. (The article is based on a lecture at the conference on the Mathematics of Gauge Theory and String Theory, University of Auckland, January 2020)","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 1-2","pages":"701-716"},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmath/haaa035","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49950719","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let u be a unipotent element in the totally positive part of a complex reductive group. We consider the intersection of the Springer fibre at u with the totally positive part of the flag manifold. We show that this intersection has a natural cell decomposition which is part of the cell decomposition (Rietsch) of the totally positive flag manifold.
{"title":"Total Positivity in Springer Fibres","authors":"G Lusztig","doi":"10.1093/qmathj/haaa021","DOIUrl":"https://doi.org/10.1093/qmathj/haaa021","url":null,"abstract":"Let u be a unipotent element in the totally positive part of a complex reductive group. We consider the intersection of the Springer fibre at u with the totally positive part of the flag manifold. We show that this intersection has a natural cell decomposition which is part of the cell decomposition (Rietsch) of the totally positive flag manifold.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 1-2","pages":"31-49"},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmathj/haaa021","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49950819","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give a characterization of Atiyah’s and Hitchin’s transverse Hilbert schemes of points on a symplectic surface in terms of bi-Poisson structures. Furthermore, we describe the geometry of hyperkähler manifolds arising from the transverse Hilbert scheme construction, with particular attention paid to the monopole moduli spaces.
{"title":"Transverse hilbert schemes, bi-hamiltonian systems and hyperkähler geometry","authors":"Roger Bielawski","doi":"10.1093/qmath/haaa059","DOIUrl":"https://doi.org/10.1093/qmath/haaa059","url":null,"abstract":"We give a characterization of Atiyah’s and Hitchin’s transverse Hilbert schemes of points on a symplectic surface in terms of bi-Poisson structures. Furthermore, we describe the geometry of hyperkähler manifolds arising from the transverse Hilbert scheme construction, with particular attention paid to the monopole moduli spaces.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 1-2","pages":"237-252"},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49950822","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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