Pub Date : 2018-06-26DOI: 10.1142/9789813229099_0001
Olivier Y Guichard
{"title":"An Introduction to the Differential Geometry of Flat Bundles and of Higgs Bundles","authors":"Olivier Y Guichard","doi":"10.1142/9789813229099_0001","DOIUrl":"https://doi.org/10.1142/9789813229099_0001","url":null,"abstract":"","PeriodicalId":166610,"journal":{"name":"The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125200796","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}
Pub Date : 2017-08-27DOI: 10.1142/9789813229099_0004
Sebastian Casalaina-Martin, Jonathan Wise
This article is based in part on lecture notes prepared for the summer school "The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles" at the Institute for Mathematical Sciences at the National University of Singapore in July of 2014. The aim is to provide a brief introduction to algebraic stacks, and then to give several constructions of the moduli stack of Higgs bundles on algebraic curves. The first construction is via a "bootstrap" method from the algebraic stack of vector bundles on an algebraic curve. This construction is motivated in part by Nitsure's GIT construction of a projective moduli space of semi-stable Higgs bundles, and we describe the relationship between Nitsure's moduli space and the algebraic stacks constructed here. The third approach is via deformation theory, where we directly construct the stack of Higgs bundles using Artin's criterion.
{"title":"An Introduction to Moduli Stacks, with a View towards Higgs Bundles on Algebraic Curves","authors":"Sebastian Casalaina-Martin, Jonathan Wise","doi":"10.1142/9789813229099_0004","DOIUrl":"https://doi.org/10.1142/9789813229099_0004","url":null,"abstract":"This article is based in part on lecture notes prepared for the summer school \"The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles\" at the Institute for Mathematical Sciences at the National University of Singapore in July of 2014. The aim is to provide a brief introduction to algebraic stacks, and then to give several constructions of the moduli stack of Higgs bundles on algebraic curves. The first construction is via a \"bootstrap\" method from the algebraic stack of vector bundles on an algebraic curve. This construction is motivated in part by Nitsure's GIT construction of a projective moduli space of semi-stable Higgs bundles, and we describe the relationship between Nitsure's moduli space and the algebraic stacks constructed here. The third approach is via deformation theory, where we directly construct the stack of Higgs bundles using Artin's criterion.","PeriodicalId":166610,"journal":{"name":"The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115385255","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}
Pub Date : 2015-09-30DOI: 10.1142/9789813229099_0003
Olivia Dumitrescu, M. Mulase
The paper aims at giving an introduction to the notion of quantum curves. The main purpose is to describe the new discovery of the relation between the following two disparate subjects: one is the topological recursion, that has its origin in random matrix theory and has been effectively applied to many enumerative geometry problems; and the other is the quantization of Hitchin spectral curves associated with Higgs bundles. Our emphasis is on explaining the motivation and examples. Concrete examples of the direct relation between Hitchin spectral curves and enumeration problems are given. A general geometric framework of quantum curves is also discussed.
{"title":"Lectures on the Topological Recursion for Higgs Bundles and Quantum Curves","authors":"Olivia Dumitrescu, M. Mulase","doi":"10.1142/9789813229099_0003","DOIUrl":"https://doi.org/10.1142/9789813229099_0003","url":null,"abstract":"The paper aims at giving an introduction to the notion of quantum curves. The main purpose is to describe the new discovery of the relation between the following two disparate subjects: one is the topological recursion, that has its origin in random matrix theory and has been effectively applied to many enumerative geometry problems; and the other is the quantization of Hitchin spectral curves associated with Higgs bundles. Our emphasis is on explaining the motivation and examples. Concrete examples of the direct relation between Hitchin spectral curves and enumeration problems are given. A general geometric framework of quantum curves is also discussed.","PeriodicalId":166610,"journal":{"name":"The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130550530","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}
Pub Date : 2014-08-02DOI: 10.1142/9789813229099_0002
L. Schaposnik
These notes have been prepared as reading material for the mini-course that the author gave at IMS, National University of Singapore, as part of the "Summer school on the moduli space of Higgs bundles".
{"title":"An Introduction to Spectral Data for Higgs Bundles","authors":"L. Schaposnik","doi":"10.1142/9789813229099_0002","DOIUrl":"https://doi.org/10.1142/9789813229099_0002","url":null,"abstract":"These notes have been prepared as reading material for the mini-course that the author gave at IMS, National University of Singapore, as part of the \"Summer school on the moduli space of Higgs bundles\".","PeriodicalId":166610,"journal":{"name":"The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134346012","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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