{"title":"Lectures on BPS states and spectral networks","authors":"Andrew Neitzke","doi":"10.1090/pcms/028/08","DOIUrl":"https://doi.org/10.1090/pcms/028/08","url":null,"abstract":"These are notes for a lecture series on BPS states and spectral networks, delivered at Park City Mathematics Institute, July 2019. The first part is a general review of the notions of BPS state and BPS index. The second part discusses the specific case of BPS states in N = (2, 2) supersymmetric field theories in two dimensions, and introduces the notion of spectral network as a way of computing the BPS indices in that context. The last part discusses the more general case of 2d and 4d BPS indices associated to surface defects in four-dimensional field theories of class S. 1. Lecture 1: What is a BPS state? BPS states appear very frequently in geometric applications of quantum field theory. The aim of this lecture is to explain rather generally what a BPS state is and some of their basic properties. In one sentence: we’ll study a representation H = H0 ⊕H1 of a certain super Lie algebra A = A0 ⊕A1, and the BPS states are the ones in irreps annihilated by nontrivial subspaces of A1. 1.1. Quantum mechanics A time-independent quantum system involves the following data: • A Hilbert space H, • A formally self-adjoint operator H : H→ H. We think of iH as generating the abelian Lie algebra (1.1.1) Lie(ISO(0, 1)) = Lie(Isom(R0,1)) ' R. Eigenvectors ofH are called bound states; each bound state thus spans a 1-dimensional irreducible representation of ISO(0, 1). The fact that H is formally self-adjoint implies that this representation is unitary. The eigenvalues of H are called bound state energies. Example 1.1.2 (Particle on the line). In your first course on quantum mechanics you study the “particle on the line.” For this example you need to fix a function V : R→ R. Then there is a time-independent quantum system T1[R,V] with: 2010 Mathematics Subject Classification. Primary ????; Secondary ????","PeriodicalId":170247,"journal":{"name":"Quantum Field Theory and Manifold Invariants","volume":"74 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122090182","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}
{"title":"Knots, polynomials, and categorification","authors":"J. Rasmussen","doi":"10.1090/pcms/028/02","DOIUrl":"https://doi.org/10.1090/pcms/028/02","url":null,"abstract":"","PeriodicalId":170247,"journal":{"name":"Quantum Field Theory and Manifold Invariants","volume":"62 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133181269","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}
{"title":"Gauge theory and a few applications to knot theory","authors":"T. Mrowka, Donghao Wang","doi":"10.1090/pcms/028/05","DOIUrl":"https://doi.org/10.1090/pcms/028/05","url":null,"abstract":"","PeriodicalId":170247,"journal":{"name":"Quantum Field Theory and Manifold Invariants","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123489109","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}
{"title":"Topological quantum field theories, knots and BPS states","authors":"P. Putrov","doi":"10.1090/pcms/028/07","DOIUrl":"https://doi.org/10.1090/pcms/028/07","url":null,"abstract":"","PeriodicalId":170247,"journal":{"name":"Quantum Field Theory and Manifold Invariants","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115891583","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}
{"title":"Lecture notes on Heegaard Floer homology","authors":"Jennifer Hom","doi":"10.1090/pcms/028/03","DOIUrl":"https://doi.org/10.1090/pcms/028/03","url":null,"abstract":"These are the lecture notes for a course on Heegaard Floer homology held at PCMI in Summer 2019. We describe Heegaard diagrams, Heegaard Floer homology, knot Floer homology, and the relationship between the knot and 3-manifold invariants.","PeriodicalId":170247,"journal":{"name":"Quantum Field Theory and Manifold Invariants","volume":"6 3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122983716","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}
{"title":"Lecture on invertible field theories","authors":"A. Debray, Søren Galatius, Martin Palmer","doi":"10.1090/pcms/028/06","DOIUrl":"https://doi.org/10.1090/pcms/028/06","url":null,"abstract":"Four lectures on invertible field theories at the Park City Mathematics Institute 2019. Cobordism categories are introduced both as plain categories and topologically enriched. We then discuss localization of categories and its relationship to classifying spaces, and state the main theorem of classification of invertible field theories in these terms. We also discuss symmetric monoidal structures and their relationship to actions of the little disk operads. In the final lecture we discuss an application of cobordism categories to characteristic classes of surface bundles. Emphasis will be on self-contained definitions and statements, referring to original literature for proofs. We also include problem sets from the three exercise sessions at the summer school, and solutions to two problems.","PeriodicalId":170247,"journal":{"name":"Quantum Field Theory and Manifold Invariants","volume":"9 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128006926","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}
{"title":"Introduction to gauge theory","authors":"Andriy Haydys","doi":"10.1090/pcms/028/01","DOIUrl":"https://doi.org/10.1090/pcms/028/01","url":null,"abstract":"This is lecture notes for a course given at the PCMI Summer School \"Quantum Field Theory and Manifold Invariants\" (July 1 -- July 5, 2019). I describe basics of gauge-theoretic approach to construction of invariants of manifolds. The main example considered here is the Seiberg--Witten gauge theory. However, I tried to present the material in a form, which is suitable for other gauge-theoretic invariants too.","PeriodicalId":170247,"journal":{"name":"Quantum Field Theory and Manifold Invariants","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129306857","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}
{"title":"Advanced topics in gauge theory: Mathematics and Physics of Higgs bundles","authors":"L. Schaposnik","doi":"10.1090/pcms/028/04","DOIUrl":"https://doi.org/10.1090/pcms/028/04","url":null,"abstract":"These notes have been prepared as reading material for the mini-course given by the author at the \"2019 Graduate Summer School\" at Park City Mathematics Institute - Institute for Advanced Study. We begin by introducing Higgs bundles and their main properties (Lecture 1), and then we discuss the Hitchin fibration and its different uses (Lecture 2). The second half of the course is dedicated to studying different types of subspaces (branes) of the moduli space of complex Higgs bundles, their appearances in terms of flat connections and representations (Lecture 3), as well as correspondences between them (Lecture 4).","PeriodicalId":170247,"journal":{"name":"Quantum Field Theory and Manifold Invariants","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130332541","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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