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Algebraic Defects and Boundaries in Integrable/Topological Systems Algebraic Defects and Boundaries in Integrable & Topological Systems by Ammar S Husain A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Nicolai Reshetikhin, Chair Professor Joel Moore, Co-Chair Professor Jonathan Wurtele Professor Alan Weinstein Fall 2017 Algebraic Defects and Boundaries in Integrable & Topological Systems Copyright 2017 by Ammar S Husain 1 Abstract Algebraic Defects and Boundaries in Integrable & Topological Systems by Ammar S Husain Doctor of Philosophy in Physics University of California, Berkeley Professor Nicolai Reshetikhin, Chair Professor Joel Moore, Co-Chair Topological and conformal field theories and integrable systems can be described by the algebraic structures of quantum groups and quantum affine algebras. Boundary conditions and defects for these theories are described via algebraic constructions from these quantum groups or quantum affine algebras. In the first third of this thesis we describe constructions associated with three dimensional topological field theories. First we compute some Brauer-Picard groups which characterize nontrivial invertible structures that can be assigned using the cobordism hypothesis with singularities. Then we give a construction of bimodule categories using the procedure of covariantization(transmutation) of coquasitriangular Hopf algebras. The first third closes with a reduction by taking K0 which results in algebraic K-theory of fusion rings. The next third describes spin chains which are governed by quantum affine algebras. The first chapter gives a proof of the asymptotic completeness of the algebraic Bethe Ansatz for the XXZ chain. Then we describe a similar reduction of taking K0 which results in algebraic K-theory of cluster algebras. The last third of this thesis describes the appearance of these defects in topological field theories constructed by the AKSZ procedure. In the three dimensional case, this gives perturbative perspectives on the theories covered in the first third of the thesis. The last chapter gives the connection with topological insulators as they fit into the general AKSZ paradigm. i Contents Contents i I Introduction1 1 Background2 1.1 Three Dimensional Topological ......................... 3 1.2 Spin Chains.................................... 5 1.3 AKSZ Formalism................................. 6 2 Overview of Dissertation7 2.1 3d Topological................................... 8 2.2 Bethe Ansatz 1+1................................. 8 2.3 AKSZ Formalism................................. 9 II Three Dimensional Topological 10 3 G-Extensions and Group Cohomology 11 3.1 Introduction.................................... 12 3.2 The essential bit of 1 Categories........................ 12 3.3 Brauer-Picard................................... 14 3.4 M¨ugerstyle G equivariantization and quasi-trivial extensions......... 16 3.5 Hopf Algebra Transmutation........................... 19 3.6 Sphericality and Anomalies ........................... 20 3.7 Quantum Group Example ............................ 20 3.8 Conclusion..................................... 22 3.9 Appendix: Full Computation .......................... 23 3.10 SO(3) fixed points ................................ 23 3.11 Replicas and Sn fixed points........................... 23 4 Bimodule Categories from Covariantization 27 4.1 Transmutation .................................. 28 4.2 Sweedler Algebra................................. 30 4.3 Logarithmic CFT................................. 32 ii 5 K-Theory of Fusion Rings 35 5.1 Introduction.................................... 36 5.2 K0 Reduction................................... 36 5.3 Algebraic K-theory review............................ 37 5.4 Examples ..................................... 38 5.5 Conclusion..................................... 41 III Spin Chains 42 6 Asymptotic Completeness 43 6.1 Introduction.................................... 44 6.2 The Six Vertex Model with Reflecting Boundary................ 44 6.3 Asymptotic Structure of Solutions to Bethe Equations............. 47 6.4 Asymptotics of Bethe Vectors .......................... 49 6.5 Conclusion..................................... 55 7 Cluster Algebras 56 7.1 Introduction.................................... 57 7.2 Cluster Categories and Algebras......................... 57 7.3 Bimodule Categories ............................... 60 7.4 Picard Groups and K0 .............................. 60 7.5 P ic and K0 of these cluster algebras....................... 63 7.6 Semiclassical Geometry.............................. 66 7.7 Conclusion..................................... 67 8 Asymptotic Representations 69 8.1 Uqb ......................................... 70 8.2 Unzipping inhomogeneities............................ 71 IV AKSZ formalism 72 9 Generalities to AKSZ Formalism 73 9.1 Supergeometry .................................. 74 9.2 Shifted Symplectic Structures .......................... 75 9.3 AKSZ Construction................................ 75 9.4 Shifted Lagrangian Correspondences ...................... 76 9.5 Gauge Fixing Lagrangians............................ 76 10 1+1D and 2+1D Examples 78 10.1 1+1 Dimensions ................................. 79 10.2 2+1 Dimensions.................................. 81 11 3+1D Topological Insulators 83 11.1 Homological Degree 0 Argument......................... 84 iii 11.2 Bulk Classical 4 dimensional abelian BF theory ................ 85 11.3 Codimension 1 Strata............................... 90 11.4 Codimension 2 Strata............................... 92 Bibliography 94 iv Acknowledgments First and foremost, appreciation go to my advisor, Nicolai Reshetikhin. I also owe a great deal to my collaborators Jon Aytac, Ilyas Bayramov and Ivan Contreras. I would also like to thank my academic siblings Ananth Sridhar, Gus Schrader, Alexander Shapiro, Harold Williams and Theo Johnson-Freyd. I would also like to thank discussions with Alex Takeda, Benjamin Gammage, Erik Aldape, Eugene Kur and Ryan Thorngren at many various stages. 1 Part I Introduction 2 Chapter 1 Background CHAPTER 1. BACKGROUND 3 Quantum groups and quantum affine algebras provide a useful tool for topological and conformal field theories and integrable systems. This is done via state-sum constructions [1], modular functors [2] and the algebraic Bethe ansatz [3,4]. Each chapter covers systems of different spacetime dimensions. Let us introduce the background for each of these systems in the simplified cases without boundaries or any other codimension 1 strata (defects). 1.1 Three Dimensional Topological The prototypical examples of systems studied by three dimensional topological field the- ories are the fractional quantum Hall effect at various levels. These sorts of topological field theories can be constructed to almost meet the Atiyah-Segal axioms [5] by the method of [1]. This thesis will mostly focus on the doubled versions which avoid this complication. This means the systems of concern will be more analogous to the spin Hall effect instead. Levin Wen Model The Levin Wen model provides a Hamiltonian realization for a Turaev-Viro model. It assigns a Hilbert space and Hamiltonian to a Riemann surface time slice. It is built from an arbitrary unitary spherical fusion category C[6]. From this data a lattice model is built such that the bulk excitations are given by simple objects of the Drinfeld center Z(C). The model is defined on a trivalent graph which is dual to a triangulation of the Riemann surface. The Hilbert space assigned to this graph is given by the finite dimensional Hilbert space spanned by the basis vectors which are labellings of edges by simple objects of the category and vertices by a basis vector of the multiplicity space for the fusion of the two incoming edges into the outgoing edge. For other orientations, this is corrected by dualizing and reversing arrows. The Hamiltonian is then cooked up with projectors as a sum on vertices and plaquettes illustrated in fig. 1.1.[7] X X H = (1 − Qv) + (1 − Bp) v p X dk B = Bk p D2 p k2I 2 X 2 D = di i2I k where Bp inserts a loop decorated by the simple object k and evaluates the resulting mor- phism as illustrated in fig. 1.2 and the dk are quantum dimensions of the specified simple objects. On the basis vectors, the first term projects to ensure that the fusion of objects on the incoming edges gives the fusion of the objects on the outgoing edges. 1.1.1 Definition The Drinfeld Center of a monoidal category C is the monoidal category of endo-pseudonatural transformations of the identity 2-functor on BC which comes from the identity functor on C. Unwinding this definition results in first of all an isomorphism from CHAPTER 1. BACKGROUND 4 Figure 1.1: Example of a plaquette in the model where i label simple objects and α in multiplicity spaces. s Figure 1.2: The action of a Bp on a plaquette id(pt) ! id(pt). That is given by a morphism of BC aka an object X of C. And also the squares need to have a filling 2-morphism φY a.k.a a morphism in C that does X⊗Y ! Y ⊗X which has to be natural in Y. This provides a model for a 2+1 dimensional field theory. The particles are in the form of constraint violations are labeled by Z(C) because the excitations are assigned to the circles. This is where the fact that the center of a spherical category gives a modular category has appeared. For further details on this relationship see [8]. Note that other lattice models provide realizations of the
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