Higher Categorical Structures in Geometry General Theory and Applications to Quantum Field Theory Dissertation zur Erlangung des Doktorgrades der Fakult¨atf¨urMathematik, Informatik und Naturwissenschaften der Universit¨atHamburg vorgelegt im Fachbereich Mathematik von Thomas Nikolaus aus Esslingen am Neckar Hamburg 2011 Als Dissertation angenommen vom Fachbereich Mathematik der Universit¨atHamburg Auf Grund der Gutachten von Prof. Dr. Christoph Schweigert und Prof. Dr. Chenchang Zhu Hamburg, den 29.06.2011 Prof. Dr. Ingenuin Gasser Leiter des Fachbereichs Mathematik Contents Introduction vii Higher categorical structures in geometry . vii Surface Holonomy and the Wess-Zumino Term . x String structures and supersymmetric sigma models . xii Chiral CFT and Dijkgraaf-Witten theory . xiv Summary of results . xvi Outline of the thesis . xvi Acknowledgements . xxi 1 Bundle Gerbes and Surface Holonomy 1 1.1 Hermitian line bundles and holonomy . 1 1.2 Gerbes and surface holonomy . 4 1.2.1 Descent of bundles . 4 1.2.2 Bundle gerbes . 6 1.2.3 Surface holonomy . 9 1.2.4 Wess-Zumino terms . 10 1.3 The representation theoretic formulation of RCFT . 11 1.3.1 Sigma models . 11 1.3.2 Rational conformal field theory . 13 1.3.3 The TFT construction of full RCFT . 14 1.4 Jandl gerbes: Holonomy for unoriented surfaces . 17 1.5 D-branes: Holonomy for surfaces with boundary . 21 1.6 Bi-branes: Holonomy for surfaces with defect lines . 22 1.6.1 Gerbe bimodules and bi-branes . 22 1.6.2 Holonomy and Wess-Zumino term for defects . 24 1.6.3 Fusion of defects . 25 2 Equivariance in Higher Geometry 29 2.1 Overview . 29 2.2 Sheaves on Lie groupoids . 32 2.2.1 Lie groupoids . 32 2.2.2 Presheaves in bicategories on Lie groupoids . 34 iii iv CONTENTS 2.2.3 Open coverings versus surjective submersions . 40 2.3 The plus construction . 42 2.4 Applications of the plus construction . 44 2.4.1 Bundle gerbes . 44 2.4.2 Jandl gerbes . 47 2.4.3 Unoriented surface holonomy . 51 2.4.4 Kapranov-Voevodsky 2-vector bundles . 56 2.5 Proof of theorem 2.2.16, part 1: Factorizing morphisms . 57 2.5.1 Strong equivalences . 57 2.5.2 τ-surjective equivalences . 59 2.5.3 Factorization . 60 2.6 Proof of theorem 2.2.16, part 2: Sheaves and strong equivalences . 61 2.7 Proof of theorem 2.2.16, part 3: Equivariant descent . 63 2.8 Proof of theorem 2.2.16, part 4: Sheaves and τ - surjective equivalences 67 2.9 Proof of the theorem 2.3.3 . 70 3 Four Equivalent Versions of Non-Abelian Gerbes 73 3.1 Outline of the chapter . 73 3.2 Preliminaries . 76 3.2.1 Lie Groupoids and Groupoid Actions on Manifolds . 76 3.2.2 Principal Groupoid Bundles . 77 3.2.3 Anafunctors . 80 3.2.4 Lie 2-Groups and crossed Modules . 84 3.3 Version I: Groupoid-valued Cohomology . 87 3.4 Version II: Classifying Maps . 89 3.5 Version III: Groupoid Bundle Gerbes . 92 3.5.1 Definition via the Plus Construction . 92 3.5.2 Properties of Groupoid Bundle Gerbes . 98 3.5.3 Classification by Cechˇ Cohomology . 102 3.6 Version IV: Principal 2-Bundles . 104 3.6.1 Definition of Principal 2-Bundles . 104 3.6.2 Properties of Principal 2-Bundles . 107 3.7 Equivalence between Bundle Gerbes and 2-Bundles . 109 3.7.1 From Principal 2-Bundles to Bundle Gerbes . 110 3.7.2 From Bundle Gerbes to Principal 2-Bundles . 120 3.8 Appendix . 128 3.8.1 Appendix: Equivariant Anafunctors and Group Actions . 128 3.8.2 Appendix: Equivalences between 2-Stacks . 131 4 A Smooth Model for the String Group 135 4.1 Recent and new models . 135 4.2 Preliminaries on gauge groups . 138 CONTENTS v 4.3 The string group as a smooth extension of G . 142 4.4 2-groups and 2-group models . 146 4.5 The string group as a 2-group . 150 4.6 Comparison of string structures . 155 4.7 Appendix: Locally convex manifolds and Lie groups . 157 4.8 Appendix: A characterization of smooth weak equivalences . 159 5 Equivariant Modular Categories via Dijkgraaf-Witten Theory 163 5.1 Motivation . 163 5.1.1 Algebraic motivation: equivariant modular categories . 163 5.1.2 Geometric motivation: equivariant extended TFT . 165 5.1.3 Summary of the results . 166 5.2 Dijkgraaf-Witten theory and Drinfel'd double . 167 5.2.1 Motivation for Dijkgraaf-Witten theory . 168 5.2.2 Dijkgraaf-Witten theory as an extended TFT . 172 5.2.3 Construction via 2-linearization . 174 5.2.4 Evaluation on the circle . 177 5.2.5 Drinfel'd double and modularity . 179 5.3 Equivariant Dijkgraaf-Witten theory . 180 5.3.1 Weak actions and extensions . 181 5.3.2 Twisted bundles . 182 5.3.3 Equivariant Dijkgraaf-Witten theory . 186 5.3.4 Construction via spans . 187 5.3.5 Twisted sectors and fusion . 190 5.4 Equivariant Drinfel'd double . 195 5.4.1 Equivariant fusion categories. 195 5.4.2 Equivariant ribbon algebras . 199 5.4.3 Equivariant Drinfel'd Double . 203 5.4.4 Orbifold category and orbifold algebra . 205 5.4.5 Equivariant modular categories . 209 5.4.6 Summary of all tensor categories involved . 211 5.5 Outlook . 212 5.6 Appendix . 213 5.6.1 Appendix: Cohomological description of twisted bundles . 213 5.6.2 Appendix: Character theory for action groupoids . 215 Bibliography 229 vi CONTENTS Introduction Higher categorical structures in geometry The following situation arises frequently in mathematics and mathematical physics: for a given smooth, finite dimensional manifold M we want to consider certain classes of geometric objects on M. The reader should keep in mind structures like metrics or symplectic forms or, more important for this thesis, objects like bundles. There are many reasons that one is interested in such objects, let us list two here: • One wants to gather information about the structure of M as a manifold. For example one can use a metric to compute holonomy groups and thereby better understand the global and local behavior of M. Another typical situation is to compute the set of isomorphism classes of G-bundles over M for a fixed Lie group G. This turns out to be an invariant of the homotopy type of M, hence can be used to distinguish manifolds that are not homotopy equivalent. • One is interested in the objects over M itself. This situation especially occurs in mathematical physics. For example in general relativity the object of interest is not the mere spacetime manifold M but a Lorentzian metric on M. Another class of examples is given by gauge theories, such as Yang-Mills-theory. The fields are given by connections on (non-abelian) bundles over M. Such fields can also play the role of background fields. For example the electromagnetic field in classical electromagnetism is given by a U(1)-bundle with connection over M that determines the equations of motion for charged particles moving through M. For bundles it is very important not only to consider the geometric objects over M, but also to take the morphisms into account, i.e. the gauge transformations. This shows that we really associate categories of objects to M. Now we do not want to restrict ourselves to one fixed manifold M, but allow different manifolds. Therefore we have to take the transformation-behavior of the geometric objects into account. More precisely we want to specialize to geometric objects that behave like bundles in so far as they can be pulled back along smooth maps f : N / M. The mathematical structure that formalizes this behavior is called a stack, see [Met03, Hei05] for a definition in the differentiable setting. Apart vii viii Introduction from associating categories to smooth manifolds and pullback functors to smooth maps, a stack has another important defining property that turns out to be crucial for geometry and central for this thesis. Namely it has to satisfy a `locality condi- tion' called the descent property. Roughly speaking this property ensures that the geometric objects can be glued together from locally defined objects. If we think of bundles again this property is clearly satisfied and can be seen as a guiding principle since the local behavior of bundles is prescribed by definition, i.e. locally they look like a product of M with a vector space, manifold, torsor etc. For a more precise discussion in the case of U(1)-bundles see section 1.2.1. In the past years it has turned out that there are certain geometric objects over M for which we do not only have to take morphisms into account, but also 2-morphisms, i.e. gauge transformations between gauge transformations. Let us give two guiding examples here: • An important class of such objects is given by bundle gerbes and bundle gerbes with connection [Bry93, Mur96, Ste00, Wal07]. See also section 1.2.2 and 2.4.1 of this thesis. In particular bundle gerbes and related objects are needed in two-dimensional non-linear sigma models with Wess-Zumino term. The role they play is analogous to the role of U(1)-bundles with connection in electromagnetism. From the mathematical side, the feature of bundle gerbes (resp. Jandl gerbes) entering here is that they allow to define surface holonomy (resp. unoriented surface holonomy). We will explain that in more detail in the next part of this introduction and in chapter 1. • Another class of examples is given by 2-principal bundles for 2-groups [Bar04, Woc08].