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Supersymmetric Field Theories and Orbifold Cohomology

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Supersymmetric Field Theories and Orbifold Cohomology SUPERSYMMETRIC FIELD THEORIES AND ORBIFOLD COHOMOLOGY A Dissertation Submitted to the Graduate School of the University of Notre Dame in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Augusto Stoffel Stephan Stolz, Director Graduate Program in Mathematics Notre Dame, Indiana April 2016 SUPERSYMMETRIC FIELD THEORIES AND ORBIFOLD COHOMOLOGY Abstract by Augusto Stoffel Using the Stolz–Teichner framework of supersymmetric Euclidean field theories (EFTs), we provide geometric interpretations of some aspects of the algebraic topology of orbifolds. We begin with a classification of 0j1-dimensional twists for EFTs over an orbifold X, and show that the collection of concordance classes of twisted EFTs over the inertia ΛX is in natural bijection with the delocalized twisted cohomology of X (which is isomorphic to its complexified K-theory). Then, turning to 1j1-dimensional considerations, we construct a (partial) twist functor over X taking as input a class in H3(X; Z). Next, we define a dimensional reduction procedure relating the 0j1-dimensional Euclidean bordism category over ΛX and its 1j1-dimensional counterpart over X, and explore some applications. As a basic example, we show that dimensional reduction of untwisted EFTs over a global quotient orbifold X==G recovers the equivariant Chern character. Finally, we describe the dimensional reduction of the 1j1-twist built earlier, showing that it has the expected relation to twisted K-theory. CONTENTS ACKNOWLEDGMENTS . iv CHAPTER 1: INTRODUCTION . 1 1.1 Supersymmetric field theories and cohomology theories . 1 1.2 Field theories over orbifolds . 4 1.3 Outline of the dissertation . 7 CHAPTER 2: STACKS IN DIFFERENTIAL GEOMETRY . 9 2.1 Sheaves and stacks on a site . 9 2.1.1 Fibered categories and presheaves . 10 2.1.2 Grothendieck topologies and descent . 12 2.1.3 The site of a fibration; stacks over stacks . 15 2.2 Differentiable stacks . 16 2.2.1 Lie groupoids and torsors . 17 2.2.2 Bibundles and Morita equivalences . 22 2.2.3 Differential geometry of stacks; gerbes . 24 2.2.4 Orbifolds; inertia . 27 2.3 Sheaf cohomology . 29 2.3.1 Calculating cohomology . 29 2.3.2 Deligne cohomology . 31 2.4 Group actions on stacks . 32 2.4.1 Basic definitions . 32 2.4.2 Quotient stacks of G-stacks . 36 CHAPTER 3: SUPERGEOMETRY . 42 3.1 Superalgebra . 42 3.2 Supermanifolds . 44 3.2.1 Basic definitions . 44 3.2.2 The functor of points formalism . 47 3.2.3 Calculus on supermanifolds . 48 3.2.4 Super Lie groups . 49 3.2.5 Superpoints, differential forms, and superconnections . 51 3.3 Euclidean structures . 53 3.3.1 Euclidean structures in dimension 1j1 . 54 3.3.2 Euclidean supercircles . 59 ii 3.4 Integration on supermanifolds . 65 3.4.1 The Berezin integral . 65 3.4.2 Domains with boundary . 67 3.4.3 A primitive integration theory on R1j1 . 68 CHAPTER 4: ZERO-DIMENSIONAL FIELD THEORIES AND TWISTED DE RHAM COHOMOLOGY . 71 4.1 Superpoints and differential forms . 71 4.2 Superconnections and twists . 75 4.3 Concordance of flat sections . 76 4.4 Twisted de Rham cohomology for orbifolds . 79 CHAPTER 5: TWISTS FOR ONE-DIMENSIONAL FIELD THEORIES . 82 5.1 Euclidean supercircles over an orbifold . 83 5.2 Gerbes and partial twists . 86 5.2.1 A super version of transgression . 87 5.2.2 The restriction to K1(X) ..................... 93 CHAPTER 6: DIMENSIONAL REDUCTION AND THE CHERN CHARAC- TER FOR ORBIFOLDS . 100 6.1 Dimensional reduction . 100 6.1.1 R==Z-equivariant bordisms . 101 6.1.2 T-equivariant bordisms . 103 6.1.3 The map BT(ΛX) ! BR==Z(ΛX) . 105 6.1.4 Global quotients . 110 6.1.5 More general kinds of equivariant bordisms . 113 6.2 The Chern character for global quotients . 114 6.2.1 The Baum–Connes Chern character . 114 6.2.2 Parallel transport and field theories . 115 6.2.3 Proof of theorem 6.4 . 117 6.3 Dimensional reduction of twists . 121 6.3.1 The underlying line bundle . 122 6.3.2 The superconnection . 123 BIBLIOGRAPHY . 128 iii ACKNOWLEDGMENTS First, I would like express my deep gratitude to my advisor Stephan Stolz for his guidance and patience, and for introducing me to so many exciting new ideas. Many thanks to the mathematics department at the University of Notre Dame, especially the topology and geometry groups, for providing a vibrant research envi- ronment. I profited much from mathematical discussions with my fellow graduate students, among whom I would like to single out Renato Bettiol, Ryan Grady, Santosh Kandel, Leandro Lichtenfelz, and Peter Ulrickson; I would also like to thank them for their friendship. I acknowledge my thesis committee, Mark Behrens, Liviu Nicolaescu, and Larry Taylor for their interest in my work, and I also thank Karsten Grove for the financial support during my sixth year. Finally, I would like to thank my wife and best friend Alana for her love, encour- agement, and support, as well as my family, Jacinta, Ireno, Amanda, and Iria for their constant concern and care, in spite of the distance. iv CHAPTER 1 INTRODUCTION The goal of this dissertation is to explore the relation between the algebraic topology of orbifolds and supersymmetric quantum field theories. Intuitively, orbifolds can be thought of as manifolds whose points are allowed to have a finite automorphism group. An example to keep in mind is the global quotient orbifold X==G arising from the action of a finite group G on a compact manifold X; morphisms between such orbifolds are induced not only by equivariant smooth maps, but also by group homomorphisms. Thus, cohomology theories for orbifolds are related to global equivariant homotopy theory, in the sense that we are dealing with invariants which depend functorially on X and G. We will focus on twisted K-theory and delocalized de Rham cohomology, as well as the Chern character relating them. On the field theory side, we employ the framework of supersymmetric Euclidean field theories due to Stolz and Teichner [38]. In our context, these field theories can be thought of as a way to encapsulate and generalize Chern–Weil theory. Before giving a more detailed overview of our result in section 1.2, we outline the main ideas in the Stolz–Teichner program. 1.1 Supersymmetric field theories and cohomology theories Let d-Bord be the category where objects are closed (d − 1)-manifolds and a morphism between two objects Yin;Yout is a d-manifold Σ together with an identification of its boundary with the disjoint union Yin q Yout, taken up to diffeomorphism relative to the boundary. The operation of disjoint union makes it into a symmetric monoidal 1 category. A topological quantum field theory, as defined by Atiyah [1], Segal [35], and others, is a symmetric monoidal functor Z : d-Bord ! C, where the target C is often taken to be the category vector spaces or some other “algebraic” category. This is a very flexible definitions that admits generalizations in many directions. Here, we interested in field theories of the supersymmetric Euclidean flavor. This means, firstly, that all bordisms are supermanifolds—a kind of mildly noncommutative manifold whose algebra of functions is Z=2-graded and commutative in the graded sense; and, secondly, all bordisms are equipped with a Euclidean structure (this induces, in particular, a flat Riemannian metric on the underlying manifold). Supersymmetric Euclidean field theories were first considered by Stolz and Teichner [39], who proposed their use as cocycles for TMF, the universal elliptic cohomology theory of topological modular forms. More specifically, they consider Euclidean field theories (EFTs) of dimension 2j1 (bordisms have local charts with 2 even and 1 odd coordinates) over a fixed manifold X. This means that besides supersymmetry and Euclidean structures, all bordisms are endowed with a smooth map to X. The collection of such objects (in fact, a groupoid) is denoted 2j1-EFT(X). Its formation is natural in X, and it is interesting to consider an equivalence relation weaker than isomorphism: two field theories are called concordant if there is a 2j1-EFT over X × R which, when restricted to X ×f0g respectively X ×f1g, recovers the two EFTs initially given. Conjecture 1.1 (Stolz and Teichner [39]). There are natural isomorphisms 2j1-EFT∗(X)=concordance ∼= TMF∗(X): A geometric construction of TMF (which, so far, is only understood in purely homotopy-theoretic terms) would be interesting for several reasons; for instance, it could help making precise heuristics from physics indicating that TMF is the correct 2 home for the T-equivariant index of families of Dirac operators on a loop space. The conjecture above is still wide open. To mention one fundamental difficulty, the excision/Mayer-Vietoris property of a cohomology theory can only possibly be reflected on the field theory side if we deal with extended (or local) field theories. However, the higher-categorical framework in which extended bordism categories are usually discussed is not amenable to supersymmetry, since the latter forces us not to consider individual bordisms, but rather always work with families. It is also interesting to consider supersymmetric EFTs of lower dimension and their relation with more classical cohomology theories. A conjecture analogous to the above states that for 1j1-EFTs the relevant cohomology theory is K-theory. In this case, a direct comparison between the field theory side and the homotopy theory side is possible (a vector bundle with connection gives rise to a supersymmetric field theory over X using the notion of super parallel transport developed by Dumitrescu [16]), but a classification of 1j1-EFTs over a manifold is a surprisingly elusive problem.
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