Preprint typeset in JHEP style - HYPER VERSION A Few Remarks On Topological Field Theory Gregory W. Moore Abstract: Physics 695, Fall 2015. September 9 - December 14, 2015. Version April 30, 2016 Contents 1. Introduction 4 2. Basic Ideas 4 2.1 More Structure 7 3. Some Basic Notions In Category Theory 9 3.1 Basic Definitions 9 3.2 Groupoids 16 3.3 Tensor Categories 17 3.4 Other Tensor Categories 18 3.5 Z2-graded vector spaces 18 3.6 Category Of Representations Of A Group 18 4. Bordism 19 4.1 Unoriented Bordism: Definition And Examples 19 4.2 The Bordism Category Bordhd−1,di 20 SO 4.3 The Oriented Bordism Category Bordhd−1,di 20 4.4 Other Bordism Categories 22 5. The Definition Of Topological Field Theory 22 6. Some General Properties 23 6.1 Unitarity 29 7. One Dimensional Topological Field Theory 29 8. Two Dimensional TFT And Commutative Frobenius Algebras 31 8.1 The Sewing Theorem 36 8.1.1 A Little Singularity Theory 38 8.1.2 Proof Of The Sewing Theorem 40 8.2 Remarks On The Unoriented Case 42 9. Computing Amplitudes 46 9.1 Summing over topologies 47 9.2 Semisimple algebras 48 10. Some examples of commutative Frobenius algebras arising in physical problems 51 10.1 Example 1: Finite Group Theory 51 10.2 Example 2: Loop Groups And The Fusion Ring Of A Rational Conformal Field Theory 53 – 1 – 10.2.1 Central Extension Of Loop Algebras 53 10.2.2 Central Extensions Of Loop Groups 55 10.2.3 The Wess-Zumino Term 55 10.2.4 Construction Of The Cocycle For LG 61 10.2.5 Integrable Highest Weight Representations 62 10.3 Example 3: The Cohomology Of A Compact Oriented Manifold 70 10.4 Example 4: Landau-Ginzburg theory 71 10.5 Example 5: Quantum cohomology 72 11. Emergent Spacetime 73 11.1 The algebra of functions on a topological space X 73 11.2 Application To 2D TFT 75 12. Quantum Mechanics And C∗ Algebras 76 12.1 Banach Algebras 77 12.2 C∗ Algebras 80 12.3 Units In Banach Algebras 82 12.4 The Spectrum Of An Element a B 83 ∈ 12.5 Commutative Banach Algebras 86 12.5.1 Characters And Spec(A) 86 12.5.2 Ideals And Maximal Ideals 86 12.5.3 The Gelfand Transform 87 12.5.4 Commutative C∗ Algebras 90 12.5.5 Application of Gelfand’s Theorem: The Spectrum Of A Self-Adjoint Element Of A 93 12.5.6 Compactness and noncompactness 93 12.6 Noncommutative Topology: The C∗-Algebra Dictionary 97 12.6.1 Hopf Algebras And Quantum Groups 99 12.7 The Irrational (And Rational) Rotation Algebras 105 12.7.1 Definition 105 12.7.2 Realization In ( ) 109 B H 12.7.3 Electrons Confined To Two-Dimensions In A Magnetic Field 110 12.7.4 Magnetic Translation Group 117 12.7.5 The Algebra For θ Rational 118 Aθ 12.7.6 Two-Dimensional Electrons On A Torus In A Magnetic Field 120 12.7.7 Band Theory 129 12.7.8 Crystallographic Symmetry And Point Group Equivariance 137 12.7.9 Electron In A Periodic Potential And A Magnetic Field 142 12.7.10The Effective Topological Field Theory 151 12.8 Deforming The Algebra Of Functions On R2n 151 12.8.1 The Moyal (or ) Product 151 ∗ 12.8.2 The Dipole Model 153 12.8.3 The Weyl Transform 154 – 2 – 12.8.4 The Wigner Function 157 12.8.5 Field Theory On A Noncommutative Space 159 12.9 Relation To Open String Theory 159 12.9.1 String Theory In A p-nutshell 159 12.9.2 Toroidal Compactification 168 12.9.3 Closed Strings And T -Duality 169 12.9.4 Relation to electric-magnetic duality 195 12.10Deformations Of Algebras And Hochschild Cohomology 203 12.10.1Poisson Manifolds 208 12.11C∗-Algebra Approach To Quantum Mechanics 208 12.11.1Positive Elements And Maps For C∗ Algebras 208 12.11.2States On A C∗-Algebra 209 12.11.3GNS Construction 211 12.11.4Operator Topologies 213 12.11.5Von Neumann Algebras And Measure Spaces 215 12.11.6The Spectral Theorem 216 12.11.7States And Operators In Classical Mechanics 218 12.11.8States And Operators In Quantum Mechanics 219 13. Boundary conditions 220 14. Open and closed 2D TFT in the semisimple case: D-branes and vector bundles 226 15. Closed strings from open strings 228 15.1 The Grothendieck group 229 16. Three Dimensions And Modular Tensor Categories 230 17. Other Generalizations 230 18. Higher Categories, Locality, and extended objects 230 19. References 233 A. Sums Over Symplectic Lattices And Theta Functions 233 A.1 Symplectic structures, complex structures, and metrics 233 A.2 Statement Of The Problem 235 A.3 Level κ Theta Functions 236 A.4 Splitting instanton sums 237 A.5 Geometrical Interpretation 239 B. Generators For OZ(Q) 239 – 3 – 1. Introduction Topological field theory is an excellent pedagogical tool for introducing both some basic ideas of physics along with some beautiful mathematical ideas. The idea of TFT arose from both the study of two-dimensional conformal field theories and from Witten’s work on the relation of Donaldson theory to N=2 supersymmetric field theory and Witten’s work on the Jones polynomial and three-dimensional quantum field theories. In conformal field theory, Graeme Segal stated a number of axioms for the definition of a CFT. These were adapted to define a notion of a TFT by Atiyah. TFT might be viewed as a basic framework for physics. It assigns Hilbert spaces, states, and transition amplitudes to topological spaces in a way that captures the most primitive notions of locality. By stripping away the many complications of “real physics” one is left with a very simple structure. Nevertheless, the resulting structure is elegant, it is related to beautiful algebraic structures which, at least in two dimensions, which have surprisingly useful consequences. This is one case where one can truly “solve the theory.” Of course, we are interested in more complicated theories. But the basic framework here can be adapted to any field theory. What changes is the geometric category under consideration. Thus, it offers one approach to the general question of “What is a quantum field theory?” 2. Basic Ideas It is possible to speak of physics in 0-dimensional spacetime. From the functional integral viewpoint this is quite natural: Path integrals become ordinary integrals. It is also very fruitful to consider string theories whose target spaces are 0-dimensional spacetimes. Nev- ertheless, in the vast majority of physical problems we work with systems in d spacetime dimensions with d> 0. We will henceforth assume d> 0. What are the most primitive things we want from a physical theory in d spacetime dimensions? In a physical theory one often decomposes spacetime into space and time as in (1). If space is a (d 1)-dimensional manifold Y then, in quantum mechanics, we associate − to it a vector space of states (Y ). H d−1 Of course, in quantum mechanics (Y ) usually has more structure - it is a Hilbert H d−1 space. But in the spirit of developing just the most primitive aspects we will not incorporate that for the moment. (The notion of a unitary TFT captures the Hilbert space, as described below.) Moreover, in a generic physical theory there are natural operators acting on this Hilbert space such as the Hamiltonian. The spectrum of the Hamiltonian and other physical observables depends on a great deal of data. Certainly they depend on the metric on spacetime since a nonzero energy defines a length scale ~c L = . E In topological field theory one ignores most of this structure, and focuses on the depen- dence of (Y ) on the topology of Y . For simplicity, we will initially assume Y is compact H without boundary. – 4 – Figure 1: A spacetime Xd = Y R. Y is (d 1)-dimensional space, possibly with nontrivial × − topology. So: In topological field theory we want to have an association: (d 1)-manifolds Y to vector spaces: Y (Y ), such that “ (Y ) is the same for − → H H homeomorphic vector spaces.” What this means is that if there is a homeomorphism ϕ : Y Y ′ (2.1) → then there is a corresponding isomorphism of vector spaces: ϕ : (Y ) (Y ′) (2.2) ∗ H →H so that composition of homeomorphisms corresponds to composition of vector space iso- morphisms. In particular, self-homeomorphisms of Y act as automorphisms of (Y ): It H therefore provides a (possibly trivial) representation of the diffeomorphism group. Now, we also want to incorporate some form of locality, at the most primitive level. Thus, if we take disjoint unions (Y Y )= (Y ) (Y ) (2.3) H 1 ∐ 2 H 1 ⊗H 2 Note that (2.3) implies that we should assign to ( ) the field of definition of our vector H ∅ space. For simplicity we will take ( )= C, although one could use other ground fields. H ∅ Remark: In algebraic topology it is quite common to assign an abelian group or vector space to a topological space. This is what the cohomology groups do, for example. But here – 5 – we see a big difference from the standard algebraic topology examples. In those examples the spaces add under disjoint union. In quantum mechanics the spaces multiply. This is the fundamental reason why many topologists refer to the topological invariants arising from topological field theories as “quantum invariants.” Finally, there is an obvious homeomorphism Y Y ′ = Y ′ Y (2.4) ∐ ∼ ∐ and hence there must be an isomorphism Ω : (Y ) (Y ′) (Y ′) (Y ) (2.5) H ⊗H →H ⊗H Figure 2: Generalizing the product structure, a d-dimensional bordism X can include topology change between the initial (d 1)-dimensional spatial slices Yin and the final spatial slice Yout.