The holomorphic s-model and its symmetries Brian R. Williams Contents 1 Introduction 3 1.1 Summary of the results...................................5 1.2 Overview...........................................5 2 Holomorphic quantum field theory7 2.1 The definition of a quantum field theory.........................9 2.1.1 Classical field theory................................9 2.1.2 Quantum field theory............................... 17 2.2 Holomorphic field theories................................ 21 2.2.1 The definition of a holomorphic theory..................... 22 2.2.2 Holomorphically translation invariant theories................. 29 2.3 Renormalization of holomorphic theories........................ 33 2.3.1 Holomorphic gauge fixing............................. 34 2.3.2 One-loop weights.................................. 36 2.3.3 A general result about chiral anomalies..................... 41 2.4 Equivariant BV quantization................................ 46 2.4.1 Classical equivariance............................... 48 2.4.2 Quantum equivariance............................... 48 2.4.3 The case of a local Lie algebra........................... 49 3 The holomorphic s-model 52 3.1 Gelfand-Kazhdan formal geometry............................ 55 3.1.1 A Harish-Chandra pair for the formal disk................... 55 3.1.2 Formal coordinates................................. 57 3.1.3 The category of formal vector bundles...................... 61 3.1.4 Gelfand-Kazhdan descent............................. 63 3.1.5 Formal characteristic classes............................ 66 3.1.6 A family of extended pairs............................ 72 3.2 Descent for extended pairs................................. 74 3.2.1 General theory of descent for L¥ pairs...................... 74 3.2.2 The flat connection from the extended pair................... 78 3.3 The classical holomorphic s-model............................ 80 1 3.3.1 The free bg system................................. 81 3.3.2 A description using L¥ spaces.......................... 84 3.4 Deformations of the holomorphic s-model....................... 87 3.4.1 Forms as local functionals............................. 89 3.4.2 Computing the deformation complex...................... 91 3.5 BV quantization of the holomorphic s-model...................... 94 3.5.1 The Wn-equivariant quantization......................... 95 3.5.2 Quantization on general manifolds via formal geometry........... 105 3.6 The local operators..................................... 107 3.6.1 The factorization algebra of observables..................... 108 3.6.2 The observables on the d-disk........................... 110 3.6.3 The sphere observables.............................. 114 3.6.4 The disk as a module................................ 124 3.6.5 A formula for the character............................ 127 4 Local symmetries of holomorphic theories 130 4.1 The current algebra on complex manifolds....................... 132 4.1.1 Definitions...................................... 132 4.1.2 Factorization Lie algebras............................. 135 4.1.3 Local cohomology................................. 137 4.1.4 The Kac-Moody factorization algebra...................... 141 4.2 Local structures of the Kac-Moody factorization algebra................ 142 4.2.1 The higher sphere algebras............................ 144 4.2.2 The strategy..................................... 145 4.2.3 An Ed algebra from tori.............................. 150 4.3 The Kac-Moody factorization algebra on general manifolds.............. 152 4.3.1 The P0-structure................................... 152 4.3.2 Factorization homology along Hopf manifolds................. 153 4.3.3 A variant of the factorization algebra...................... 156 4.4 Universal Grothendieck-Riemann-Roch from BV quantization............ 158 4.4.1 The classical family................................. 160 4.4.2 BV quantization in families............................ 161 4.4.3 Local to global................................... 165 4.4.4 A module for the higher Kac-Moody....................... 167 4.5 Holomorphic diffeomorphisms.............................. 168 4.5.1 Holomorphic vector fields............................. 168 4.5.2 Gelfand-Fuks cohomology............................ 173 4.5.3 The local cohomology of holomorphic vector fields.............. 175 4.5.4 An explicit description of the local cocycles................... 179 5 Appendix 182 5.1 The dg model for punctured affine space........................ 182 2 Chapter 1 Introduction The topic of this thesis is a mathematically rigorous analysis of a class of quantum field theories that depend on the complex structure of a manifold in an analogous way that topological theories depend on the smooth structure. Topological field theories have gained much interest in the world of mathematics due to their elegant functorial descriptions, as well as their applications to geometry and topology. Holomorphic theories, as we will define them, often contain strictly more information than their topological counterparts. We will show how such theories also admit their own mathematical description using techniques of homological algebra and renormalization combined with the theory of factorization algebras. This approach to perturbative quantum field theory is largely based on Costello’s program devel- oped in [Cos11] which provides a rigorous construction of the path integral where the quantum parameterh ¯ is treated formally. The central mathematical players in this setup are the observables of a quantum field theory. Physically speaking, the fields j are the objects describing a theory and the observables are the so-called measurements, or functions on the space of fields, j 7! O(j), one can perform on a system. These measurements appear in quantum theory through their expectation values Z hOi = Dj e−S(j)/¯h O(j), j2Fields where the integral is over the the fields j 2 Fields, and Dj e−S(j)/¯h is the a priori ill-defined path integral measure. The main objective of [Cos11] is to give a mathematical definition of this path integral measure largely inspired by a combination of the Wilsonian perspective of effective field theory together with Batalin-Vilkovisky approach to studying gauge theories. Building off of this foundation, the work of Costello-Gwilliam in [CG17,CG] sets up the mathematical theory of observables in quantum field theory using the language of factorization algebras. The idea of using factorization algebras to describe the observables of a theory was initiated in Beilinson-Drinfeld’s work on chiral algebras [BD04]. Their goal, in part, was to provide a geo- metric interpretation of vertex algebras in chiral conformal field theory. While the setting that Costello-Gwilliam work in is much more general than the world of conformal field theory, the 3 types of quantum field theories we study in this thesis can be viewed as a much more direct generalization of the chiral conformal field theory situation. In my previous work [Wil17, GGW, GWb, GWa] I have shown how many elements of conformal field theory, and bosonic string the- ory, have rigorous interpretations and extensions within the BV formalism. Throughout the the- sis we stress similarities with structures present in higher dimensional holomorphic theories with those of two-dimensional chiral conformal field theory. For instance, the state space of a holomor- phic theory is a natural dg module for the higher sphere algebra. In the other direction, it seems possible to write down a full algebraic structure that the local operators of a higher dimensional holomorphic theory, akin to that of a vertex algebra. These “higher dimensional” vertex algebras are outside of the scope of this thesis, but we hope to return to them in future publications. While the class of field theories we consider are very natural from a mathematical point of view, they also provide a unique perspective into well-studied physical theories. Often times, topo- logical theories provide a useful simplification, or approximation, of supersymmetric field theo- ries. Perhaps the most well-known example of this are the A and B-models of topological string theory. These are topological theories that appear as “twists” of the two-dimensional (2, 2)- supersymmetric s-model and have shown to be an extremely useful window into supersym- metric string theory. One issue with this is that for low numbers of supersymmetry, topological twists may not exist (for example, four-dimensional N = 1). On the other hand, holomorphic twists almost always exist. In this thesis, we make the argument that holomorphic theories in the BV formalism can provide insight into various supersymmetric theories while at the same time admit salient mathematical structures that are readily studied by well-known techniques in homological algebra. The primary holomorphic theory we focus on in this thesis is the holomorphic s-model. The classi- cal theory pertains to the moduli space of holomorphic maps between complex manifolds Y ! X. In complex dimension one, this theory has appeared as the holomorphic twist of the (0, 2)- supersymmetric s-model by Witten [Wit07] and was rigorously shown to recover the Witten genus by Costello in [Cosa]. In higher dimensions, we find a more sensitive invariant control- ling the quantization of the model, generalizing the geometric condition for a manifold to be String. This s-model appears naturally in many physical theories, most notably as reductions of (twists of) higher dimensional supersymmetric and supergravity