A MODEL FOR COMPLEX ANALYTIC EQUIVARIANT ELLIPTIC COHOMOLOGY FROM QUANTUM FIELD THEORY DANIEL BERWICK-EVANS AND ARNAV TRIPATHY Abstract. We construct a global geometric model for complex analytic equivariant elliptic cohomology for all compact Lie groups. Cocycles are specified by functions on the space of fields of the two-dimensional sigma model with background gauge fields and N = (0; 1) supersymmetry. We also consider a theory of free fermions valued in a representation whose partition function is a section of a determinant line bundle. We identify this section with a cocycle representative of the (twisted) equivariant elliptic Euler class of the representation. Finally, we show that the moduli stack of U(1)-gauge fields carries a multiplication compatible with the complex analytic group structure on the universal (dual) elliptic curve, with the Euler class providing a choice of coordinate. This provides a physical manifestation of the elliptic group law central to the homotopy- theoretic construction of elliptic cohomology. Contents 1. Introduction 1 2. Motivation: Classical fields, Q-cohomology, and localization 9 3. Classical fields as a superstack 13 4. Twisted sectors and equivariant de Rham complexes 20 5. Deforming twisted sectors and analytic functions 32 6. A cocycle model for equivariant elliptic cohomology 38 7. Free fermions and the equivariant elliptic Euler class 45 8. Elliptic group laws from U(1)-gauge fields 56 Appendix A. Supermanifolds and superstacks 59 Appendix B. Supergeometry of the equivariant de Rham complex 73 Appendix C. The de Rham model for equivariant elliptic cohomology over C 82 References 85 1. Introduction Elliptic cohomology grew out of a combination of Quillen's deep work on the role arXiv:1805.04146v2 [math.AT] 24 Aug 2020 of formal group laws in stable homotopy theory [Qui69] and Ochanine's construction of elliptic genera [Och87, Seg88, LRS95]. In the late 1980s, Witten showed that certain 2- dimensional supersymmetric quantum field theories have deformation invariants valued in elliptic genera [Wit87, Wit88]. Shortly thereafter, Segal suggested the existence of a map from a (yet to be constructed) space of 2-dimensional supersymmetric field theories to elliptic cohomology, realizing these quantum field theories as geometric cocycles [Seg88, Seg07]. This spurred some dazzling developments in homotopy theory led by Hopkins and collaborators [Hop94, AHS01, Hop02, GH04, AHS04, Goe09, Lur09, AHR10, Lur16, Lur18, Lur19]. One highlight is the construction of the universal elliptic cohomology of topological modular forms (TMF) with its string orientation. This orientation lifts the universal elliptic genus (now called the Witten genus) to a map of ring spectra, analogous to how the Spin orientation in real K-theory lifts the A^-genus. Date: August 25, 2020. 1 2 DANIEL BERWICK-EVANS AND ARNAV TRIPATHY Despite all this progress, Segal's proposed connection between field theories and elliptic cohomology remains unrealized. The original ideas have been refined considerably by Stolz and Teichner, leading to a conjecture [ST11, Conjecture 1.17] which we loosely summarize as follows. Conjecture 1.1 (Segal, Stolz{Teichner). For any smooth manifold M, there exists a nat- ural isomorphism 2−dimensional quantum field theories ∼ (1) =deformation ! TMF(M) with N = (0; 1) supersymmetry over M that realizes deformation classes of field theories as classes in TMF(M). Taken at face value, this is extremely surprising: quantum field theories are objects that belong to the world of analysis and differential geometry, whereas elliptic cohomol- ogy is constructed using sophisticated algebraic and homotopy-theoretic techniques. The statement is powerful in either direction. Using the left-hand side to understand the right- hand side would provide geometric cocycles for elliptic cohomology, as was Segal's original motivation. Conversely, the right-hand side gives a homotopy theoretic characterization of deformation classes of two-dimensional field theories, which would be a tool of great interest to high-energy theorists, e.g., see [GJFW19, GPPV18]. To date, the primary evidence for (1) comes from analogous statements relating K- theory and one-dimensional supersymmetric quantum field theories, i.e., supersymmetric quantum mechanics. Indeed, Stolz and Teichner [ST04, HST10] constructed a space of supersymmetric quantum mechanical theories that represents the spectrum KO, giving a parallel to (1). The physics argument for the index theorem [Wit82, AG83] has also been deeply influential, where the index map in K-theory is identified with a quantization map in supersymmetric quantum mechanics. This point of view leads to the slogan \the Witten genus is the index of the Dirac operator on loop space." To date, this can only be understood at a physics level of rigor [Wit88]. However, considerable insights were made early on [Tau89, BT89] and progress on the analytic underpinnings continues [KM13, KM15, Mel13]. In short, Conjecture 1.1 asks for an upgrade of certain structures in supersymmetric quantum mechanics to ones in supersymmetric quantum field theory. The mathematics of quantum field theory has been fraught with a myriad of technical difficulties, which is largely why (1) is so difficult. Indeed, there are swaths of quantum field theory for which there is currently no hope for rigorous foundations. Other pieces (e.g., free theories) are well- understood by mathematicians but usually not very interesting to physicists. Conjecture 1.1 identifies a Goldilocks zone: a class of quantum field theories for which one might hope that rigorous foundations exist, and yet|in view of the complexities in TMF|must be highly nontrivial. An equivariant refinement of Conjecture 1.1. Supposing one could construct the space of quantum field theories on the left side of (1), there remains another significant challenge: writing down a map. Fortunately, there is a universal property characterizing TMF. Roughly, it is the global sections of a sheaf of elliptic cohomology theories on the moduli stack of elliptic curves. For a fixed family of elliptic curves E, this sheaf defines an elliptic cohomology theory hE that is characterized by its theory of Chern classes. For c the Chern class of the tautological line, consider pullbacks ∗ ∗ ∗ 1 p1 ;p2 ;m 1 1 1 1 1 (2) c 2 hE(CP ) −! hE(CP × CP ); p1; p2; m: CP × CP ! CP ; along projections p1; p2 and the multiplication map m that classifies the tensor product of tautological lines. This gives a universal formula for the Chern class of a tensor product of line bundles in terms of a power series F , ∗ ∗ ∗ 1 1 m c = F (p1c; p2c) 2 hE(CP × CP );F 2 hE(pt)[[x; y]]: A MODEL FOR EQUIVARIANT ELLIPTIC COHOMOLOGY FROM QUANTUM FIELD THEORY 3 By definition of the elliptic cohomology theory hE, F is the formal group law of the elliptic curve with choice of coordinate determined by c. This leads to an important question: where in 2-dimensional physics do we find elliptic (formal) group laws? One place is U(1)-gauge theory, where classical fields are principal U(1)-bundles with connection. Such bundles can be tensored, and this endows moduli spaces of U(1)-gauge fields with a multiplication. In particular, the flat U(1)-bundles on a genus 1 Riemann surface can be identified with degree zero holomorphic line bundles with group operation given by tensor product. This complex analytic group is canonically isomorphic to an elliptic curve. The appearance of elliptic curves as groups|rather than merely their elliptic formal group laws|suggests a connection with equivariant elliptic cohomology. The coefficients for Grojnowski's complex analytic U(1)-equivariant elliptic cohomol- ogy is the sheaf of functions on the dual elliptic curve, EllU(1)(pt) 'OE_ [Gro07]. Natu- rality for the homomorphisms m; p1; p2 : U(1) × U(1) ! U(1) then lifts the formal group law (2) to multiplication on the curve itself, expressed in terms of its sheaf of functions. An important structure in the theory is a decompletion of (2) in the sense of Atiyah{Segal completion [AS69]. Namely, Ando constructed a `-twisted class completion Ell` (pt) −! h (BU(1)) = h ( 1) (3) U(1) E E CP 2 2 cU(1) 7! c 1 that refines the non-equivariant Chern classes c 2 hE(CP ) for each elliptic curve [And00]. _ Geometrically, twisted means that cU(1) is a section of a line bundle over E . Restricting to an elliptic curve curve E and trivializing this line bundle near the identity e 2 E identifies cU(1) with a choice of coordinate c as in (2). Returning to the discussion of U(1)-gauge theory, the sheaf EllU(1)(pt) 'OE_ can be identified with functions on the moduli space of flat U(1)-bundles on a genus 1 Riemann surface. In physics, this sheaf arises as functions on a moduli space of classical vacua or dimensionally reduced fields in U(1)-gauge theory on genus 1 Riemann surfaces. There is a well-known line bundle over this moduli space that encodes the anomaly of a family of free fermion theories. These theories are built out of the @¯-operator on an elliptic curve twisted by a degree zero holomorphic line bundle, and the anomaly line bundle is the Quillen determinant line bundle of these @¯-operators [Fre87, x4]. Furthermore, this line bundle has a holomorphic section determined by the partition function of the quantum field theory. This partition function can be identified with the Weierstrass σ-function, which is a particular θ-function that features prominently in the string orientation of topological modular forms [AHS01, AHR10]. This discussion leads to a refinement of Conjecture 1.1. In the following, TMFG denotes the derived global sections of a refinement of the sheaf EllG to one over Z; see [Lur19, GM20]. Conjecture 1.2. For a smooth manifold M with the action of a compact Lie group G, there exists a natural isomorphism 8 2−dimensional quantum field theories 9 < = ∼ (4) with N = (0; 1) supersymmetry and =deformation ! TMFG(M) : background G flavor symmetry over M ; that realizes deformation classes of field theories as classes in TMFG(M).