On equivariant topological modular forms

This document expands our structural knowledge of topological modular forms TMF in two directions: the first, by extending the functoriality inherent to the definition of TMF, and the second, being tools to calculate the effect that endomorphisms of TMF have on homotopy groups. These structural statements allow us to lift classical operations on modular forms, such as Adams operations, Hecke operators, and Atkin-Lehner involutions, to stable operations on TMF. Some novel applications of these operations are then found, including a derivation of some congruences of Ramanujan in a purely homotopy theoretic manner, improvements upon known bounds of Maeda's conjecture, as well as some applications in homotopy theory. These techniques serve as teasers for the potential of these operations.

We reformulate the question of the absence of global anomalies of heterotic string theory mathematically in terms of a certain natural transformation text {TMF} ^bullet rightarrow (I_mathbb {Z}Omega ^{text {string} })^{bullet -20}, from topological modular forms to the Anderson dual of string bordism groups, using the Segal–Stolz–Teichner conjecture. We will show that this natural transformation vanishes, implying that heterotic global anomalies are always absent. The fact that text {TMF} ^{21}(text {pt} )=0 plays an important role in the process. Along the way, we also discuss how the twists of text {TMF} can be described under the Segal–Stolz–Teichner conjecture, by using the result of Freed and Hopkins concerning anomalies of quantum field theories. The paper contains separate introductions for mathematicians and for string theorists, in the hope of making the content more accessible to a larger audience. The sections are also demarcated cleanly into mathematically rigorous parts and those which are not.

We construct a complex analytic version of an equivariant cohomology theory which appeared in a paper of Rezk, and which is roughly modelled on the Borel-equivariant cohomology of the double free loop space. The construction is defined on finite, torus-equivariant CW complexes and takes values in coherent holomorphic sheaves over the moduli stack of complex elliptic curves. Our methods involve an inverse limit construction over all finite-dimensional subcomplexes of the double free loop space, following an analogous construction of Kitchloo for single free loop spaces. We show that, for any given complex elliptic curve $\mathcal {C}$, the fiber of our construction over $\mathcal {C}$ is isomorphic to Grojnowski's equivariant elliptic cohomology theory associated to $\mathcal {C}$.

We present a calculation, which shows how the moduli of complex analytic elliptic curves arises naturally from the Borel cohomology of an extended moduli space of $U(1)$-bundles on a torus. Furthermore, we show how the analogous calculation, applied to a moduli space of principal bundles for a $K(\mathbb{Z},2)$ central extension of $U(1)^d$ give rise to Looijenga line bundles. We then speculate on the relation of these calculations to the construction of complex analytic equivariant elliptic cohomology.

This paper starts with an exposition of descent-theoretic techniques in the study of Picard groups of [math] –ring spectra, which naturally lead to the study of Picard spectra. We then develop tools for the efficient and explicit determination of differentials in the associated descent spectral sequences for the Picard spectra thus obtained. As a major application, we calculate the Picard groups of the periodic spectrum of topological modular forms [math] and the nonperiodic and nonconnective [math] . We find that [math] is cyclic of order [math] , generated by the suspension [math] (a result originally due to Hopkins), while [math] . In particular, we show that there exists an invertible [math] –module which is not equivalent to a suspension of [math] .

These lecture notes are based on Yang’s talk at the MATRIX program Geometric R-Matrices: from Geometry to Probability, at the University of Melbourne, Dec. 18–22, 2017, and Zhao’s talk at Perimeter Institute for Theoretical Physics in January 2018. We give an introductory survey of the results in Yang and Zhao (Quiver varieties and elliptic quantum groups, 2017. arxiv1708.01418). We discuss a sheafified elliptic quantum group associated to any symmetric Kac-Moody Lie algebra. The sheafification is obtained by applying the equivariant elliptic cohomological theory to the moduli space of representations of a preprojective algebra. By construction, the elliptic quantum group naturally acts on the equivariant elliptic cohomology of Nakajima quiver varieties. As an application, we obtain a relation between the sheafified elliptic quantum group and the global affine Grassmannian over an elliptic curve.

Automorphic forms and cohomology theories on Shimura curves of small discriminant

Calculating descent for 2-primary topological modular forms

A classification of complex rank 3 vector bundles on [formula omitted]

The purpose of this note is to introduce the arithmetic, study of the universal elliptic curve over Igusa curves. Specifically, its Hasse-WeilL-function is computed in terms of modular forms and is shown to have interesting zeros. Explicit examples are presented for which the Birch and Swinnerton-Dyer conjecture is verified.

Visualizing elements of order two in the Weil–Châtelet group

A differential equation satisfied by the arithmetic Kodaira–Spencer class

Representations of loop groups, Dirac operators on loop space, and modular forms

This paper is devoted to the classical problem of the inversion of ultraelliptic integrals given by basic holomorphic differentials on a curve of genus 2. Basic solutions F and G of this problem are obtained from a single-valued 4-periodic meromorphic function on the Abelian covering W of the universal hyperelliptic curve of genus 2. Here W is the nonsingular analytic curve W = {u =(u1, u3) ∈ ℂ2: σ(u) = 0}, where σ(u) is the two-dimensional sigma function. We show that G(z) = F(ξ(z)), where z is a local coordinate in a neighborhood of a point of the smooth curve W and ξ(z) is the smooth function in this neighborhood given by the equation σ(u1, ξ(u1)) = 0. We obtain differential equations for the functions F(z), G(z), and ξ(z), recurrent formulas for the coefficients of the series expansions of these functions, and a transformation of the function G(z) into the Weierstrass elliptic function ℘ under a deformation of a curve of genus 2 into an elliptic curve.

This paper concerns the relations between the relative Manin–Mumford conjecture and Pink’s conjecture on unlikely intersections in mixed Shimura varieties. The variety under study is the 4-dimensional Poincare biextension attached to a universal elliptic curve. A detailed list of its special subvarieties is drawn up, providing partial verifications of Pink’s conjecture in this case, and two open problems are stated in order to complete its proof.