Elliptic cohomology

Elliptic cohomology
Name	Elliptic cohomology
Caption	Modular parameter space for elliptic curves
Field	Algebraic topology
Introduced	1980s
Founders	Michael Hopkins, Haynes Miller, Paul Landweber, Igor Krichever

Elliptic cohomology is a family of generalized cohomology theories arising in algebraic topology that connect the theory of elliptic curves, modular forms, and formal group laws with stable homotopy theory. It refines classical theories such as K-theory and complex cobordism by encoding arithmetic and geometric data of moduli spaces of elliptic curves and links to the Witten genus and two-dimensional conformal field theorys. Elliptic cohomology has led to interactions among figures and institutions including Michael Hopkins, Graeme Segal, Edward Witten, Paul Landweber, Mark Mahowald, Haynes Miller, Igor Krichever, Jacob Lurie, Oxford University, Institute for Advanced Study, and Massachusetts Institute of Technology.

Introduction

The subject emerged from attempts to construct cohomology theories that carry richer arithmetic than singular cohomology or ordinary cohomology, and more refined multiplicative structure than ordinary K-theory or complex bordism (MU). Early work by Landweber, Stong, and collaborators produced criteria for producing homology theories from formal group laws, influencing later constructions by Hopkins, Miller, Witten, and Segal. Elliptic cohomology associates to each elliptic curve over a base scheme a spectrum in the sense of stable homotopy theory, while modularity phenomena relate to deep results in the theory of modular curves, Hecke operators, and arithmetic geometry studied at places such as Harvard University and Princeton University.

Historical development and motivation

Motivations trace to attempts by Witten to interpret the index theorem in two-dimensional quantum field theories and to define the Witten genus as an invariant of string manifolds, prompting links to modular forms and elliptic genera. The Landweber exact functor theorem, developed by Landweber, Ravenel, and Stong, gave necessary machinery later used by Hopkins and Miller to construct spectra with prescribed formal group laws. Contributions by Krichever connected integrable systems and Baker–Akhiezer functions to elliptic genera, while work at University of Chicago and Northwestern University explored chromatic phenomena through the Nilpotence Theorem of Devinatz, Hopkins, and Smith. Subsequent developments by Lurie framed derived algebraic geometry and sheaves on the moduli stack of elliptic curves as foundations for sheafified elliptic cohomology, with institutional advances at California Institute of Technology and NHM.

Mathematical definition and constructions

Constructions use the language of formal group laws, Brown–Peterson cohomology, and the Landweber exact functor theorem to produce complex-orientable cohomology theories from elliptic curves and coordinates. One approach produces spectra such as the Topological modular forms spectrum via derived global sections of a sheaf of E-infinity ring spectrums on the moduli stack of elliptic curves, a program developed by Hopkins, Miller, and later formalized by Lurie using derived algebraic geometry and higher topos theory. Other constructions include analytic models via theta functions and Krichever’s construction using algebraic curves and Baker–Akhiezer functions, and p-adic approaches using Lubin–Tate and Morava E-theory techniques developed by Morava and Ravenel. Theories such as TMF and variants with level structure incorporate actions of Hecke algebras and congruence subgroups like SL2(Z), Gamma0(N), and Gamma1(N).

Relationship to modular forms and formal groups

Elliptic cohomology realizes the connection between cohomology operations and the theory of modular forms by mapping characteristic classes to modular forms and by encoding the formal group law of an elliptic curve in the complex orientation of the cohomology theory. The Witten genus lands in rings of modular forms studied by Serre and Deligne, while level structures relate to the arithmetic of modular curves such as X0(N) and X1(N), and to the representation theory of Monster group studied in Moonshine contexts. Formal group techniques from Hazewinkel and Quillen underpin the chromatic filtration studied by Ravenel, Hopkins, and Smith, placing elliptic cohomology at chromatic height two and linking it to Morava stabilizer group actions.

Examples and computations

Concrete computations include calculations of elliptic cohomology rings for classifying spaces like BU(n), BSU(n), and finite groups such as Cyclic groups and Symmetric groups, with input from the Atiyah–Hirzebruch spectral sequence and Adams spectral sequence technology developed by Adams and Novikov. Known spectra include TMF, TMF(Γ), and variants localized at primes studied by Mahowald, Rezk, Hill, and Lawson. Calculations often exploit the character map to K-theory, and compare to invariants from elliptic genus computations by Ochanine and Bott–Taubes. Specific computations at primes p = 2, 3 involve work by Shimomura, Yabe, and Hopkins–Mahowald–Sadofsky.

Connections to physics and string theory

Elliptic cohomology emerged from physical insights by Witten into supersymmetric sigma models and from attempts to rigorously capture partition functions and anomalies in two-dimensional conformal field theorys and string theory compactifications studied at CERN and Caltech. The Witten genus provides a bridge from string manifold invariants to modular forms, informing work on anomaly cancellation and D-brane charges analyzed using K-theory by Minasian and Moore. Elliptic cohomology’s role in moonshine phenomena links to vertex operator algebra constructions by Frenkel, Lepowsky, and Meurman and to modularity observed in Monstrous Moonshine studied by Conway and Norton.

Generalizations and variants

Generalizations include equivariant elliptic cohomology developed by Ginzburg, Kapranov, and Vasserot connecting to representation theory of loop groups and affine Lie algebras, sheaf-theoretic versions in derived algebraic geometry by Lurie, and p-adic or higher-height analogues such as Morava E-theory and higher chromatic theories studied by Ravenel and Hopkins. Level-structure refinements, equivariant refinements for compact Lie groups like U(n) and Tn, and connections to factorization homology and topological quantum field theory continue to expand the landscape, engaging researchers at institutions including Princeton University, University of Cambridge, and University of Chicago.

Category:Algebraic topology