nLab elliptic spectrum

Idea

An elliptic spectrum is a spectrum which represents an elliptic cohomology theory.

Definition

For $E$ a ring spectrum, write $E^\bullet(\ast)$ for its coefficient ring and generally $E^\bullet(X)$ for its generalized cohomology ring over any $X$.

Definition

An elliptic spectrum is a triple consisting of

1. an elliptic curve $A$ over the coefficient ring $E^\bullet(\ast)$;

2. an even periodic ring spectrum $E$;

3. an equivalence

   $$Spec E^\bullet(BU(1)) \stackrel{\simeq}{\longrightarrow} Pic_A^0$$

   between the algebraic spectrum of the $E$-cohomology ring over the classifying space for complex line bundles (see at complex oriented cohomology theory) and the formal Picard group $Pic_A^0$ of $A$.

This is due to (Ando-Hopkins-Strickland01, def. 1.2). See for instance also (Gepner 05, def. 15).

Remark

(role of the formal Picard scheme)

Originally (and still in many or even most references), def. is stated with the formal Picard group $Pic_A^0$ replaced by the formal completion $\hat A$ of $A$ at its neutral element.

These two versions of the definition in itself are equivalent, since elliptic curves are self-dual abelian varieties equipped with a canonical isomorphism $\hat A \simeq Pic^0_A$exhibited by the Poincaré line bundle.

But for the development of the theory, notably for application to equivariant elliptic cohomology, for the relation of elliptic cohomology to loop group representations etc., it is crucial to understand that $E^\bullet(B U(1))$ is the space of sections of a line bundle over a (formal) moduli space of line bundles on, in turn, the elliptic curve, instead of directly on the elliptic curve itself.

Indeed, generally for $G$ a compact Lie group, we have that $E^\bullet(B G)$ is the space of sections of the WZW model-line bundle for conformal blocks (the prequantum line bundle of the $G$-Chern-Simons theory) on the (formal) moduli space of flat connections on $G$-principal bundles over the elliptic curve. This is the central statement at equivariant elliptic cohomology. As the appearance of the WZW model here shows, this is also crucial for understanding the role of elliptic spectra in quantum field theory/string theory, see at equivariant elliptic cohomology – Interpretation in Quantum field theory/String theory for more on this.

Moreover, understanding $Spec E^\bullet(BU(1))$ as being about moduli of line bundles on the elliptic curve is crucial for understanding the generalization of the concept of elliptic spectra, for instance to K3-spectra. This is indicated in the following table

moduli spaces of line n-bundles with connection on $n$-dimensional $X$

$n$	Calabi-Yau n-fold	line n-bundle	moduli of line n-bundles	moduli of flat/degree-0 n-bundles	Artin-Mazur formal group of deformation moduli of line n-bundles	complex oriented cohomology theory	modular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory
$n = 0$		unit in structure sheaf	multiplicative group/group of units		formal multiplicative group	complex K-theory
$n = 1$	elliptic curve	line bundle	Picard group/Picard scheme	Jacobian	formal Picard group	elliptic cohomology	3d Chern-Simons theory/WZW model
$n = 2$	K3 surface	line 2-bundle	Brauer group	intermediate Jacobian	formal Brauer group	K3 cohomology
$n = 3$	Calabi-Yau 3-fold	line 3-bundle		intermediate Jacobian		CY3 cohomology	7d Chern-Simons theory/M5-brane
$n$				intermediate Jacobian

Related concepts

• elliptic cohomology, elliptic genus, tmf

• Eilenberg-MacLane spectrum

• K3-spectrum

References

The concept of elliptic spectrum was introduced in

• Matthew Ando, Michael Hopkins, Neil Strickland, Elliptic spectra, the Witten genus, and the theorem of the cube, Inventiones Mathematicae, 146:595–687, 2001, (pdf, doi:10.1007/s002220100175)

A brief review is for instance in of

• David Gepner, section 1 of Homotopy topoi and equivariant elliptic cohomology, 2005.

Survey includes

• Jacob Lurie, A Survey of Elliptic Cohomology

• Charles Rezk, Elliptic cohomology and elliptic curves, Felix Klein Lectures, Bonn 2015 (web)