nLab
elliptic spectrum

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Idea

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

Definition

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

Definition

An elliptic spectrum is a triple consisting of

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

  2. an even periodic ring spectrum EE;

  3. an equivalence

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

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

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

Remark

Originally (and still in many or even most references), def. 1 is stated with the formal Picard group Pic A 0Pic_A^0 replaced by the formal completion A^\hat A of AA 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 APic X 0A\simeq Pic_X^0exhibited 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 (BU(1))E^\bullet(B U(1)) is the space of sections of a line bundle over a (formal) moduli space of line bundles on the elliptic curve, instead of on the elliptic curve itself.

Indeed, generally for GG a compact Lie group, then E (BG)E^\bullet(B G) is the space of sections of the WZW model-line bundle on the (formal) moduli space of flat connections on GG-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 SpecE (BU(1))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 nn-dimensional XX

nnCalabi-Yau n-foldline n-bundlemoduli of line n-bundlesmoduli of flat/degree-0 n-bundlesArtin-Mazur formal group of deformation moduli of line n-bundlescomplex oriented cohomology theorymodular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory
n=0n = 0unit in structure sheafmultiplicative group/group of unitsformal multiplicative groupcomplex K-theory
n=1n = 1elliptic curveline bundlePicard group/Picard schemeJacobianformal Picard groupelliptic cohomology3d Chern-Simons theory/WZW model
n=2n = 2K3 surfaceline 2-bundleBrauer groupintermediate Jacobianformal Brauer groupK3 cohomology
n=3n = 3Calabi-Yau 3-foldline 3-bundleintermediate JacobianCY3 cohomology7d Chern-Simons theory/M5-brane
nnintermediate Jacobian

References

The concept of elliptic spectrum was introduced in

A brief review is for instance in of

Survey includes

Revised on December 15, 2016 11:28:28 by Urs Schreiber (83.208.22.80)