# nLab elliptic genus

Contents

### Context

#### Index theory

index theory, KK-theory

noncommutative stable homotopy theory

partition function

genus, orientation in generalized cohomology

## Definitions

operator K-theory

K-homology

# Contents

## Idea

An elliptic genus is a genus in elliptic cohomology (Landweber-Ravenel-Stong 93). In analogy to how there is a “universal elliptic cohomology”, namely tmf, there is a universal elliptic genus – the Witten genus. This arises as the large volume limit of the partition function of the superstring whose target space is the given manifold.

## Definition

The original definition of elliptic genus is due to (Ochanine 87) (see the review (Ochanine 09)) and says that an genus of oriented manifolds is called an elliptic genus if it vanishes on manifolds which are projective spaces of the form $\mathbb{C}P(\xi)$ for $\xi$ an even-dimensional complex vector bundle over an oriented closed manifold.

The terminomology elliptic for this was motivated by the central theorem of (Ochanine 87) which says that every genus $\phi$ satisfying this condition has a logarithm $log_\phi$ of the form

$log_\phi(u) = \int_{0^u} (1- 2 \delta t^2+ \epsilon t^4)^{-1/2}$

for some constants $\delta, \epsilon$. Hence for non-degenerate choices of parameters ($\delta^2 \neq \epsilon$ and $\epsilon \neq 0$) in the square root this is the expansion at 0 of an elliptic function.

So the logarithm here is an elliptic integral? and that was the original reason for the term “elliptic genus”.

## Examples

### Degenerate case: Signature genus

The degenerate case with parameters $\delta = \epsilon = 1$ (as above) is the signature genus.

### Degenerate case: $\hat A$-genus

The degenerate case with parameters $\delta = - \frac{1}{8}$ and $\epsilon = 0$ (as above) is the A-hat genus.

### Universal case: Witten genus

Given an elliptic genus with non-degenerate parameters $\delta, \epsilon \in \mathbb{C}$ (as above, see also at j-invariant), the Jacobi quartic Riemann surface which is given by the equation

$Y^2 = X^4 - 2 \delta X^2 + \epsilon$

is naturally parameterized by the upper half plane. Under this identification obe may think of $\epsilon$ and $\delta$ as functions of moduli of elliptic curves and concretely as modular forms for the subgroup $\Gamma_0(2)$ of that of Möbius transformations.

Viewed this way the collection of all elliptic genera provides a single genus with coefficients in this ring $MF_\bullet(\Gamma_0(2))$ of modular forms

$w \colon \Omega^{SO}_\bullet \longrightarrow MF_\bullet(\Gamma_0(2))$

(such that postcomposition with evaluation on any elliptic curve parameterized by the given value of $\delta$ and $\epsilon$ produces the corrponding elliptic genus).

This “universal” elliptic genus is the Witten genus.

## Properties

### Integrality on Spin-manifolds

On manifolds with spin structure the elliptic genus takes values in integral series $\mathbb{Z}[ [q] ]$.

### Relation to partition functions of superstring

The partition function of a type II superstring as a function depending on the modulus of the worldsheet elliptic curve yields an elliptic genus (Witten 87). (The analog for the heterotic string is hence called the Witten genus with values in the “universal elliptic cohomology” theory, tmf).

For equivariant/gauged string sigma-models the elliptic genus should take values in equivariant elliptic cohomology, see at gauged WZW mode – Partition function in elliptic cohomology.

$d$partition function in $d$-dimensional QFTsuperchargeindex in cohomology theorygenuslogarithmic coefficients of Hirzebruch series
0push-forward in ordinary cohomology: integration of differential formsorientation
1spinning particleDirac operatorKO-theory indexA-hat genusBernoulli numbersAtiyah-Bott-Shapiro orientation $M Spin \to KO$
endpoint of 2d Poisson-Chern-Simons theory stringSpin^c Dirac operator twisted by prequantum line bundlespace of quantum states of boundary phase space/Poisson manifoldTodd genusBernoulli numbersAtiyah-Bott-Shapiro orientation $M Spin^c \to KU$
endpoint of type II superstringSpin^c Dirac operator twisted by Chan-Paton gauge fieldD-brane chargeTodd genusBernoulli numbersAtiyah-Bott-Shapiro orientation $M Spin^c \to KU$
2type II superstringDirac-Ramond operatorsuperstring partition function in NS-R sectorOchanine elliptic genusSO orientation of elliptic cohomology
heterotic superstringDirac-Ramond operatorsuperstring partition functionWitten genusEisenstein seriesstring orientation of tmf
self-dual stringM5-brane charge
3w4-orientation of EO(2)-theory

The notion of elliptic genus was introduced in

• Serge Ochanine, Sur les genres multiplicatifs definis par des integrales elliptiques, Topology, Vol. 26, No. 2, 1987 (pdf)

A quick review is in

• Serge Ochanine, What is… an elliptic genus?, Notices of the AMS, volume 56, number 6 (2009) (pdf)

More review (and in the context of the lift to the spin orientation of Tate K-theory) is in

• Matthias Kreck, Stefan Stolz, section 2 of $HP^2$-bundles and elliptic homology, Acta Math, 171 (1993) 231-261 (pdf)

• Charles Thomas, section 1 of Elliptic cohomology, Kluwer Academic, 2002

The relation of this to elliptic cohomology was understood in

• Peter Landweber, Elliptic Cohomology and Modular Forms, in Elliptic Curves and Modular Forms in Algebraic Topology, Lecture Notes in Mathematics Volume 1326, 1988, pp 55-68 (LandweberEllipticModular.pdf?)

• Peter Landweber, Douglas Ravenel, Robert Stong, Periodic cohomology theories defined by elliptic curves, in Haynes Miller et al (eds.), The Cech centennial: A conference on homotopy theory, June 1993, AMS (1995) (pdf)

The interpretation of the elliptic genus/Witten genus as the partition function of the type II superstring/heterotic string is due to

• Edward Witten, Elliptic genera and quantum field theory, Comm. Math. Phys. Volume 109, Number 4 (1987), 525-536. (EUCLID)

The integrality of the elliptic genus and elliptic homology on Spin-manifolds is due to

• D.V. Chudnovsky, G.V. Chudnovsky, Elliptic modular functions and elliptic genera, Topology, Volume 27, Issue 2, 1988, Pages 163–170

Similar elliptic genera of $N=2$ $d = 2$ field theories and Landau-Ginzburg models are discussed in

• Edward Witten, On the Landau-Ginzburg Description of $N=2$ Minimal Models, Int.J.Mod.Phys.A9:4783-4800,1994 (arXiv:hep-th/9304026)

• Toshiya Kawai, Yasuhiko Yamada, Sung-Kil Yang, Elliptic Genera and $N=2$ Superconformal Field Theory (arXiv:hep-th/9306096)

More on this is in

• Sujay K. Ashok, Jan Troost, A Twisted Non-compact Elliptic Genus, JHEP 1103:067,2011 (arXiv:1101.1059)

Refinement of the Ochanine genus to a homomorphism of ring spectra (in analogy to the lift of the Witten genus to the string orientation of tmf) is considered in