# 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.

### Rigidity theorem

$d$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

### Elliptic cohomology

#### General

The concept of elliptic cohomology originates around:

and in the universal guise of topological modular forms in:

• Michael Hopkins, Algebraic topology and modular forms in Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), pages 291–317, Beijing, 2002. Higher Ed. Press (arXiv:math/0212397)

Surveys:

Textbook accounts:

#### Equivariant elliptic cohomology

Relation to Kac-Weyl characters of loop group representations

The case of twisted ad-equivariant Tate K-theory:

#### Via derived $E_\infty$-geometry

Formulation of (equivariant) elliptic cohomology in derived algebraic geometry/E-∞ geometry (derived elliptic curves):

### Elliptic genera

#### General

The general concept of elliptic genus originates with:

Early development:

Review:

• Peter Landweber, Elliptic genera: An introductory overview In: P. Landweber (eds.) Elliptic Curves and Modular Forms in Algebraic Topology, Lecture Notes in Mathematics, vol 1326. Springer (1988) (doi:10.1007/BFb0078036)

• Kefeng Liu, Modular forms and topology, Proc. of the AMS Conference on the Monster and Related Topics, Contemporary Math. (1996) (pdf, pdf, doi:10.1090/conm/193)

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

The Stolz conjecture on the Witten genus:

The Jacobi form-property of the Witten genus:

• Matthew Ando, Christopher French, Nora Ganter, The Jacobi orientation and the two-variable elliptic genus, Algebraic and Geometric Topology 8 (2008) p. 493-539 (pdf)

For the Ochanine genus:

#### Equivariant elliptic genera

The statement, with a string theory-motivated plausibility argument, is due to Witten 87.

The first proof was given in:

Reviewed in:

• Raoul Bott, On the Fixed Point Formula and the Rigidity Theorems of Witten, Lectures at Cargése 1987. In: ’t Hooft G., Jaffe A., Mack G., Mitter P.K., Stora R. (eds) Nonperturbative Quantum Field Theory. NATO ASI Series (Series B: Physics), vol 185. Springer (1988) (doi:10.1007/978-1-4613-0729-7_2)

Further proofs and constructions:

On manifolds with SU(2)-action:

• Anand Dessai, The Witten genus and $S^3$-actions on manifolds, 1994 (pdf, pdf)

#### Twisted elliptic genera

Discussion of elliptic genera twisted by a gauge bundle, i.e. for string^c structure):

### Elliptic genera as super $p$-brane partition functions

The interpretation of elliptic genera (especially the Witten genus) as the partition function of a 2d superconformal field theory (or Landau-Ginzburg model) – and especially of the heterotic string (“H-string”) or type II superstring worldsheet theory has precursors in

and then strictly originates with:

Review in:

#### Formulations

##### Via super vertex operator algebra

Formulation via super vertex operator algebras:

and for the topologically twisted 2d (2,0)-superconformal QFT (the heterotic string with enhanced supersymmetry) via sheaves of vertex operator algebras in

based on chiral differential operators:

##### Via Dirac-Ramond operators on free loop space

Tentative interpretation as indices of Dirac-Ramond operators as would-be Dirac operators on smooth loop space:

##### Via conformal nets

Tentative formulation via conformal nets:

#### Conjectural interpretation in tmf-cohomology

The resulting suggestion that, roughly, deformation-classes (concordance classes) of 2d SCFTs with target space $X$ are the generalized cohomology of $X$ with coefficients in the spectrum of topological modular forms (tmf):

and the more explicit suggestion that, under this identification, the Chern-Dold character from tmf to modular forms, sends a 2d SCFT to its partition function/elliptic genus/supersymmetric index:

This perspective is also picked up in Gukov, Pei, Putrov & Vafa 18.

Discussion of the 2d SCFTs (namely supersymmetric SU(2)-WZW-models) conjecturally corresponding, under this conjectural identification, to the elements of $\mathbb{Z}/24$ $\simeq$ $tmf^{-3}(\ast) = \pi_3(tmf)$ $\simeq$ $\pi_3(\mathbb{S})$ (the third stable homotopy group of spheres):

#### Occurrences in string theory

##### H-string elliptic genus

Further on the elliptic genus of the heterotic string being the Witten genus:

The interpretation of equivariant elliptic genera as partition functions of parametrized WZW models in heterotic string theory:

Speculations on physics aspects of lifting the Witten genus to topological modular forms:

##### M5-brane elliptic genus

On the M5-brane elliptic genus:

A 2d SCFT argued to describe the KK-compactification of the M5-brane on a 4-manifold (specifically: a complex surface) originates with

Discussion of the resulting elliptic genus (2d SCFT partition function) originates with:

Further discussion in:

##### M-string elliptic genus

On the elliptic genus of M-strings inside M5-branes:

##### E-string elliptic genus

On the elliptic genus of E-strings as wrapped M5-branes:

• J. A. Minahan, D. Nemeschansky, Cumrun Vafa, N. P. Warner, E-Strings and $N=4$ Topological Yang-Mills Theories, Nucl. Phys. B527 (1998) 581-623 (arXiv:hep-th/9802168)

• Wenhe Cai, Min-xin Huang, Kaiwen Sun, On the Elliptic Genus of Three E-strings and Heterotic Strings, J. High Energ. Phys. 2015, 79 (2015). (arXiv:1411.2801, doi:10.1007/JHEP01(2015)079)