elliptic genus in nLab


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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 ℂP(ξ)\mathbb{C}P(\xi) for ξ\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 ϕ\phi satisfying this condition has a logarithm log ϕlog_\phi of the form

log ϕ(u)=∫ 0 u(1−2δt 2+ϵt 4) −1/2 log_\phi(u) = \int_{0^u} (1- 2 \delta t^2+ \epsilon t^4)^{-1/2}

for some constants δ,ϵ\delta, \epsilon. Hence for non-degenerate choices of parameters (δ 2≠ϵ\delta^2 \neq \epsilon and ϵ≠0\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”.

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

The degenerate case with parameters δ=−18\delta = - \frac{1}{8} and ϵ=0\epsilon = 0 (as above) is the A-hat genus.

is naturally parameterized by the upper half plane. Under this identification obe may think of ϵ\epsilon and δ\delta as functions of moduli of elliptic curves and concretely as modular forms for the subgroup Γ 0(2)\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 •(Γ 0(2))MF_\bullet(\Gamma_0(2)) of modular forms

w:Ω • SO⟶MF •(Γ 0(2)) 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 δ\delta and ϵ\epsilon produces the corrponding elliptic genus).

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

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

On equivariant elliptic cohomology and positive energy representations of loop groups:

Relation to Kac-Weyl characters of loop group representations

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

See also:

Via derived E ∞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)

The identification of elliptic genera, via fiber integration/Pontrjagin-Thom collapse, as complex orientations of elliptic cohomology (sigma-orientation/string-orientation of tmf/spin-orientation of Tate K-theory):

For the Ochanine genus:

Equivariant elliptic genera

Genera in equivariant elliptic cohomology and the rigidity theorem for 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:

Twisted elliptic genera

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

Elliptic genera as super pp-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:

With emphasis on orbifold CFTs:

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:

In relation to error-correcting codes:

  • Kohki Kawabata, Shinichiro Yahagi, Elliptic genera from classical error-correcting codes [[arXiv:2308.12592]]
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 XX are the generalized cohomology of XX 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 ℤ/24\mathbb{Z}/24 ≃\simeq tmf −3(*)=π 3(tmf) tmf^{-3}(\ast) = \pi_3(tmf) ≃\simeq π 3(𝕊)\pi_3(\mathbb{S}) (the third stable homotopy group of spheres):

Discussion properly via (2,1)-dimensional Euclidean field theory:

See also:

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:

Proposals 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:

On the elliptic genus of E-strings as M2-branes ending on M5-branes: