noncommutative topology, noncommutative geometry
noncommutative stable homotopy theory
genus, orientation in generalized cohomology
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.
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
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”.
The degenerate case with parameters $\delta = \epsilon = 1$ (as above) is the signature genus.
The degenerate case with parameters $\delta = - \frac{1}{8}$ and $\epsilon = 0$ (as above) is the A-hat 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
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
(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.
On manifolds with spin structure the elliptic genus takes values in integral series $\mathbb{Z}[ [q] ]$.
(Chudnovsky-Chudnovsky 88, Landweber 88, section 5 Kreck-Stolz 93, Hovey 91)
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.
partition functions in quantum field theory as indices/genera/orientations in generalized cohomology theory:
The notion of elliptic genus was introduced in
A quick review is in
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
The integrality of the elliptic genus and elliptic homology on Spin-manifolds is due to
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)
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
Last revised on October 27, 2016 at 11:55:37. See the history of this page for a list of all contributions to it.