Question

What is elliptic genera in physics? Reading many relevant papers, they just defined elliptic genus as sort of partition function. I try to find useful materials to explain it, but I couldn't find it. Can you give some intuition for this terminology in physical view?

This post imported from StackExchange Physics at 2014-09-06 13:34 (UCT), posted by SE-user phy_math

Comments

It is a mathematical term. See here: en.wikipedia.org/wiki/Genus_of_a_multiplicative_sequence The elliptic genus of a manifold when its arguments are specialised gives the Euler characteristic and the Hirzebruch signature.

This post imported from StackExchange Physics at 2014-09-06 13:34 (UCT), posted by SE-user suresh

Answer 1

There is a mathematical definition of genus in general and of elliptic genus in particular, which you may or may not find enlightning. It says something like that a genus is an assignment of some quantity to a manifold, such that the quantity depends suitably only on the cobordims class of the manifold and such that the disjoint union of two manifolds is taken to the sum and the product of two manifolds to the product of that quantity.

So then you may ask: is there a physical interpretation of this? Where would such genera appear in physics? And here the striking answer is: at least the important genera turn out to be the assignments that take a manifold, regard it as the target spacetime in which some spinning particle or superparticle or spinning string or superstring or some other brane propagates, and then assigns the quantity which is the partition function of the worldvolume theory of that little brane propagating in that manifold. 

This was Edward Witten's big insight, part of what won his Fields medal, regarding the Ochanine elliptic genus, which he effectively understood to be the partition function of the type II superstring. Then he checked what the partition function of the heterotic superstring would give and found this way a new genus, now named after him: the Witten genus.

Since then, genera are being identified all over the place as partition functions of supersymmetric QFTs. One may also think of these are being the indices of the supercharge and hence often they are called indices. Check out this table.

Answer 2

I recommend reading Witten's paper Elliptic genera and quantum field theory on the topic. His abstract is as follows:

It is shown that in elliptic cohomology - as recently formulated in the mathematics literature - the supercharge of the supersymmetric nonlinear sigma model plays a role similar to the role of the Dirac operator in \(K\)-theory. This leads to several insights concerning elliptic cohomology and string theory. Some of the relevant constructions have been done previously by Schellekens and Warner in a different context.

His paper is available online at ProjectEuclid at http://projecteuclid.org/download/pdf_1/euclid.cmp/1104117076.