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What is the elliptic genus (see also Witten index) in string/M-theory and (susy gauge)field theory constructions out of them? What does it tell us heuristically and what is its relation to the partition function?

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Below is a summary of my very limited understanding of what the elliptic genus is. I'll first give you the mathematical definition, followed by an explanation of how it appears naturally in physics. As a first exposure to the subject, it's perhaps best to not consider the most general definitions. In what follows, I will only explain what the elliptic genus of a (compact) Calabi-Yau manifold $M$ is (instead of a general vector bundle over $M$).

Mathematical description:

One of the most basic topological invariants of a manifold is it's Euler characteristic $\chi(M)$. This is defined as the alternating sum of the dimensions of the (co)homology groups. But for our purposes the more useful description is the one found by using the Hirzebruch-Riemann-Roch theorem, which says if the Chern class of $M$ splits as $$c(M) = \prod_{i=1}^D (1+x_i)$$ then $$\chi(M) = \int_M \prod_{i=1}^Dx_i,$$ namely it's just the integral of the top Chern form (also known as the Euler class).

The $\chi_y$ genus of manifold is a generalization of this which instead of being a pure number, depends on a (complex) parameter $y$:

$$\chi_y(M) := \int_M \prod_{i=1}^Dx_i\frac{1-ye^{-x_i}}{1-e^{-x_i}}$$ which we can see reduces to the Euler characteristic when $y=1$.

The elliptic genus $\mathcal{E}_M(\tau,z)$ is again a further generalization, now made to depend on two parameters, $\tau$ and $z$ (or $q$ and $y$ if we let $q = e^{2\pi i \tau}$ and $y=e^{2\pi i z}$). It can be defined using this complicated formula:

$$\mathcal{E}_M(\tau,z) = \int_M \prod_{i=1}^D\Big[x_i\frac{1-ye^{-x_i}}{1-e^{-x_i}} \prod_{n=1}^{\infty} \frac{1-q^n y e^{-x_i}}{1-q^ne^{-x_i}}\frac{1-q^n y^{-1} e^{x_i}} {1-q^n e^{x_i}} \Big]y^{\zeta(0)} .$$

As is apparent from the formula, we have the specializations $$\mathcal{E}_M(y=1, q) = \chi(M), \,\,\,\,\,\,\ \mathcal{E}_M(y, q=0) = y^{\zeta(0)}\chi_y(M).$$

Obviously, someone didn't pull this complicated formula for the elliptic genus out of a hat. Instead it can be viewed as the ordinary $\chi_y$ genus of an infinite-dimensional manifold, the loop space of $M$.

An interesting property is that for a Calabi-Yau $D$-fold, $\mathcal{E}_M(\tau,z)$ is a weak Jacobi form of weight $0$.

What does this have anything to do with physics? Supersymmetric sigma models provide an answer:

Physical interpretation:

What's quite amazing is that all three quantities defined above (and many others) appear as indices of supersymmetric sigma models. Here's how:

Let $M$ be a Riemannian manifold. The Lagrangian of the supersymmetric quantum mechanics model is given by

$$ L = \frac{1}{2}g_{ij} \dot{\phi^i} \dot{\phi^j} + \frac{i}{2}g_{ij} \bar{\psi}^i D_t \psi^j + \frac{1}{4}R_{ijkl}\psi^i \bar{\psi}^j \psi^k \bar{\psi}^l,$$

which when we canonically quantize, gives us a supersymmetric system. The Hilbert space of this system is the space of differential forms on $M$ and a basic property is that there is a conserved fermion number operator $F$ which coincides with the operator that gives eigenvalue $p$ when acting on a $p$-form. Further, $F$ splits as $F= F_+ + F_-,$ depending on the type of fermion used to add to the form degree. The usual partition function is given by $Z(\beta) = \text{Tr} e^{-\beta H}$, which is typically difficult to compute exactly and is not deformation invariant. However, there is a nicer supersymmetric version, known as the Witten index given by $$ \text{Tr} (-1)^F e^{-\beta H}.$$ It can be shown using basic properties of supersymmetry that the Witten index is equal to the $$\text{number of bosonic ground states} - \text{number of fermionic ground states}$$ and in particular is independent of $\beta$. Exploiting this independence, one can compute a path integral and show that in fact this is exactly the same as the Euler characteristic of $M$: $$\text{Tr} (-1)^F e^{-\beta H} = \chi(M).$$ This is very interesting: It gives physical meaning to a topological invariant!

Now we have a similar generalization of this index to get the $\chi_y$ genus. If one considers the index $$\text{Tr}(-1)^{F_+}(-y)^{F_-} e^{-\beta H}$$ one recovers the $\chi_y$ genus of the manifold (so this index is still independent of $\beta$).

To get the elliptic genus, one has to increase the dimension of the model by one. One now considers a $(1+1)$-dimensional quantum field theory with a similar (suitably generalized) Lagrangian to the one above. If one computes (a somewhat generalized version of) the index above, $\text{Tr}(-1)^{F_+}(-y)^{F_-} e^{-\beta H}$ in the Ramond-Ramond sector of this quantum field theory, one finds that it's no longer independent of $\beta$ and upon doing the path integral in the semi-classical approximation (exact due to supersymmetry) one recovers the expression for the elliptic genus.

(I've skipped some details in the last part, and also some motivational aspects of the definitions, which I hope to add later).

A useful reference which can be used as a starting point to understand this in more detail is "Mirror Symmetry" by Vafa, Zaslow et. al. (Chapter 10 in particular)

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  • $\begingroup$ In Chapter 10 where exactly? It is the one about d=1 QM. Does not see too much related. $\endgroup$
    – Marion
    May 26, 2015 at 13:31
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    $\begingroup$ Chapter 10 discusses the relationship between the Euler characteristic and the Witten index (pages 206-211) and also the general properties of supersymmetric theories. The elliptic genus is a generalization of these results, so it's good (almost necessary) to have these concepts down first. $\endgroup$ May 26, 2015 at 19:42
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There are obviously differeing genus types according to which partition function in d-dimensional QFT.

At the very outset of $0$-d QFT the index is the push forward in ordinary de Rahm Cohomology; in other words, the integration of differential forms. The genus as you put it, is non-existant here.

In $1$-d QFT the index of the Dirac Operator is $K0$-homology. Genus in this context is the so-called $\hat{A}$ genus; this has been proved via the Atiyah–Singer index theorem to not always be an integer. In terms of an operator index, it is given by composing the action of the covariant derivative on sections of a Riemannian Manifold. It is an elliptic operator. The index of this operator is called the $\hat{A}$, A-genus.

The partition function within the same $1$-d QFT as the endpoint of type II superstring has a supercharge of $\text{Spin}^c$ Dirac operator twisted by Chan-Paton gauge field, the index in cohomology is the D-brane charge, the genus here is the so called Todd genus. The todd genus is the index of the Dolbeault operator.

For $2$-d Type-II Superstring, the index in cohomology theory is the Superstring partition function; whose genus is the elliptis genus. The genus is elliptic if it vanishes on manifolds that are projective spaces of the form $\mathbb{C}P(\xi)$ where $\xi$ is an even-dimensional complex vector bundle over an oriented closed manifold.

The genus described above describes the he partition function of a type II superstring as a function depending on the modulus of the worldsheet elliptic curve which yields an elliptic genus (Witten, as you describe above).

EDIT: References.

The original description of the Witten genus from string theory is due to

  • Edward Witten, Elliptic Genera And Quantum Field Theory ,
    Commun.Math.Phys. 109 525 (1987).

  • Edward Witten, The Index Of The Dirac Operator In Loop Space Proc. of Conf. on Elliptic Curves and Modular Forms in Algebraic Topology
    Princeton (1986).

These two publications are fallouts from the insightful publications below;

  • 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.
  • 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).

That a spin structure makes the Witten genus take values in an integral series is due to

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

That it takes a rational string structure to make the elliptic genus land in modular forms was noticed within

  • Don Zagier, Note on the Landweber-Stong elliptic genus 1986.
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  • $\begingroup$ Do you have some introductory reference that would help me a little bit? Thanks a lot. $\endgroup$
    – Marion
    May 6, 2015 at 11:30
  • $\begingroup$ @Marion Sure, please see edits. $\endgroup$
    – Autolatry
    May 6, 2015 at 11:50

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