I am giving a talk on String theory to a math undergraduate audience. I am looking for a nice and suprising mathematical computation, maybe just a surprising series expansion, which is motivated by string theory and which can be motivated and explained relatively easily. Examples of what I have in mind are the results in Dijkgraaf's "Mirror symmetry and the elliptic curve", or the "genus expansion" of the MacMahon function (aka DT/GW for affine three-space), but I am not sure I can fit either into the time I have. Any thoughts?
[Math] String theory “computation” for math undergrad audience
mp.mathematical-physicsstring-theory
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For fixed integers $g,n$, any projective scheme $X$ over a field $k$, and a linear map $\beta:\operatorname{Pic}(X)\to\mathbb Z$, the space $\overline{M}_{g,n}(X,\beta)$ of stable maps is well defined as an Artin stack with finite stabilizer, no matter the characteristic of $k$. You can even replace $k$ by $\mathbb Z$ if you like.
Now if $X$ is a smooth projective scheme over $R=\mathbb Z[1/N]$ for some integer $N$, then $\overline{M}_{g,n}(X,\beta) \times_R \mathbb Z/p\mathbb Z$ is a Deligne-Mumford stack for almost all primes $p$. For such $p$, $\overline{M}_{g,n}(X,\beta) \times_R \mathbb Z/p\mathbb Z$ has a virtual fundamental cycle, and so you have well-defined Gromov-Witten invariants. This holds for all but finitely many $p$. Nothing about $\mathbb C$ here, that is my point, the construction is purely algebraic and very general.
It is when you say "Hodge structures" then you better work over $\mathbb C$, unless you mean $p$-Hodge structures.
As far as mirror symmetry in characteristic $p$, much of it is again characteristic-free. For example Batyrev's combinatorial mirror symmetry for Calabi-Yau hypersurfaces in toric varieties is simply the duality between reflexive polytopes. You can do that in any characteristic, indeed over $\mathbb Z$ if you like.
Let me add something to what David and Urs have written already, since the way those two answers are shaping up, perhaps what I'm about to say does not get mentioned.
One of the most interesting applications of generalised geometry in string theory is in the study of supersymmetric flux compactifications. Ten-dimensional superstring theories have a well-defined limit (=the effective theory of massless states) which corresponds to ten-dimensional supergravity theories. One way to view these theories is as variational problems for certain geometric PDEs which generalise the Einstein-Maxwell equations. The dynamical variables consist of a lorentzian ten-dimensional metric and some extra fields, depending on the theory in question. One set of of fields common to the ten- and eleven-dimensional supergravity theories are $p$-forms obeying possibly nonlinear versions of Maxwell equations. In the Physics literature these $p$-form fields are called fluxes and the geometric data consisting of the lorentzian manifold, the fluxes and any other fields all subject to the field equations are known as supergravity backgrounds.
These supergravity backgrounds are actually not just lorentzian manifolds, but in fact they are spin and part of the baggage of the supergravity theory is a connection (depending on the fluxes,...) on the spinor bundle, which defines a notion of parallel transport. Parallel spinor fields are known as (supergravity) Killing spinors and backgrounds admitting Killing spinors are called supersymmetric.
One way to make contact with the 4-dimensional physics of everyday experience is to demand that the ten-dimensional geometry be of the form $M \times K$, where $M$ is a four-dimensional lorentzian spacetime (usually a lorentzian spaceform: Minkowski, de Sitter or anti de Sitter spacetimes) and $K$ a compact six-dimesional riemannian manifold, known as the compactification manifold.
When all fields, except the metric, are set to zero, the connection agrees with the spin lift of the Levi-Civita connection and supersymmetric backgrounds of this type are lorentzian Ricci-flat manifolds admitting parallel spinor fields. If we demand that they be metrically a product $M \times K$ as described above, then a typical solution is $M$ being Minkowski spacetime and $K$ a six-dimensional manifold admitting parallel spinors; that is, a Calabi-Yau manifold, by which I mean simply a manifold with holonomy contained in $SU(3)$. This result, which today seems quite unassuming, was revolutionary when it was first discovered in the 1985 paper of Candelas, Horowitz, Strominger and Witten. That paper is responsible for the interest of physicists in Calabi-Yau manifolds and ensuing rapprochement between physicists and algebraic geometers, the fruits of which we're still reaping today.
But Calabi-Yau compactifications are in fact very special from the physics point of view: since most of the fields in the theory (especially the fluxes) have been turned off. Generalised geometry enters in the search for more realistic "flux compactifications". One of the fields which all ten-dimensional supergravity theories have in common is the $B$-field (also called Kalb-Ramond field). One of Hitchin's motivations for the introduction of generalised geometry was to give a natural geometric meaning to the $B$-field. For example, the automorphism group of the Courant algebroid $T \oplus T^*$ is the semidirect product of the group of diffeomorphisms and $B$-field transformations.
More generally, I think that it is still true that all known supersymmetric flux compactifications $M \times K$ (even allowing for warped metrics) of ten-dimensional supergravity theories are such that $K$ is a generalised Calabi-Yau manifold. The fluxes turn out to be related to the pure spinors in the definition of a GCY structure. There are many papers on this subject and perhaps a good starting point is this review by Mariana Graña.
There are other uses of generalised geometry in string theory, e.g., the so-called doubled field theory formalism, as in this recent paper of Chris Hull and Barton Zwiebach.
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Two counting problems -- from my own very biased and personal viewpoint -- that can perhaps be motivated:
But these can't beat calculating an actual partition function (as in Richard Eager's answer), unless you're trying to emphasize mathiness.