dg.differential-geometrymp.mathematical-physicsstring-theory
What role do generalized geometries (in terms of Dirac structures, for instance, symplectic, Poisson, complex, and generalized complex structures in the sense of Hitchin, Cavalcanti, and Gualtieri) play in string theory?
EDIT: More generally, what role to Dirac structures (subbundles of the generalized tangent bundle $TM \bigoplus T^*M$ which are maximally isotropic to the natural pairing and closed under the Courant bracket) play?
What you seem to be thinking isn't really statistical physics, but effective field theory (EFT). Loosely, EFTs take a more fundamental theory, average something out, and give you a field theory that works in some domain of applicability.
Electromagnetism in matter is an EFT because it takes electromagnetism in vacuum with lots of point sources (atoms, molecules, electrons, ...) and, because we can't solve E&M with $10^{23}$ sources exactly, we average over them, making some simplifying assumptions, and get a new theory. In this case, E&M in matter looks almost exactly like E&M in vacuum, but with some extra undetermined constants ('permittivity', etc.) which are determined by 'microscopic' physics, but we usually just measure because it's very hard to calculate them. (It looks just like in vacuum by construction--you can make less aggressive assumptions and get nonlinear theories that apply, e.g., in very high frequency fields.)
Similarly, classical E&M in vacuum is an EFT of quantum electrodynamics, where we average out the quantum fluctuations.
The picture of particles as "composite" particles that behave as particles that show up as single fields in a Lagrangian (e.g,. scalar mesons in a Klein-Gordon equation) are effective field theories that average out the internal degrees of freedom.
It turns out that basically every theory we know of is an EFT in some way (other than string theory, which, not being a field theory, cannot really be an EFT). In particular, the standard model is an EFT. It is believed (for good reasons, beyond the scope of this answer--read and understand any intro string theory book for a basic answer!) that the standard model is an EFT of string theory. That is, we take string theory, average out the 'stringy' degrees of freedom, and we get out the standard model.
Part of the reason we believe this is because we can get both GR and normal quantum mechanics (and various quantum field theories) in 4d in this manner from string theory. We can also get many field theories that look very much like the standard model but have different particle contents. But we can't yet get the standard model exactly out.
We do however, since we can get GR out in a limit of string theory, have a good enough description of it to start to understand stringy effects of gravity. In particular, in normal GR, it is well-known that black holes behave as "thermodynamic systems" in some sense. This was first pointed out in a famous paper by Hawking! It seemed that black holes behaved like thermodynamic systems in the sense that certain dynamical variables satisfied suspiciously identical equations to the laws of thermodynamics.
In particular, the surface area seemed to 'be' the entropy of a thermodynamic system. (For details, see: http://www.scholarpedia.org/article/Bekenstein-Hawking_entropy)
For various reasons, it was taken fairly seriously that black holes should be actual thermodynamic systems. One big reason was that it was found (also due to Hawking) that black holes should emit thermal (i.e., black body) radiation as if they were thermodynamic systems in equilibrium. The radiation emitted (Hawking radiation) was identical to radiation emitted from a blackbody at a temperature that you'd expect if the black hole was a legitimate thermodynamical system.
This was puzzling, because the entropy associated with a black hole was extremely large, but it is a well-known result (see Wald or Misner, Thorne, and Wheeler's intro GR books) that the event horizon of a black hole is featureless. In other words, any 'patch' of event horizon is identical to any other. In other words, there is only one state in classical GR: a featureless event horizon shielding a featureless singularity. (If you're concerned about other particles inside giving the black hole a large entropy, the particles inside cannot normally have an entropy high enough to account for this, and at any rate you can always consider the state after some finite (far-away) time when everyone has hit the point-singularity, which, being a point, has no structure.)
So this is a puzzle. Classical GR says there is only one state. Semi-classical GR, and classical GR assuming it's a real thermodynamic system, both say there are a gigantic number. What's going on here? What are the microscopic states?
Realizing that we have to consider microscopic states makes us see that GR is not the right picture to do this. GR is the already averaged theory. Just like a naive picture of a gas in terms of its extensive quantities--temperature, pressure, volume,...--seems to tell you there is only one state: a gas described by (T,P,V,...).
It turns out, string theory (and no other theory!) lets us calculate the entropy of black holes by counting microscopic states! The difficulty is that, unlike a gas where the states are |particle(x,y,z,spin,....)>, the string states are really screwed up looking states that have absolutely no classical analog (and I don't understand well enough to describe simply). But we can count them, and when we superimpose them we get a classical black hole plus quantum corrections, when we average them to get an EFT we get a classical black hole, and when we count them we get its entropy!
So, yes, there is a connection between strings and statistical physics. That connection is averaging over stringy states gives you an EFT, and one we can calculate is when that EFT is GR, and we can use it to calculate black hole entropy by counting stringy states explicitly.
Edit: References included due to discussion in the comments:
"Exploring Black Holes" by Taylor and Wheeler. Basic black hole stuff, including discussion of shell coordinates (coordinates of observer at fixed $r=r_0$ away from the black hole) which show $dr_{\text{shell}}/dt_{\text{shell}} \rightarrow c$ as $r\rightarrow 2M = r_{\text{horizon}}$, and who observes things in what systems.
"General Relativity" by Wald and "Quantum Field Theory in Curved Spacetime" by Wald. Intermediate to advanced level discussion of black holes, horizons, black hole entropy, and black hole thermodynamics in the first. In the second, more formal qft in curved spacetime, Hawking radiation, and entropy.
"The Thermodynamics of Black Holes" by Wald (http://arxiv.org/abs/gr-qc/9912119). A review-level article on thermodynamics of black holes, which contains a more lengthy discussion of the quantum mechanics, statistical mechanical aspects, as well as various approaches to solving the "entropy problem." Many of the references in this paper are also very good.
"Black Holes as Effective Geometries" by Balasubramanian, de Boer, El-Showk, Messamah (http://arxiv.org/abs/0811.0263). A research article showing the possible calculating the entropy in a way that lets us understand the geometry of the individual microstates explicitly. Difficult, but the intro parts should be reasonably accessible.
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.
Best Answer
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.