The local Gromov-Witten theory of curves -- Jim Bryan

<p>

In <a href=http://arxiv.org/abs/math.AG/0411037>math.AG/0411037</a>,  
Rahul Pandharipande and I
completely solve the all genus, equivariant, Gromov-Witten theory for the
class of threefolds given as the total space of a rank two bundle over a
curve.

<p>

In the case where the threefold is equivariantly Calabi-Yau, our solution
has a closed form in terms of a Q-deformation of representations of the
symmetric group. This closed formula has been used by physicists to verify a
duality between topological string theory and BPS states of black holes and
to discover a new duality with a Q-deformed 2D Yang-Mills
theory:
<a href=http://arxiv.org/abs/hep-th/0411280>hep-th/0411280</a>.

<p>

In the case of P^1 x C^2, our theory also has a
remarkable and completely unexpected equivalence with the equivariant
quantum cohomology of the Hilbert scheme of points in the affine plane,
recently computed by Okounkov and Pandharipande:
<a href=http://arxiv.org/abs/math.AG/0411210>math.AG/0411210</a>.