The String Coffee Table

Given some homogeneous polynomial

of degree $d$, one can, roughly, associate two different sorts of 2-dimensional field theories with it.

1) On the one hand we can consider sigma-models whose target is the projective variety ($\to$) $X$ of zeros of this polynomial. If this happens to be a Calabi-Yau we can consider the A- or B-model topological string on that target.

2) On the other hand, one can regard $W$ as the superpotential of a Landau-Ginzburg model ($\to$).

In the first case, for the B-model string, the corresponding category of branes is $D^b(\mathrm{Coh}(X))$, the bounded derived category of coherent sheaves on $X$ ($\to$).

In the second case, the category of branes looks superficially different. Let me just call this the category of Landau-Ginzburg B-branes.

Now, we can think of both these models as different points in one and the same moduli space of a $N=(\mathrm{2,2})$ gauged linear sigma-model (GLSM). There is a certain parameter, called $r$, parameterizing this model, and in the limit that $r$ tends to plus or minus infinity, the GLSM tends to the nonlinear $\sigma$-model on $X=\{x_i\mid W(x_1,\cdots ,x_N)=0\}$ or the Landau-Ginzburg model with superpotential $W$, respectively.

What Hori and Herbst are trying to do is to use this gauged linear sigma model to flow the category of Landau-Ginzburg B-branes through moduli space to the derived category of coherent sheaves on $X$, thus realizing the equivalence of these two categories by means of a “physical” system.

That both categories are in fact equivalent (when suitable assumptions hold which I am glossing over), was shown in

Dmitri Orlov
Derived categories of coherent sheaves and triangulated categories of singularities
math.AG/0503632,

theorem 3.11.

Related results have been discussed in

Yujiro Kawamata
Log Crepant Birational Maps and Derived Categories
math.RT/0510187 .

As far as I understood, Hori and Herbst expected that in fact the category of branes of the full GLSM is, too, equivalent to both of the above categories. Hoewever, it turns out that this equivalence has so far only been shown for special choices of some other parameter, called $\theta$.

I am wondering if this should be worrisome. Wouldn’t it be natural for the category of branes of the GLSM to be larger (and non-equivalent) to the category of branes obtained in the limiting case $r\to \pm \mathrm{\infty }$?

Hori proceeded by spelling out lots of details at the level of Lagrangians, which I won’t even try to reproduce in total.

I’ll just indicate enough details to see the two parameters $r$ and $\theta$ appearing.

The GLSM involves a twisted chiral gauge superfield $V$ (I think), which appears in the Lagrangian in terms of its superderivative

The gauge kinetic term of this field in the Lagranian is

There is also a matter kinetic term of the form

and a superpotential term for these matter fields

where $W$ is our polynomial from above.

Finally, and that’s where the two parameters come in, there is an F-term

where the complex parameter $t$ has real and imaginary part given by the two parameters in question

One can see that for $r\to \mathrm{\infty }$ the matter fields $X_i$ localize on the zeros of $W$, thus leading to a nonlinear sigma-model on $X$.