How do you get the spectral curve from a Calabi-Yau?

Question

In N=2 Quantum Field Theories and Their BPS Quivers by Alim, Cecotti, Córdova, Espahbodi, Rastogi and Vafa the authors give a recipe which constructs from a pair $(C,\phi)$ consisting of a Riemann surface $C$ and a meromorphic quadratic differential $\phi$ on $C$ a (non-compact) Calabi-Yau 3-fold $X$.

Essentially, $X$ is cut out from the total space of the bundle $E=T^*C\oplus T^*C\oplus T^*C$ by the equation $uv=y^2-\phi(z)$ where $z$ is a local coordinate on $C$ and $(u,v,y)$ are coordinates on the fibre $E_z$. An important feature is that $X$ contains the spectral curve of $(C,\phi)$, i.e., the curve $\Sigma\subset T^*C$ which is a branched double cover of $C$ branched at the critical points of $\phi$.

From poking around physics literature that I don't understand, I get the sense that this construction is in some sense post hoc and the natural construction is to go the other way. That is, one should start with (some particular type of) CY $X$ and extract from it the pair $(C,\phi)$. Hence my question:

Is there a natural recipe which takes a CY $X$ and returns the pair $(C,\phi)$, inverse to the construction above?

Any insights or references to the literature would be greatly appreciated.

Answer 1

In general there is no way to extract a spectral curve from a Calabi-Yau threefold.

In the study of strings on Calabi-Yaus, one object of interest is the periods, i.e. integrals of the holomorphic top form. We are particularly interested in the behavior of these period as the complex structure of the Calabi-Yau is varied.

One is interested in these questions for honest, compact Calabi-Yaus but non-compact examples are often studied for a variety of reasons, most notably because many questions have explicit answers.

Typically, the non-compact examples in question are local singularities that have been smoothed by complex structure deformation. Importantly, the singularities of interest are restricted to be those that could in principle arise at a finite distance in the moduli space of a compact Calabi-Yau (see this paper). (Practically speaking this often bounds the degrees appearing in equations cutting out the singularity.)

The constructions discussed in the referenced paper are merely a very simple class of examples. Due to the nature of that setup, the problem reduces to studying the Jacobian of a curve (we did NOT invent the construction! It is twenty years old.). The construction can be generalized to build non-compact Calabi-Yaus from n-fold branched covers (embedding general Hitchin systems into the problem).

However many more complicated examples of non-compact Calabi-Yaus exist where the period geometry does not reduce (in any known way) to the Jacobian of a curve. A key phrase to look for in the the literature is "geometric engineering." For some recent classification results, see e.g. this paper, which investigates singularities defined by a complete intersection of two hypersurfaces.