I would like to know if there is any setting where the two notions of
central charge of 2D conformal field theories,
Calabi-Yau dimension of fractionally Calabi-Yau categories
can be understood as "one and the same".
One motivation (not very serious) is that
on the one hand there are CY categories of dimension $(h-2)/h$ where $h$ is the Coxeter number of a Dynkin diagram,
on the other hand, Coxeter numbers often appear in central charges.
Given a $N=(2,2)$ two dimensional superconformal field theory (SCFT), one can construct two topological field theories called the $A$ and $B$ models. To each of these topological field theories, one should be able to associate a ($A_\infty$) triangulated category of boundary conditions, called category of branes. Thus corresponding to $A$ and $B$ models one obtains the categories of $A$- and $B$-branes. These categories are Calabi-Yau categories of dimension $c/3$ where $c$ is the central charge of the $N=(2,2)$ superconformal field theory. This Calabi-Yau property should be a consequence of the existence of the spectral flow in the $N=(2,2)$ superconformal algebra (and so I should make a technical hypothesis on the $N=(2,2)$ SCFT: the differences between left and right R-charges should be integers).
When I have written "should be", it means that they are rather standard facts for some physicists but that I don't necessarely know how to make and prove general rigorous statements. For more information on this topic, one might want to have a look at the book http://www.claymath.org/library/monographs/cmim04.pdf
But there are many examples where the $A$ and $B$ models and the corresponding categories of $A$ and $B$-branes are known and where the central charge of the maybe not yet rigorously constructed $N=(2,2)$ SCFT is also known.
Examples:
1) If $(X, J, \omega)$ is a Calabi-Yau manifold of complex dimension $d$, with $J$ a complex structure and $\omega$ a Kähler form then one should be able to construct a $N=(2,2)$ SCFT (the two dimensional sigma model of target $(X,J,\omega)$), of central charge $c=3d$. The corresponding category of $A$-branes is the Fukaya catgeory of $(X,\omega)$ and the category of $B$-branes is the derived category of coherent sheaves on $(X,J)$, which are indeed Calabi-Yau categories of dimension $d$.
2) If $W(x_1,...,x_n)$ is a quasihomogeneous polynomial, i.e. satisfying $W(\lambda^{w_1}x_1,...,\lambda^{w_n}x_n)=\lambda^d W(x_1,...,x_n)$ for some $w_1,...,w_n,d$ positive integers then one should be able to construct a $N=(2,2)$ SCFT (the Landau-Ginzburg model of superpotential $W$), of central charge $c=3 \sum_{i=1}^n(1-2 \frac{w_i}{d})$. I don't know what is the category of $A$-branes in general but the category of $B$-branes is the category of matrix factorizations of $W$, which is Calabi-Yau of dimension $\sum_{i=1}^n(1-2 \frac{w_i}{d})$. In 1), the dimension of the Calabi-Yau categories was an integer but it is in general a rational number in 2).
If $W$ is an $ADE$ surface singularity, then $c=3(1-\frac{2}{h})$ where $h$ is the $ADE$ Coxeter number and the corresponding $N=(2,2)$ SCFT is believed to be the ADE $N=(2,2)$ minimal model. It is possible to show that the category of matrix factorizations of an $ADE$ surface singularity is equivalent to the derived category of representations of the $ADE$ quiver, which is a standard example of Calabi-Yau category of dimension $1-\frac{2}{h}$.
Remark: as the comment by Hugh Thomas shows, the terminology "central charge" can be confusing because any central element in an algebra could be called central charge and there are many different algebras and central elements... The question is about the central charge of a $2d$ CFT, i.e. the central element of the Virasoro algebra. The central charge to which Hugh Thomas refers to is different but also appears naturally in the story: it is part of the information which should be obtained from the $N=(2,2)$ SCFT and should determine which of the branes of the topological theories are in fact branes of the superconformal theory. Its name central charge refers to the fact that in some cases (when the $N=(2,2)$ SCFT is used as internal space for a string compactification), it is really a central charge in a $N=2$ four dimensional supersymmetry algebra (corresponding to the four non-compact dimensions of the string compactification).
