Let's consider a 0-dimensional $N \times N$ Hermitean one matrix model. It is defined by a potential V(M). Its partition function is $$Z = \int_{H_{N}} dM e^{-\frac{1}{g}V(M)}$$ where $H_{N}$ is the space of $N \times N$ Hermitean matrices and g is the coupling constant. I usually think at such a matrix model as a 0+0 dimensional QFT : the matrix M is a field over a space-time reduced to a point.

Let's consider the large N limit of this matrix model : $N \longrightarrow \infty$, $g \longrightarrow 0$, $\lambda = gN$ fixed. In this limit, the free energy $F= log(Z)$ has an expansion $$F = \sum_{g=0}^{\infty} F_{g} \lambda^{2g-2}.$$ In this limit, the repartition of eigenvalues of $M$ converges to a deterministic measure with density $y(x)$ function of the eigenvalue x. The graph of this function is a real curve in the x-y plane called the (real) spectral curve. (In the simplest case where the potential V is quadratic, this curve is a half-circle, it is the well known Wigner's semicircle law).

In general, the real spectral curve is defined by an algebraic equation in x,y. In particular, it is possible to consider this equation with x,y complex to obtain a non-compact Riemann surface that I call the complex spectral curve. So in the large $N$ limit, we have emergence of some geometry via the real or complex spectral curves.

My question is:

Is there a precise holographic description of this emergence of geometry?

In usual holography, we have a duality between a 1+(d-1) QFT and a 1+d quantum gravity theory with emergence of one space-like direction. If I look at the matrix model as a 0+0 QFT then in particular I have no time and it is not clear for me what an holographic description should be. If there is some kind of emergence of the spectral curve, is it of the real or of the complex one?

Things related but which does not answer the question:

The existence of a relation between matrix models and quantum gravity can be guessed from the similarity between the large N expansion of the free energy F and the genus expansion of the free energy of a string theory (the perturbative expansion of the matrix models can be written in terms of ribbon graphs, it is the same argument as the 't Hooft one on the possibility of a relation between gauge theory in the large color limit and string theory).

Dijkgraak, Vafa and all have shown that starting from the complex spectral curve, it is possible to construct a non-compact Calabi-Yau 3-fold whose B-model topological string has for genus expansion the large N expansion of the matrix model.

There is a long story of relations between matrix models and 2d quantum gravity, which I don't know very well (essentially, the combinatorics problems of triangulations of surfaces have a matrix model interpretation because of the ribbon graph form of the large N expansion of the matrix model). An additional question would be :

Is the relation between matrix models and 2d quantum gravity an example of holography?

(Intuitively it should be but detailed formulation is not clear to me: for example, it seems we pass from 0 to 2 dimensions...)



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