3
$\begingroup$

It is known that the partition function $$ \mathcal{Z}_1=\int dH e^{-N{\rm Tr}(H^2)}e^{-NV(H)},$$ where the integral is over $N\times N$ hermitian matrices $H$, with the potential $$ V(H)=\sum_{j\ge 1}\frac{t_j}{j}{\rm Tr}(H^j),$$ is related to an integrable hierarchy of differential equations in the $t$ variables. This can be reduced to eigenvalues as $$ \mathcal{Z}_1\propto\int_{-\infty}^\infty dx \Delta^2(x)\prod_j e^{-Nx_j^2}e^{-NV(x_j)},$$ where $\Delta(x)$ is the Vandermonde.

Suppose now I change this in the following way $$ \mathcal{Z}_2=\int dM e^{-N{\rm Tr}(MM^\dagger)}e^{-NV(MM^\dagger)},$$ where $M$ is not hermitian and $V$ is the same as above. This can also be reduced to eigenvalues as $$ \mathcal{Z}_2\propto\int_0^\infty dx \Delta^2(x)\prod_j e^{-Nx_j}e^{-NV(x_j)}.$$

The question is: is $\mathcal{Z}_2$ also related to an integrable hierarchy?

Improve this question
$\endgroup$
2
  • $\begingroup$ can you specify the differential equations you have in mind? $\endgroup$
    – Adrien Hardy
    Commented Dec 12, 2015 at 19:45
  • 1
    $\begingroup$ @AdrienHardy The function $\mathcal{Z}_1$ is related to the so-called KP hierarchy (for Kadomtsev-Petviashvili) $\endgroup$
    – Marcel
    Commented Dec 13, 2015 at 13:40

1 Answer 1

Reset to default
1
$\begingroup$

Yes. This model is sometimes referred to as the "complex matrix model." The partition function is known to be a tau-function of the forced Toda chain hierarchy and, as a consequence, solves the KP hierarchy.

Improve this answer
$\endgroup$
4
  • $\begingroup$ This one?: arxiv.org/pdf/hep-th/9203043v1.pdf $\endgroup$
    – AHusain
    Commented Apr 12, 2016 at 19:14
  • $\begingroup$ Do you have a reference? $\endgroup$
    – Marcel
    Commented Apr 12, 2016 at 23:04
  • $\begingroup$ @AHusain As far as I remember this one does not consider the complex matrix model. However, the integrable properties of the complex matrix model are closely related to the properties of the Hermitian matrix model, thus, one can use the same approach. $\endgroup$
    – Sasha
    Commented Apr 13, 2016 at 0:21
  • $\begingroup$ I don't know a good reference, however, idea of the proof of the Toda integrability for the complex matrix model is the same as for the Hermitian matrix model. Namely, you can represent a partition function as a determinant $$ {\mathcal Z}_2 =det_{i,j=1}^N A_{ij}$$ of the matrix of the form $$A_{ij}=\frac{\partial^{i+j-2} }{\partial t_1^{i+j-2}} A$$ where $$A=\int_{0}^\infty dxe^{-Nx-NV(x)}.$$ $\endgroup$
    – Sasha
    Commented Apr 13, 2016 at 0:37

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .