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I am a physicist working on a project dedicated to the quantisation of commuting matrix models. In the appropriate formalism this problem is reduced to a quantisation in a curved space -- the space of commuting matrices. The general prescription for quantisation in curved space involves ambiguity of the Hamiltonian operator proportional to the scalar curvature of the curved space - hence my question.

A set of $p$ commuting $n\times n$ hermitian matrices $X^{\mu}$ for $\mu=1,\dots p$, is parametrised in terms of a set of $p$ diagonal matrices $\Lambda^{\mu}$ and an unitary matrix $U$ via:

$X^{\mu}=U\,\Lambda^{\mu}\,U^{\dagger}~~$ for $~\mu=1\dots p$,

clearly not all degrees of U contribute to this parametrisation, for example a reparametrisation $U'= D\,U$, where $D$ is a diagonal unitary matrix would result in the same set of commuting matrices. In other words only the elements of the quotient space $U(n)\,/\,U(1)^n$, which is the maximal flag manifold $F_n$, contribute to the parametrisation. The metric on the resulting curved manifold can be calculated as a pull-back of the metric on the space of hermitian matrices defined as:

$ds_{X}^2=Tr\,\left( dX^{\mu}dX^{\mu}\right) $ ,

Using that $U^{\dagger}d X^{\mu} U=d\Lambda^{\mu}+[\theta,\Lambda^{\mu}]~~$, where $\theta$ is the Maurer-Cartan form $\theta=U^{\dagger}dU$, one can write the induced metric as:

$ds^2=\sum\limits_{i=1}^n(d\vec\lambda_i)^2+2\sum\limits_{i<j}(\vec\lambda_i-\vec\lambda_j)^2\theta_{ij}\bar{\theta}_{ij}~~$, where $~~\vec \lambda_i =(\Lambda^1_{ii},\dots,\Lambda^p_{ii})$ .

Now I need the Riemann curvature of the above metric. It seems that it is convenient to work in tetrad formalism, using tetrads $E_{ij}=|\vec\lambda_i-\vec\lambda_j|\,\theta_{ij}$, for $i<j$. The problem is that $d E_{ij}$ will now contain a term proportional to $(\theta_{ii}-\theta_{jj})\wedge\theta_{ij}$ and since $\theta_{ii}$ are not part of the basis the spin curvature cannot be written easily without using the explicit parametrisation of $U(n)$. Intuitively, I know that the scalar curvature should depend only on the lambdas ($\vec\lambda_i$), and I have verified that explicitly for $SU(2)$ and $SU(3)$, however a general result seems to require some invariant way to express the pullback of the term $(\theta_{ii}-\theta_{jj})\wedge\theta_{ij}$ on the submanifold spanned by the off diagonal $\theta$'s.

I was wondering if mathematicians have explored the manifold of commuting hermitian matrices. In fact even a reference to a convenient parametrisation of the maximal flag manifold $F_n$ would greatly help me in deriving a general expression for the scalar curvature. Any comments/suggestions are welcomed.

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  • $\begingroup$ What do you mean by "p-commuting"? I never heard of this terminology. $\endgroup$ Apr 29 '14 at 0:13
  • $\begingroup$ Sorry, I simply meant a number of $p$ commuting matrices ... $\endgroup$
    – vesofilev
    Apr 29 '14 at 0:38
  • $\begingroup$ I edited my post to erase the confusing dashes... $\endgroup$
    – vesofilev
    Apr 29 '14 at 0:51
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    $\begingroup$ Do you assume that the commuting matrices are "regular" in the sense that each has all distinct eugenvalues? Otherwise, you get partial flag manifolds as well. $\endgroup$ Apr 29 '14 at 1:47
  • $\begingroup$ Yes, you do assume that they are regular. The singular case is highly suppressed because the Vandermonde determinant vanishes in this case and the weight in the partition function is proportional to it. $\endgroup$
    – vesofilev
    Apr 29 '14 at 1:53
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See here for curvature calculations of flag manifolds.

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I actually solved the problem. The key idea is to use the fact that the metric depends on the internal manifold (the flag) only though the Maurer-Cartan forms and hence the scalar curvature cannot depend on the position in the internal manifold. One can then expand the elements of $SU(N)$ near the origin. Keep the metric exact in terms of the lambdas and to second order in the flag directions. Once can then perform explicit calculations of the scalar curvature.

In case someone needs this in the future this is the result for the scalar curvature:

$R=-4(p-1)\sum\limits_{i\neq j}\frac{1}{(\vec\lambda_i-\vec\lambda_j)^2}-3\sum\limits_{i\neq j\neq k}\frac{(\vec\lambda_i-\vec\lambda_j).(\vec\lambda_i-\vec\lambda_k)}{(\vec\lambda_i-\vec\lambda_j)^2(\vec\lambda_i-\vec\lambda_k)^2}$,

where $p$ is the number of commuting matrices and $\vec\lambda_i$ are the eigenvalues.

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