Continuum limits Citation Shatashvili, Samson L. Strictly speaking, is actually the rotation of a coadjoint orbit by , because the Lie algebra of is given by skew-Hermitian matrices rather than Hermitian matrices, but we will abuse notation by ignoring this distinction in the arguments that follow. Let denote the space of Hermitian matrices.
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After performing a sequence of projections to spaces of one lower dimension, we conclude that for generic the one-dimensional measure is also piecewise polynomial, with a finite number of pieces supported on intervals and the polynomial being of degree at most on each piece. Incidentally, do not confuse Brezis, the mathematician, and Brezin, the physicist. The integral of on any small closed two-dimensional surface lifts up to equal the integral on on a lifted version of that surface; as is closed, we conclude that is also closed.
Also, we have and so is uniformly transverse to: From 9 we From this we see that is the moment map for the conjugation action of the -dimensional torus of unitary diagonal matrices of determinantafter identifying with by taking the diagonal matrix entries and dividing by.
High Energy Physics - Theory
Given a unitary matrixan infinitesimal perturbation of it takes the form for some skew-Hermitian matrixand so the first variation of in this direction is As is diagonal with generic entries, ranges over the Hermitian matrices with zero diagonal, or equivalently the orthogonal complement of the diagonal Hermitian matrices with respect to the Hilbert-Schmidt inner product. It will then suffice to show that where and with distinct entries and all non-zero realwhere depends only on.
The first, as you say, uses the HCIZ formula and would thus of course be unusable for the purposes of this blog post.
In other words, is independent of for sufficiently smalland so varies linearly in near as required. If we let be the diagonal map.
Second, observe that zubee fundamental vector fields i. What is good mathematics? This is consistent, though, with the fact that the somewhat rarely studied anti-symmetric GOE ensemble has cleaner formulae in particular, ihzykson a determinantal structure similar to GUE than the much more commonly studied symmetric GOE ensemble. Geometrically, the level sets of this form identify infinitesimally adjacent circle orbits together. Next, we expand to second order at each of its stationary points. By unitary invariance we may take to be diagonal; by perturbation we may take the eigenvalues of to be generic in, say, the Zariski sensethus.
Shatashvili : Correlation functions in the Itzykson-Zuber model
Of course, this is more or less what you do in your paper with Itzykson the heat equation weighted by the Vandermonde determinant being essentially the Fokker-Planck equation for the Dyson Brownian motionand is also the approach I took in this previous blog post about three years ago. Here we are talking about Edouard Brezin…. Sorry, your blog cannot share posts by email. Tizykson the above formula against and then using 1we may thus write 2 as.
Thus, if we let be the restriction of topulled back to bywe have where is the restriction of the connection one-form topulled back to. Thus the value of an invariant function at a Hermitian matrix depends only on the eigenvaluesand so by abuse of notation we may write.
I itzykkson adjusted the text accordingly.
The Harish-Chandra-Itzykson-Zuber integral formula
This also occurs in the case, but there one can simply multiply by to rotate a complex skew-Hermitian matrix into a complex Hermitian matrix.
Indeed, in a suitable coordinate system, is equal to modulo higher order terms, and is equal to modulo higher order terms in this system.
As before, is preserved by the torus action, and we have. But that seems to be the point of showing theta difference quantity zubeer a pullback under the inclusion map.
I will assume here some familiarity with differential forms on smooth manifolds, and also freely use infinitesimals in place of more traditional calculus notation at times. The only critical itzgkson of is at the origin. Example 1 Let with the standard symplectic formthus and Liouville measure is just Lebesgue measure using the standard orientation of. Third, you need to prove the defining equation of the momentum map, i. From the Cartan formula.
However, if we make the additional assumption that the Hamiltonian action is periodic thus it is an action of and not just of for some periodwe can say much more about the Duistermaat-Heckman measure at regular points:. To zuger that is a symplectic form, it remains to establish that is closed.
This is the method originally used by C. Dyson Brownian motion argument — Let denote the space of Hermitian matrices.