where is integrated over the Haar probability measure of the unitary group and is a non-zero complex parameter, as the expression
when the eigenvalues of are simple, where denotes the Vandermonde determinant
and is the constant
There are at least two standard ways to prove this formula in the literature. One way is by applying the Duistermaat-Heckman theorem to the pushforward of Liouville measure on the coadjoint orbit (or more precisely, a rotation of such an orbit by ) under the moment map , and then using a stationary phase expansion. Another way, which I only learned about recently, is to use the formulae for evolution of eigenvalues under Dyson Brownian motion (as well as the closely related formulae for the GUE ensemble), which were derived in this previous blog post. Both of these approaches can be found in several places in the literature (the former being observed in the original paper of Duistermaat and Heckman, and the latter observed in the paper of Itzykson and Zuber as well as in this later paper of Johansson), but I thought I would record both of these here for my own benefit.
The Harish-Chandra-Itzykson-Zuber formula can be extended to other compact Lie groups than . At first glance, this might suggest that these formulae could be of use in the study of the GOE ensemble, but unfortunately the Lie algebra associated to corresponds to real anti-symmetric matrices rather than real symmetric matrices. 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. This is consistent, though, with the fact that the (somewhat rarely studied) anti-symmetric GOE ensemble has cleaner formulae (in particular, having a determinantal structure similar to GUE) than the (much more commonly studied) symmetric GOE ensemble.
— 1. Dyson Brownian motion argument —
Let denote the space of Hermitian matrices. We place a Haar measure on this space; the exact normalisation of this measure will ultimately not be relevant (it will create a number of factors which will eventually cancel each other out). Define an invariant function on to be a function which is invariant with respect to conjugations, thus for all and . Thus the value of an invariant function at a Hermitian matrix depends only on the eigenvalues , and so by abuse of notation we may write
where is now a function on the Weyl chamber
By the Riesz representation theorem, there must be some density function with the property that
where is Lebesgue measure on , and is an invariant function with sufficient regularity and decay (e.g. smooth and exponentially decaying will certainly suffice). To compute this density function, we can exploit the explicit formulae for the GUE ensemble. As discussed in this previous blog post, the GUE ensemble is a probability measure on with probability density
for some normalisation constant (the exact choice of which depends on how one normalised ), and the eigenvalues of GUE have the probability density
Computing the expectation of for a GUE matrix using these formulae for an invariant function with sufficient regularity and decay, we conclude that
and hence (since )
for any invariant function with sufficient regularity and decay.
which is the bilinear form associated to the heat flow on for time . We will evaluate this integral in two different ways. On the one hand, we can expand the integral as
where and . Next, conjugating by an arbitrary unitary matrix and then integrating over Haar measure on , we can rewrite this as
Now we compute (2) another way. For fixed , the integral
can be interpreted as the expectation of where is a copy of GUE. By the Brezin-Hikami-Johansson formula (see Theorem 7 of these notes), the eigenvalues of are distributed according to the density
and so the previous integral can be written as
(Note: in the paper of Johansson, two proofs of the Brezin-Hikami-Johansson formula were given; one via the Harish-Chandra-Itzykson-Zuber formula discussed in this post, and the other via solving the equations of Dyson Brownian motion. The former proof of course cannot be invoked here as it would be circular, but the latter proof, which is the one used in the notes linked to above, can be used without risk of circularity.)
Comparing this against (3), we conclude that
This equation was derived for all positive real , but by analytic continuation it is then true for all non-zero complex . Replacing by we obtain the Harish-Chandra-Itzykson-Zuber integral formula.
— 2. The Duistermaat-Heckman theorem —
The Duistermaat-Heckman theorem concerns a significantly more general situation than the one appearing in the Harish-Chandra-Itzykson-Zuber integral formula, namely that of a symplectic manifold with a torus action that is associated to a moment map. To describe this theorem, it is simplest to begin with the one-dimensional setting of Hamiltonian circle actions on a symplectic manifold. (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.)
Recall that a symplectic manifold is a smooth manifold equipped with a symplectic form , that is to say a smooth anti-symmetric two-form on which is both non-degenerate (thus whenever is a vector field that is non-vanishing at ) and closed (thus ). Symplectic manifolds are necessarily even dimensional (because odd-dimensional anti-symmetric real matrices automatically have a zero eigenvalue and are thus degenerate). If is a -dimensional manifold, the Liouville measure on that manifold is defined as the volume form , where we use to denote the -fold wedge product of with itself, and (by abuse of notation) we identify volume forms with measures. (Note that wedge product is a commutative operation on even-order forms such as .)
for all smooth vector fields . From the non-degeneracy of , we see that vanishes precisely at the critical (or stationary) points of the Hamiltonian .
From the Cartan formula
for the Lie derivative of a form along a vector field , we see in the case of the symplectic form and Hamiltonian vector field that
and so the symplectic form is preserved by the vector field . From the product rule we conclude that Liouville measure is also preserved, a fact known as Liouville’s theorem in Hamiltonian mechanics:
We can exponentiate the Hamiltonian vector field to obtain a one-parameter group of smooth maps for :
This is a smooth action of the additive group , and by the preceding discussion, these maps are symplectomorphisms and in particular preserve Liouville measure, thus one can think of as a homomorphism from to the symplectomorphism group of .
Now suppose that is compact, so that the Liouville measure is finite. We can then pushforward Liouville measure by the Hamiltonian to create a measure on , which we will call the Duistermaat-Heckman measure associated to this Hamiltonian. At present, the Hamiltonian can be an arbitrary smooth function, and so the Duistermaat-Heckman measure is also more or less completely arbitrary. However, we do at least have Sard’s theorem, which asserts that almost every (in the sense of Lebesgue measure) point in is a regular value for in the sense that it is not the image of a critical point of . In the neighbourhood of each regular point, an application of the inverse function theorem shows that the Duistermaat-Heckman measure is smooth (or more precisely, a smooth multple of Lebesgue.
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 period ), we can say much more about the Duistermaat-Heckman measure at regular points:
Proposition 1 (Duistermaat-Heckman theorem for circle actions) Let be a -dimensional compact symplectic manifold for some , and let be a Hamiltonian associated to a periodic action of for some period . Then, in a sufficiently small neighbourhood of any given regular value of the Duistermaat-Heckman measure is a polynomial multiple of Lebesgue measure, with the polynomial being of degree at most .
In particular, if has only finitely many critical points, then the Duistermaat-Heckman measure is a piecewise polynomial multiple of Lebesgue measure on .
Let us illustrate this theorem with some key examples. We begin with a near-example, in which the compactness hypothesis is dropped.
Example 1 Let with the standard symplectic form , thus
and Liouville measure is just Lebesgue measure (using the standard orientation of ). Let be the Hamiltonian
for some and . The only critical point of is at the origin. Then the associated Hamiltonian vector field is
and so (using complex notation )
In particular, the Hamiltonian action is periodic with period . The Duistermaat-Heckman measure is supported on the half-line , with being the only non-regular value, and one can compute that it is times Lebesgue measure on this half-line, or equivalently it is the pushforward of Lebesgue measure on under the map .
Example 2 We can make a higher-dimensional (but still non-compact) version of the above example. Let , and let be given the standard symplectic form
so that Liouville measure is again Lebesgue measure. If we let be the Hamiltonian
then the only critical point is at the origin, and the associated Hamiltonian vector field is
and so (using the complex notation ) we have
We thus see that this flow will also be periodic if the are commensurate (i.e. linear multiples of each other), and that is the only non-regular value. As for the Duistermaat-Heckman measure, it is the pushforward of Lebesgue measure on the orthant by the map , which one easily verifies to be a polynomial multiple of Lebesgue measure on , with the polynomial being of degree .
Now we give an example that dates back to Archimedes (two millennia before the formal development of symplectic geometry!).
Example 3 (Archimedes sphere and cylinder) Let be the unit sphere, which we can coordinatise using the usual spherical coordinates , as
with the Riemannian metric
and a symplectic form
and associated Liouville measure . We take the Hamiltonian to be the vertical coordinate function
and the Hamiltonian vector field is then the rotation vector field around the vertical axis:
As such, the exponential map is the anti-clockwise rotation by radians around the vertical axis:
which is periodic with period , and the only stationary points are at the north and south poles (so the only non-regular values of are and ), and the Duistermaat-Heckman measure is times Lebesgue measure on . This latter fact is equivalent to the famous theorem of Archimedes that a sphere and its circumscribing cylinder have the same surface area on horizontal slices.
Now we prove Proposition 1. Let be a regular value of . Then, for all sufficiently close to , is also regular, and is a -dimensional manifold on which acts freely, thus one can view as a circle bundle, with the circles being the actions of . By (4), the Hamiltonian vector field, which is tangent to these circles, is symplectically orthogonal to the tangent bundle of . Thus, if we quotient out by the circle action, we obtain a -dimensional manifold , and the symplectic form restriced to descends to a smooth anti-symmetric form on , which is non-degenerate because is non-degenerate. 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. Thus is a -dimensional symplectic manifold, known as the symplectic reduction (or Marsden-Weinstein symplectic quotient) of by at the value .
Now consider an infinitesimal -dimensional parallelepiped in . This can be lifted up (non-uniquely, and modulo higher order corrections) to an infinitesimal -dimensional parallelepiped in ; applying the action of the Hamiltonian vector field for an infinitesimal time , and also letting vary in an infinitesimal interval (using some arbitrary smooth connection to identify together different fibres of arising fromthis interval), we obtain (modulo higher order corrections) a -dimensional parallelepiped in . Applying the Liouville measure to this parallelepiped, we see that the volume of this parallelepiped is (modulo higher order corrections) equal to times the volume of the original parallelepiped with respect to the Liouville measure on . (Indeed, in a suitable coordinate system, is equal to modulo higher order terms, and is equal to modulo higher order terms in this system). Integrating over the action of , and then dividing out by , we conclude that the Radon-Nikodym derivative of the Duistermaat-Heckman measure at with respect to Lebesgue measure, is equal to times the volume of the symplectic reduction . It thus suffices to show that for near , this volume is a polynomial of degree at most in .
We now use a little bit of de Rham cohomology. Recall the product rule
for smooth differential forms of order respectively. One corollary of this is that the wedge product of a closed form and an exact form is exact. Indeed, if , then
and so is exact. In particular, if one modifies the closed form by an exact form (for some -form ), then is modified by an exact volume form, and so is unchanged. Thus, this integral only depends on the cohomology class of , and so by abuse of notation it can be written as
where we are now implicitly using the product structure on the de Rham cohomology ring arising from the above observation.
From Cartan’s formula (5) we see that the Lie derivative of a closed form is exact, and so the cohomology class of is unaffected by perturbative diffeomorphisms. By the inverse function theorem, is diffeomorphic to for sufficiently close to , and so by further abuse of notation we can identify all such with (and now view as an element of ) and write the preceding integral as
To show that this expression is polynomial in of degree at most , it thus suffices to show that the cohomology class varies linearly in for sufficiently close to .
We now work in a tubular neighbourhood of for some small . This is a smooth circle bundle, and so we can place a smooth connection on this bundle, which we can represent as a connection one-form on , that is to say a smooth one-form on which is invariant with respect to the circle action (thus ) and such that . Indeed, one can build such a horizontal connection locally around the neighbourhood of any given circle orbit, and then patch such connections together using a smooth partition of unity. Geometrically, the level sets of this form identify infinitesimally adjacent circle orbits together.
Using the non-degenerate form , we can build the dual vector field to the connection one-form , so that . As and are both invariant with respect to the circle action, then is too:
Also, we have
and so is uniformly transverse to :
In particular, a flow along for a sufficiently small time will identify with .
We can use the Cartan formula (5) to compute how the flow along the vector field transforms the symplectic form :
Thus, if we let be the restriction of to , pulled back to by , we have
where is the restriction of the connection one-form to , pulled back to .
By construction, is the pullback of from to . The connection one-form is not such a pullback, because it has a non-zero contraction with . However, annihilates and so is the pullback of a -form on . As
we conclude on quotienting out by the circle action that
as the right-hand side is an exact form on , we conclude that
In other words, is independent of for sufficiently small , and so varies linearly in near as required. This completes the proof of Proposition 1.
Remark 1 The above argument in fact shows that is the (negative of the) Chern class of , viewed as a circle bundle over .
The above arguments can be extended to higher-dimensional actions than circle actions. Given a torus acting smoothly on a compact symplectic manifold , each tangent vector of the torus gives rise to a vector field on . We say that this torus is associated to a moment map taking values in the dual Lie algebra of the torus if is smooth and one has
for all . The Duistermaat-Heckman measure associated to is the finite measure on obtained by pushing forward Liouville measure by . A point in is said to be a critical point if is not of full rank, and a point in is a regular value if it is not the image of a critical point under . Again, Sard’s theorem guarantees that almost every point in is regular. We then have the higher-dimensional version of Proposition 1:
Theorem 2 (Duistermaat-Heckman theorem, general case) Let be a -dimensional compact symplectic manifold for some , and be a -dimensional torus acting on with an associated moment map . Then, in a sufficiently small neighbourhood of any given regular value of , the Duistermaat-Heckman measure is a polynomial multiple of Haar measure on , with the polynomial being of degree at most .
This theorem can be proven by a modification of the techniques used to prove Proposition 1; we sketch the details here. As before, we can form the symplectic reduction of at any regular value of by quotienting out by the torus action, obtaining a -dimensional symplectic manifold. If one selects a Haar measure on (and thus on and ), we then see as before that the Radon-Nikodym derivative of Duistermaat-Heckman measure at a regular value with respect to Haar measure of is equal to the volume of the symplectic manifold , times the volume of the torus with respect to the Haar measure on . Arguing as before, it then suffices to show that varies linearly in for sufficiently close to a regular value . As before, we can view for some sufficiently small open neighbourhood of as a torus bundle, and so one can again create a connection one-form on ; but now it is no longer a scalar one-form, but takes values in ; it is invariant with respect to the action of the torus, and obeys the identity
for all . This generates a -valued vector field that is symplectically dual to , thus
for all . As before, is preserved by the torus action, and we have
for all and . In particular, we see that maps to for any sufficiently small . As before, Cartan’s formula yields that
and repeating the previous arguments then shows that is independent of for sufficiently close to and any fixed , where denotes the directional derivative in the variable along the direction. This gives the desired local linearity of , giving Theorem 2.
— 3. Stationary phase —
for any Hermitian , all complex , and some depending only on . For, if this identity held, then by sending we see that the coefficient of the Taylor series of would equal . But we may expand this determinant as
so by Taylor expansion the coefficient is
where the outer summation is over all natural numbers that sum to .
Consider a summand in which for some . Then we see that this summand changes sign if we swap and . For this reason we see that we may restrict attention to the case when the are all distinct; as the sum to , we conclude that is a permutation of . We can then rearrange the above sum as
which factorises as , yielding the Harish-Chandra-Itzykson-Zuber integral formula.
It remains to prove (12). By unitary invariance we may take to be diagonal; by perturbation we may take the eigenvalues of to be generic (in, say, the Zariski sense), thus
for some and generic. By analytic continuation we may take to be imaginary. It will then suffice to show that
where and with distinct entries and all non-zero real , where depends only on . By subtracting a constant from we may take to be trace zero, and similarly for (and then generic relative to this constraint). The right-hand side can be expanded as
As for the left side, we introduce the coadjoint orbit
which, as the eigenvalues of are generic, is a smooth manifold of dimension , which has a transitive action of on it. (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.) If we let be the diagonal map
for all generic vectors of trace zero, all non-zero , and some Haar measure on (i.e. a non-zero -invariant Radon measure), and some constant depending only on . Note that (7) is asserting an exact formula for the Fourier transform of the pushforward measure (or of its one-dimensional projection ).
Note that as is assumed to have trace zero, all elements of have trace zero as well, so actually takes values in the hyperplane .
We now seek to interpret as a symplectic manifold and as a moment map for a torus action, in order to view as a Duistermmat-Heckman measure. We first need to construct a symplectic form on the coadjoint orbit , known as the Kirrilov-Kostant–Souriau form, as follows. Note that at any element of , the tangent vectors to at (which can be viewed as Hermitian matrices) take the form for some skew-Hermitian matrix . The symplectic form at is then defined by the formula
(The sign conventions are sometimes reversed in the literature; note that and are Hermitian and so this expression is real-valued.) Using the cyclic properties of trace, we have the identity
whence we see that the above form is well-defined (the dependence of on factors through , and similarly for ). It is clearly a smooth anti-symmetric form, and is also invariant with respect to the conjugation action. By working with explicit matrix coefficients at and using the hypotheses that the are distinct, one can soon verify that is non-degenerate at , and hence non-degenerate everywhere by the invariance. To verify that is a symplectic form, it remains to establish that is closed. This can be done by direct computation, but it turns out to be slicker to delay the verification of the symplectic nature of until some further facts about and have been established.
Now we express as a moment map (and, as a byproduct, conclude the closed nature of ). For any , we consider the scalar map , which can be written as
Differentiating this along an arbitrary vector field we see that
In particular, is closed:
On the other hand, as is preserved by the action, and is the vector field for the infinitesimal generator conjugation with respect to the unitary diagonal matrix for infinitesimal , we see that preserves :
Applying Cartan’s formula (5) we conclude that
for all . By unitary invariance again we conclude that for any skew-Hermitian , where . As the span the tangent space, we obtain , and so is closed and thus symplectic as required.
Now that is known to be a symplectic manifold, we may form the Liouville measure ; as is invariant under the conjugation action, is also, so this is a Haar measure. 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 determinant , after identifying with by taking the diagonal matrix entries and dividing by . This makes a Duistermaat-Heckman measure, and thus a multiple of Lebesgue measure on by polynomial of degree at most at every regular value of .
Now we work out what the regular values of are. If is a critical point of , then by the above calculations we must have for some non-trivial . The centraliser of a non-constant diagonal matrix consists of block diagonal matrices, so this only occurs when is block-diagonal. As has the same eigenvalues as , each block of then has eigenvalues that are a subset of . Taking partial traces, we then conclude that lies on a hyperplane of the form
for some and some and . Outside of these hyperplanes, we have regular points. The set of block diagonal matrices in can easily be verified to have zero measure (it has strictly lower dimension than ), so the Duistermaat-Heckman measure is thus a piecewise polynomial multiple of Lebesgue measure on which is smooth everywhere except at the hyperplanes (10). Note that these hyperplanes (10) partition into a finite number of polytopes. 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. (This fact can also be established directly from Proposition 1 in the case when is commensurate as well as generic.) In particular, if one differentiates (in the distributional sense) this one-dimensional measure times, one obtains a distribution that is supported on a finite number of points (depending on ), and so its Fourier transform takes the form for some polynomials . Undoing the differentiation, we thus conclude that
for some polynomials and some distinct reals . This is already some way towards what our goal (7), but we still need to sort out exactly what the and are. There are a number of ways to do this (e.g. one can use the Atiyah-Bott–Berline-Vergne localization theorem, or the machinery of symplectic cobordism), but we will use the method of stationary phase instead. The idea is to obtain an asympotic of the form
when (keeping fixed); this asymptotic with the correct main term, combined with the expression (11) with a general main term but no error term, gives the formula (7) with the correct main term and no error term (the point being that it is not possible for an expression of the form to decay to zero at infinity without actually vanishing identically, as can be seen for instance by using frequency localisation operators to separate the contribution of each of the frequencies ).
where is the phase
Following the method of statinonary phase, we now study how the stationary points of . Given a unitary matrix , an infinitesimal perturbation of it takes the form for some skew-Hermitian matrix , and 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 . In order for to be stationary at , it is thus necessary and sufficient for to be diagonal, which (as is a generic diagonal matrix) only occurs when is a diagonal matrix. Thus we see that is only stationary at the permutation matrices , at which point it takes the value of .
Next, we expand to second order at each of its stationary points. We begin with an expansion near the identity matrix . We use exponential coordinates , where is a small skew-Hermitian matrix, so that
Taylor expanding the exponential, we obtain
Writing , this becomes
and hence by the skew-Hermitian nature of
We thus see that the Hessian of at in exponential coordinates has a determinant square root (using the branch in the upper half-plane, which is what is needed for the asymptotics) which is for some absolute constant . Similarly, the determinant square root of the Hessian of at any other permutation matrix can be computed to be . Using stationary phase expansions (see e.g. Chapter IX of Stein’s “Harmonic analysis“) we obtain the desired expansion (12).