†

# Arbitrary rotation invariant random matrix ensembles and supersymmetry:
orthogonal and unitary–symplectic case

Mario Kieburga)†, Johan Grönqvistb) and Thomas Guhra) a)Universität Duisburg-
Essen, Lotharstraße 1, 47048 Duisburg, Germany
b)Matematisk Fysik, LTH, Lunds Universitet, Box 118, 22100 Lund, Sweden
<EMAIL_ADDRESS>
###### Abstract

Recently, the supersymmetry method was extended from Gaussian ensembles to
arbitrary unitarily invariant matrix ensembles by generalizing the
Hubbard–Stratonovich transformation. Here, we complete this extension by
including arbitrary orthogonally and unitary–symplectically invariant matrix
ensembles. The results are equivalent to, but the approach is different from
the superbosonization formula. We express our results in a unifying way. We
also give explicit expressions for all one–point functions and discuss
features of the higher order correlations.

###### pacs:

02.30.Px, 05.30.Ch, 05.30.-d, 05.45.Mt

††: J. Phys. A: Math. Gen.

##  1 Introduction

In random matrix theory, supersymmetry is an indispensable tool [1, 2, 3, 4].
Recently, this method was extended from Gaussian probability densities to
arbitrary rotation invariant ones. Presently, there are two approaches
referred as superbosonization. The first approach is a generalization of the
Hubbard–Stratonovich transformation for rotation invariant random matrix
ensembles [5]. The basic idea is the introduction of a proper
Dirac–distribution in superspace, extending earlier work in the context of
scattering theory [6], universality considerations [7], field theory [8, 9]
and quantum chromodynamics [10]. The second approach is the superbosonization
formula developed in Refs. [11, 12]. It is an identity for integrals over
superfunctions on rectangular supermatrices which are rotation invariant under
an ordinary group.

Here, we further extend the generalized Hubbard–Stratonovich transformation to
the orthogonal and the unitary symplectic symmetry class in a unifying way. To
this end, we use an analog of the Sekiguchi differential operator for ordinary
matrix Bessel–functions. We also aim at a presentation which is mathematically
more sound than the one in Ref. [5].

The article is organized as follows. The problem is posed in Sec. 2. We give
an outline of the calculation in Sec. 3. In Sec. 4, we present the generalized
Hubbard–Stratonovich transformation. In Sec. 5, we carry out the calculation
for arbitrary ensembles as far as possible. Then, we restrict the computation
to the three classical symmetry classes. We, thereby, extend the
supersymmetric Ingham–Siegel integral [5]. In Sec. 6, we give a more compact
expression of the generating function in terms of supermatrix
Bessel–functions. We show that the generating function is independent of the
chosen representation for the characteristic function. The one–point and
higher correlation functions are expressed as eigenvalue integrals in Sec. 7.
In the appendices, we present details of the calculations.

##  2 Posing the problem

We consider a sub-vector space $\mathfrak{M}_{N}$ of the hermitian $N\times
N$–matrices ${\rm Herm\,}(2,N)$. ${\rm Herm\,}(\beta,N)$ is the set of real
orthogonal ($\beta=1$), hermitian ($\beta=2$) and quaternionic self-adjoint
($\beta=4$) matrices and $\beta$ is the Dyson-index. We use the complex
$2\times 2$ dimensional matrix representation for quaternionic numbers
$\mathbb{H}$. The results can easily be extended to other representations of
the quaternionic field. For the relation between the single representations,
we refer to a work by Jiang [13].

The object of interest is an arbitrary sufficiently integrable probability
density $P$ on $\mathfrak{M}_{N}$. Later, we assume that $P$ is an invariant
function under the action of the group

${\rm U\,}^{(\beta)}(N)=\left\\{\begin{array}[]{ll}{\rm O}(N)&,\ \beta=1\\\
{\rm U\,}(N)&,\ \beta=2\\\ {\rm USp}(2N)&,\ \beta=4\end{array}\right.$ (2.1)

and $\mathfrak{M}_{\gamma_{2}N}={\rm Herm\,}(\beta,N)$. Here, we introduce
$\gamma_{2}=1$ for $\beta\in\\{1,2\\}$ and $\gamma_{2}=2$ for $\beta=4$ and,
furthermore, $\gamma_{1}=2\gamma_{2}/\beta$ and
$\tilde{\gamma}=\gamma_{1}\gamma_{2}$. These constants will play an important
role in the sequel.

We are interested in the $k$–point correlation functions

$R_{k}(x)=\mathbf{d}^{k}\int\limits_{\mathfrak{M}_{N}}P(H)\prod\limits_{p=1}^{k}\tr\delta(x_{p}\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{N}-H)d[H]$ (2.2)

with the $k$ energies $x={\rm diag\,}(x_{1},\ldots,x_{k})$. Here, $\mathbf{d}$
is the inverse averaged eigenvalue degeneracy of an arbitrary matrix
$H\in\mathfrak{M}_{N}$. The measure $d[H]$ is defined as in Ref. [14], it is
the product of all real and imaginary parts of the matrix entries. For
example, we have $\mathbf{d}=1/2$ for $\mathfrak{M}_{2N}={\rm Herm\,}(4,N)$
and $\mathbf{d}=1$ for no eigenvalue degeneracy as for $\mathfrak{M}_{N}={\rm
Herm\,}(\beta,N)$ with $\beta\in\\{1,2\\}$. We use in Eq. (2.2) the
$\delta$–distribution which is defined by the matrix Green’s function. The
definition of the $k$–point correlation function (2.2) differs from Mehta’s
[15]. The two definitions can always be mapped onto each other as explained
for example in Ref. [4].

We recall that it is convenient to consider the more general function

$\widehat{R}_{k}\left(x^{(L)}\right)=\mathbf{d}^{k}\int\limits_{\mathfrak{M}_{N}}P(H)\prod\limits_{p=1}^{k}\tr[(x_{p}+L_{p}\imath\varepsilon)\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{N}-H]^{-1}d[H]$ (2.3)

where we have suppressed the normalization constant. The quantities $L_{j}$ in
$x^{(L)}={\rm
diag\,}(x_{1}+L_{1}\imath\varepsilon,\ldots,x_{k}+L_{k}\imath\varepsilon)$ are
elements in $\\{\pm 1\\}$. We define $x^{\pm}={\rm
diag\,}(x_{1}\pm\imath\varepsilon,\ldots,x_{k}\pm\imath\varepsilon)$.
Considering the Fourier transformation of (2.2) we have

$\displaystyle r_{k}(t)$ $\displaystyle=$
$\displaystyle(2\pi)^{-k/2}\int\limits_{\mathbb{R}^{k}}R_{k}(x)\prod\limits_{p=1}^{k}\exp\left(\imath
x_{p}t_{p}\right)d[x]=$ (2.4) $\displaystyle=$
$\displaystyle\left(\frac{\mathbf{d}}{\sqrt{2\pi}}\right)^{k}\int\limits_{\mathfrak{M}_{N}}P(H)\prod\limits_{p=1}^{k}\tr\exp\left(\imath
Ht_{p}\right)d[H]\ .$

The Fourier transformation of (2.3) yields

$\displaystyle\widehat{r}_{k}(t)$ $\displaystyle=$
$\displaystyle(2\pi)^{-k/2}\int\limits_{\mathbb{R}^{k}}\widehat{R}_{k}\left(x^{(L)}\right)\prod\limits_{p=1}^{k}\exp\left(\imath
x_{p}t_{p}\right)d[x]=$ (2.5) $\displaystyle=$
$\displaystyle\prod\limits_{p=1}^{k}\left[-L_{p}\
2\pi\imath\Theta(-L_{p}t_{p})\exp\left(\varepsilon
L_{p}t_{p}\right)\right]r_{k}(t)$

where $\Theta$ is the Heavyside–distribution.

As in Ref. [5], the $k$–point correlation function is completely determined by
Eq. (2.3) with $L_{p}=-1$ for all $p$ if the Fourier transform (2.4) is entire
in all entries, i.e. analytic in all entries with infinite radius of
convergence. We obtain such a Fourier transform if the $k$–point correlation
function $R_{k}$ is a Schwartz–function on $\mathbb{R}^{k}$ with the property

$\int\limits_{\mathbb{R}^{k}}|R_{k}(x)|\prod\limits_{p=1}^{k}\exp\left(\tilde{\delta}x_{p}\right)d[x]<\infty\quad,\quad\forall\tilde{\delta}\in\mathbb{R}\
.$ (2.6)

This set of functions is dense in the set of Schwartz–functions on
$\mathbb{R}^{k}$ without this property. The notion dense refers to uniform
convergence. This is true since every Schwartz–function times a Gaussian
distribution $\exp\left(-\epsilon\sum\limits_{p=1}^{k}x_{p}^{2}\right)$,
$\epsilon>0$, is a Schwartz–function and fulfils Eq. (2.6). We proof that
$r_{k}$, see Eq. (2.4), is indeed entire in all entries for such $k$–point
correlation functions. To this end, we consider the function

$r_{k\delta}(t)=\int\limits_{\mathfrak{B}_{\delta}}R_{k}(x)\prod\limits_{p=1}^{k}\exp\left(\imath
x_{p}t_{p}\right)d[x],$ (2.7)

where $\mathfrak{B}_{\delta}$ is the closed $k$-dimensional real ball with
radius $\delta\in\mathbb{R}^{+}$. Due to the Paley–Wiener theorem [16],
$r_{k\delta}$ is for all $\delta\in\mathbb{R}^{+}$ entire analytic. Let
$\mathfrak{B}_{\tilde{\delta}}^{\mathbb{C}}$ be another $k$-dimensional
complex ball with radius $\tilde{\delta}\in\mathbb{R}^{+}$. Then, we have

$\underset{\delta\to\infty}{\lim}\underset{t\in\mathfrak{B}_{\tilde{\delta}}^{\mathbb{C}}}{\sup}|r_{k\delta}(t)-r_{k}(t)|\leq\underset{\delta\to\infty}{\lim}\int\limits_{\mathbb{R}^{k}\setminus\mathfrak{B}_{\delta}}|R_{k}(x)|\prod\limits_{p=1}^{k}\exp\left(\tilde{\delta}x_{p}\right)d[x]=0\
.$ (2.8)

The limit of $r_{k\delta}$ to $r_{k}$ is uniform on every compact support on
$\mathbb{C}^{k}$. Thus, $r_{k}$ is entire analytic.

The modified correlation function $\widehat{R}_{k}$ for all choices of the
$L_{p}$ can be reconstructed by Eq. (2.5). In Sec. 7, we extend the results by
a limit–value–process in a local convex way to non-analytic functions.

We derive $\widehat{R}_{k}\left(x^{-}\right)$ from the generating function

$Z_{k}\left(x^{-}+J\right)=\int\limits_{\mathfrak{M}_{N}}P(H)\prod\limits_{p=1}^{k}\frac{\det[H-(x_{p}^{-}+J_{p})\leavevmode\hbox{\small
1\kern-3.8pt\normalsize
1}_{N}]}{\det[H-(x_{p}^{-}-J_{p})\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{N}]}d[H]$ (2.9)

by differentiation with respect to the source variables [17]

$\widehat{R}_{k}\left(x^{-}\right)=\left(\frac{\mathbf{d}}{2}\right)^{k}\left.\frac{\partial^{k}}{\prod_{p=1}^{k}\partial
J_{p}}Z_{k}\left(x^{-}+J\right)\right|_{J=0}$ (2.10)

where $x^{-}+J=x^{-}\otimes\leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}_{4}+{\rm diag\,}(J_{1},\ldots,J_{k})\otimes{\rm
diag\,}(-\leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}_{2},\leavevmode\hbox{\small 1\kern-3.8pt\normalsize 1}_{2})$. By
definition, $Z_{k}$ is normalized to unity at $J=0$.

##  3 Sketch of our approach

To provide a guideline through the detailed presentation to follow in the
ensuing Sections, we briefly sketch the main ideas as in Ref. [5] and as
further extended in the present contribution.

To express the generating function (2.9) as an integral in superspace, we
write the determinants as Gaussian integrals over vectors of ordinary and
Grassmann variables. We then perform the ensemble average which is equivalent
to calculating the characteristic function

$\Phi(K)=\int P(H)\exp(\imath\tr HK)d[H]$ (3.1)

of the probability density. The rotation invariance of $P(H)$ carries over to
$\Phi(K)$. The ordinary matrix $K$ contains the abovementioned vectors of
ordinary and Grassmann variables as dyadic matrices. It has a dual matrix $B$
in superspace whose entries are all scalarproducts of these vectors. The
reduction in the degrees of freedom is fully encoded in this duality, as the
dimensions of $K$ and $B$ scale with $N$ and $k$, respectively. The crucial
identity

$\tr K^{m}={\rm Str\,}B^{m},\quad\forall m\in\mathbb{N},$ (3.2)

yields the supersymmetric extension of the rotation invariant characteristic
function,

$\Phi(K)=\Phi(\tr K,\tr K^{2},...)=\Phi({\rm Str\,}B,{\rm
Str\,}B^{2},...)=\Phi(B)\ ,$ (3.3)

which is now viewed as a function in ordinary and superspace. We rewrite it by
inserting a proper Dirac–distribution in superspace,

$\displaystyle\Phi(B)$ $\displaystyle=$
$\displaystyle\int\Phi(\rho)\delta(\rho-B)d[\rho]$ (3.4) $\displaystyle\sim$
$\displaystyle\int\int\Phi(\rho)\exp[\imath{\rm
Str\,}(\rho-B)\sigma]d[\rho]d[\sigma]\ ,$ (3.5)

where the supermatrix $\rho$ and $\sigma$ are introduced as integration
variables. The vectors of ordinary and Grassmann variables now appear as in
the conventional Hubbard–Stratonovich transformation and can hence be
integrated out in the same way. We are left with the integrals over $\rho$ and
$\sigma$. If we do the integral over $\rho$ we arrive at the result

$Z_{k}\left(x^{-}+J\right)\sim\int Q(\sigma){\rm
Sdet\,}^{-N/\gamma_{1}}(\sigma-x^{-}-J)d[\sigma].$ (3.6)

for the generating function. The superfunction $Q$ is the superspace Fourier
transform of $\Phi$ and plays the role of a probability density in superspace,

$Q(\sigma)=\int\Phi(\rho)\exp(\imath{\rm Str\,}\rho\sigma)d[\rho]\ .$ (3.7)

If we choose to integrate over $\sigma$ instead, we obtain another
representation of the generating function

$Z_{k}\left(x^{-}+J\right)\sim\int\Phi(\rho)I(\rho)\exp[-\imath{\rm
Str\,}\rho(x^{-}+J)]d[\rho]\ ,$ (3.8)

which still contains the characteristic function. The distribution $I(\rho)$
appears. It is the supersymmetric version of the Ingham–Siegel integral. It is
a rotation invariant function resulting from the Fourier transformation of the
superdeterminant in Eq. (3.6).

One way to proceed further is to diagonalize the supermatrix $\rho$ and to
integrate over the angles. We may omit Efetov–Wegner terms and have

$Z_{k}\left(x^{-}+J\right)\sim\int\Phi(r)I(r)\varphi(-\imath r,x^{-}+J)d[r],$
(3.9)

where $\varphi$ is a supermatrix Bessel–function. The differentiation with
respect to $J$ gives $\widehat{R}_{k}$. We can introduce other signatures of
$L$ by Fourier transformation of Eq. (3.8) and identification with Eq. (2.5).
Eventually, we find the correlation functions $R_{k}$.

##  4 Generalized Hubbard–Stratonovich transformation

In Sec. 4.1, we express the determinants in Eq. (2.9) as Gaussian integrals
and introduce the characteristic function of the matrix ensemble. In Sec. 4.2,
we qualitatively present the duality between ordinary and superspace which is
quantitatively discussed in Sec. 4.3. Then, we restrict the matrix ensembles
to the classical symmetry classes. In Sec. 4.4, we investigate the
diagonalization of the dyadic matrix $K$ appearing from the Gaussian
integrals. The ambiguity of the supersymmetric extension of the characteristic
function is discussed in Sec. 4.5. In Sec. 4.6, we present the symmetries of
the appearing supermatrices. In Sec. 4.7, we replace the dyadic supermatrix in
the supersymmetric extended characteristic function with a symmetric
supermatrix discussed in the section before.

###  4.1 Average over the ensemble and the characteristic function

To formulate the generating function as a supersymmetric integral, we consider
a complex Grassmann algebra $\Lambda=\bigoplus\limits_{j=0}^{2Nk}\Lambda_{j}$
with $Nk$-pairs $\\{\zeta_{jp},\zeta_{jp}^{*}\\}_{j,p}$ of Grassmann variables
[18]. We define the $k$ anticommuting vectors and their adjoint

$\zeta_{p}=(\zeta_{1p},\ldots,\zeta_{Np})^{T}\ \ \ {\rm and}\ \ \
\zeta_{p}^{\dagger}=(\zeta_{1p}^{*},\ldots,\zeta_{Np}^{*})\ ,$ (4.1)

respectively. For integrations over Grassmann variables, we use the
conventions of Ref. [14]. We also consider $k$ $N$–dimensional complex vectors
$\\{z_{p},z_{p}^{\dagger}\\}_{1\leq p\leq k}$. In the usual way, we write the
determinants as Gaussian integrals and find for Eq. (2.9)

$\displaystyle Z_{k}(x^{-}+J)$ $\displaystyle=$
$\displaystyle(-\imath)^{Nk}\int\limits_{\mathfrak{M}_{N}}\int\limits_{\mathfrak{C}_{kN}}d[\zeta]d[z]d[H]P(H)\times$
(4.2) $\displaystyle\times$ $\displaystyle{\rm
exp}\left(\imath\sum\limits_{p=1}^{k}\left\\{\zeta_{p}^{\dagger}[H-(x_{p}^{-}+J_{p})\leavevmode\hbox{\small
1\kern-3.8pt\normalsize
1}_{N}]\zeta_{p}+z_{p}^{\dagger}[H-(x_{p}^{-}-J_{p})\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{N}]z_{p}\right\\}\right)$

where
$d[\zeta]=\prod\limits_{p=1}^{k}\prod\limits_{j=1}^{N}d\zeta_{jp}d\zeta_{jp}^{*}$,
$d[z]=\prod\limits_{p=1}^{k}\prod\limits_{j=1}^{N}dz_{jp}dz_{jp}^{*}$ and
$\mathfrak{C}_{kN}=\mathbb{C}^{kN}\times\Lambda_{2Nk}$. Using

$\sum\limits_{p=1}^{k}\left(\zeta_{p}^{\dagger}H\zeta_{p}+z_{p}^{\dagger}Hz_{p}\right)=\tr
H\widetilde{K}$ (4.3)

with

$\widetilde{K}=\sum\limits_{p=1}^{k}\left(z_{p}z_{p}^{\dagger}-\zeta_{p}\zeta_{p}^{\dagger}\right)$
(4.4)

leads to

$\displaystyle Z_{k}(x^{-}+J)$ $\displaystyle=$
$\displaystyle(-\imath)^{Nk}\int\limits_{\mathfrak{C}_{kN}}\mathcal{F}P\left(\hat{\pi}(\mathfrak{M}_{N};\widetilde{K})\right)\times$
(4.5) $\displaystyle\times$ $\displaystyle{\rm
exp}\left(-\imath\sum\limits_{p=1}^{k}\left[(x_{p}^{-}+J_{p})\zeta_{p}^{\dagger}\zeta_{p}+(x_{p}^{-}-J_{p})z_{p}^{\dagger}z_{p}\right]\right)d[\zeta]d[z]\
.$

where the integration over $H$ is the Fourier transformation of the
probability density $P$,

$\mathcal{F}P\left(\hat{\pi}(\mathfrak{M}_{N};\widetilde{K})\right)=\int\limits_{\mathfrak{M}_{N}}P(H)\exp\left(\imath\tr
H\widetilde{K}\right)d[H]\ .$ (4.6)

This Fourier transform is called characteristic function and is denoted by
$\Phi$ in Ref. [5] and in Eq. (3.1). The projection operator
$\hat{\pi}(\mathfrak{M}_{N})$ onto the space $\mathfrak{M}_{N}$ is crucial.
For $\mathfrak{M}_{\gamma_{2}N}={\rm Herm\,}(\beta,N)$ the projection operator
is

$\hat{\pi}\left({\rm
Herm\,}(\beta,N);\widetilde{K}\right)=\frac{1}{2}\left[\widetilde{K}+\widehat{Y}(\widetilde{K})\right]$
(4.7)

with

$\widehat{Y}(\widetilde{K})=\left\\{\begin{array}[]{ll}\widetilde{K}^{T}&,\
\beta=1\\\ \widetilde{K}&,\ \beta=2\\\ \left(Y_{{\rm
s}}\otimes\leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}_{N}\right)\widetilde{K}^{T}\left(Y_{{\rm
s}}^{T}\otimes\leavevmode\hbox{\small 1\kern-3.8pt\normalsize 1}_{N}\right)&,\
\beta=4\end{array}\right.$ (4.8)

and the symplectic unit

$Y_{{\rm s}}=\left[\begin{array}[]{cc}0&1\\\ -1&0\end{array}\right]\ ,$ (4.9)

where $\leavevmode\hbox{\small 1\kern-3.8pt\normalsize 1}_{N}$ is the $N\times
N$–unit matrix. The transposition in Eq. (4.8) can also be replaced by the
complex conjugation due to $\widetilde{K}^{\dagger}=\widetilde{K}$. The
projection onto the set of diagonal matrices
$\bigoplus\limits_{j=1}^{N}\mathbb{R}$ is

$\hat{\pi}\left(\bigoplus_{j=1}^{N}\mathbb{R};\widetilde{K}\right)={\rm
diag\,}\left(\widetilde{K}_{11},\widetilde{K}_{22},\ldots,\widetilde{K}_{NN}\right)\
.$ (4.10)

###  4.2 Duality between ordinary and superspace

Is it always possible to find a supermatrix representation for the
characteristic function $\mathcal{F}P$ such that Eq. (4.5) has an integral
representation over supermatrices as it is known [5, 12] for rotation
invariant $P$ on $\mathfrak{M}_{\gamma_{2}N}={\rm Herm\,}(\beta,N)$? The
integral (4.5) is an integral over the supervectors
$v_{j}=(z_{j1}^{*},\ldots,z_{jk}^{*},-\zeta_{j1}^{*},\ldots,-\zeta_{jk}^{*})^{T}$
and their adjoint
$v_{j}^{\dagger}=(z_{j1},\ldots,z_{jk},\zeta_{j1},\ldots,\zeta_{jk})$. The
adjoint “$\dagger$” is the complex conjugation with the supersymmetric
transposition and “$T$” is the ordinary transposition. The entries of the
matrix $\widetilde{K}$ are $v_{n}^{\dagger}v_{m}$. If we do not use any
symmetry of the matrix ensemble, we can write these scalar products of
supervectors as supertraces

$v_{n}^{\dagger}v_{m}={\rm Str\,}v_{m}v_{n}^{\dagger}\ .$ (4.11)

Then, we can transform each of these supertraces with a Dirac–distribution to
an integral over a $(k+k)\times(k+k)$–supermatrix. We defined the
Dirac–distribution in superspace as in Refs. [19, 10]. The ambiguity discussed
in Ref. [20] occurring by such a transformation is discussed in the
subsections 4.5 and 6.3.

The procedure above is tedious. Using the symmetries of the ensemble
($\mathcal{F}P,\mathfrak{M}_{N}$), we can reduce the number of integrals in
superspace. We will see that the number of commuting real integrals and of
Grassmannian integrals is $2k^{2}+2k^{2}$ ($\beta=2$) or $4k^{2}+4k^{2}$
($\beta\in\\{1,4\\}$) for a rotation invariant matrix ensembles on ${\rm
Herm\,}(\beta,N)$. If there is not a symmetry the number of integrals has not
been reduced. One has to integrate over $N(N+1)$ ordinary hermitian $k\times
k$–matrices and their corresponding anticommuting parameters if the
transformation above is used.

###  4.3 Analysis of the duality between ordinary and superspace

We consider an orthonormal basis $\\{A_{n}\\}_{1\leq n\leq d}$ of
$\mathfrak{M}_{N}$ where $d$ is the dimension of $\mathfrak{M}_{N}$. We use
the trace $\tr{A_{n}A_{m}}=\delta_{nm}$ as the scalar product and recall that
$\mathfrak{M}_{N}$ is a real vector space. Every element of this basis is
represented as

$A_{n}=\sum\limits_{j=1}^{N}\lambda_{jn}e_{jn}e_{jn}^{\dagger}\ \ \ {\rm
with}\ \ \ \sum\limits_{j=1}^{N}\lambda_{jn}^{2}=1\ .$ (4.12)

Here, $e_{jn}$ are the normalized eigenvectors of $A_{n}$ to the eigenvalues
$\lambda_{jn}$. Then, we construct every matrix $H\in\mathfrak{M}_{N}$ in this
basis

$H=\sum\limits_{n=1}^{d}h_{n}A_{n}\ .$ (4.13)

We find for the characteristic function

$\displaystyle\mathcal{F}P\left(\hat{\pi}(\mathfrak{M}_{N};\widetilde{K})\right)$
$\displaystyle=$
$\displaystyle\int\limits_{\mathfrak{M}_{N}}P\left(\sum\limits_{n=1}^{d}h_{n}A_{n}\right){\rm
exp}\left(\imath\sum\limits_{n=1}^{d}h_{n}\tr A_{n}\widetilde{K}\right)d[H]=$
(4.14) $\displaystyle=$
$\displaystyle\mathcal{F}P\left(\sum\limits_{n=1}^{d}\tr\left(\widetilde{K}A_{n}\right)A_{n}\right)\
.$

With help of Eq. (4.12) and an equation analogous to (4.11), the
characteristic function is

$\mathcal{F}P\left(\hat{\pi}(\mathfrak{M}_{N};\widetilde{K})\right)=\mathcal{F}P\left(\sum\limits_{n=1}^{d}{\rm
Str\,}\left(\sum\limits_{j=1}^{N}\lambda_{jn}Ve_{jn}e_{jn}^{\dagger}V^{\dagger}\right)A_{n}\right)$
(4.15)

with $V=(v_{1},\ldots,v_{N})$. We see that the matrix $\widetilde{K}$ is
projected onto

$K=\hat{\pi}(\mathfrak{M}_{N};\widetilde{K})$ (4.16)

where the projection is the argument of the characteristic function in Eq.
(4.14). The matrices in the supertraces of (4.15) can be exchanged by
$(k+k)\times(k+k)$–supermatrices with the Delta–distributions described above.
If the ensemble has no symmetry then we have reduced the number of
supermatrices to the dimension of $\mathfrak{M}_{N}$. Nevertheless, we can
find a more compact supersymmetric expression of the matrix $K$ such that the
number of the resulting integrals only depends on $k$ but not on $N$. This is
possible if $K$ is a dyadic matrix of vectors where the number of vectors is
independent of $N$ and the probability distribution only depends on invariants
of $H$. The ensembles with $\mathfrak{M}_{\gamma_{2}N}={\rm Herm\,}(\beta,N)$
and a probability density $P$ invariant under the action of ${\rm
U\,}^{(\beta)}(N)$ fulfil these properties. It is known [5, 12] that these
cases have a very compact supersymmetric expression. Furthermore, these
ensembles are well analyzed for Gaussian–distributions with help of the
Hubbard–Stratonovitch transformation [1, 3, 2].

In the present context, the cases of interest are
$\mathfrak{M}_{\gamma_{2}N}={\rm Herm\,}(\beta,N)$ with a probability density
$P$ invariant under the action ${\rm U\,}^{(\beta)}(N)$. We need this symmetry
to simplify Eq. (4.15). Let $N\geq\gamma_{1}k$. This restriction also appears
in the superbosonization formula [12]. If $N<\gamma_{1}k$, one has to be
modify the calculations below. For the superbosonization formula, Bunder,
Efetov, Kravtsov, Yevtushenko, and Zirnbauer [20] presented such a
modification.

The symmetries of a function $f$ carry over to its Fourier transform
$\mathcal{F}f$. Thus, the characteristic function $\mathcal{F}P$ is invariant
under the action of ${\rm U\,}^{(\beta)}(N)$. Let $\widetilde{K}_{0}$ be an
arbitrary ordinary hermitian matrix in the Fourier transformation (4.6) of the
probability density. We assume that the characteristic function is analytic in
the eigenvalues of $\widetilde{K}_{0}$. Then, we expand $\mathcal{F}P$ as a
power series in these eigenvalues. Since the characteristic function is
rotation invariant every single polynomial in this power series of a
homogeneous degree is permutation invariant. With help of the fundamental
theorem of symmetric functions [21] we rewrite these polynomials in the basis
of elementary polynomials. This is equivalent to writing these polynomials in
the basis of the traces $\tr\left[\hat{\pi}\left({\rm
Herm\,}(\beta,N),\widetilde{K}_{0}\right)\right]^{m}$, $m\in\mathbb{N}$. The
analytic continuation of $\mathcal{F}P$ from $\widetilde{K}_{0}$ to
$\widetilde{K}$ yields that the characteristic function in (4.6) only depends
on $\tr\left[\hat{\pi}\left({\rm
Herm\,}(\beta,N),\widetilde{K}\right)\right]^{m}$, $m\in\mathbb{N}$.

Defining the matrix

$V^{\dagger}=(z_{1},\ldots,z_{k},Yz_{1}^{*},\ldots,Yz_{k}^{*},\zeta_{1},\ldots,\zeta_{k},Y\zeta_{1}^{*},\ldots,Y\zeta_{k}^{*})$
(4.17)

and its adjoint

$V=(z_{1}^{*},\ldots,z_{k}^{*},Yz_{1},\ldots,Yz_{k},-\zeta_{1}^{*},\ldots,-\zeta_{k}^{*},Y\zeta_{1},\ldots,Y\zeta_{k})^{T}$
(4.18)

with

$Y=\left\\{\begin{array}[]{ll}\leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}_{N}&,\ \beta=1\\\ 0&,\ \beta=2\\\ Y_{{\rm
s}}^{T}\otimes\leavevmode\hbox{\small 1\kern-3.8pt\normalsize 1}_{N}&,\
\beta=4\end{array}\right.,$ (4.19)

we find

$K=\hat{\pi}\left({\rm
Herm\,}(\beta,N);\widetilde{K}\right)=\frac{1}{\tilde{\gamma}}V^{\dagger}V\ .$
(4.20)

The crucial identity

$\tr(V^{\dagger}V)^{m}={\rm Str\,}(VV^{\dagger})^{m}$ (4.21)

holds for all $\beta$. It connects ordinary and superspace. For $\beta=2$, a
proof can be found in Ref. [5]. In A, we show that the equation

${\rm Str\,}V_{1}V_{2}={\rm Str\,}V_{2}V_{1}$ (4.22)

holds for all rectangular matrices of the form

$V_{1}=\left[\begin{array}[]{cc}\overbrace{A_{1}}^{a}&\overbrace{B_{1}}^{b}\hskip
0.85358pt\\}c\\\ C_{1}&D_{1}\hskip 5.97508pt\\}d\end{array}\right]\ \ \ {\rm
and}\ \ \
V_{2}=\left[\begin{array}[]{cc}\overbrace{A_{2}}^{c}&\overbrace{B_{2}}^{d}\hskip
0.85358pt\\}a\\\ C_{2}&D_{2}\hskip 5.69054pt\\}b\end{array}\right]$ (4.23)

where $A_{j}$ and $D_{j}$ have commuting entries and $B_{j}$ and $C_{j}$
anticommuting ones. This implies in particular that Eq. (4.21) holds for all
$\beta$. Hence, we reduced the amount of supermatrices corresponding to
$\widetilde{K}$ in Eq. (4.15) to one $(2k+2k)\times(2k+2k)$–supermatrix. In
Ref. [5], the characteristic function $\Phi$ was, with help of Eq. (4.21),
extended to superspace. We follow this idea and, then, proceed with the
Dirac–distribution mentioned above.

###  4.4 Problems when diagonalizing $K$

In Ref. [5], two approaches of the duality relation between ordinary and
superspace were presented. The first approach is the duality equation (4.21)
for $\beta=2$. In our article, we follow this idea. In the second approach,
the matrix $K$ was diagonalized. With the eigenvalues of $K$, a projection
operator was constructed for the definition of a reduced probability density
according to the probability density $P$.

The latter approach fails because $K$ is only diagonalizable if it has no
degeneracy larger than $\gamma_{2}$. Moreover for diagonalizable $K$, one can
not find an eigenvalue $\lambda=0$. This is included in the following
statement which we derive in E.

###### Statement 4.1

Let $N,\widetilde{N}\in\mathbb{N}$, $H^{(0)}\in{\rm Herm\,}(\beta,N)$,
$l\in\mathbb{R}^{\widetilde{N}}$ and $\\{\tau_{q}\\}_{1\leq
q\leq\widetilde{N}}$ $\gamma_{2}N$–dimensional vectors consisting of Grassmann
variables $\tau_{q}=(\tau_{q}^{(1)},\ldots,\tau_{q}^{(\gamma_{2}N)})^{T}$.
Then, the matrix

$H=H^{(0)}+\sum\limits_{q=1}^{\widetilde{N}}l_{q}\left[\tau_{q}\tau_{q}^{\dagger}+\widehat{Y}\left(\tau_{q}^{*}\tau_{q}^{T}\right)\right]$
(4.24)

can not be diagonalized $H=U{\rm
diag\,}(\lambda_{1},\ldots,\lambda_{N})U^{\dagger}$ by a matrix $U$ with the
properties

$U^{\dagger}U=UU^{\dagger}=\leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}_{N}\ ,\ \ U^{*}=\widehat{Y}(U)$ (4.25)

and the body of $U$ lies in ${\rm U\,}^{(\beta)}(N)$ iff $H^{(0)}$ has
degeneracy larger than $\gamma_{2}$. Moreover, $H$ has no eigenvalue
$\lambda\in\mathbb{R}$.

In our particular case, $K$ can not be diagonalized for $k<N-1$. Hence, we do
not follow the second approach of Ref. [5]. We emphasize that none of the
other results in Ref. [5] is affected as they are proven by the correct first
approach which we pursue here.

###  4.5 Ambiguity of the characteristic function in the supersymmetric
extension

In this section, we discuss the problem that the extension of the
characteristic function $\mathcal{F}P$ from ordinary matrices to supermatrices
is not unique. This results from the fact that symmetric supermatrices
comprise two kinds of eigenvalues, i.e. bosonic and fermionic eigenvalues.
Whereas ordinary symmetric matrices have only one kind of eigenvalues. In the
supertraces, these two different kinds are differently weighted by a minus
sign. To illustrate this problem, we also give a simple example.

The rotation invariance of $\mathcal{F}P$ enables us to choose a
representation $\mathcal{F}P_{0}$ of $\mathcal{F}P$ acting on an arbitrary
number of matrix invariants

$\mathcal{F}P_{0}\left(\tr K^{m}|m\in\mathbb{N}\right)=\mathcal{F}P(K)\ .$
(4.26)

For this representation, a unique superfunction exists defined by

$\Phi_{0}(\sigma)=\mathcal{F}P_{0}\left({\rm
Str\,}\sigma^{m}|m\in\mathbb{N}\right)$ (4.27)

where

$\mathcal{F}P_{0}\left({\rm
Str\,}B^{m}|m\in\mathbb{N}\right)=\mathcal{F}P_{0}\left(\tr
K^{m}|m\in\mathbb{N}\right)$ (4.28)

with $B=\tilde{\gamma}^{-1}VV^{\dagger}$. However, the choice of the
representation $\mathcal{F}P_{0}$ is not unique. The question arises whether
it is a well defined object. It is clear that two representations
$\mathcal{F}P_{0}$ and $\mathcal{F}P_{1}$ are equal on ${\rm Herm\,}(\beta,N)$
due to the Cayley–Hamilton theorem,

$\mathcal{F}P_{0}(H)=\mathcal{F}P_{1}(H)\ ,\ H\in{\rm Herm\,}(\beta,N).$
(4.29)

The Cayley–Hamilton theorem states that there is a polynomial which is zero
for $H$. Thus, $H^{M}$ with $M>N$ is a polynomial in $\\{H^{n}\\}_{1\leq n\leq
N}$. Plugging an arbitrary symmetric supermatrix $\sigma$ into the
corresponding superfunctions $\Phi_{0}$ and $\Phi_{1}$ we realize that the
choices are not independent such that

$\Phi_{0}(\sigma)\neq\Phi_{1}(\sigma)$ (4.30)

holds for some $\sigma$.

For example with $N=2$, $k=1$ and $\beta=2$, let the characteristic function
$\mathcal{F}P(H)=\mathcal{F}P_{0}\left(\tr H^{3}\right)$. We get with help of
the Cayley–Hamilton theorem

$\mathcal{F}P_{1}\left(\tr H^{2},\tr H\right)=\mathcal{F}P_{0}\left(2\tr H\tr
H^{2}-\tr^{3}H\right)=\mathcal{F}P_{0}\left(\tr H^{3}\right)=\mathcal{F}P(H)\
.$ (4.31)

Let the set of ${\rm U\,}^{(\beta)}(p/q)$–symmetric supermatrices be

$\displaystyle\left\\{\sigma\in{\rm
Mat}(\tilde{\gamma}p/\tilde{\gamma}q)\left|\sigma^{\dagger}=\sigma,\
\sigma^{*}=\widehat{Y}_{{\rm S}}(\sigma)\right.\right\\}{\rm\ and}$ (4.32)
$\displaystyle\widehat{Y}_{{\rm
S}}(\sigma)=\left\\{\begin{array}[]{ll}\left[\begin{array}[]{cc}\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{2p}&0\\\ 0&Y_{{\rm
s}}\otimes\leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}_{q}\end{array}\right]\sigma\left[\begin{array}[]{cc}\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{2p}&0\\\ 0&Y_{{\rm
s}}^{T}\otimes\leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}_{q}\end{array}\right]&,\ \beta=1,\\\ \sigma^{*}&,\ \beta=2,\\\
\left[\begin{array}[]{cc}Y_{{\rm s}}\otimes\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{p}&0\\\ 0&\leavevmode\hbox{\small
1\kern-3.8pt\normalsize
1}_{2q}\end{array}\right]\sigma\left[\begin{array}[]{cc}Y_{{\rm
s}}^{T}\otimes\leavevmode\hbox{\small 1\kern-3.8pt\normalsize 1}_{p}&0\\\
0&\leavevmode\hbox{\small 1\kern-3.8pt\normalsize 1}_{2q}\end{array}\right]&,\
\beta=4,\end{array}\right.$ (4.44)

with respect to the supergroups

${\rm U\,}^{(\beta)}(p/q)=\left\\{\begin{array}[]{ll}{\rm
UOSp\,}^{(+)}(p/2q)&,\ \beta=1\\\ {\rm U\,}(p/q)&,\ \beta=2\\\ {\rm
UOSp\,}^{(-)}(2p/q)&,\ \beta=4\end{array}\right.\ .$ (4.45)

${\rm Mat}(\tilde{\gamma}p/\tilde{\gamma}q)$ is the set of
$(\tilde{\gamma}p+\tilde{\gamma}q)\times(\tilde{\gamma}p+\tilde{\gamma}q)$–supermatrices
with the complex Grassmann algebra
$\bigoplus\limits_{j=0}^{8k^{2}}\Lambda_{j}$. The definition of the two
representations ${\rm UOSp\,}^{(\pm)}$ of the supergroup ${\rm UOSp\,}$ can be
found in Refs. [22, 14]. We refer to the classification of Riemannian
symmetric superspaces by Zirnbauer [23].

We consider a ${\rm U\,}(1/1)$–symmetric supermatrix $\sigma$. This yields for
the supersymmetric extension of Eq. (4.31)

$\mathcal{F}P_{0}\left(2{\rm Str\,}\sigma{\rm Str\,}\sigma^{2}-{\rm
Str\,}^{3}\sigma\right)\neq\mathcal{F}P_{0}\left({\rm
Str\,}\sigma^{3}\right)=\mathcal{F}P_{0}\left(\frac{1}{4}\left(3\frac{{\rm
Str\,}^{2}\sigma^{2}}{{\rm Str\,}\sigma}+{\rm Str\,}^{3}\sigma\right)\right)\
.$ (4.46)

One obtains the last equation with a theorem similar to the Cayley–Hamilton
theorem. More specificly, there exists a unique polynomial equation of order
two

$\sigma^{2}-\frac{{\rm Str\,}\sigma^{2}}{{\rm
Str\,}\sigma}\sigma-\frac{1}{4}\left({\rm Str\,}^{2}\sigma-\frac{{\rm
Str\,}^{2}\sigma^{2}}{{\rm Str\,}^{2}\sigma}\right)=0\ ,$ (4.47)

for a ${\rm U\,}(1/1)$–symmetric supermatrix $\sigma$.

The resulting integral in Sec. 5 for the generating function
$Z_{k}|_{\mathfrak{M}_{N}={\rm Herm\,}(\beta,N)}$ is invariant under the
choice of $\Phi_{0}$. This is proven in Sec. 6.3. Such an ambiguity of the
supersymmetric extension of the characteristic function was also investigated
by the authors of Ref. [20]. They avoided the question of the definition of a
Dirac–distribution on superspace by the superbosonization formula. We
introduce for the supersymmetric extension from Eq. (4.28) to Eq. (4.27) a
Dirac–distribution depending on the representation of the superfunction.

###  4.6 Symmetries of the supermatrices

We find for a chosen representation $\mathcal{F}P_{0}$

$Z_{k}(x^{-}+J)=(-\imath)^{k_{2}N}\int\limits_{\mathfrak{C}^{k_{2}N}}\Phi_{0}(B)\exp\left[-\imath{\rm
Str\,}(x^{-}+J)B\right]d[\zeta]d[z]\ .$ (4.48)

Here, we introduce $k_{2}=\gamma_{2}k$, $k_{1}=\gamma_{1}k$ and
$\tilde{k}=\tilde{\gamma}k$. We will simplify the integral (4.48) to integrals
over $k_{1}$ eigenvalues in the Boson–Boson block and over $k_{2}$ eigenvalues
in the Fermion–Fermion block.

For every $\beta$, we have

$B^{\dagger}=B\ ,$ (4.49)

i.e. $B$ is self-adjoint. The complex conjugation yields

$B^{*}=\left\\{\begin{array}[]{ll}\widetilde{Y}B\widetilde{Y}^{T}\qquad,\
\beta\in\\{1,4\\}\\\ \widetilde{Y}B^{*}\widetilde{Y}^{T}\qquad,\
\beta=2\end{array}\right.$ (4.50)

with the $(2k+2k)\times(2k+2k)$–supermatrices

$\left.\widetilde{Y}\right|_{\beta=1}=\left[\begin{array}[]{ccc}0&\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{k}&0\\\ \leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{k}&0&0\\\ 0&0&Y_{{\rm
s}}\otimes\leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}_{k}\end{array}\right]\qquad,\qquad\left.\widetilde{Y}\right|_{\beta=4}=\left[\begin{array}[]{ccc}Y_{{\rm
s}}\otimes\leavevmode\hbox{\small 1\kern-3.8pt\normalsize 1}_{k}&0&0\\\
0&0&\leavevmode\hbox{\small 1\kern-3.8pt\normalsize 1}_{k}\\\
0&\leavevmode\hbox{\small 1\kern-3.8pt\normalsize 1}_{k}&0\end{array}\right]$
(4.51)

and $\left.\widetilde{Y}\right|_{\beta=2}={\rm
diag\,}(1,0,1,0)\otimes\leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}_{k}$. We notice that for the unitary case $B$ is effectively a
$(k+k)\times(k+k)$–supermatrix, i.e. half the dimension. With help of the
properties (4.49) and (4.50) we construct the supermatrix sets

$\widetilde{\Sigma}_{0}(\beta,k)=\left\\{\sigma\in{\rm
Mat}(2k/2k)\left|\sigma^{\dagger}=\sigma,\
\sigma^{*}=\left\\{\begin{array}[]{ll}\widetilde{Y}\sigma\widetilde{Y}^{T}&,\
\beta\in\\{1,4\\}\\\ \widetilde{Y}\sigma^{*}\widetilde{Y}^{T}&,\
\beta=2\end{array}\right\\}\right.\right\\}\ .$ (4.52)

A matrix in $\widetilde{\Sigma}_{0}(\beta,k)$ fulfils the odd symmetry (4.50).
We transform this symmetry with the unitary transformations

$U|_{\beta=1}=\frac{1}{\sqrt{2}}\left[\begin{array}[]{ccc}\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{k}&\leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}_{k}&0\\\ -\imath\leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}_{k}&\imath\leavevmode\hbox{\small 1\kern-3.8pt\normalsize 1}_{k}&0\\\
0&0&\sqrt{2}\ \leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}_{2k}\end{array}\right]\ \ ,\ \
U|_{\beta=4}=\frac{1}{\sqrt{2}}\left[\begin{array}[]{ccc}\sqrt{2}\
\leavevmode\hbox{\small 1\kern-3.8pt\normalsize 1}_{2k}&0&0\\\
0&\leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}_{k}&\leavevmode\hbox{\small 1\kern-3.8pt\normalsize 1}_{k}\\\
0&-\imath\leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}_{k}&\imath\leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}_{k}\end{array}\right],$ (4.53)

$U|_{\beta=2}=\leavevmode\hbox{\small 1\kern-3.8pt\normalsize 1}_{4k}$,
according to the Dyson–index, arriving at the well–known symmetries of
symmetric supermatrices [23], see also Eq. (4.32). Defining the sets
$\Sigma_{0}(\beta,k)=U\widetilde{\Sigma}_{0}(\beta,k)U^{\dagger}$, we remark
that the body of the Boson–Boson block of any element in these sets is a
matrix in ${\rm Herm\,}(\beta,k_{1})$. The body of the Fermion–Fermion block
of any matrix in $\Sigma_{0}(\beta,k)$ lies in ${\rm Herm\,}(4/\beta,k_{2})$.

We introduce a generalized Wick–rotation $e^{\imath\psi}$ to guarantee the
convergence of the supermatrix integrals. The usual choice of a Wick–rotation
is $e^{\imath\psi}=\imath$ for investigations of Gaussian probability
densities [5, 1, 2]. Here, general Wick–rotations [14] are also of interest.
Probability densities which lead to superfunction as $\exp\left(-{\rm
Str\,}\sigma^{4}\right)$ do not converge with the choice $\imath$. Thus, we
consider the modified sets

$\Sigma_{\psi}(\beta,k)=\widehat{\Psi}_{\psi}\Sigma_{0}(\beta,k)\widehat{\Psi}_{\psi}\
.$ (4.54)

with $\widehat{\Psi}_{\psi}={\rm diag\,}(\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{2k},e^{\imath\psi/2}\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{2k})$. Let $\Sigma_{\psi}^{0}(\beta,k)$ be the set
of supermatrices which contains only zero and first order terms in the
Grassmann variables.

In the sequel, we restrict our calculations to superfunctions which possess a
Wick–rotation such that the integrals below are convergent. We have not
further explored the set of superfunctions with this property, but we know
that this set has to be very large and sufficient for our purposes. For
example, superfunctions of the form

$\Phi_{0}(\sigma)=\widetilde{\Phi}(\sigma)\exp\left(-{\rm
Str\,}\sigma^{2n}\right),\quad n\in\mathbb{N},$ (4.55)

fulfil this property if ${\rm ln}\widetilde{\Phi}(\sigma)$ does not increase
as fast as ${\rm Str\,}\sigma^{2n}$ at infinity.

###  4.7 Transformation to supermatrices by a Dirac–distribution

Following Refs. [6, 5, 10], $\Phi_{0}(B)$ can be written as a convolution in
the space of supermatrices $\Sigma_{\psi}^{0}(\beta,k)$ with a
Dirac–distribution. We have

$\displaystyle Z_{k}(x^{-}+J)$ $\displaystyle=$
$\displaystyle(-\imath)^{k_{2}N}\int\limits_{\mathfrak{C}_{k_{2}N}}\int\limits_{\Sigma_{\psi}^{0}(\beta,k)}\Phi_{0}(\rho)\delta\left(\rho-
UBU^{\dagger}\right)d[\rho]\times$ (4.56) $\displaystyle\times$
$\displaystyle\exp\left[-\imath{\rm Str\,}(x^{-}+J)B\right]d[\zeta]d[z]$

where the measure is defined as

$d[\rho]=d[\rho_{1}]d[\rho_{2}]\underset{1\leq n\leq
k_{1}}{\prod\limits_{1\leq m\leq k_{2}}}d\eta_{nm}d\eta_{nm}^{*}\ .$ (4.57)

Here, $\\{\eta_{nm},\eta_{nm}^{*}\\}$ are pairs of generators of a Grassmann
algebra, while $\rho_{1}$ is the Boson–Boson and $\rho_{2}$ is the
Fermion–Fermion block without the phase of the Wick–rotation. Since $\rho_{1}$
and $\rho_{2}$ are in ${\rm Herm\,}(\beta,k_{1})$ and ${\rm
Herm\,}(4/\beta,k_{2})$, respectively, we use the real measures for
$d[\rho_{1}]$ and $d[\rho_{2}]$ which are defined in Ref. [14]. We exchange
the Dirac–distribution by two Fourier transformations as in Refs. [5, 10].
Then, Eq. (4.56) becomes

$\displaystyle Z_{k}(x^{-}+J)$ $\displaystyle=$
$\displaystyle(-\imath)^{k_{2}N}2^{2k(k-\tilde{\gamma})}\int\limits_{\mathfrak{C}_{k_{2}N}}\int\limits_{\Sigma_{-\psi}^{0}(\beta,k)}\mathcal{F}\Phi_{0}(\sigma)\times$
(4.58) $\displaystyle\times$ $\displaystyle\exp\left[\imath{\rm
Str\,}B\left(U^{\dagger}\sigma U-x^{-}-J\right)\right]d[\sigma]d[\zeta]d[z]$

where the Fourier transform of $\Phi_{0}$ is

$\mathcal{F}\Phi_{0}(\sigma)=\int\limits_{\Sigma_{\psi}^{0}(\beta,k)}\Phi_{0}(\rho)\exp\left(-\imath{\rm
Str\,}\rho\sigma\right)d[\rho]\ .$ (4.59)

We write the supertrace in the exponent in Eq. (4.58) as a sum over
expectation values

${\rm Str\,}B\left(U^{\dagger}\sigma
U-x^{-}-J\right)=\frac{1}{\tilde{\gamma}}\sum\limits_{j=1}^{N}\tr\Psi_{j}^{\dagger}\left(U^{\dagger}\sigma
U-x^{-}-J\right)\Psi_{j}$ (4.60)

with respect to the real, complex or quaternionic supervectors

$\Psi_{j}^{\dagger}=\left\\{\begin{array}[]{ll}\left\\{z_{jn},z_{jn}^{*},\zeta_{jn},\zeta^{*}_{jn}\right\\}_{1\leq
n\leq k}&,\ \beta=1\\\ \left\\{z_{jn},0,\zeta_{jn},0\right\\}_{1\leq n\leq
k}&,\ \beta=2\\\ \left\\{\left[\begin{array}[]{c}z_{jn}\\\
z_{j+N,n}\end{array}\right],\left[\begin{array}[]{c}-z_{j+N,n}^{*}\\\
z_{jn}^{*}\end{array}\right],\left[\begin{array}[]{c}\zeta_{jn}\\\
\zeta_{j+N,n}\end{array}\right],\left[\begin{array}[]{c}-\zeta_{j+N,n}^{*}\\\
\zeta_{jn}^{*}\end{array}\right]\right\\}_{1\leq n\leq k}&,\
\beta=4\end{array}\right.$ (4.61)

The integration over one of these supervectors yields

$\int\limits_{\mathfrak{C}_{k_{2}}}{\rm
exp}\left[\frac{\imath}{\tilde{\gamma}}\tr\Psi_{j}^{\dagger}\left(U^{\dagger}\sigma
U-x^{-}-J\right)\Psi_{j}\right]d[\Psi_{j}]=\imath^{k_{2}}{\rm
Sdet\,}^{-1/\gamma_{1}}\mathfrak{p}\left(\sigma-x^{-}-J\right)\ .$ (4.62)

$\mathfrak{p}$ projects onto the non-zero matrix blocks of
$\Sigma_{-\psi}(\beta,k)$ which are only $(k+k)\times(k+k)$–supermatrices for
$\beta=2$. $\mathfrak{p}$ is the identity for $\beta\in\\{1,4\\}$. The Eq.
(4.62) is true because $U$ commutes with $x^{-}+J$. Then, Eq. (4.58) reads

$Z_{k}(x^{-}+J)=2^{2k(k-\tilde{\gamma})}\int\limits_{\Sigma_{-\psi}^{0}(\beta,k)}\mathcal{F}\Phi_{0}(\sigma){\rm
Sdet\,}^{-N/\gamma_{1}}\mathfrak{p}\left(\sigma-x^{-}-J\right)d[\sigma]\ .$
(4.63)

Indeed, this result coincides with Ref. [5] for $\beta=2$ where the Fourier
transform $\mathcal{F}\Phi_{0}(\sigma)$ was denoted by $Q(\sigma)$. Eq. (4.63)
reduces for Gaussian ensembles with arbitrary $\beta$ to expressions as in
Refs. [3] and [2]. The integral is well defined because $\varepsilon$ is
greater than zero and the body of the eigenvalues of the Boson–Boson block is
real. The representation (4.63) for the generating function can also be
considered as a random matrix ensemble lying in the superspace.

Eq. (4.63) is one reason why we called this integral transformation from the
space over ordinary matrices to supermatrices as generalized
Hubbard–Stratonovich transformation. If the probability density $P$ is
Gaussian then we can choose $\Phi_{0}$ also as a Gaussian. Thus, this
transformation above reduces to the ordinary Hubbard–Stratonovich
transformation and the well-known result (4.63).

##  5 The supersymmetric Ingham–Siegel integral

We perform a Fourier transformation in superspace for the convolution integral
(4.63) and find

$Z_{k}(x^{-}+J)=2^{2k(k-\tilde{\gamma})}\int\limits_{\Sigma_{\psi}^{0}(\beta,k)}\Phi_{0}(\rho)I_{k}^{(\beta,N)}(\rho)\exp\left[-\imath{\rm
Str\,}\rho\left(x^{-}+J\right)\right]d[\rho]\ .$ (5.1)

Here, we have to calculate the supersymmetric Ingham–Siegel integral

$I_{k}^{(\beta,N)}(\rho)=\int\limits_{\Sigma_{-\psi}^{0}(\beta,k)}\exp\left(-\imath{\rm
Str\,}\rho\sigma^{+}\right){\rm
Sdet\,}^{-N/\gamma_{1}}\mathfrak{p}\sigma^{+}d[\sigma]$ (5.2)

with $\sigma^{+}=\sigma+\imath\varepsilon\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{4k}$.

Ingham [24] and Siegel [25] independently calculated a version of (5.2) for
ordinary real symmetric matrices. The case of hermitian matrices was discussed
in Ref. [26]. Since we were unable to find the ordinary Ingham–Siegel integral
also for the quaternionic case, we give the result here. It is related to
Selbergs integral [27]. Let $R\in{\rm Herm\,}(\beta,m)$, $\varepsilon>0$ and a
real number $n\geq m-1+2/\beta$, then we have

$\displaystyle\int\limits_{{\rm Herm\,}(\beta,m)}\exp\left(-\imath\tr
RS^{+}\right){\det}^{-n/\gamma_{1}}S^{+}d[S]=\imath^{-\beta
mn/2}G_{n-m,m}^{(\beta)}\displaystyle{\det}^{\lambda}R\ \Theta(R)$ (5.3)

where $S^{+}=S+\imath\varepsilon\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{\gamma_{2}m}$, the exponent is

$\lambda=\frac{n-m}{\gamma_{1}}-\frac{\gamma_{1}-\gamma_{2}}{2}$ (5.4)

and the constant is

$G_{n-m,m}^{(\beta)}=\left(\frac{\gamma_{2}}{\pi}\right)^{\beta
m(n-m+1)/2-m}\prod\limits_{j=n-m+1}^{n}\frac{2\pi^{\beta
j/2}}{\Gamma\left(\beta j/2\right)}\ .$ (5.5)

$\Gamma(.)$ is the Euler gamma–function and $\Theta(.)$ is the Heavyside-
function for matrices which is defined as

$\Theta(R)=\left\\{\begin{array}[]{ll}1&,\ R{\rm\ is\ positive\ definite}\\\
0&,\ {\rm else}\end{array}\right.\ .$ (5.6)

The ordinary Ingham–Siegel integral was recently used in the context of
supersymmetry by Fyodorov [26]. The integral was extended to the superspace
$\Sigma_{\pi/2}^{0}(2,k)$ in Ref. [5]. In this article, we need a
generalization to all $\Sigma_{-\psi}^{0}(\beta,k)$, in particular
$\beta=1,4$.

The integral (5.2) is invariant under the action of ${\rm
U\,}^{(\beta)}(k_{1}/k_{2})$. Thus, it is convenient to consider
$I(r,\varepsilon)$, where $r={\rm
diag\,}(r_{11},\ldots,r_{\tilde{k}1},r_{12},\ldots,r_{\tilde{k}2})$ is the
diagonal matrix of eigenvalues of $\rho$ and contains nilpotent terms. The
authors of Ref. [10] claimed in their proof of Theorem 1 in Chapter 6 that the
diagonalization at this point of the calculation yields Efetov–Wegner terms.
These terms do not appear in the $\rho_{2}$ integration because we do not
change the integration variables, i.e. the integration measure $d[\rho]$
remains the same. For the unitary case, see Ref. [5]. We consider the
eigenvalues of $\rho$ as functions of the Cartesian variables. We may
certainly differentiate a function with respect to the eigenvalues if we keep
track of how these differential operators are defined in the Cartesian
representation.

As worked out in C.1, the supersymmetric Ingham–Siegel integral (5.2) reads

$I_{k}^{(\beta,N)}(\rho)=\displaystyle
C{\det}^{\kappa}r_{1}\Theta(r_{1}){\det}^{k}r_{2}\exp\left(-e^{\imath\psi}\varepsilon\tr
r_{2}\right)\left[D_{k_{2}r_{2}}^{(4/\beta)}\left(\imath
e^{\imath\psi}\gamma_{1}\varepsilon\right)\right]^{N}\frac{\delta(r_{2})}{|\Delta_{k_{2}}(r_{2})|^{4/\beta}}\
.$ (5.7)

The constant is

$C=\displaystyle\left(-\frac{e^{-\imath\psi}}{\gamma_{1}}\right)^{k_{2}N}\left(-\frac{\tilde{\gamma}}{2\pi}\right)^{k_{1}k_{2}}\left(\frac{2\pi}{\gamma_{1}}\right)^{k_{2}}\left(\frac{\pi}{\gamma_{1}}\right)^{2k_{2}(k_{2}-1)/\beta}\frac{G_{Nk_{1}}^{(\beta)}}{g_{k_{2}}^{(4/\beta)}}$
(5.8)

with

$\displaystyle
g_{k_{2}}^{(4/\beta)}=\frac{1}{k_{2}!}\prod\limits_{j=1}^{k_{2}}\frac{\pi^{2(j-1)/\beta}\Gamma\left(2/\beta\right)}{\Gamma\left(2j/\beta\right)}\
.$ (5.9)

while the exponent is given by

$\kappa=\frac{N}{\gamma_{1}}+\frac{\gamma_{2}-\gamma_{1}}{2}$ (5.10)

and the differential operator

$D_{k_{2}r_{2}}^{(4/\beta)}\left(\imath
e^{\imath\psi}\gamma_{1}\varepsilon\right)=\frac{1}{\Delta_{k_{2}}(r_{2})}\det\left[r_{a2}^{N-b}\left(\frac{\partial}{\partial
r_{a2}}+(k_{2}-b)\frac{2}{\beta}\frac{1}{r_{a2}}-e^{\imath\psi}\gamma_{1}\varepsilon\right)\right]_{1\leq
a,b\leq k_{2}}$ (5.11)

is the analog to the Sekiguchi differential operator [28]. We derived it in B.

The complexity of $D_{k_{2}r_{2}}^{(4/\beta)}(\imath
e^{\imath\psi}\varepsilon)$ makes Eq. (5.7) cumbersome, a better
representation is desirable. To simplify Eq. (5.7), we need the following
statement which is shown in C.2.

###### Statement 5.1

We consider two functions $F,f:{\rm
Herm\,}(4/\beta,k_{2})\rightarrow\mathbb{C}$ invariant under the action of
${\rm U\,}^{(4/\beta)}(k_{2})$ and Schwartz–functions of the matrix
eigenvalues. Let $F$ and $f$ have the relation

$F(\rho_{2})=f(\rho_{2})\det\rho_{2}^{N/\gamma_{1}-k}{\rm\ \ for\ all\
}\rho_{2}\in{\rm Herm\,}(4/\beta,k_{2})\ .$ (5.12)

Then, we have

$\displaystyle\int\limits_{\mathbb{R}^{k_{2}}}\int\limits_{{\rm
Herm\,}(4/\beta,k_{2})}F(r_{2}){\det}^{k}r_{2}|\Delta_{k_{2}}(r_{2})|^{4/\beta}\exp\left(\imath\tr
r_{2}\sigma_{2}\right){\det}^{N/\gamma_{1}}\left(e^{-\imath\psi}\sigma_{2}+\imath\varepsilon\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{\tilde{k}}\right)d[\sigma_{2}]d[r_{2}]=$
$\displaystyle=w_{1}f(0)=\int\limits_{\mathbb{R}^{k_{2}}}F(r_{2})|\Delta_{k_{2}}(r_{2})|^{4/\beta}\left[\frac{w_{2}\exp\left(\varepsilon
e^{\imath\psi}\tr
r_{2}\right)}{|\Delta_{k_{2}}(r_{2})|^{4/\beta}}\prod\limits_{j=1}^{k_{2}}\left(\frac{\partial}{\partial
r_{j2}}\right)^{N-k_{1}}\delta(r_{j2})\right]d[r_{2}]$ (5.13)

where the constants are

$\displaystyle
w_{1}=\left(\frac{2\pi}{\gamma_{1}}\right)^{k_{2}}\left(\frac{\pi}{\gamma_{1}}\right)^{2k_{2}(k_{2}-1)/\beta}\frac{\left(\imath^{N}e^{-\imath\psi
N}\right)^{k_{2}}}{g_{k_{2}}^{(4/\beta)}}\prod_{b=1}^{k_{2}}\prod\limits_{a=1}^{N}\left(\frac{a}{\gamma_{1}}+\frac{b-1}{\gamma_{2}}\right)$
(5.14) $\displaystyle
w_{2}=\frac{(-1)^{k_{1}k_{2}}}{g_{k_{2}}^{(4/\beta)}}\left(\frac{2\pi}{\gamma_{1}}\right)^{k_{2}}\left(\frac{\pi}{\gamma_{1}}\right)^{2k_{2}(k_{2}-1)/\beta}\left[\frac{(-\imath)^{N}e^{-\imath\psi
N}}{\left(N-k_{1}\right)!\gamma_{1}^{N}}\right]^{k_{2}}\prod_{j=0}^{k_{2}-1}\frac{\Gamma\left(N+1+2j/\beta\right)}{\Gamma\left(1+2j/\beta\right)}\
.$ (5.15)

This statement yields for the supersymmetric Ingham–Siegel integral

$I_{k}^{(\beta,N)}(\rho)=\displaystyle
W\Theta(r_{1})\frac{{\det}^{\kappa}r_{1}}{|\Delta_{k_{2}}(r_{2})|^{4/\beta}}\prod\limits_{j=1}^{k_{2}}\left(\frac{\partial}{\partial
r_{j2}}\right)^{N-k_{1}}\delta(r_{j2})$ (5.16)

where the constant reads

$\displaystyle W$ $\displaystyle=$
$\displaystyle\left(\frac{\tilde{\gamma}}{2\pi}\right)^{k_{1}k_{2}}\left(\frac{2\pi}{\gamma_{1}}\right)^{k_{2}}\left(\frac{\pi}{\gamma_{1}}\right)^{2k_{2}(k_{2}-1)/\beta}\left[\frac{\left(-e^{-\imath\psi}\right)^{N}}{\left(N-k_{1}\right)!\gamma_{1}^{N}}\right]^{k_{2}}\times$
(5.17) $\displaystyle\times$
$\displaystyle\frac{G_{Nk_{1}}^{(\beta)}}{g_{k_{2}}^{(4/\beta)}}\prod_{j=0}^{k_{2}-1}\frac{\Gamma\left(N+1+2j/\beta\right)}{\Gamma\left(1+2j/\beta\right)}\
.$

We further simplify this formula for $\beta=1$ and $\beta=2$. The powers of
the Vandermonde–determinant $\Delta_{k_{2}}^{4/\beta}(r_{2})$ are polynomials
of degree $k_{2}\times 2(k_{2}-1)/\beta$. The single power of one eigenvalue
derivative must be $2(k_{2}-1)/\beta$ if we substitute these terms in Eq.
(5.16) by partial derivatives of the eigenvalues, for details see C.2. Hence,
this power is a half-integer for $\beta=4$. Also, $\Delta_{k_{2}}(r_{2})$ has
no symmetric term where all eigenvalues have the same power. Therefore, we can
not simplify the quaternionic case in the same manner.

We use the identities

$\displaystyle\prod\limits_{j=1}^{n}\frac{\partial^{n-1}}{\partial
x_{j}^{n-1}}\Delta_{n}^{2}(x)$ $\displaystyle=$
$\displaystyle(-1)^{n(n-1)/2}n!\left[(n-1)!\right]^{n}\ ,$ (5.18)
$\displaystyle\prod\limits_{j=1}^{n}\frac{\partial^{2(n-1)}}{\partial
x_{j}^{2(n-1)}}\Delta_{n}^{4}(x)$ $\displaystyle=$ $\displaystyle
n!\left[(2n-2)!\right]^{n}\prod\limits_{j=0}^{n-1}(2j+1)$ (5.19)

and find

$\displaystyle I_{k}^{(1,N)}(\rho)$ $\displaystyle=$ $\displaystyle
2^{-k(k-2)}\left[\frac{2\pi e^{-\imath\psi N}}{(N-2)!}\right]^{k}\times$
(5.20) $\displaystyle\times$ $\displaystyle\Theta(r_{1})\det
r_{1}^{(N-1)/2}\prod\limits_{j=1}^{k}\left(-\frac{\partial}{\partial
r_{j2}}\right)^{N-2}\delta(r_{j2})$

and

$\displaystyle I_{k}^{(2,N)}(\rho)$ $\displaystyle=$
$\displaystyle(-1)^{k(k+1)/2}2^{-k(k-1)}\left[\frac{2\pi e^{-\imath\psi
N}}{(N-1)!}\right]^{k}\times$ (5.21) $\displaystyle\times$
$\displaystyle\Theta(r_{1})\det
r_{1}^{N}\prod\limits_{j=1}^{k}\left(-\frac{\partial}{\partial
r_{j2}}\right)^{N-1}\delta(r_{j2})\ .$

For $\beta=4$, we summarize the constants and have

$\displaystyle I_{k}^{(4,N)}(\rho)$ $\displaystyle=$ $\displaystyle
2^{-k(k-2)}\left[\frac{2\pi e^{-\imath\psi N}}{(N-k)!}\right]^{2k}\times$
(5.22) $\displaystyle\times$ $\displaystyle\Theta(r_{1})\det
r_{1}^{N+1/2}\frac{4^{k}k!}{\pi^{k}|\Delta_{2k}(r_{2})|}\prod\limits_{j=1}^{2k}\left(-\frac{\partial}{\partial
r_{j2}}\right)^{N-k}\delta(r_{j2})\ .$

These distributions are true for superfunctions whose Fermion–Fermion block
dependence is as in Eq. (5.12). Eqs. (5.20) and (5.21) can be extended to
distributions on arbitrary Schwartz–functions which is not the case for Eq.
(5.22). The constants in Eqs. (5.20) and (5.21) must be the same due to the
independence of the test–function.

###### Statement 5.2

Equations (5.20) and (5.21) are true for rotation invariant superfunctions
$\Phi_{0}$ which are Schwartz–functions in the Fermion–Fermion block entries
along the Wick–rotated real axis.

We derive this statement in C.3.

Indeed, the Eq. (5.21) is the same as the formula for the supersymmetric
Ingham–Siegel integral for $\beta=2$ in Ref. [5]. Comparing both results, the
different definitions of the measures have to be taken into account. We also
see the similarity to the superbosonization formula [9, 8, 12, 11, 20, 10] for
$\beta\in\\{1,2\\}$. One can replace the partial derivative in Eq. (5.20) and
(5.21) by contour integrals if the characteristic function $\Phi_{0}$ is
analytic. However for $\beta=4$, more effort is needed. For our purposes, Eqs.
(5.7) and (5.22) are sufficient for the quaternionic case. In the unitary
case, the equivalence of Eq. (5.21) with the superbosonization formula was
confirmed with help of Cauchy integrals by Basile and Akemann. [10]

##  6 Final representation of the generating function and its independence of
the choice for $\Phi_{0}$

In Sec. 6.1, we present the generating function as a supersymmetric integral
over eigenvalues and introduce the supersymmetric Bessel–functions. In Sec.
6.2, we revisit the unitary case and point out certain properties of the
generating function. Some of these properties, independence of the
Wick–rotation and the choice of $\Phi_{0}$, are also proven for the orthogonal
and unitary–symplectic case in Sec. 6.3.

###  6.1 Eigenvalue integral representation

The next step of the calculation of the generating function $Z_{k}(x^{-}+J)$
is the integration over the supergroup. The function
$\Phi_{0}(\rho)I_{k}^{(\beta,N)}(\rho)$ is invariant under the action of ${\rm
U\,}^{(\beta)}(k_{1}/k_{2})$.

We define the supermatrix Bessel–function

$\varphi_{k_{1}k_{2}}^{(\beta)}(s,r)=\int\limits_{{\rm
U\,}^{(\beta)}(k_{1}/k_{2})}\exp\left({\rm Str\,}sUrU^{\dagger}\right)d\mu(U)$
(6.1)

as in Refs. [29, 14]. We choose the normalization

$\displaystyle\int\limits_{\Sigma^{0}_{\psi}(\beta,k)}f(\sigma)\exp\left({\rm
Str\,}\sigma
x\right)d[e^{-\imath\psi/2}\eta]d[e^{\imath\psi}\sigma_{2}]d[\sigma_{1}]=$
(6.2) $\displaystyle=$
$\displaystyle\int\limits_{\mathbb{R}^{k_{1}}}\int\limits_{\mathbb{R}^{k_{2}}}f(s)\varphi_{k_{1}k_{2}}^{(\beta)}(s,x)\left|B_{k}^{(\beta)}(s_{1},e^{\imath\psi}s_{2})\right|d[e^{\imath\psi}s_{2}]d[s_{1}]+{\rm
b.t.}$

which holds for every rotation invariant function $f$. This normalization
agrees with Refs. [30, 31, 29, 5, 14]. The boundary terms (${\rm b.t.}$)
referred to as Efetov–Wegner terms [32, 33, 10] appear upon changing the
integration variables [34] or, equivalently, upon partial integration [14].
The Berezinian is

$B_{k}^{(\beta)}(s_{1},e^{\imath\psi}s_{2})=\displaystyle\frac{\Delta_{k_{1}}^{\beta}(s_{1})\Delta_{k_{2}}^{4/\beta}(e^{\imath\psi}s_{2})}{V_{k}^{2}(s_{1},e^{\imath\psi}s_{2})}$
(6.3)

where
$V_{k}(s_{1},e^{\imath\psi}s_{2})=\prod\limits_{n=1}^{k_{1}}\prod\limits_{m=1}^{k_{2}}\left(s_{n1}-e^{\imath\psi}s_{m2}\right)$
mixes bosonic and fermionic eigenvalues. These Berezinians have a
determinantal structure

$B_{k}^{(\beta)}(s_{1},e^{\imath\psi}s_{2})=\left\\{\begin{array}[]{ll}\displaystyle\det\left[\frac{1}{s_{a1}-e^{\imath\psi}s_{b2}}\
,\ \frac{1}{(s_{a1}-e^{\imath\psi}s_{b2})^{2}}\right]\underset{1\leq b\leq
k}{\underset{1\leq a\leq 2k}{}}&,\ \beta=1\\\
\displaystyle{\det}^{2}\left[\frac{1}{s_{a1}-e^{\imath\psi}s_{b2}}\right]_{1\leq
a,b\leq k}&,\ \beta=2\\\ \displaystyle
B_{k}^{(1)}(e^{\imath\psi}s_{2},s_{1})&,\ \beta=4\end{array}\right.\ .$ (6.4)

For $\beta=2$ this formula was derived in Ref. [32]. The other cases are
derived in D. We notice that this determinantal structure is similar to the
determinantal structure of the ordinary Vandermonde–determinant raised to the
powers $2$ and $4$. This structure was explicitly used [15] to calculate the
$k$–point correlation function of the GUE and the GSE.

We find for the generating function

$\displaystyle Z_{k}(x^{-}+J)$ $\displaystyle=$ $\displaystyle
2^{2k(k-\tilde{\gamma})}e^{\imath\psi
k_{1}}\int\limits_{\mathbb{R}^{k_{1}}}\int\limits_{\mathbb{R}^{k_{2}}}\Phi_{0}(r)I_{k}^{(\beta,N)}(r)\times$
(6.5) $\displaystyle\times$
$\displaystyle\varphi_{k_{1}k_{2}}^{(\beta)}(-\imath
r,x^{-}+J)\left|B_{k}^{(\beta)}(r_{1},e^{\imath\psi}r_{2})\right|d[r_{2}]d[r_{1}]+{\rm
b.t.}\ .$

The normalization of $Z_{k}$ is guaranteed by the Efetov–Wegner terms. When
setting $(k-l)$ with $l<k$ of the source variables $J_{p}$ to zero then we
have

$\left.Z_{k}(x^{-}+J)\right|_{J_{l}=\ldots=J_{k}=0}=Z_{l-1}(\tilde{x}^{-}+\widetilde{J})\
,$ (6.6)

$\tilde{x}={\rm diag\,}(x_{1},\ldots,x_{l-1}),\ \widetilde{J}={\rm
diag\,}(J_{1},\ldots,J_{l-1})$, by the integration theorems in Ref. [1, 35,
36, 37, 3, 14]. This agrees with the definition (2.9).

###  6.2 The unitary case revisited

To make contact with the discussion in Ref. [5], we revisit the unitary case
using the insight developed here.

For a further calculation we need the explicit structure of the supersymmetric
matrix Bessel–functions. However, the knowledge of these functions is limited.
Only for certain $\beta$ and $k$ we know the exact structure. In particular
for $\beta=2$ the supermatrix Bessel–function was first calculated in Ref.
[32, 30] with help of the heat equation. Recently, this function was re-
derived by integrating the Grassmann variables in Cartesian coordinates [14],

$\displaystyle\varphi_{kk}^{(2)}(-\imath
r,x^{-}+J)=\displaystyle\frac{\imath^{k}\exp\left(-\varepsilon{\rm
Str\,}r\right)}{2^{k^{2}}\pi^{k}}\times$
$\displaystyle\times\frac{\det\left[\exp\left(-\imath
r_{m1}(x_{n}-J_{n})\right)\right]_{1\leq m,n\leq k}\det\left[\exp\left(\imath
e^{\imath\psi}r_{m2}(x_{n}+J_{n})\right)\right]_{1\leq m,n\leq
k}}{\sqrt{B_{k}^{(2)}(r_{1},e^{\imath\psi}r_{2})B_{k}^{(2)}\left(x-J,x+J\right)}}$
(6.7)

with $x\pm J={\rm diag\,}(x_{1}\pm J_{1},\ldots,x_{k}\pm J_{k})$ and the
positive square root of the Berezinian

$\displaystyle\sqrt{B_{k}^{(2,2)}(r_{1},e^{\imath\psi}r_{2})}=\displaystyle\det\left[\frac{1}{r_{a1}-e^{\imath\psi}r_{b2}}\right]_{1\leq
a,b\leq
k}=(-1)^{k(k-1)/2}\frac{\Delta_{k}(s_{1})\Delta_{k}(e^{\imath\psi}s_{2})}{V_{k}(s_{1},e^{\imath\psi}s_{2})}\
.$ (6.8)

Due to the structure of $\varphi_{kk}^{(2)}$ and $B_{k}^{(2)}$, we write the
generating function for $\beta=2$ as an integral over $\Phi_{0}$ times a
determinant [5]

$\displaystyle Z_{k}(x^{-}+J)$ $\displaystyle=$
$\displaystyle(-1)^{k(k+1)/2}\displaystyle{\det}^{-1}\left[\frac{1}{x_{a}-x_{b}-J_{a}-J_{b}}\right]_{1\leq
a,b\leq
k}\int\limits_{\mathbb{R}^{k}}\int\limits_{\mathbb{R}^{k}}\Phi_{0}(r)\times$
(6.9) $\displaystyle\times$
$\displaystyle\det\left[\mathfrak{F}_{N}(\tilde{r}_{mn},\tilde{x}_{mn})\Theta(r_{m1})\exp\left(-\varepsilon{\rm
Str\,}\tilde{r}_{mn}\right)\right]_{1\leq m,n\leq k}d[r_{2}]d[r_{1}]+{\rm
b.t.}$

where $\tilde{r}_{mn}={\rm diag\,}\left(r_{m1},e^{\imath\psi}r_{n2}\right)$,
$\tilde{x}_{mn}={\rm diag\,}\left(x_{m}-J_{m},x_{n}+J_{n}\right)$ and

$\mathfrak{F}_{N}(\tilde{r}_{mn},\tilde{x}_{mn})=\frac{\imath
r_{m1}^{N}\exp\left(-\imath{\rm
Str\,}\tilde{r}_{mn}\tilde{x}_{mn}\right)}{(N-1)!(r_{m1}-e^{\imath\psi}r_{n2})}\left(-e^{-\imath\psi}\frac{\partial}{\partial
r_{n2}}\right)^{N-1}\delta(r_{n2})\ .$ (6.10)

Then, the modified $k$–point correlation function is

$\displaystyle\qquad\quad\widehat{R}_{k}(x^{-})$ $\displaystyle=$
$\displaystyle\int\limits_{\mathbb{R}^{k}}\int\limits_{\mathbb{R}^{k}}\Phi_{0}(r)\times$
(6.11) $\displaystyle\times$
$\displaystyle\det\left[\mathfrak{F}_{N}(\tilde{r}_{mn},x_{mn})\Theta(r_{m1})\exp\left(-\varepsilon{\rm
Str\,}\tilde{r}_{mn}\right)\right]_{1\leq m,n\leq k}d[r_{2}]d[r_{1}]+{\rm
b.t.}$

and the $k$–point correlation function is

$R_{k}(x)=\int\limits_{\mathbb{R}^{k}}\int\limits_{\mathbb{R}^{k}}\Phi_{0}(r)\det\left[\frac{\mathfrak{F}_{N}(\tilde{r}_{mn},x_{mn})}{2\pi\imath}\right]_{1\leq
m,n\leq k}d[r_{2}]d[r_{1}]+{\rm b.t.}\ .$ (6.12)

We defined $x_{mn}={\rm diag\,}(x_{m},x_{n})$. The boundary terms comprise the
lower correlation functions. The $k$–point correlation function for $\beta=2$
is a determinant of the fundamental function

$R^{({\rm
fund})}(x_{m},x_{n})=\int\limits_{\mathbb{R}}\int\limits_{\mathbb{R}}\Phi_{0}(r)\frac{\mathfrak{F}_{N}(r,x_{mn})}{2\pi\imath}dr_{2}dr_{1}$
(6.13)

if there is one characteristic function $\mathcal{F}P_{0}$ with a
supersymmetric extension $\Phi_{0}$ factorizing for diagonal supermatrices,

$\Phi_{0}(r)={\rm Sdet\,}{\rm
diag\,}\left[\widehat{\Phi}_{0}(r_{11}),\ldots,\widehat{\Phi}_{0}(r_{k1}),\widehat{\Phi}_{0}\left(e^{\imath\psi}r_{12}\right),\ldots,\widehat{\Phi}_{0}\left(e^{\imath\psi}r_{k2}\right)\right]\
,$ (6.14)

with $\widehat{\Phi}_{0}:\mathbb{C}\rightarrow\mathbb{C}$. For example, the
shifted Gaussian ensemble in App. F of Ref. [5] is of such a type.

In Eq. (6.13) we notice that this expression is independent of the generalized
Wick–rotation. Every derivative of the fermionic eigenvalue $r_{2}$ contains
the inverse Wick–rotation as a prefactor. Moreover, the Wick–rotation in the
functions are only prefactors of $r_{2}$. Thus, an integration over the
fermionic eigenvalues $r_{2}$ in Eq. (6.11) cancels the Wick–rotation out by
using the Dirac–distribution. Also, this integration shows that every
representation of the characteristic function gives the same result, see
Theorem 6.1 in the next subsection. However, the determinantal structure with
the fundamental function in Eq. (6.13) depends on a special choice of
$\Phi_{0}$.

###  6.3 Independence statement

For $\beta=1$ and $\beta=4$ we do not know the ordinary matrix Bessel–function
explicitly. Hence, we can not give such a compact expression as in the case
$\beta=2$. On the other hand, we can derive the independence of the
Wick–rotation and of the $\Phi_{0}$ choice of the generating function.

###### Statement 6.1

The generating function $Z_{k}$ is independent of the Wick–rotation and of the
choice of the characteristic functions supersymmetric extension $\Phi_{0}$
corresponding to a certain matrix ensemble $(P,{\rm Herm\,}(\beta,N))$.

Derivation:
We split the derivation in two parts. The first part regards the Wick–rotation
and the second part yields the independence of the choice of $\Phi_{0}$.

Due to the normalization of the supermatrix Bessel–function (6.2),
$\varphi_{k_{1}k_{2}}^{(\beta)}(-\imath r,x^{-}+J)$ only depends on
$e^{\imath\psi}r_{2}$. The same is true for $\Phi_{0}$. Due to the property

$D_{k_{2}r_{2}}^{(4/\beta)}\left(\imath
e^{\imath\psi}\gamma_{1}\varepsilon\right)=e^{\imath
k_{2}\psi}D_{k_{2},e^{\imath\psi}r_{2}}^{(4/\beta)}\left(\imath\gamma_{1}\varepsilon\right)\
,$ (6.15)

the Ingham–Siegel integral in the form (5.7) times the phase
$e^{\imath(k_{1}-k_{2})\psi}$ only depends on $e^{\imath\psi}r_{2}$ and
$e^{-\imath\psi}\partial/\partial r_{2}$. The additional phase comes from the
$\rho$–integration. Thus, we see the independence of the Wick–rotation because
of the same reason as in the $\beta=2$ case.

Let $\Phi_{0}$ and $\Phi_{1}$ be two different supersymmetric extensions of
the characteristic function $\mathcal{F}P$. Then these two superfunctions only
depend on the invariants $\\{{\rm Str\,}\sigma^{m_{j}}\\}_{1\leq j\leq l_{0}}$
and $\\{{\rm Str\,}\sigma^{n_{j}}\\}_{1\leq j\leq l_{1}}$,
$m_{j},n_{j},l_{0},l_{1}\in\mathbb{N}$. We consider $\Phi_{0}$ and $\Phi_{1}$
as functions of $\mathbb{C}^{l_{0}}\rightarrow\mathbb{C}$ and
$\mathbb{C}^{l_{1}}\rightarrow\mathbb{C}$, respectively. Defining the function

$\Delta\Phi(x_{1},\ldots,x_{M})=\Phi_{0}(x_{m_{1}},\ldots,x_{m_{l_{0}}})-\Phi_{1}(x_{n_{1}},\ldots,x_{n_{l_{1}}}),$
(6.16)

where $M={\rm max}\\{m_{a},n_{b}\\}$, we notice with the discussion in Sec.
4.5 that

$\Delta\Phi(x_{1},\ldots,x_{M})|_{x_{j}=\tr H^{j}}=0$ (6.17)

for every hermitian matrix $H$. However, there could be a symmetric
supermatrix $\sigma$ with

$\Delta\Phi(x_{1},\ldots,x_{M})|_{x_{j}={\rm Str\,}\sigma^{j}}\neq 0.$ (6.18)

With the differential operator

$\mathfrak{D}_{r}=\left[D_{k_{2}r_{2}}^{(4/\beta)}\left(\imath
e^{\imath\psi}\gamma_{1}\varepsilon\right)\right]^{N-k_{1}}\frac{\varphi_{k_{1}k_{2}}^{(\beta)}(-\imath
r,x^{-}+J)}{V_{k}(r_{1},e^{\imath\psi}r_{2})},$ (6.19)

we consider the difference of the generating functions

$\displaystyle\Delta Z_{k}(x^{-}+J)$ $\displaystyle=$ $\displaystyle
Z_{k}(x^{-}+J)|_{\Phi_{0}}-Z_{k}(x^{-}+J)|_{\Phi_{1}}=$ (6.20)
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{k_{1}}}|\Delta_{k_{2}}(r_{1})|^{\beta}{\det}^{\kappa}r_{1}\Theta(r_{1})\left.\mathfrak{D}_{r}\Delta\Phi(x)|_{x_{j}={\rm
Str\,}r^{j}}\right|_{r_{2}=0}d[r_{1}]$

Here, we omit the Efetov–Wegner terms. The differential operator is invariant
under the action of the permutation group $S(k_{2})$ on the fermionic block
${\rm Herm\,}(4/\beta,k_{2})$. Hence, we find

$\displaystyle\left.\mathfrak{D}_{r}\Delta\Phi(x)|_{x_{j}={\rm
Str\,}r^{j}}\right|_{r_{2}=0}$ $\displaystyle=$
$\displaystyle\left.\underset{|a|\leq
k_{2}(N-k_{1})}{\sum\limits_{a\in\\{0,\ldots,N-k_{1}\\}^{M}}}d_{a}(r)\prod\limits_{j=1}^{M}\frac{\partial^{a_{j}}}{\partial
x_{j}^{a_{j}}}\Delta\Phi(x)|_{x_{j}={\rm Str\,}r^{j}}\right|_{r_{2}=0}=$
(6.21) $\displaystyle=$ $\displaystyle\underset{|a|\leq
k_{2}(N-k_{1})}{\sum\limits_{a\in\\{0,\ldots,N-k_{1}\\}^{M}}}d_{a}(r_{1})\prod\limits_{j=1}^{M}\frac{\partial^{a_{j}}}{\partial
x_{j}^{a_{j}}}\Delta\Phi(x)|_{x_{j}=\tr r^{j}}=$ $\displaystyle=$
$\displaystyle 0,$

where $d_{a}$ are certain symmetric functions depending on the eigenvalues
$r$. At $r_{2}=0$ these functions are well-defined since the supermatrix
Bessel–functions and the term $V_{k}^{-1}(r_{1},e^{\imath\psi}r_{2})$ are
$C^{\infty}$ at this point. Thus, we find that

$\Delta Z_{k}(x^{-}+J)=0.$ (6.22)

This means that the generating function is independent of the supersymmetric
extension of the characteristic function. $\square$

##  7 One–point and higher order correlation functions

We need an explicit expression or some properties of the supermatrix
Bessel–function to simplify the integral for the generating function. For
$k=1$ we know the supermatrix Bessel–functions for all $\beta$. The simplest
case is $\beta=2$ where we take the formula (6.12) with $k=1$ and obtain

$R_{1}(x)=R^{({\rm
fund})}(x,x)=\int\limits_{\mathbb{R}}\int\limits_{\mathbb{R}}\Phi_{0}(r)\frac{\mathfrak{F}_{N}\left(r,x\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{2}\right)}{2\pi\imath}dr_{2}dr_{1}\ .$ (7.1)

Since the Efetov–Wegner term in the generating function is just unity there
are no boundary terms in the level density. For $\beta\in\\{1,4\\}$ we use the
supermatrix Bessel–function [29, 38, 14]

$\displaystyle\varphi_{21}^{(1)}(-\imath r,x^{-}+J)$ $\displaystyle=$
$\displaystyle\frac{-2J}{\pi}\exp\left[-\imath{\rm
Str\,}r(x^{-}+J)\right]\times$ (7.2) $\displaystyle\times$
$\displaystyle\left[\imath{\rm
Str\,}r+J\left(r_{11}-e^{\imath\psi}r_{2}\right)\left(r_{21}-e^{\imath\psi}r_{2}\right)\right]\
.$

We find

$\displaystyle\widehat{R}_{1}(x^{-})=\displaystyle-\imath\int\limits_{\mathbb{R}^{2}}\int\limits_{\mathbb{R}}\Phi_{0}(r)\det
r_{1}^{(N-1)/2}{\rm
Str\,}r\frac{|r_{11}-r_{21}|}{(r_{11}-e^{\imath\psi}r_{2})^{2}(r_{21}-e^{\imath\psi}r_{2})^{2}}\times$
$\displaystyle\times\displaystyle\exp\left(-\imath x^{-}{\rm
Str\,}r\right)\Theta(r_{1})\frac{1}{(N-2)!}\left(-e^{-\imath\psi}\frac{\partial}{\partial
r_{2}}\right)^{N-2}\delta(r_{2})d[r_{1}]dr_{2}$ (7.3)

for $\beta=1$ and

$\displaystyle\widehat{R}_{1}(x^{-})=\displaystyle-4\imath\int\limits_{\mathbb{R}}\int\limits_{\mathbb{R}^{2}}\Phi_{0}(r)r_{1}^{2N+1}{\rm
Str\,}r\frac{e^{\imath\psi}r_{12}-e^{\imath\psi}r_{22}}{(r_{1}-e^{\imath\psi}r_{12})^{2}(r_{1}-e^{\imath\psi}r_{22})^{2}}\times$
$\displaystyle\times\exp\left(-\imath x^{-}{\rm
Str\,}r\right)\Theta(r_{1})\frac{\det
e^{\imath\psi}r_{2}}{(2N+1)!}\left(4e^{-2\imath\psi}D_{2,r_{2}}^{(1)}\right)^{N}\frac{\delta(r_{12})\delta(r_{22})}{e^{\imath\psi}r_{12}-e^{\imath\psi}r_{22}}d[r_{2}]dr_{1}$
(7.4)

for $\beta=4$. The differential operator has the explicit form

$D_{2,r_{2}}^{(1)}=\frac{\partial^{2}}{\partial r_{12}\partial
r_{22}}-\frac{1}{2}\frac{1}{r_{12}-r_{22}}\left(\frac{\partial}{\partial
r_{12}}-\frac{\partial}{\partial r_{22}}\right)\ .$ (7.5)

For the level density we have

$\displaystyle
R_{1}(x)=\displaystyle-\frac{1}{2\pi}\int\limits_{\mathbb{R}^{2}}\int\limits_{\mathbb{R}}\Phi_{0}(r)\det
r_{1}^{(N-1)/2}\exp\left(-\imath x{\rm Str\,}r\right){\rm
Str\,}r\frac{|r_{11}-r_{21}|}{(r_{11}-e^{\imath\psi}r_{2})^{2}(r_{21}-e^{\imath\psi}r_{2})^{2}}\times$
$\displaystyle\times\displaystyle\left(\Theta(r_{1})+\Theta(-r_{1})\right)\frac{1}{(N-2)!}\left(-e^{-\imath\psi}\frac{\partial}{\partial
r_{2}}\right)^{N-2}\delta(r_{2})d[r_{1}]dr_{2}$ (7.6)

for $\beta=1$ and

$\displaystyle
R_{1}(x)=\displaystyle-\frac{2}{\pi}\int\limits_{\mathbb{R}}\int\limits_{\mathbb{R}^{2}}\Phi_{0}(r)r_{1}^{2N+1}\exp\left(-\imath
x{\rm Str\,}r\right){\rm
Str\,}r\frac{e^{\imath\psi}r_{12}-e^{\imath\psi}r_{22}}{(r_{1}-e^{\imath\psi}r_{12})^{2}(r_{1}-e^{\imath\psi}r_{22})^{2}}\times$
$\displaystyle\times\frac{\det
e^{\imath\psi}r_{2}}{(2N+1)!}\left(4e^{-2\imath\psi}D_{2,r_{2}}^{(1)}\right)^{N}\frac{\delta(r_{12})\delta(r_{22})}{e^{\imath\psi}r_{12}-e^{\imath\psi}r_{22}}d[r_{2}]dr_{1}$
(7.7)

for $\beta=4$. The equations (7.4) to (7.7) comprise all level–densities for
arbitrary matrix ensembles invariant under orthogonal and unitary–symplectic
rotations. As probability densities which do not factorize are included, these
results considerably extend those obtained by orthogonal polynomials.

For higher order correlation functions we use the definition (2.3) and the
definition of the matrix Green’s function. With help of the quantities $L={\rm
diag\,}(L_{1},\ldots,L_{k})\in\\{\pm 1\\}^{k}$ and
$\widehat{L}=L\otimes\leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}_{2\tilde{\gamma}}$, this yields

$\displaystyle R_{k}(x)=\displaystyle
2^{2k(k-\tilde{\gamma})}\int\limits_{\mathbb{R}^{k_{1}}}\int\limits_{\mathbb{R}^{k_{2}}}\Phi_{0}(r)\underset{\epsilon\searrow
0}{\lim}\sum\limits_{L\in\\{\pm 1\\}^{k}}\prod\limits_{j=1}^{k}L_{j}\
\frac{I_{k}^{(\beta,N)}\left(\widehat{L}r\right)\exp\left(-\varepsilon{\rm
Str\,}\widehat{L}r\right)}{\left(2\pi\imath
e^{-\imath\psi\gamma_{1}}\right)^{k}}\times$
$\displaystyle\times\displaystyle\left.\left(\prod\limits_{j=1}^{k}-\frac{1}{2}\frac{\partial}{\partial
J_{j}}\right)\varphi_{k_{1}k_{2}}^{(\beta)}(-\imath
r,x^{(0)}+J)\right|_{J=0}\left|B_{k}^{(\beta)}(r_{1},e^{\imath\psi}r_{2})\right|d[r_{2}]d[r_{1}]+{\rm
b.t.}$ (7.8)

for analytic correlation functions. We extend this formula to all rotation
invariant ensembles by the universality of the integral kernel. First, we make
a limit of a uniformly convergent series of Schwartz–functions analytic in the
real components of its entries to every arbitrary Schwartz–function describing
a matrix ensemble. The Schwartz–functions are dense in a weak sense in the
sets of Lebesgue–integrable Functions $L^{p}$ and the tempered distributions.
Thus, we integrate Eq. (7.8) with an arbitrary Schwartz–function on
$\mathbb{R}^{k}$ and take the limit of a series of Schwartz–functions
describing the ensembles to a tempered distribution which completes the
extension.

##  8 Remarks and conclusions

We extended the method of the generalized Hubbard–Stratonovich transformation
to arbitrary orthogonally and unitary–symplectically invariant random matrix
ensembles. Due to a duality between ordinary and supersymmetric matrix spaces,
the integral for the $k$–point correlation function is over a superspace. This
integral was reduced to an eigenvalue integral for all probability densities,
including those which do not factorize. The results are in terms of the
characteristic function. Thus, the characteristic function has to be
calculated for the ensemble in question. Since the matrix Bessel–functions of
the ordinary orthogonal and unitary–symplectic group [39, 29, 40] and, thus,
the supermatrix Bessel–functions of ${\rm UOSp\,}(2k/2k)$ are not known
explicitly beyond $k=1$, we can not further simplify our results. However, we
found the previously unknown determinantal structure of the Berezinian of
${\rm UOSp\,}(2k/2k)$.

Up to the restriction $N\geq k_{1}$, formula (7.8) is exact for every $k$, $N$
and rotation invariant ensemble. Thus, it can serve not only as starting point
for universality considerations [7], but for all other studies.

The expressions for the supersymmetric Ingham–Siegel integrals (5.20), (5.21)
and (5.22) confirm the equivalence of the superbosonization formula [20, 11,
12] with our derivation. A work for a proof of this equivalence for all
$\beta$’s is in progress. The comparison of the superbosonization formula [12,
11] with Eq. (5.1) shows that the crucial difference lies in the integration
domain. However, the Dirac–distribution and the partial derivatives in the
fermionic part imply a representation as a contour integral which is
equivalent to the compact space used in the superbosonization formula.

## Acknowledgements

We thank H. Kohler for clarifying remarks on relation between the ordinary
matrix Bessel–functions and the Jack–polynomials as well as on the Sekiguchi
differential operators. We are also grateful to S. Mandt, H.-J. Sommers and
M.R. Zirnbauer for fruitful discussions. A big thank you goes to P. Heinzner
and E. Vishnyakova for helpful advice on the Paley–Wiener theorem. We thank
the referee for helpful remarks. We acknowledge financial support from the
Deutsche Forschungsgemeinschaft within Sonderforschungsbereich Transregio 12
“Symmetries and Universality in Mesoscopic Systems” (M.K. and T.G.) and from
Det Svenska Vetenskapsrådet (J.G.).

##  Appendix A Circularity of the supertrace for rectangular supermatrices

The circularity for rectangular matrices of pure commuting entries or
anticommuting entries was derived by Berezin [18]. Since we have not found the
general theorem for arbitrary rectangular supermatrices, we give the trivial
statement.

###### Statement A.1

Let the matrices $V_{1}$ and $V_{2}$ be the same as in Eq. (4.23). Then, we
have

${\rm Str\,}V_{1}V_{2}={\rm Str\,}V_{2}V_{1}$ (1.1)

Derivation:
We recall the circularity of the trace for rectangular matrices of commuting
elements $\tr A_{1}A_{2}=\tr A_{2}A_{1}$ and its anticommuting analogue $\tr
B_{1}B_{2}=-\tr B_{2}B_{1}$ which has been proven by Berezin [18]. We make the
simple calculation

$\displaystyle{\rm Str\,}V_{1}V_{2}$ $\displaystyle=$ $\displaystyle\tr
A_{1}A_{2}+\tr B_{1}C_{2}-\tr C_{1}B_{2}-\tr D_{1}D_{2}$ (1.2)
$\displaystyle=$ $\displaystyle\tr A_{2}A_{1}-\tr C_{2}B_{1}+\tr
B_{2}C_{1}-\tr D_{2}D_{1}$ $\displaystyle=$ $\displaystyle{\rm
Str\,}V_{2}V_{1}$

$\square$

For our purposes we must prove

$\tr(V^{\dagger}V)^{m}={\rm Str\,}(VV^{\dagger})^{m}\ .$ (1.3)

We define $V_{1}=V^{\dagger}$ and $V_{2}=(VV^{\dagger})^{m-1}V$ and get
$a=2k$, $b=2k$, $c=\gamma_{2}N$ and $d=0$. Applying corollary A.1 and
reminding that $\tr A={\rm Str\,}A$ for a matrix of commuting elements and
identification with the Boson–Boson block, we have the desired result (1.3).

##  Appendix B A matrix–Bessel version of the Sekiguchi differential operator

We derive a version for the Sekiguchi differential operator for the ordinary
matrix Bessel–functions $\varphi_{N}^{(\beta)}(y,x)$ on the connection between
the Jack–polynomials and the ordinary matrix Bessel–functions.

The Sekiguchi differential operator is defined as [28]

$\displaystyle
D_{Nz}(u,\beta)=\Delta_{N}^{-1}(z)\det\left[z_{a}^{N-b}\left(z_{a}\frac{\partial}{\partial
z_{a}}+(N-b)\frac{\beta}{2}+u\right)\right]_{1\leq a,b\leq N}=$
$\displaystyle=\Delta_{N}^{-1}(z)\det\left[\frac{\beta}{2}\left(z_{a}\frac{\partial}{\partial
z_{a}}+u\right)z_{a}^{N-b}+\left(1-\frac{\beta}{2}\right)z_{a}^{N-b}\left(z_{a}\frac{\partial}{\partial
z_{a}}+u\right)\right]_{1\leq a,b\leq N}\ .$ (2.1)

Here, $u$ is a boost and the expansion parameter to generate the elementary
polynomials in the Cherednik operators, for more explicit information see Ref.
[41]. Let $J_{N}^{(\beta)}(n,z)$ the Jack–polynomial with the partition
$n_{1}\geq\ldots\geq n_{N}$ and the standard parameter
$\alpha=\frac{2}{\beta}$ in Macdonald’s [42] notation. The Jack–polynomials
are eigenfunctions with respect to $D_{Nz}(u,\beta)$

$D_{Nz}(u,\beta)J_{N}^{(\beta)}(n,z)=\prod\limits_{a=1}^{N}\left[n_{a}+(N-a)\frac{\beta}{2}+u\right]J_{N}^{(\beta)}(n,z)\
.$ (2.2)

The aim is to find a similar differential operator for the ordinary matrix
Bessel–function $\varphi_{N}^{(\beta)}(y,x)$ such that

$\displaystyle
D_{Nx}^{(\beta)}(B)\varphi_{N}^{(\beta)}\left(\frac{y}{\gamma_{2}},x\right)$
$\displaystyle=$
$\displaystyle\prod\limits_{a=1}^{N}\imath\left(y_{a}+B\right)\varphi_{N}^{(\beta)}\left(\frac{y}{\gamma_{2}},x\right)=$
(2.3) $\displaystyle=$
$\displaystyle{\det}^{1/\gamma_{2}}\imath(y+B\leavevmode\hbox{\small
1\kern-3.8pt\normalsize
1}_{\gamma_{2}N})\varphi_{N}^{(\beta)}\left(\frac{y}{\gamma_{2}},x\right).$

###### Statement B.1

The differential operator which fulfils Eq. (2.3) is

$D_{Nx}^{(\beta)}(B)=\Delta_{N}^{-1}(x)\det\left[x_{a}^{N-b}\left(\frac{\partial}{\partial
x_{a}}+(N-b)\frac{\beta}{2}\frac{1}{x_{a}}+\imath B\right)\right]_{1\leq
a,b\leq N}\ .$ (2.4)

Derivation:
Kohler [43] has presented a connection between the Jack–polynomials and the
matrix Bessel–functions. Let

$z_{a}=e^{\imath\frac{2\pi}{L}x_{a}}\ \ \ {\rm and}\ \ \
n_{a}=\frac{L}{2\pi}y_{a}-\left(\frac{N+1}{2}-a\right)\frac{\beta}{2}$ (2.5)

then it is true

$\varphi_{N}^{(\beta)}\left(\frac{y}{\gamma_{2}},x\right)=\underset{L\to\infty}{\rm
lim}\left(\frac{\Delta_{N}(z)}{\Delta_{N}(x)\Delta_{N}(y)}\right)^{\beta/2}\prod\limits_{a=1}^{N}z_{a}^{-\beta(N-1)/4}J_{N}^{(\beta)}(n,z)\
.$ (2.6)

We expand the determinant in Eq. (2.1) and have

$\displaystyle D_{Nz}(u,\beta)=$
$\displaystyle=\Delta_{N}^{-1}(z)\sum\limits_{m\in\\{0,1\\}^{N}}\prod\limits_{a=1}^{N}\left[\frac{\beta}{2}\left(z_{a}\frac{\partial}{\partial
z_{a}}+u\right)\right]^{m_{a}}\Delta_{N}(z)\prod\limits_{a=1}^{N}\left[\left(1-\frac{\beta}{2}\right)\left(z_{a}\frac{\partial}{\partial
z_{a}}+u\right)\right]^{1-m_{a}}.$ (2.7)

Using the substitution (2.5) and

$\widetilde{\Delta}(x)=\prod\limits_{1\leq a<b\leq
N}2\imath\sin\left(\frac{\pi}{L}(x_{a}-x_{b})\right)\exp\left(\imath\pi\frac{x_{a}+x_{b}}{L}\right)\
,$ (2.8)

we consider the limit

$\displaystyle\underset{L\to\infty}{\lim}\left(\frac{2\pi\imath}{L}\right)^{N}D_{Nz}(u,\beta)=$
$\displaystyle=\underset{L\to\infty}{\lim}\frac{1}{\widetilde{\Delta}(x)}\sum\limits_{m\in\\{0,1\\}^{N}}\prod\limits_{a=1}^{N}\left[\frac{\beta}{2}\left(\frac{\partial}{\partial
x_{a}}+\imath\frac{2\pi
u}{L}\right)\right]^{m_{a}}\widetilde{\Delta}(x)\times$
$\displaystyle\times\prod\limits_{j=1}^{N}\left[\left(1-\frac{\beta}{2}\right)\left(\frac{\partial}{\partial
x_{a}}+\imath\frac{2\pi u}{L}\right)\right]^{1-m_{a}}=$
$\displaystyle=\Delta_{N}^{-1}(x)\sum\limits_{m\in\\{0,1\\}^{N}}\prod\limits_{a=1}^{N}\left[\frac{\beta}{2}\left(\frac{\partial}{\partial
x_{a}}+\imath
B\right)\right]^{m_{a}}\Delta_{N}(x)\left[\left(1-\frac{\beta}{2}\right)\left(\frac{\partial}{\partial
x_{a}}+\imath B\right)\right]^{1-m_{a}}=$
$\displaystyle=\Delta_{N}^{-1}(x)\det\left[\frac{\beta}{2}\left(\frac{\partial}{\partial
x_{a}}+\imath
B\right)x_{a}^{N-b}+\left(1-\frac{\beta}{2}\right)x_{a}^{N-b}\left(\frac{\partial}{\partial
x_{a}}+\imath B\right)\right]_{1\leq a,b\leq N}=$
$\displaystyle=\Delta_{N}^{-1}(x)\det\left[x_{a}^{N-b}\left(\frac{\partial}{\partial
x_{a}}+(N-b)\frac{\beta}{2}\frac{1}{x_{a}}+\imath B\right)\right]_{1\leq
a,b\leq N}\ .$ (2.9)

Here, we defined a boost $B=\underset{L\to\infty}{\lim}2\pi u/L$ . The
eigenvalue in Eq. (2.2) is in the limit

$\underset{L\to\infty}{\lim}\left(\frac{2\pi\imath}{L}\right)^{N}\prod\limits_{a=1}^{N}\left[n_{a}+(N-a)\frac{\beta}{2}+u\right]=\prod\limits_{a=1}^{N}\imath\left(y_{a}+B\right)={\det}^{1/\gamma_{2}}\imath(y+B\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{\gamma_{2}N})\ .$ (2.10)

We assume that Eq. (2.6) is a uniformly convergent limit. Thus, we combine
(2.6), (2.9) and (2.10) with Eq. (2.2) and find Eq. (2.4). $\square$

Indeed, $D_{Nx}^{(\beta)}(B)$ is for the unitary case, $\beta=2$,

$D_{Nx}^{(2)}(B)=\Delta_{N}^{-1}(x)\prod\limits_{a=1}^{N}\left(\frac{\partial}{\partial
x_{a}}+\imath B\right)\Delta_{N}(x)\ .$ (2.11)

##  Appendix C Calculation of the supersymmetric Ingham–Siegel integral

In C.1, we compute the Ingham–Siegel integral. We derive the statements 5.1
and 5.2 in C.2 and C.3, respectively.

###  C.1 Decomposition of the Boson–Boson and Fermion–Fermion block
integration

We split $\sigma$ in its Boson–Fermion block structure

$\mathfrak{p}\sigma=\left[\begin{array}[]{cc}\sigma_{1}&e^{-\imath\psi/2}\sigma_{\eta}^{\dagger}\\\
e^{-\imath\psi/2}\sigma_{\eta}&e^{-\imath\psi}\sigma_{2}\end{array}\right]\ .$
(3.1)

The following calculation must be understand in a weak sense. We first
integrate over a conveniently integrable function and, then, perform the
integral transformations. Hence, we understand $I_{k}^{(\beta,N)}$ as a
distribution where we must fix the underlying set of test–functions. For our
purposes, we need Schwartz–functions analytic in the real independent
variables.

Since the superdeterminant of
$\mathfrak{p}\left(\sigma+\imath\varepsilon\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{4k}\right)$ is

${\rm
Sdet\,}\mathfrak{p}\sigma^{+}=\frac{\det\left(\sigma_{1}+\imath\varepsilon\leavevmode\hbox{\small
1\kern-3.8pt\normalsize
1}_{\tilde{k}}\right)}{\det\left[e^{-\imath\psi}\sigma_{2}+\imath\varepsilon\leavevmode\hbox{\small
1\kern-3.8pt\normalsize
1}_{\tilde{k}}-e^{-\imath\psi}\sigma_{\eta}\left(\sigma_{1}+\imath\varepsilon\leavevmode\hbox{\small
1\kern-3.8pt\normalsize
1}_{\tilde{k}}\right)^{-1}\sigma_{\eta}^{\dagger}\right]}$ (3.2)

we shift $\sigma_{2}$ by analytic continuation to
$\sigma_{2}+\sigma_{\eta}\left(\sigma_{1}+\imath\varepsilon\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{\tilde{k}}\right)^{-1}\sigma_{\eta}^{\dagger}$ and
obtain

$\displaystyle I_{k}^{(\beta,N)}(\rho)$ $\displaystyle=$
$\displaystyle\int\limits_{\Sigma_{-\psi}^{0}(\beta,k)}\displaystyle{\rm
exp}\left(-\imath\tr r_{1}\sigma_{1}+\imath\tr
r_{2}\sigma_{2}+\imath\tr\left[r_{2}\sigma_{\eta}\left(\sigma_{1}+\imath\varepsilon\leavevmode\hbox{\small
1\kern-3.8pt\normalsize
1}_{\tilde{k}}\right)^{-1}\sigma_{\eta}^{\dagger}\right]\right)\times$ (3.3)
$\displaystyle\times$ $\displaystyle\exp\left(\varepsilon{\rm
Str\,}r\right)\left[\frac{{\det}\left(e^{-\imath\psi}\sigma_{2}+\imath\varepsilon\leavevmode\hbox{\small
1\kern-3.8pt\normalsize
1}_{\tilde{k}}\right)}{{\det}\left(\sigma_{1}+\imath\varepsilon\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{\tilde{k}}\right)}\right]^{N/\gamma_{1}}d[\sigma]\
.$

An integration over the Grassmann variables yields

$\displaystyle I_{k}^{(\beta,N)}(\rho)$ $\displaystyle=$
$\displaystyle\left(\frac{-\imath\tilde{\gamma}}{2\pi}\right)^{k_{1}k_{2}}\exp\left(\varepsilon{\rm
Str\,}r\right){\det}^{k}r_{2}\times$ (3.4) $\displaystyle\times$
$\displaystyle\int\limits_{{\rm Herm\,}(\beta,k_{1})}\exp\left(-\imath\tr
r_{1}\sigma_{1}\right){\det}\left(\sigma_{1}+\imath\varepsilon\leavevmode\hbox{\small
1\kern-3.8pt\normalsize
1}_{\tilde{k}}\right)^{-N/\gamma_{1}-k}d[\sigma_{1}]\times$
$\displaystyle\times$ $\displaystyle\int\limits_{{\rm
Herm\,}(4/\beta,k_{2})}\exp\left(\imath\tr
r_{2}\sigma_{2}\right){\det}\left(e^{-\imath\psi}\sigma_{2}+\imath\varepsilon\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{\tilde{k}}\right)^{N/\gamma_{1}}d[\sigma_{2}]\ .$

With help of Eq. (5.3) we have

$\displaystyle I_{k}^{(\beta,N)}(\rho)$ $\displaystyle=$
$\displaystyle\imath^{-k_{2}N}G_{Nk_{1}}^{(\beta)}\left(-\frac{\tilde{\gamma}}{2\pi}\right)^{k_{1}k_{2}}\displaystyle{\det}^{\kappa}r_{1}\Theta(r_{1})\exp\left(-e^{\imath\psi}\varepsilon\tr
r_{2}\right)\times$ (3.5) $\displaystyle\times$
$\displaystyle{\det}^{k}r_{2}\int\limits_{{\rm
Herm\,}(4/\beta,k_{2})}\exp\left(\imath\tr
r_{2}\sigma_{2}\right){\det}^{N/\gamma_{1}}\left(e^{-\imath\psi}\sigma_{2}+\imath\varepsilon\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{\tilde{k}}\right)d[\sigma_{2}]\ .$

The remaining integral over the Fermion–Fermion block $\sigma_{2}$,

$\displaystyle\mathfrak{I}(r_{2})=\exp\left(-e^{\imath\psi}\varepsilon\tr
r_{2}\right)\int\limits_{{\rm Herm\,}(4/\beta,k_{2})}\exp\left(\imath\tr
r_{2}\sigma_{2}\right){\det}^{N/\gamma_{1}}\left(\sigma_{2}+\imath
e^{\imath\psi}\varepsilon\leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}_{\tilde{k}}\right)d[\sigma_{2}]\ ,$ (3.6)

is up to a constant a differential operator with respect to $r_{2}$ times the
Dirac–distribution of $r_{2}$ because the determinant term is for
$\beta\in\\{1,2\\}$ a polynomial in $\sigma_{2}$ and for $\beta=4$ we use
Cramers–degeneracy. We give several representations of this distribution.

We first start with an eigenvalue–angle decomposition of
$\sigma_{2}=Us_{2}U^{\dagger}$ where $s_{2}$ is diagonal and $U\in{\rm
U\,}^{(4/\beta)}(k_{2})$. Integrating over the group ${\rm
U\,}^{(4/\beta)}(k_{2})$, Eq. (3.6) becomes

$\displaystyle\mathfrak{I}(r_{2})$ $\displaystyle=$
$\displaystyle\displaystyle\exp\left(-e^{\imath\psi}\varepsilon\tr
r_{2}\right)g_{k_{2}}^{(4/\beta)}\times$ (3.7) $\displaystyle\times$
$\displaystyle\int\limits_{\mathbb{R}^{k_{2}}}\varphi_{k_{2}}^{(4/\beta)}(r_{2},s_{2}){\det}^{N/\gamma_{1}}\left(s_{2}+\imath
e^{\imath\psi}\varepsilon\leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}_{\tilde{k}}\right)|\Delta_{k_{2}}(s_{2})|^{4/\beta}d[s_{2}].$

For more information about the ordinary matrix Bessel–function

$\varphi_{k_{2}}^{(4/\beta)}(r_{2},s_{2})=\int\limits_{{\rm
U\,}^{(4/\beta)}(k_{2})}\exp\left(\imath\tr
r_{2}Us_{2}U^{\dagger}\right)d\mu(U)$ (3.8)

with normalized Haar–measure $d\mu(U)$ see in Ref. [39, 40]. The constant
$g_{n}^{(\beta)}$ is defined by

$\int\limits_{{\rm
Herm\,}(\beta,n)}f(H)d[H]=g_{n}^{(\beta)}\int\limits_{\mathbb{R}^{n}}f(E)|\Delta_{n}(E)|^{\beta}d[E]$
(3.9)

independent of a sufficiently integrable function $f$ which is invariant under
the action of ${\rm U\,}^{(\beta)}(n)$. The Gaussian distribution is such a
function. For the left hand side we obtain

$\int\limits_{{\rm Herm\,}(\beta,n)}\exp\left(-\tr
H^{2}\right)d[H]=\gamma_{2}^{-n(2n-1)/2}2^{-\beta n(n-1)/4}\pi^{n/2+\beta
n(n-1)/4}\ .$ (3.10)

The integral on the right hand side is equal to

$\int\limits_{\mathbb{R}^{n}}\exp\left(-\gamma_{2}\sum\limits_{j=1}^{n}E_{j}^{2}\right)|\Delta_{n}(E)|^{\beta}d[E]=\left\\{\begin{array}[]{ll}2^{-n(n-5)/4}\prod\limits_{j=1}^{n}\Gamma\left(\frac{j}{2}+1\right)&,\
\beta=1,\\\
2^{-n(n-1)/2}\pi^{n/2}\prod\limits_{j=1}^{n}\Gamma\left(j+1\right)&,\
\beta=2,\\\
2^{-n(2n-1/2)}\pi^{n/2}\prod\limits_{j=1}^{n}\Gamma\left(2j+1\right)&,\
\beta=4,\end{array}\right.$ (3.11)

see Mehta’s book [15]. Thus, we have

$g_{n}^{(\beta)}=\frac{1}{n!}\prod\limits_{j=1}^{n}\frac{\pi^{\beta(j-1)/2}\Gamma\left(\beta/2\right)}{\Gamma\left(\beta
j/2\right)}\ .$ (3.12)

This constant is the quotient of the volumes of the permutation group $S(n)$
and of the flag manifold ${\rm U\,}^{(\beta)}(n)/[{\rm U\,}^{(\beta)}(1)]^{n}$
with the volume element defined as in Ref. [44] denoted by ${\rm Vol}_{B}$.

We plug the differential operator of B (2.3) into Eq. (3.7) and have

$\displaystyle\mathfrak{I}(r_{2})$ $\displaystyle=$ $\displaystyle
g_{k_{2}}^{(4/\beta)}\exp\left(-e^{\imath\psi}\varepsilon\tr
r_{2}\right)(\imath\gamma_{1})^{-k_{2}N}\times$ (3.13) $\displaystyle\times$
$\displaystyle\displaystyle\left[D_{k_{2}r_{2}}^{(4/\beta)}\left(\imath
e^{\imath\psi}\gamma_{1}\varepsilon\right)\right]^{N}\int\limits_{\mathbb{R}^{k_{2}}}\phi_{k_{2}}^{(4/\beta)}(r_{2},s_{2})|\Delta_{k_{2}}(s_{2})|^{4/\beta}d[s_{2}]\
.$

The integration over the eigenvalues leads to the Dirac–distribution

$\displaystyle\mathfrak{I}(r_{2})$ $\displaystyle=$
$\displaystyle\displaystyle\left(\frac{2\pi}{\gamma_{1}}\right)^{k_{2}}\left(\frac{\pi}{\gamma_{1}}\right)^{2k_{2}(k_{2}-1)/\beta}\frac{\exp\left(-e^{\imath\psi}\varepsilon\tr
r_{2}\right)}{g_{k_{2}}^{(4/\beta)}}(\imath\gamma_{1})^{-k_{2}}\times$ (3.14)
$\displaystyle\times$
$\displaystyle\displaystyle\left[D_{k_{2}r_{2}}^{(4/\beta)}\left(\imath
e^{\imath\psi}\gamma_{1}\varepsilon\right)\right]^{N}\frac{\delta(r_{2})}{|\Delta_{k_{2}}(r_{2})|^{4/\beta}}$

and we find the representation for the supersymmetric Ingham–Siegel integral
(5.7).

###  C.2 Derivation of statement 5.1

The boost $\imath e^{\imath\psi}\varepsilon$ in the determinant can simply be
shifted away because of

$D_{k_{2}r_{2}}^{(4/\beta)}\left(\imath
e^{\imath\psi}\gamma_{1}\varepsilon\right)\exp\left(\varepsilon
e^{\imath\psi}\tr r_{2}\right)=\exp\left(\varepsilon e^{\imath\psi}\tr
r_{2}\right)D_{k_{2}r_{2}}^{(4/\beta)}(0)=\exp\left(\varepsilon
e^{\imath\psi}\tr r_{2}\right)D_{k_{2}r_{2}}^{(4/\beta)}$ (3.15)

and Eq. (3.14). Let $\mathfrak{S}$ the set of ${\rm
U\,}^{(4/\beta)}(k_{2})$–invariant Schwartz–functions on ${\rm
Herm\,}(4/\beta,k_{2})\rightarrow\mathbb{C}$. The ordinary matrix
Bessel–functions are complete and orthogonal in $\mathfrak{S}$ with the
sesquilinear scalar product

$\langle
f|f^{\prime}\rangle=\int\limits_{\mathbb{R}^{k_{2}}}f^{*}(x)f^{\prime}(x)|\Delta_{k_{2}}(x)|^{4/\beta}d[x]\
.$ (3.16)

The completeness and the orthogonality are

$\displaystyle\langle\phi_{k_{2}}^{(4/\beta)}(x)|\phi_{k_{2}}^{(4/\beta)}(x^{\prime})\rangle$
$\displaystyle=$
$\displaystyle\int\limits_{\mathbb{R}^{k_{2}}}|\phi_{k_{2}}^{(4/\beta)}(y)\rangle\langle\phi_{k_{2}}^{(4/\beta)}(y)|\
|\Delta_{k_{2}}(y)|^{4/\beta}d[y]=$ (3.17) $\displaystyle=$
$\displaystyle\int\limits_{\mathbb{R}^{k_{2}}}\phi_{k_{2}}^{(4/\beta)}(y,x)\phi_{k_{2}}^{(4/\beta)*}(y,x^{\prime})|\Delta_{k_{2}}(y)|^{4/\beta}d[y]=$
$\displaystyle=$ $\displaystyle C_{k}^{(\beta)}\frac{1}{k_{2}!}\sum_{p\in
S(k_{2})}\frac{\prod\limits_{j=1}^{k_{2}}\delta(x_{j}-x_{p(j)}^{\prime})}{|\Delta_{k_{2}}(x)|^{2/\beta}|\Delta_{k_{2}}(x^{\prime})|^{2/\beta}}$

where $S(n)$ is the permutation group of $n$ elements. We defined the constant

$C_{k}^{(\beta)}=\left(\frac{2\pi}{\gamma_{1}}\right)^{k_{2}}\left(\frac{\pi}{\gamma_{1}}\right)^{2k_{2}(k_{2}-1)/\beta}\left(g_{k_{2}}^{(4/\beta)}\right)^{-2}\
.$ (3.18)

Thus, we write $D_{k_{2}r_{2}}^{(4/\beta)}$ in the Bessel–function basis

$\displaystyle\qquad\qquad D_{k_{2}}^{(4/\beta)}$ $\displaystyle=$
$\displaystyle{C_{k}^{(\beta)}\
}^{-2}\int\limits_{\mathbb{R}^{k_{2}}}|\phi_{k_{2}}^{(4/\beta)}(y)\rangle\langle\phi_{k_{2}}^{(4/\beta)}(y)|\
|\Delta_{k_{2}}(y)|^{4/\beta}d[y]\times$ (3.19) $\displaystyle\times$
$\displaystyle
D_{k_{2}x}^{(4/\beta)}\int\limits_{\mathbb{R}^{k_{2}}}|\phi_{k_{2}}^{(4/\beta)}(y^{\prime})\rangle\langle\phi_{k_{2}}^{(4/\beta)}(y^{\prime})|\
|\Delta_{k_{2}}(y^{\prime})|^{4/\beta}d[y^{\prime}]=$ $\displaystyle=$
$\displaystyle{C_{k}^{(\beta)}\
}^{-1}\int\limits_{\mathbb{R}^{k_{2}}}{\det}(i\gamma_{1}y)^{1/\gamma_{1}}\phi_{k_{2}}^{(4/\beta)}(y,x)\phi_{k_{2}}^{(4/\beta)*}(y,x^{\prime})|\Delta_{k_{2}}(y)|^{4/\beta}d[y]$

with the action on a function $f\in\mathfrak{S}$

$\displaystyle D_{k_{2}}^{(4/\beta)}|f\rangle$ $\displaystyle=$
$\displaystyle{C_{k}^{(\beta)}\
}^{-1}\int\limits_{\mathbb{R}^{k_{2}}}\int\limits_{\mathbb{R}^{k_{2}}}{\det}(i\gamma_{1}y)^{1/\gamma_{1}}\phi_{k_{2}}^{(4/\beta)}(y,x)\phi_{k_{2}}^{(4/\beta)*}(y,x^{\prime})f(x^{\prime})\times$
(3.20) $\displaystyle\times$
$\displaystyle|\Delta_{k_{2}}(x^{\prime})|^{4/\beta}|\Delta_{k_{2}}(y)|^{4/\beta}d[x^{\prime}]d[y]\
.$

Due to this representation of the Sekiguchi differential operator analog,
$\imath^{k_{2}}D_{k_{2}}^{(4/\beta)}$ is symmetric with respect to the scalar
product (3.16)

$\langle
f|\imath^{k_{2}}D_{k_{2}}^{(4/\beta)}|f^{\prime}\rangle=\langle\imath^{k_{2}}D_{k_{2}}^{(4/\beta)}f|f^{\prime}\rangle\
.$ (3.21)

Let $L$ be a real number. Then, we easily see with help of Eq. (2.4)

$D_{k_{2}x}^{(4/\beta)}\det
x^{L/\gamma_{1}}=\prod\limits_{b=1}^{k_{2}}\left(L+\frac{2}{\beta}b-\frac{2}{\beta}\right)\det
x^{(L-1)/\gamma_{1}}\ .$ (3.22)

Since the property (3.21), we obtain for a function $f\in\mathfrak{S}$

$\displaystyle\int\limits_{\mathbb{R}^{k_{2}}}\det
x^{L/\gamma_{1}}|\Delta_{k_{2}}(x)|^{4/\beta}D_{k_{2}x}^{(4/\beta)}f(x)d[x]=$
(3.23) $\displaystyle=$
$\displaystyle(-1)^{k_{2}}\int\limits_{\mathbb{R}^{k_{2}}}f(x)|\Delta_{k_{2}}(x)|^{4/\beta}D_{k_{2}x}^{(4/\beta)}\det
x^{L/\gamma_{1}}d[x]=$ $\displaystyle=$
$\displaystyle(-1)^{k_{2}}\prod\limits_{b=1}^{k_{2}}\left(L+\frac{2}{\beta}b-\frac{2}{\beta}\right)\int\limits_{\mathbb{R}^{k_{2}}}f(x)|\Delta_{k_{2}}(x)|^{4/\beta}\det
x^{(L-1)/\gamma_{1}}d[x]\ .$

The boundary terms of the partial integration do not appear because $f$ is a
Schwartz–function and $D_{k_{2}x}^{(4/\beta)}$ has the representation (3.19).

Let $F$ and $f$ be the functions of statement 5.1. Then, we calculate

$\displaystyle\int\limits_{\mathbb{R}^{k_{2}}}\int\limits_{{\rm
Herm\,}(4/\beta,k_{2})}F(r_{2}){\det}^{k}r_{2}|\Delta_{k_{2}}(r_{2})|^{4/\beta}\exp\left(\imath\tr
r_{2}\sigma_{2}\right){\det}^{N/\gamma_{1}}\left(e^{-\imath\psi}\sigma_{2}+\imath\varepsilon\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{\tilde{k}}\right)d[\sigma_{2}]d[r_{2}]=$
$\displaystyle=\int\limits_{\mathbb{R}^{k_{2}}}\int\limits_{{\rm
Herm\,}(4/\beta,k_{2})}f(r_{2}){\det}^{N/\gamma_{1}}r_{2}|\Delta_{k_{2}}(r_{2})|^{4/\beta}\exp\left(\imath\tr
r_{2}\sigma_{2}\right)\times$
$\displaystyle\times{\det}^{N/\gamma_{1}}\left(e^{-\imath\psi}\sigma_{2}+\imath\varepsilon\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{\tilde{k}}\right)d[\sigma_{2}]d[r_{2}]=$
$\displaystyle=\left(\frac{-\imath
e^{-\imath\psi}}{\gamma_{1}}\right)^{k_{2}N}g_{k_{2}}^{(4/\beta)}\int\limits_{\mathbb{R}^{k_{2}}}\int\limits_{\mathbb{R}^{k_{2}}}f(r_{2})\exp\left(\varepsilon
e^{\imath\psi}\tr r_{2}\right)|\Delta_{k_{2}}(r_{2})|^{4/\beta}\times$
$\displaystyle\times{\det}^{N/\gamma_{1}}s_{2}|\Delta_{k_{2}}(s_{2})|^{4/\beta}\left(D_{k_{2}s_{2}}^{(4/\beta)}\right)^{N}\phi_{k_{2}}^{(4/\beta)}(r_{2},s_{2})d[s_{2}]d[r_{2}]=$
$\displaystyle=(\imath
e^{-\imath\psi})^{k_{2}N}g_{k_{2}}^{(4/\beta)}\prod\limits_{a=1}^{N}\prod\limits_{b=1}^{k_{2}}\left(\frac{a}{\gamma_{1}}+\frac{b-1}{\gamma_{2}}\right)\times$
$\displaystyle\times\int\limits_{\mathbb{R}^{k_{2}}}\int\limits_{\mathbb{R}^{k_{2}}}f(r_{2})\exp\left(\varepsilon
e^{\imath\psi}\tr
r_{2}\right)|\Delta_{k_{2}}(r_{2})|^{4/\beta}|\Delta_{k_{2}}(s_{2})|^{4/\beta}\phi_{k_{2}}^{(4/\beta)}(r_{2},s_{2})d[s_{2}]d[r_{2}]=$
$\displaystyle=\left(\frac{2\pi}{\gamma_{1}}\right)^{k_{2}}\left(\frac{\pi}{\gamma_{1}}\right)^{2k_{2}(k_{2}-1)/\beta}\frac{\left(\imath
e^{-\imath\psi}\right)^{k_{2}N}}{g_{k_{2}}^{(4/\beta)}\gamma_{1}^{k_{2}N}}\prod_{j=0}^{k_{2}-1}\frac{\Gamma\left(N+1+2j/\beta\right)}{\Gamma\left(1+2j/\beta\right)}f(0)\
.$ (3.24)

The second equality in Eq. (5.13) is true because of

$f(0)=\left.\prod\limits_{j=1}^{k_{2}}\frac{1}{\left(N-k_{1}\right)!}\left(\frac{\partial}{\partial
r_{j2}}\right)^{N-k_{1}}\left[f(r_{2})\exp\left(\varepsilon e^{\imath\psi}\tr
r_{2}\right)\det r_{2}^{N/\gamma_{1}-k}\right]\right|_{r_{2}=0}.$ (3.25)

The function in the bracket is $F$ times the exponential term ${\rm
exp}\left(\varepsilon e^{\imath\psi}\tr r_{2}\right)$.

###  C.3 Derivation of statement 5.2

We have to show

$\displaystyle\int\limits_{{\rm Herm\,}(4/\beta,k_{2})}\int\limits_{{\rm
Herm\,}(4/\beta,k_{2})}F(\rho_{2}){\det}^{k}\rho_{2}\exp\left(\imath\tr\rho_{2}\sigma_{2}\right){\det}^{N/\gamma_{1}}\sigma_{2}d[\sigma_{2}]d[\rho_{2}]\sim$
(3.26) $\displaystyle\sim$
$\displaystyle\int\limits_{\mathbb{R}^{k_{2}}}F(r_{2})\prod\limits_{j=1}^{k}\left(-\frac{\partial}{\partial
r_{j2}}\right)^{N-2/\beta}\delta(r_{j2})d[r_{2}]$

for every rotation invariant Schwartz–function $F:{\rm
Herm\,}(4/\beta,k_{2})\rightarrow\mathbb{C}$ and $\beta\in\\{1,2\\}$. Due to

$\displaystyle\int\limits_{{\rm Herm\,}(4/\beta,k_{2})}\exp\left(\imath\tr
r_{2}\sigma_{2}\right){\det}\sigma_{2}^{N/\gamma_{1}}d[\sigma_{2}]$
$\displaystyle\sim$
$\displaystyle\int\limits_{\mathbb{R}}\int\limits_{\mathbb{R}^{4(k_{2}-1)/\beta}}y^{N}{\rm
exp}\left[\imath r_{k_{2}2}\tr(y\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{\tilde{\gamma}}+v^{\dagger}v)\right]d[v]dy\times$
(3.27) $\displaystyle\times$ $\displaystyle\int\limits_{{\rm
Herm\,}(4/\beta,k_{2}-1)}\exp\left(\imath\tr\tilde{r}_{2}\tilde{\sigma}_{2}\right){\det}\tilde{\sigma}_{2}^{(N+2/\beta)/\gamma_{1}}d[\tilde{\sigma}_{2}]$

with the decompositions $r_{2}={\rm
diag\,}\left(\tilde{r}_{2},r_{k_{2}2}\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{\tilde{\gamma}}\right)$ and

$\sigma_{2}=\left[\begin{array}[]{cc}\tilde{\sigma}_{2}&v\\\
v^{\dagger}&y\leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}_{\tilde{\gamma}}\end{array}\right]\ ,$ (3.28)

we make a complete induction. Thus, we reduce the derivation to

$\int\limits_{\mathbb{R}}\int\limits_{\mathbb{R}}\int\limits_{\mathbb{R}^{4(k_{2}-1)/\beta}}f(x)x^{k_{1}}y^{N}{\rm
exp}\left[\imath
x\tr(y+v^{\dagger}v)\right]d[v]dydx\sim\int\limits_{\mathbb{R}}f(x)\frac{\partial^{N-2/\beta}}{\partial
x^{N-2/\beta}}\delta(x)d[x]$ (3.29)

where $f:\mathbb{R}\rightarrow\mathbb{C}$ is a Schwartz–function. The function

$\tilde{f}(y)=\int\limits_{\mathbb{R}}f(x)x^{k_{1}}\exp\left(\imath
xy\right)dx$ (3.30)

is also a Schwartz–function. Hence, we compute

$\displaystyle\int\limits_{\mathbb{R}}\int\limits_{\mathbb{R}}\int\limits_{\mathbb{R}^{4(k_{2}-1)/\beta}}f(x)x^{k_{1}}y^{N}{\rm
exp}\left[\imath x\tr(y+v^{\dagger}v)\right]d[v]dydx=$ (3.31) $\displaystyle=$
$\displaystyle\int\limits_{\mathbb{R}}\int\limits_{\mathbb{R}^{4(k_{2}-1)/\beta}}\tilde{f}\left[\tr(y+v^{\dagger}v)\right]y^{N}d[v]dy=$
$\displaystyle=$
$\displaystyle\int\limits_{\mathbb{R}}\int\limits_{\mathbb{R}^{4(k_{2}-1)/\beta}}y^{N-2(k_{2}-1)/\beta}\left(-\frac{\partial}{\partial
y}\right)^{2(k_{2}-1)/\beta}\tilde{f}\left(\tr(y+v^{\dagger}v)\right)d[v]dy\sim$
$\displaystyle\sim$
$\displaystyle\int\limits_{\mathbb{R}}\int\limits_{\mathbb{R}^{+}}\tilde{v}^{2(k_{2}-1)/\beta-1}\left(-\frac{\partial}{\partial\tilde{v}}\right)^{2(k_{2}-1)/\beta}\tilde{f}\Bigl{(}\tr(y+\tilde{v})\Bigr{)}y^{N-2(k_{2}-1)/\beta}d\tilde{v}dy\sim$
$\displaystyle\sim$ $\displaystyle\int\limits_{\mathbb{R}}\tilde{f}\left(\tr
y\right)y^{N-2(k_{2}-1)/\beta}dy\sim$ $\displaystyle\sim$
$\displaystyle\int\limits_{\mathbb{R}}f(x)x^{k_{1}}\left(-\frac{\partial}{\partial
x}\right)^{N-2(k_{2}-1)/\beta}\delta(x)dx\sim$ $\displaystyle\sim$
$\displaystyle\int\limits_{\mathbb{R}}f(x)\frac{\partial^{N-2/\beta}}{\partial
x^{N-2/\beta}}\delta(x)d[x]\ ,$

which is for $\beta\in\\{1,2\\}$ well–defined.

##  Appendix D Determinantal structure of the ${\rm UOSp\,}(2k/2k)$–Berezinian

###### Statement D.1

Let $k\in\mathbb{N}$, $x_{1}\in\mathbb{C}^{2k}$ and $x_{2}\in\mathbb{C}^{k}$.
$x_{1}$ and $x_{2}$ satisfy the condition

$x_{a1}-x_{b2}\neq 0\ \ ,\ \forall a\in\\{1,\ldots,2k\\}\ \wedge\
b\in\\{1,\ldots,k\\}\ .$ (4.1)

Then, we have

$\frac{\Delta_{2k}(x_{1})\Delta_{k}^{4}(x_{2})}{V_{k}^{2}(x_{1},x_{2})}=(-1)^{k(k-1)/2}\det\left[\left\\{\frac{1}{x_{a1}-x_{b2}}\right\\}\underset{{1\leq
b\leq k}}{\underset{1\leq a\leq
2k}{}},\left\\{\frac{1}{(x_{a1}-x_{b2})^{2}}\right\\}\underset{{1\leq b\leq
k}}{\underset{1\leq a\leq 2k}{}}\right].$ (4.2)

We prove this theorem by complete induction.
Derivation:
We rearrange the determinant by exchanging the columns

$\displaystyle\det\left[\left\\{\frac{1}{x_{a1}-x_{b2}}\right\\}\underset{{1\leq
b\leq k}}{\underset{1\leq a\leq 2k}{}}\ ,\
\left\\{\frac{1}{(x_{a1}-x_{b2})^{2}}\right\\}\underset{{1\leq b\leq
k}}{\underset{1\leq a\leq 2k}{}}\right]=$
$\displaystyle=(-1)^{k(k-1)/2}\det\left[\frac{1}{x_{a1}-x_{b2}}\ ,\
\frac{1}{(x_{a1}-x_{b2})^{2}}\right]\underset{{1\leq b\leq k}}{\underset{1\leq
a\leq 2k}{}}\ .$ (4.3)

Thus, the minus sign in Eq. (4.2) cancels out.

We find for $k=1$

$\det\left[\begin{array}[]{cc}\displaystyle\frac{1}{x_{11}-x_{2}}&\displaystyle\frac{1}{(x_{11}-x_{2})^{2}}\\\
\displaystyle\frac{1}{x_{21}-x_{2}}&\displaystyle\frac{1}{(x_{21}-x_{2})^{2}}\end{array}\right]=\frac{(x_{11}-x_{21})}{(x_{11}-x_{2})^{2}(x_{21}-x_{2})^{2}}\
.$ (4.4)

We assume that this theorem is for $k-1$ true. Let

$\displaystyle s$ $\displaystyle=$
$\displaystyle\left[\frac{1}{x_{a1}-x_{b2}}\ ,\
\frac{1}{(x_{a1}-x_{b2})^{2}}\right]\underset{{1\leq b\leq k}}{\underset{1\leq
a\leq 2k}{}}=\left[\begin{array}[]{cc}s_{1}&w\\\ v&s_{2}\end{array}\right]\ ,$
(4.7) $\displaystyle s_{1}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}\displaystyle\frac{1}{x_{11}-x_{12}}&\displaystyle\frac{1}{(x_{11}-x_{12})^{2}}\\\
\displaystyle\frac{1}{x_{21}-x_{12}}&\displaystyle\frac{1}{(x_{21}-x_{12})^{2}}\end{array}\right]\
,$ (4.10) $\displaystyle s_{2}$ $\displaystyle=$
$\displaystyle\left[\frac{1}{x_{a1}-x_{b2}}\ ,\
\frac{1}{(x_{a1}-x_{b2})^{2}}\right]\underset{{2\leq b\leq k}}{\underset{3\leq
a\leq 2k}{}}\ ,$ (4.11) $\displaystyle v$ $\displaystyle=$
$\displaystyle\left[\frac{1}{x_{a1}-x_{12}}\ ,\
\frac{1}{(x_{a1}-x_{12})^{2}}\right]_{3\leq a\leq 2k}\ {\rm and}$ (4.12)
$\displaystyle w$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}\displaystyle\frac{1}{x_{11}-x_{b2}}&\displaystyle\frac{1}{(x_{11}-x_{b2})^{2}}\\\
\displaystyle\frac{1}{x_{21}-x_{b2}}&\displaystyle\frac{1}{(x_{21}-x_{b2})^{2}}\end{array}\right]_{2\leq
b\leq k}\ .$ (4.15)

Then, we have

$\det s=\det
s_{1}\det(s_{2}-vs_{1}^{-1}w)\overset{(D.4)}{=}\frac{(x_{11}-x_{21})}{(x_{11}-x_{12})^{2}(x_{21}-x_{12})^{2}}\det(s_{2}-vs_{1}^{-1}w)\
.$ (4.16)

The matrix in the determinant is equal to

$(s_{2}-vs_{1}^{-1}w)^{T}=\left[\begin{array}[]{c}\displaystyle\frac{(x_{11}-x_{a1})(x_{21}-x_{a1})(x_{12}-x_{b2})^{2}}{(x_{a1}-x_{12})^{2}(x_{11}-x_{b2})(x_{21}-x_{b2})}\frac{1}{x_{a1}-x_{b2}}\\\
\\\
\displaystyle\frac{(x_{11}-x_{a1})(x_{21}-x_{a1})(x_{12}-x_{b2})}{(x_{a1}-x_{12})^{2}(x_{11}-x_{b2})^{2}(x_{21}-x_{b2})^{2}}\frac{P_{ab}}{(x_{a1}-x_{b2})^{2}}\end{array}\right]\underset{{2\leq
b\leq k}}{\underset{3\leq a\leq 2k}{\underset{}{\underset{}{\underset{}{}}}}}$
(4.17)

where $P_{ab}$ is a polynomial

$\displaystyle
P_{ab}=(x_{a1}-x_{b2})(x_{11}-x_{b2})(x_{12}-x_{b2})-(x_{a1}-x_{12})(x_{11}-x_{b2})(x_{21}-x_{b2})-$
$\displaystyle-(x_{21}-x_{b2})(x_{a1}-x_{b2})(x_{11}-x_{12})=$
$\displaystyle=(x_{11}-x_{b2})(x_{21}-x_{b2})(x_{12}-x_{b2})+$
$\displaystyle+(x_{a1}-x_{b2})\left[(x_{11}+x_{21})(x_{12}+x_{b2})-2x_{11}x_{21}-2x_{12}x_{b2}\right]=$
$\displaystyle=A_{b}^{(1)}+(x_{a1}-x_{b2})A_{b}^{(2)}\ .$ (4.18)

The polynomials $A_{b}^{(1)}$ and $A_{b}^{(2)}$ are independent of the index
$a$. Due to the multilinearity and the skew symmetry of the determinant, the
result is

$\det
s=\frac{(x_{11}-x_{21})}{(x_{11}-x_{12})^{2}(x_{21}-x_{12})^{2}}\frac{\prod\limits_{a=3}^{2k}(x_{11}-x_{a1})(x_{21}-x_{a1})\prod\limits_{b=2}^{k}(x_{12}-x_{b2})^{4}}{\prod\limits_{a=3}^{2k}(x_{a1}-x_{12})^{2}\prod\limits_{b=2}^{k}(x_{11}-x_{b2})^{2}(x_{21}-x_{b2})^{2}}\det
s_{2}$ (4.19)

which completes the induction. $\square$

##  Appendix E Derivation of statement 4.1

Let $\lambda$ be the wanted eigenvalue and is a commuting variable of the
Grassmann algebra constructed from the
$\\{\tau_{q}^{(p)},\tau_{q}^{(p)*}\\}_{p,q}$. Then, we split this eigenvalue
in its body $\lambda^{(0)}$ and its soul $\lambda^{(1)}$, i.e.
$\lambda=\lambda^{(0)}+\lambda^{(1)}$. Let $v$ the $\gamma_{2}N$–dimensional
eigenvector of $H$ such that

$Hv=\lambda v{\rm\ \ and\ \ }v^{\dagger}v=1\ .$ (5.1)

In this equation, we recognize in the lowest order of Grassmann variables that
$\lambda^{(0)}$ is an eigenvalue of $H^{(0)}$. Then, let $\lambda^{(0)}$ be an
eigenvalue of the highest degeneracy $\delta$ of $H^{(0)}$, i.e. $\delta={\rm
dim\ ker}(H^{(0)}-\lambda^{(0)}\leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}_{N})$. Without loss of generality, we assume that $H^{(0)}$ is diagonal and
the eigenvalue $\lambda^{(0)}$ only appears in the upper left
$\delta\times\delta$–matrix block,

$H^{(0)}=\left[\begin{array}[]{cc}\lambda^{(0)}\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{\delta}&0\\\
0&\widetilde{H}^{(0)}\end{array}\right]\ .$ (5.2)

We also split the vectors in $\delta$ and $N-\delta$ dimensional vectors

$v^{(0)}=\left[\begin{array}[]{c}v_{1}\\\ v_{2}\end{array}\right]{\rm\ \ and\
\ }\tau_{q}=\left[\begin{array}[]{c}\tau_{q1}\\\ \tau_{q2}\end{array}\right]\
.$ (5.3)

Thus, we find the two equations from (5.1)

$\displaystyle T_{11}v_{1}-\lambda^{(1)}v_{1}+T_{12}v_{2}$ $\displaystyle=$
$\displaystyle 0\ ,$ (5.4) $\displaystyle
T_{21}v_{1}+\left[\widetilde{H}^{(0)}-\lambda\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{N-\delta}+T_{22}\right]v_{2}$ $\displaystyle=$
$\displaystyle 0$ (5.5)

where
$T_{nm}=\sum\limits_{q=1}^{\widetilde{N}}l_{q}\left[\tau_{qn}\tau_{qm}^{\dagger}+\widetilde{Y}\left(\tau_{qn}^{*}\tau_{qm}^{T}\right)\right]$.
Eq. (5.5) yields

$v_{2}=-\left[\widetilde{H}^{(0)}-\lambda\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{N-\delta}+T_{22}\right]^{-1}T_{21}v_{1}\ .$ (5.6)

Hence, the body of $v_{2}$ is zero and we have for Eq. (5.4)

$T_{11}v_{1}-\lambda^{(1)}v_{1}-T_{12}\left[\widetilde{H}^{(0)}-\lambda\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{N-\delta}+T_{22}\right]^{-1}T_{21}v_{1}=0\ .$
(5.7)

If the degeneracy is $\delta>\gamma_{2}$, we consider a $\delta$–dimensional
real vector $w\neq 0$ such that $w^{\dagger}v_{1}=0$. Then, we get for the
lowest order in the Grassmann variables of Eq. (5.7) times $w^{\dagger}$

$w^{\dagger}T_{11}v_{1}^{(0)}=0$ (5.8)

where $v_{1}^{(0)}$ is the body of $v_{1}$. The entries of $w^{\dagger}T_{11}$
are linearly independent. Thus, the body of $v_{1}$ is also zero. This
violates the second property of (5.1).

Let the degeneracy $\delta=\gamma_{2}$. Then, $v_{1}$ is
$\gamma_{2}$-dimensional and is normalizable. For $\beta=4$, we have the
quaternionic case and the matrix before $v_{1}$ in Eq. (5.7) is a diagonal
quaternion. Hence, it must be true

$\lambda^{(1)}\leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}_{\gamma_{2}}=T_{11}-T_{12}\left[\widetilde{H}^{(0)}-\lambda\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{N-\delta}+T_{22}\right]^{-1}T_{21}\ .$ (5.9)

Considering the second order term in the Grassmann variables of Eq. (5.9),
$\lambda$’s second order term is $T_{11}$ for $\beta\in\\{1,2\\}$ and $\tr
T_{11}/2$ for $\beta=4$. Eq. (5.9) is unique solvable by recursive
calculation. We plug the right hand side of Eq. (5.9) into the $\lambda^{(1)}$
on the same side and repeat this procedure. Hence, we define the operator

$\displaystyle O(\mu)$ $\displaystyle=$
$\displaystyle\frac{1}{\gamma_{2}}\tr\left\\{T_{11}-T_{12}\left[\widetilde{H}^{(0)}-(\lambda^{(0)}+\mu)\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{N-\delta}+T_{22}\right]^{-1}T_{21}\right\\}{\rm\
and}$ (5.10) $\displaystyle O^{n+1}(\mu)$ $\displaystyle=$ $\displaystyle
O\left[O^{n}(\mu)\right]\ .$ (5.11)

Then, $\lambda^{(1)}=O^{n}(\lambda^{(1)})$ is true for arbitrary
$n\in\mathbb{N}$. The recursion is finished for $n_{0}\in\mathbb{N}$ if
$\lambda^{(1)}=O^{n_{0}}(\lambda^{(1)})=O^{n_{0}}(0)$. Due to the Grassmann
variables, this recursion procedure eventually terminates after the
$(\gamma_{2}N\widetilde{N}/2)$’th time. Thus, the eigenvalue $\lambda$ depends
on Grassmann variables and is not a real number.

## References

## References

  * [1] K.B. Efetov.  Adv. Phys., 32:53, 1983.
  * [2] J.J.M. Verbaarschot, H.A. Weidenmüller, and M.R. Zirnbauer.  Phys. Rep., 129:367, 1985.
  * [3] K.B. Efetov.  Supersymmetry in Disorder and Chaos.  Cambridge University Press, Cambridge, 1st edition, 1997.
  * [4] T. Guhr, A. Müller-Groeling, and H.A. Weidenmüller.  Phys. Rep., 299:189, 1998.
  * [5] T. Guhr.  J. Phys., A 39:13191, 2006.
  * [6] N. Lehmann, D. Saher, V.V. Sokolov, and H.-J. Sommers.  Nucl. Phys., A 582:223, 1995.
  * [7] G. Hackenbroich and H.A. Weidenmüller.  Phys. Rev. Lett., 74:4118, 1995.
  * [8] K.B. Efetov, G. Schwiete, and K. Takahashi.  Phys. Rev. Lett., 92:026807, 2004.
  * [9] K.B. Efetov and V.R. Kogan.  Phys. Rev., B 70:195326, 2004.
  * [10] F. Basile and G. Akemann.  JHEP, page 0712:043, 2007.
  * [11] H.-J. Sommers.  Acta Phys. Pol., B 38:1001, 2007.
  * [12] P. Littelmann, H.-J. Sommers, and M.R. Zirnbauer.  Commun. Math. Phys., 283:343, 2008.
  * [13] T. Jiang.  J. Math. Phys., 46:052106, 2005.
  * [14] M. Kieburg, H. Kohler, and T. Guhr.  J. Math. Phys., 50:013528, 2009.
  * [15] M.L. Mehta.  Random Matrices and the statistical Theory of Energy Levels.  Academic Press Inc., New York, 1st edition, 1967.
  * [16] L. Hörmander.  The Analysis of Linear Partial Differential Operators.  Springer, Berlin, Heidelberg, New York, 1976.
  * [17] M.R. Zirnbauer.  The Supersymmetry Method of Random Matrix Theory, Encyclopedia of Mathematical Physics, eds. J.-P. Franoise, G.L. Naber and Tsou S.T., Elsevier, Oxford, 5:151, 2006.
  * [18] F.A. Berezin.  Introduction to Superanalysis.  D. Reidel Publishing Company, Dordrecht, 1st edition, 1987.
  * [19] B. DeWitt.  Supermanifolds.  Cambridge University Press, Cambridge, 1st edition, 1984.
  * [20] J.E. Bunder, K.B. Efetov, K.B. Kravtsov, O.M. Yevtushenko, and M.R. Zirnbauer.  J. Stat. Phys., 129:809, 2007.
  * [21] B.L. van der Waerden.  Algebra I.  Springer, Berlin, Heidelberg, New York, 8th edition, 1971.
  * [22] H. Kohler and T. Guhr.  J. Phys., A 38:9891, 2005.
  * [23] M.R. Zirnbauer.  J. Math. Phys., 37:4986, 1996.
  * [24] A.E. Ingham.  Proc. Camb. Phil. Soc., 29:271, 1933.
  * [25] C.L. Siegel.  Ann. Math., 36:527, 1935.
  * [26] Y.V. Fyodorov.  Nucl. Phys., B 621:643, 2002.
  * [27] M.L. Mehta.  Random Matrices.  Academic Press Inc., New York, 2nd edition, 1991.
  * [28] A. Okounkov and G. Olshanski.  Math. Res. Letters, 4:69, 1997.
  * [29] T. Guhr and H. Kohler.  J. Math. Phys., 43:2741, 2002.
  * [30] T. Guhr.  Commun. Math. Phys., 176:555, 1996.
  * [31] T. Guhr.  Ann. Phys. (N.Y.), 250:145, 1996.
  * [32] T. Guhr.  J. Math. Phys., 32:336, 1991.
  * [33] T. Guhr.  J. Math. Phys., 34:2523, 1993.
  * [34] M.J. Rothstein.  Trans. Am. Math. Soc., 299:387, 1987.
  * [35] F. Wegner, 1983.
  * [36] F. Constantinescu.  J. Stat. Phys., 50:1167, 1988.
  * [37] F. Constantinescu and H.F. de Groote.  J. Math. Phys., 30:981, 1989.
  * [38] E. Brezin and S. Hikami.  J. Phys. A: Math. Gen., 36:711, 2003.
  * [39] T. Guhr and H. Kohler.  J. Math. Phys., 43:2707, 2002.
  * [40] M. Bergère and B. Eynard, 2008.  arxiv:0805.4482v1 [math-ph].
  * [41] B. Feigin, M. Jimbo, T. Miwa, and E. Mukhin.  Internat. Math. Res. Notices, 23:1223, 2002.
  * [42] I.G. Macdonald.  Symmetric Functions and Hall polynomials.  Oxford University Press, Oxford, 2nd edition, 1995.
  * [43] H. Kohler, 2007.  arxiv:0801.0132v1 [math-ph].
  * [44] K. Życzkowski and H.-J. Sommers.  J. Phys., A 36:10115, 2003.