Derivation of the Supersymmetric Harish Chandra Integral for UOSp(k 1 /2k 2 )

Transcription

1 Derivation of the Supersymmetric Harish Chandra Integral for UOSpk 1 /2k 2 Thomas Guhr 1 and Heiner Kohler 2 1 Matematisk Fysik, LTH, Lunds Universitet, Box 118, Lund, Sweden 2 Departamento de Teoría de Materia Condensada, Universidad Autónoma, Madrid, Spain Dated: June 23, 2004 The previous supersymmetric generalization of the unitary Harish Chandra integral prompted the conjecture that the Harish Chandra formula should have an extension to superspaces. We prove this conjecture for the unitary orthosymplectic supermanifold UOSpk 1/2k 2. To this end, we construct and solve an eigenvalue equation. I. INTRODUCTION Harish Chandra 1 gave a closed formula for a class of group integrals. Let G be a compact semi simple Lie group and let a and b fixed elements in the Cartan subalgebra H 0 of G. Harish Chandra s formula then reads U G exp tru 1 aub dµu = 1 W w W exp tr wab ΠaΠwb, 1.1 where dµu stands for the properly normalized invariant measure and Πa for the product of all positive roots of H 0. Moreover, W is the Weyl reflection group of G and W is the number of its elements. We notice that the integrals 1.1 should not be confused with Gelfand s spherical functions 2 5. They are defined by a group integral which looks at first sight just like the one in Eq. 1.1, however, for Gelfand s spherical functions, a and b are not in the Cartan subalgebra. Thus, Gelfand s spherical functions are very different objects. Only in the case of G = SUN, the Harish Chandra integral 1.1 coincides with the unitary spherical function of Gelfand. It is known as the Itzykson Zuber integral 6. A very handy diffusion equation method was developed in Ref. 6 to derive this unitary case. A supersymmetric generalization of the Itzykson Zuber integral, i.e. the extension of the Harish Chandra integral to the case of the unitary supermanifold Uk 1 /k 2, was first obtained in Ref. 7 by generalizing the Itzykson Zuber diffusion equation method to supersymmetry. In its most general form, this integral was obtained in Ref. 8 as an application of Gelfand Tzetlin coordinates for Uk 1 /k 2 and also in Ref. 9 by employing the methods as given in Ref. 7. Serganova 10 and Zirnbauer 11 conjectured that Harish Chandra s formula should not only have a supersymmetric extension in the unitary case, but also generalize to all classical supermanifolds SG which can be viewes as certain extensions of the ordinary Lie groups. Thus, one would expect a result of the following form to hold, u SG exp trg u 1 sur dµu = 1 SW w SW exp trg wsr πsπwr, 1.2 where s and r are in the Cartan subalgebra SH 0 of SG. The Weyl reflection group SW and the root system πs have to be properly generalized to superspace, SW is the number of elements in SW. The extension of the Harish Chandra integral to the case of the unitary supermanifold Uk 1 /k 2 is certainly of the form 1.2. The most interesting remaining case is the unitary orthosymplectic supermanifold UOSpk 1 /2k 2. For this case, we present a proof of the conjecture 1.2 in this note. In Sec. II, we state and derive the supersymmetric Harish Chandra integral for UOSpk 1 /2k 2. We summarize and conclude in Sec. III. II. THE SUPERMANIFOLD INTEGRAL AND ITS DERIVATION After briefly summarizing properties of the supermanifold UOSpk 1 /2k 2 and introducing our notation in Sec. II A, we state the supermanifold integral in Sec. II B. The solution is sketched in the following two sections. In Sec. II C, we formulate an eigenvalue equation for the integral. It is solved by separation in Sec. II D. A. The supermanifold UOSpk 1/2k 2 Kac 12,13 gave a classification of the classical superalgebras similar to Cartan s classification of the Lie algebras in ordinary space. In principle, to each classical superalgebra a supergroup is associated by the exponential mapping.

2 However, the classification pattern of the supergroups is usually somewhat coarser 14,15. If one omits the supergroup stemming from the exceptional superalgebras, one is left with only four different types of subgroups of the general linear supergroup GLk 1 /k 2, namely the unitary supergroup Uk 1 /k 2 the orthosymplectic supergroup OSpk 1 /2k 2 and the supergroups associated with the strange superalgebras Pk and Qk. The supergroup OSpk 1 /2k 2 consists of the elements u in GLk 1 /2k 2 which leave invariant the metric L = diag 1 k1, J, such that u T Lu = L. Here, J is the symplectic metric [ J = τ k2 = k2 1 k2 0 ], with τ 1 = [ ] Especially important in physical applications are the supergroups Uk 1 /k 2 and the supermanifold UOSpk 1 /2k 2, formed by the elements of OSpk 1 /2k 2 which fulfill the additional condition u u = 1. It contains the compact ordinary groups Ok 1 and USp2k 2 as subgroups. However, it is strictly speaking not a compact real form of OSpk 1 /2k 2, see App. A and Refs. 16,17. While the fermionic dimension 2k 2 of UOSpk 1 /2k 2 is always even, the bosonic dimension k 1 can be even or odd, eventually resulting in some slight differences for the supergroup integral. One can also define the superalgebra uospk 1 /2k 2 connected as usual to the supergroup UOSpk 1 /2k 2 via the exponential mapping 15. For σ uospk 1 /2k 2 we have u = expσ UOSpk 1 /2k 2. These generators σ span the superalgebra. An element of this superalgebra can be written as σ = σo σ a σ at. 2.3 σ a σ a The k 1 k 1 matrix σ o is antisymmetric, it is in the algebra ok 1 and generates the ordinary group Ok 1. The 2k 2 2k 2 matrix σ usp is in the algebra usp2k 2 and generates the ordinary group USp2k 2, i.e. in the basis defined by Eq. 2.1 it is of the form [ ] σ usp σ usp,1 σ = usp, σ usp,2 σ usp σ usp,1t Here σ usp,1 is a skew hermitean matrix and σ usp,2 is complex symmetric. The k 2 k 1 matrix σ a in Eq. 2.3 contains k 1 k 2 independent complex anticommuting variables, it is in the sector usopk 1 /2k 2 ok 1 usp2k 2. The asterix denotes the complex conjugate of the second kind for Grassmann variables. One uses the supertrace denoted by trg to define an invariant bilinear form on the superalgebra. Although the algebra uospk 1 /2k 2 defined by the matrices σ in Eq. 2.3 is closely related to the algebra of the orthosymplectic group ospk 1 /2k 2 we avoid the notation of a real form of ospk 1 /2k 2 since it can not be derived from ospk 1 /2k 2, R by an involutive automorphism 17. Particularly important in the present context is the Cartan subalgebra uosp 0 k 1 /2k 2 of uospk 1 /2k 2. As in the theory of Lie algebrae in ordinary space there is a difference for the orthogonal group in even or odd dimension. We introduce the notation [k 1 /2] for the integer part of k 1 /2. Then, for even bosonic dimension [k 1 /2] = k 1 /2, the elements of uosp 0 k 1 /2k 2 are the matrices σ 0 = diag s 11 τ 1,..., s [k1/2]1τ 1, is 12,...,is k22, is 12,..., is k22, 2.5 while for odd bosonic dimension [k 1 /2] = k 1 1/2, the Cartan subalgebra uosp 0 k 1 /2k 2 consists of the matrices σ 0 = diag s 11 τ 1,..., s [k1/2]1τ 1, 0, is 12,..., is k22, is 12,..., is k Thus, uosp 0 k 1 /2k 2 is the direct sum of the Cartan subalgebras ok 1 and usp2k 2. We could now work with the superalgebra as defined above. However, we redefine it by a procedure similar to a Wick rotation. In most of the physics literature, this is done to ensure convergence of the integrals in superspace, see for example Ref. 21. Although the Wick rotation is not necessary in the present context, we decided to redefine the superalgebra to keep with the notation in the physics literature and in previous work on the unitary supergroup 7 9. Hence, we proceed as follows. The Killing Cartan form of the supermatrices defined in Eqs. 2.5 and 2.6 is not negative definite. We define another set of diagonal supermatrices by [ ] i 0 s = σ 0 1 0, 2.7

3 such that always trg s 2 > 0. In the sequel we use exclusively the set of supermatrices s. We denote it also by uosp 0 k 1 /2k 2. By the same token we refer to its orbits under the group action σ = u 1 su, u UOSpk 1 /2k 2 as the superalgebra uospk 1 /2k 2. We emphasize once more that this redefinition has no direct impact in the present context. A remark is in order for the mathematically oriented reader. The object UOSpk 1 /2k 2 defined above is not included in the usual classification of symmetric superspaces. However, it still is a group in the naive, but in the present context crucial, sense that it describes generalized rotations in a superspace. It belongs to a class of supermanifolds which have been coined cs manifolds in Ref. 18. We refer the reader to the literature for further reading on the classification of supermanifolds 12 16,19,20 and to App. A. For this reason we refer to UOSpk 1 /2k 2 defined above always as supermanifold. 3 B. Statement of the supermanifold integral We can now write formula 1.2 for the case of the supermanifold UOSpk 1 /2k 2 more explicitly. For two fixed elements s and r of the Cartan subalgebra uosp 0 k 1 /2k 2, we have exp itrg u 1 sur dµu u UOSpk 1/2k 2 = 1 det[cos2sp1 r q1 ] + det[i sin2s p1 r q1 ] det[ 2i sin2s p2 r q2 ] [k 1 /2]!k 2! s r 2.8 for k 1 even and u UOSpk 1/2k 2 exp itrg u 1 sur dµu = 1 det[i sin2s p1 r q1 ] det[2 cos2s p2 r q2 ] [k 1 /2]!k 2! s r 2.9 for k 1 odd. We introduced the function s which is given by s = 2 k2 p<q s2 p1 s2 q1 p<q s2 p2 s2 q2 k 2 s p2 p,q s2 p1 + s2 q for even bosonic dimension k 1 and by s = 2 k2 p<q s2 p1 s2 q1 p<q s2 p2 s2 q2 [k 1/2] s p1 p,q s2 p1 + s2 q for odd bosonic dimension k 1. These two formulae differ only in the last terms of the numerators. Formulae 2.8 and 2.9 contain, as special cases, the ordinary orthogonal and unitary symplectic Harish Chandra integrals for G = SOk 1 and for G = USp2k 2, if we set k 2 = 0 or k 1 = 0, respectively. Thus, the derivation of the supersymmetric integral to follow also includes a rederivation of those ordinary integrals. For equal bosonic and fermionic dimension, formula 2.8 was conjectured in Ref. 11 and used to calculate the correlation functions in a certain circular random matrix ensemble. We mention in passing that the invariant measure dµu and its normalization relate to non trivial questions of certain boundary contributions in superanalysis 22 which are highly important in applications. In the present context, however, we do not need to go into this. C. Eigenvalue equation The main idea for the derivation of formulae 2.8 and 2.9 is to properly modify the supersymmetric extension 7 of the Itzykson Zuber diffusion equation method 6 to the present case. It turns out that it is somewhat more convenient to construct the eigenvalue equation associated with the diffusion equation. The two equations are related by Fourier expansion. Such an eigenvalue equation for the ordinary case of SUN as originally discussed by Itzykson and Zuber 6 was constructed by Brézin 23. Berezin and Karpelevich 24 had studied such an eigenvalue equation to calculate the twofold group integral named after them, see also Ref. 25. To construct the eigenvalue equation needed to derive

4 formulae 2.8 and 2.9, we adjust the steps made in Refs. 26,27, where a supersymmetric eigenvalue equation was employed for the extension of the Berezin Karpelevich integral. We introduce the Laplace operator over the superalgebra uospk 1 /2k 2 = 1 2 k 1 p<q σ o2 pq 2k 2 + p,q 1 + δ pq 4 σ pq usp σ pq usp k 1 k 2 q=1 σ pq a σ pq a Its eigenfunctions are the plane waves expitrg σρ with both matrices σ, ρ uospk 1 /2k 2. Thus, we have expitrg σρ = trg ρ 2 expitrg σρ We now diagonalize both matrices according to σ = u 1 su and ρ = v 1 rv where u and v are in the supermanifold UOSpk 1 /2k 2 and s and r are in the Cartan subalgebra uosp 0 k 1 /2k 2, i.e. given by Eq. 2.5 or Eq. 2.6, respectively. Integrating both sides over v and using the invariance of the measure dµv, we arrive at the radial eigenvalue equation s χ k12k 2 s, r = trg r 2 χ k12k 2 s, r, 2.14 where, now using u instead of v again, the function χ k12k 2 s, r = exp itrg u 1 sur dµu 2.15 u UOSpk 1/2k 2 is the integral we want to calculate. The operator s in Eq is the radial part of. The term radial refers to the Cartan subalgebra. This usage which is common in mathematics should not lead to confusions with the radial operators used, for example, in Refs where quite different spaces were studied. To obtain the radial operator s, we need the Jacobian, or Berezinian, of the variable transformation σ = u 1 su. This Berezinian is given by the the functions B 2 k 12k 2 s of Eqs and It was not possible for us to find out where this Berezinian was first obtained, and we do not claim originality for its calculation. In any case, to make the paper self contained, we sketch the calculation in Appendix B. Hence, the radial part of the Laplacean over uospk 1 /2k 2 reads s = 1 2 [k 1/2] 1 Bk 2 B 2 k 12k 2 s s 12k 2 s + 1 p1 s p1 2 k 2 1 Bk 2 B 2 k 12k 2 s s 12k 2 s p2 s p2 The number of bosonic eigenvalues is [k 1 /2], i.e. identical for the pairs uospk 1 /2k 2 and uospk 1 + 1/2k 2 with k 1 even. However, the operator s is not the same in these two cases, because the functions 2.10 and 2.11 differ. D. Solution by separation The Laplacean 2.16 is separable. We make an ansatz for the supermanifold integral which separates off the square roots of the Berezinians, χ k12k 2 s, r = A tedious but straightforward calculation then yields the trivial eigenvalue equation for the function ω β k 12k 2 s, r. Here we introduced the gradient s =,...,,,..., s 11 s 12 ω k12k 2 s, r s r s 2 ω k 12k 2 s, r = trg r 2 ω k12k 2 s, r s [k1/2]1 s k22, 2.19 which also defines the flat Laplacean / s 2 appearing in the eigenvalue equation A crucial feature of the square root of the Berezinian enters the derivation of Eq It satisfies the harmonic equation s 2 B k 12k 2 s = 0, 2.20

5 which we prove in Appendix C. Any linear combination of products of exponentials solves the eigenvalue equation However, as the supermanifold integral and the eigenvalue equations are obviously invariant under permutations of the variables in the ok 1 sector s p1, p = 1,...,[k 1 /2] or, equivalently, r p1, p = 1,...,[k 1 /2] and under permutations of the variables in the uosp2k 2 sector is p2, p = 1,...,k 2 or, equivalently, ir p2, p = 1,...,k 2, the desired solution must have the same property. Moreover, there is a symmetry under a parity transformation for the variables is p2, p = 1,...,k 2 and ir p2, p = 1,...,k 2. That is, the solution must be invariant under the substitution s p2 s p2 and r p2 r p2. For k 1 odd, the same symmetry must hold also for s p1, p = 1,...,[k 1 /2] and r p1, p = 1,...,[k 1 /2] respectively. By these symmetries the solution of Eq for k 1 odd is up to normalization uniquely determined, ω k12k 2 s, r = 1 [k 1 /2]!k 2! det[i sin2s p1r q1 ] det[2 cos2s p2 r q2 ] For k 1 even, the part antisymmetric under the parity transformation has to be kept and we find ω k12k 2 s, r = 1 det[cos2sp1 r q1 ] + det[i sin2s p1 r q1 ] det[ 2i sin2s p2 r q2 ] [k 1 /2]!k 2! These results give, together with the ansatz 2.17, the desired supermanifold integrals 2.8 and 2.9. As already mentioned, our derivation of the supersymmetric group integral contains as special cases a rederivation of the ordinary orthogonal and unitary symplectic Harish Chandra integrals for k 2 = 0 or k 1 = 0, respectively. 5 III. SUMMARY AND CONCLUSIONS We calculated the supersymmetric Harish Chandra integral for the unitary orthosymplectic supermanifold, thereby proving a conjecture 10,11. Our derivation uses a diffusion equation or, equivalentely, eigenvalue equation method. It is based on the separability of the Laplacean. Our present contribution is a further extension of this technique, which, to the best of our knowledge, has previously only been used for group integrals over the unitary group: orginally, it was introduced for the Harish Chandra integral over the ordinary unitary group 6,23, then extended for the supersymmetric Harish Chandra integral over the unitary supermanifold 7. Already in 1958, Berezin and Karpelevich 24 had developed such a technique for an integral over two unitary groups, see also Ref. 25. This was also extended to the supersymmetric case 26,27. Here, we considered the unitary orthosymplectic supermanifold and adjusted the eigenvalue equation method to this case. As the unitary orthosymplectic supermanifold contains the ordinary orthogonal and unitary symplectic groups as subgroups and special cases, we automatically also extended the eigenvalue equation method to these two ordinary groups. We are aware of only two methods which could be an alternative: character expansions and Gelfand Tzetlin coordinates. Balantekin developed the character expansion method for the unitary ordinary and supermanifold 32,33 and obtained various group integrals. Recently, this method was further extended and employed in Ref. 34. Similar considerations are also of interest if one studies the Itzykson Zuber integral for matrices of large dimension 35. Moreover, character expansions could also be developed for the calculation of certain integrals over the ordinary orthogonal and unitary symplectic group 36, but Harish Chandra integrals have so far not been tackled with this approach. Gelfand Tzetlin coordinates 37,38 allow one to compute the ordinary 39 and supersymmetric 8 Itzykson Zuber integral directly, i.e. without using a diffusion or eigenvalue equation. This method has not been applied yet to work out Harish Chandra integrals for the ordinary orthogonal or unitary symplectic or the supersymmetric unitary orthosymplectic manifold. However, it has been employed for Gelfand s spherical functions Considering all the cases, in which non trivial group integrals could be obtained for the first time or in which known results could be rederived faster, the diffusion or eigenvalue equation method used and further extended here shows a remarkably wide range of applicability. Acknowledgments TG and HK acknowledge financial support from the Swedish Research Council and from the RNT Network of the European Union with Grant No. HPRN CT , respectively. HK also thanks the division of Mathematical Physics, LTH, for its hospitality during his visits to Lund.

6 6 APPENDIX A: ON THE SUPERMANIFOLD UOSpk 1/2k 2 The real form of a complex supergroup G is obtained by acting with some involutive antilinear automorphism s onto the group. Then the set G s of fixed points of s is a real form of G. Since the number of antilinear involutive automorphisms is limited, so is the number of inequivalent real forms. Indeed, the real forms of all complex superalgebrae and therefore of the supergroups have been classified in 16,17. In particular, there it has been shown that the complex supergroup OSpk 1 /2k 2 has only two real forms. They are most conveniently characterised by the restriction of G s to its even part G s0. For OSpk 1 /2k 2 this even part G s0 has to be a product of some real forms of Ok 1, C and Sp2k 2, C. The only two possibilities for G s0 are Or, k 1 r Spk 2, R and in addition, for k 1 even, O k 1 Spr, k 2 r. The corresponding automorphisms can also be found in 16,17. That is, both real forms are non compact supergroups. Since wish to generalize Harish Chandra s formula for ordinary compact groups we have to relax the definition above and seek for a class of automorphisms of the algebra that allow for compact manifolds. Therefore we consider automorphisms t which are involutive only on the even elements t 2 g 0 = g 0 and antiinvolutive t 2 g 0 = g 0 on the odd elements. For ospk 1 /2k 2 such an automorphism is given by tg = g with g ospk 1 /2k 2. Now we can construct a functor from the pair ospk 1 /2k 2, t to a supermanifold. The so constructed object is, however not a real supermanifold since the functor is not defined on the set of real supercommutative algebrae. It falls into the class of cscomplex supersymmetric manifolds 18. It is, however a frequently arising object in physics 21,40 and has an intuitive meaning in terms of the matrices as defined in Sec. II A with the usual anticommutator as product. APPENDIX B: CALCULATION OF THE BEREZINIAN We use the standard procedure of obtaining the metric tensor g whose superdeterminant is the square of the Berezinian. The variation of the element σ = u 1 su uospk 1 /2k 2 reads dσ = u 1 ds + [s, δũ]u, where δũ = duu 1 B1 is also in the superalgebra uospk 1 /2k 2. Thus, the invariant length element is given by trg dσ 2 = trg ds + [s, δũ] 2 = trg ds 2 + trg [s, δũ] 2 = [k 1/2] k 2 ds 2 p1 + ds 2 p2 + n α o n 2 δũ o n 2 + n α usp n 2 δũ usp n 2 + n α a n 2 δũ a n 2. B2 In the last step, we expanded the traces, the metric g can then be read of from the coefficients in front of the squared variation differentials. We split the contribution from the commutator in three terms and introduced a new index n, labelling the roots and the variation differentials stemming from δũ. There are three types of roots, corresponding to the ok 1 and the usp2k 2 subalgebras and the remaining sector uospk 1 /2k 2 ok 1 usp2k 2 containing the anticommuting degrees of freedom. For even bosonic dimension 2k 1, there are 2k 1 k 1 1 roots α o n of o2k 1, given by ±s p1 ± s q1 with independent signs and p < q. For odd bosonic dimension 2k 1 + 1, there are 2k 1 additional roots ±s p1 which are needed for the complete root system of o2k The roots α usp n of usp2k 2 are ±is p2 ± is q2 with independent signs and p < q, and furthermore ±2is p2, all together 2k2 2 roots. Finally, we have the 2k 1 k 2 roots α a n from uospk 1 /2k 2 ok 1 usp2k 2, which read ±s p1 ± is q2 with independent signs and indices p, q. For odd bosonic dimension we have 2k 1 additional roots α a n = ±is p2. Collecting everything, the superdeterminant detg g of the metric g is the product of all roots from ok 1 and usp2k 2, divided by the product of all roots from uospk 1 /2k 2 ok 1 usp2k 2. The square root of detg g then gives the Berezinians 2.10 and APPENDIX C: SQUARE ROOT OF THE BEREZINIAN AS A HARMONIC FUNCTION The result 2.20 is crucial for the separation ansatz and for the derivation of the ensuing eigenvalue equations. It is tedious, but elementary to prove it by explicit calculation. We consider even k 1, the case of odd k 1 is treated in the same way. Using relations such as 1 s 2 p1 s2 q1 p q = 0 and p q t s 2 p1 s 2 p1 s2 q1 s2 p1 s2 t1 = 0 C1

7 7 we find Similarly, we obtain [k 1 1/2] s 1 s k 2 s 2 p1 s 2 p2 s = p,q,t s = p,q,t 4s 2 p1 s 2 p1 + s2 q2 s2 p1 + s2 t2 p,q 6 s 2 p1 + + s2 q2 p,q p q,t 4s 2 p2 s 2 p2 + s2 q1 s2 p2 + s2 t1 4s 2 p1 8s 2 p1 s 2 p1 s2 q1 s2 p1 + s2 t2 s 2 p1 +. C2 s2 q2 2 p q,t 4s 2 q2 8s 2 p2 s 2 p2 s2 q2 s2 p2 + s2 t1 2 s 2 p,q p1 + + s2 q2 s 2 p,q p1 +. C3 s2 q2 2 Combining these two intermediate results, we arrive at 1 s s 2 B k 12k 2 s = 4s 2 t2 s 2 p q,t p1 + s2 t2 s2 q1 + s2 t2 8s 2 p1 s 2 p1 s2 q1 s2 p1 + s2 t2 + 4s 2 t1 s 2 p q,t t1 + s2 p2 s2 t1 + s2 q2 8s 2 p2 s 2 p2 s2 q2 s2 p2 + s2 t1 + 8s 2 p1 s 2 p,q p s 2 q1 s2 q2 2 s 2 q1 + 8 s2 p2 2 s 2 p,q p1 + s2 q2 = 4s 2 t2 s 2 p q,t p1 + s2 t2 s2 q1 + s2 t2 4s 2 p1 s 2 p1 s2 q1 s2 p1 + s2 t2 4s 2 q1 s 2 q1 s2 p1 s2 q1 + s2 t2 + 4s 2 t1 s 2 t1 + s2 p2 s2 t1 + s2 q2 4s 2 p2 s 2 p2 s2 q2 s2 p2 + s2 t1 4s 2 q2 s 2 q2 s2 p2 s2 q2 + s2 t1 which is Eq p q,t = 0, C4

8 8 1 Harish-Chandra, Am. J. Math. 79, I.M. Gelfand, Dokl. Akad. Nauk. SSSR S. Helgason, Groups and Geometric Analysis, San Diego: Academic Press, Harish-Chandra, Am. J. Math. 80, M.A. Olshanetsky and A.M. Perelomov, Phys. Rep. 94, C. Itzykson and J.B. Zuber, J. Math. Phys. 21, Phys. Rep. 129, T. Guhr, J. Math. Phys. 32, T. Guhr, Commun. Math. Phys. 176, J. Alfaro, R. Medina and L. Urrutia, J. Math. Phys. 36, V. Serganova, private communication, Berkeley M.R. Zirnbauer, J. Phys. A29, V.C. Kac, Comm. Math. Phys. 53, V.C. Kac, Advances in Math. 26, V. Rittenberg, A Guide to Lie Superalgebras, Lecture Notes in Physics 79, Berlin: Springer Verlag, F.A. Berezin, Introduction to Superanalysis, MPAM9, Dordrecht: D. Reidel Publishing, V. Serganova, Funct. Analysis and its Applications 17, M. Parker, Jour. Math. Phys. 21, J. Bernstein, in: Quantum Fields and Strings II, p.41 P. Deligne, P. Etingof, D. S. Freed eds. AMS, D. A. Leites, Usp. Mat. Nauk. 35, M. Zirnbauer, Jour. Math. Phys. 37, K. Efetov, Adv. Phys. 32, M.J. Rothstein, Trans. Am. Math. Soc. 299, E. Brézin, in: Two dimensional quantum gravity and random surfaces, p. 37, D.J. Gross, T. Piran and S. Weinberg eds., Singapore: World Scientific, F.A. Berezin and F.I. Karpelevich, Dokl. Akad. NAUK SSSR 118, B. Hoogenboom, Ark. Mat. 20, T. Guhr and T. Wettig, J. Math. Phys. 37, A.D. Jackson, M.K. Sener and J.J.M. Verbaarschot, Nucl. Phys. B506, T. Guhr and H. Kohler, math-ph/ T. Guhr and H. Kohler, J. Math. Phys. 43, T. Guhr and H. Kohler, math-ph/ T. Guhr and H. Kohler, J. Math. Phys. 43, A.B. Balantekin, Phys. Rev. D62, ; hep-th/ A.B. Balantekin, Phys. Rev. E64, ; cond-mat/ B. Schlittgen and T. Wettig, math-ph/ P. Zinn Justin and J.B. Zuber, math-ph/ A.B. Balantekin and P. Cassak J. Math. Phys. 43, I.M. Gelfand and M.L. Tzetlin, Dokl. Akad. Nauk. 71, A.O. Barut and R. Raczka, Theory of Group Representations and Applications, Warszawa: Polish Scientific Publishers, S.L. Shatashvili, Commun. Math. Phys. 154, J. J. M. Verbaatschot, H. Weidenmüller, M. Zirnbauer Phys. Rep. 129,