O(N) colour-flavour transformations and characteristic polynomials of real random matrices

The fermionic, bosonic and supersymmetric variants of the colour-flavour transformation are derived for the orthogonal group. These transformations are then used to calculate the ensemble averages of characteristic polynomials of real random matrices. 1 Statement of main results Since the pioneering work of Wigner, many physical systems have been successfully studied with the help of random matrix models. Among these asymmetric real random matrices arising in applications in neural networks [1], quantum chaos [2] and QCD [3] are known to be the most difficult. In fact, until the very recent breakthrough [4, 5, 6, 7, 8], the eigenvalue correlation functions of real and complex eigenvalues were not accessible even for Gaussian matrices and calculating the ensemble averages of eigenvalue statistics in the complex plane for a sufficiently general class of real random matrices remains a challenging problem. The mathematical difficulties in calculating the eigenvalue correlation functions of real random matrices are mainly due to the fact that their eigenvalues are either real or pairwise complex conjugate and the mathematical tools available to study real random matrices are very limited, especially when compared to those available for complex matrices. In this paper we derive several integral transformations dealing with integrations over real orthogonal matrices which we believe might be useful in the above context. These integral transformations are known under the name of the Colour-Flavour Transformations. The Colour-Flavor Transformations (CFT) are certain types of integral transformations based on Howe’s dual pair theory. They were was first derived by Zirnbauer [9] in 1996 and since then have became a standard tool in mesoscopic physics, random matrix theory, lattice QCD as well as other fields. The CFTs were originally derived for U(N) and Sp(2N), and later generalized to other classical groups [10, 11, 12, 13]. However, both the fermionic and bosonic variants of the O(N) CFT appearing in the literature [12, 13] do not seem to be reproduced in the correct form, possibly suffering from typos. In this paper, we first correct 1 the bosonic and fermionic versions of the O(N) CFT and then derive the supersymmetric version, which is a new result. Fermionic O(N) CFT