A simple computation of Z(2k)

We present a new simple proof of Euler’s formulas for Z(2k), where k= 1,2,3,.... The computation is done using only the defining properties of the Bernoulli polynomials and summing a telescoping series, and the same method also yields integral formulas for Z(2k+ 1)

Morse-Bott functions on orthogonal groups

We make a detailed study of various (quadratic and linear) Morse-Bott trace functions on the orthogonal groups $O(n)$. We describe the critical loci of the quadratic trace function Tr$(AXBX^T)$ and determine their indices via perfect fillings of tables associated with the multiplicities of the eigenvalues of $A$ and $B$. We give a simplified treatment of T. Frankel's analysis of the linear trace function on $SO(n)$, as well as a combinatorial explanation of the relationship between the mod $2$ Betti numbers of $SO(n)$ and those of the Grassmannians $\mathbb{G}(2k,n)$ obtained from this analysis. We review the basic notions of Morse-Bott cohomology in a simple case where the set of critical points has two connected components. We then use these results to give a new Morse-theoretic computation of the mod $2$ Betti numbers of $SO(n)$.Comment: 28 page

A renormalization group computation of the critical exponents of hierarchical spin glasses

The infrared behaviour of a non-mean field spin-glass system is analysed, and the critical exponent related to the divergence of the correlation length is computed at two loops within the epsilon-expansion technique with two independent methods. Both methods yield the same result confirming that the infrared behaviour of the theory if well-defined and the underlying ideas of the Renormalization Group hold also in such non-mean field disordered model. By pushing such calculation to high orders in epsilon, a consistent and predictive non-mean field theory for such disordered system could be established