Abstract
The standard U(N) and SU(N) integrals are calculated in the large N limit. Our main finding is that for an important class of integrals this limit is different for two groups. We describe the critical behaviour of SU(N) models and discuss implications of our results for the large N behaviour of SU(N) lattice gauge theories at finite temperatures and non-zero baryon chemical potential. The key ingredients of our approach are 1) expansion of the integrals into a sum over irreducible representations and 2) calculation of sums over partitions of r of products of dimensions of two different representations of a symmetric group Sr.
Highlights
Let U ∈ G, where G = U(N), SU(N) and A, B are two arbitrary N × N matrices
In this paper we evaluate the following integral over G in the large N limit for some particular choices of A and B
Which can be solved exactly and which shows that the large N limit differs for two groups
Summary
If one takes A = eμI and B = e−μI the expression in the exponent of the integrand is nothing but the leading term in the expansion of the quark determinant at large masses, where TrU can be interpreted as the Polyakov loop Such broad applications make it necessary to have a deep understanding of the properties of the integral in different regimes. N limit of sums over all partitions λ This is simple for examples we consider in this paper when the sum over λ can be reduced to the calculation of the following expression. We explore the simplest case with B = 0 and an arbitrary matrix A. which can be solved exactly and which shows that the large N limit differs for two groups.
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