Partition functions in superstring theory and SQCD

Abstract

In this thesis we apply the methods of partition functions to massive superstring spectra and the moduli spaces, or spaces of zero-energy configurations, of supersymmetric QCD gauge theories. In the first part of this thesis we consider the massive covariant perturbative superstring spectra of compactifications of the type I open superstring preserving 4, 8 or 16 supercharges. There are an enormous number of ways in which the required amount of symmetry can be obtained, but here we concentrate on the ‘universal’ states that are present in every possible compactification preserving that amount of supersymmetry. For each super-Poincare representation we derive the multiplicity generating function, or the power series counting the number of times that representation occurs at each mass level, and from these we derive empirically the stable pattern or leading Regge trajectory that these multiplicity generating functions approach in the limit of large spin. For the mathematically tractable and phenomenologically relevant case of 4 supercharges we also derive these power series analytically and see that they agree with the empirical ones. In the second part we introduce the type of partition functions called Hilbert series, which count the number of algebraically or linearly independent polynomials at each graded level of a graded algebraic structure such as a (graded) ring, module or ideal. In supersymmetric gauge theories the algebraic structure is the chiral ring which is generated by the gauge-invariant operators of the theory. The specific theories we consider are supersymmetric generalizations of QCD, or SQCD, with exceptional or related (by sequence or folding of the Dynkin diagram or Higgsing) gauge groups with specified numbers of flavours of matter in specific representations. We show, as for theories with classical gauge groups, that the moduli spaces are Calabi-Yau manifolds and also demonstrate relations between the Hilbert series of SQCD theories related by Higgsing on one or more flavours of matter in specific representations.