Spectral sum rules and Selberg's integral

Using Selberg's integral formula we derive all Leutwyler-Smilga type sum rules for one and two flavors, and for each of the three chiral random matrix ensembles. In agreement with arguments from effective field theory, all sum rules for $N_f = 1$ coincide for the three ensembles. The connection between spectral correlations and the low-energy effective partition function is discussed.Comment: 10 pages, SUNY-NTG-94/

Chiral condensate and Dirac spectrum of one- and two-flavor QCD at nonzero θ\theta-angle

In the $\epsilon$-domain of QCD we have obtained exact analytical expressions for the eigenvalue density of the Dirac operator at fixed $\theta \ne 0$ for both one and two flavors. These results made it possible to explain how the different contributions to the spectral density conspire to give a chiral condensate at fixed $\theta$ that does not change sign when the quark mass (or one of the quark masses for two flavors) crosses the imaginary axis, while the chiral condensate at fixed topological charge does change sign. From QCD at nonzero density we have learnt that the discontinuity of the chiral condensate may move to a different location when the spectral density increases exponentially with the volume with oscillations on the order of the inverse volume. This is indeed what happens when the product of the quark masses becomes negative, but the situation is more subtle in this case: the contribution of the "quenched" part of the spectral density diverges in the thermodynamic limit at nonzero $\theta$, but this divergence is canceled exactly by the contribution from the zero modes. We conclude that the zero modes are essential for the continuity of the chiral condensate and that their contribution has to be perfectly balanced against the contribution from the nonzero modes. Lattice simulations at nonzero $\theta$-angle can only be trusted if this is indeed the case.Comment: 9 pages, 8 figures, Contribution to the Proceedings of Lattice201

Gradient flows without blow-up for Lefschetz thimbles

We propose new gradient flows that define Lefschetz thimbles and do not blow up in a finite flow time. We study analytic properties of these gradient flows, and confirm them by numerical tests in simple examples.Comment: 31 pages, 11 figures, (v2) conclusion part is expande