Thimble regularization at work: from toy models to chiral random matrix theories

We apply the Lefschetz thimble formulation of field theories to a couple of different problems. We first address the solution of a complex 0-dimensional phi^4 theory. Although very simple, this toy-model makes us appreciate a few key issues of the method. In particular, we will solve the model by a correct accounting of all the thimbles giving a contribution to the partition function and we will discuss a number of algorithmic solutions to simulate this (simple) model. We will then move to a chiral random matrix (CRM) theory. This is a somehow more realistic setting, giving us once again the chance to tackle the same couple of fundamental questions: how many thimbles contribute to the solution? how can we make sure that we correctly sample configurations on the thimble? Since the exact result is known for the observable we study (a condensate), we can verify that, in the region of parameters we studied, only one thimble contributes and that the algorithmic solution that we set up works well, despite its very crude nature. The deviation of results from phase quenched results highlights that in a certain region of parameter space there is a quite important sign problem. In view of this, the success of our thimble approach is quite a significant one.Comment: 33 pages, 8 figures. Some extra references have been added and subsection 3.1 has been substantially expanded. Some extra comments on numerics have also been added in subsection 4.4. Appendix A and appendix B.1 now features some more detail

Fiscal and monetary interaction under monetary policy uncertainty

Despite the recent increasing number of studies on monetary policy uncertainty, its role on the strategic interactions between fiscal and monetary policies has not been fully explored. Our paper aims to fill this gap by tackling this issue by evaluating the consequences produced by multiplicative uncertainty in such a context.Monetary-fiscal policy interactions, paramenter uncertainty, symbiosis, monetary policy attenuation

Convergence of Ricci flow solutions to Taub-NUT

We study the Ricci flow starting at an SU(2) cohomogeneity-1 metric $g_{0}$ on $\mathbb{R}^{4}$ with monotone warping coefficients and whose restriction to any hypersphere is a Berger metric. If $g_{0}$ has bounded Hopf-fiber, curvature controlled by the size of the orbits and opens faster than a paraboloid in the directions orthogonal to the Hopf-fiber, then the flow converges to the Taub-NUT metric $g_{\mathsf{TNUT}}$ in the Cheeger-Gromov sense in infinite time. We also classify the long-time behaviour when $g_{0}$ is asymptotically flat. In order to identify infinite-time singularity models we obtain a uniqueness result for $g_{\mathsf{TNUT}}$.Comment: 49 pages, final version. Accepted in Commun. Partial. Differ. Eq

Type-II singularities and long-time convergence of rotationally symmetric Ricci flows

In this thesis we study singularity formation and long-time behaviour of families of cohomogeneity one Ricci flows. In Chapter 1 we analyse the Ricci flow on R n+1, with n ≥ 2, starting at some complete bounded curvature SO(n + 1)-invariant metric g0. We prove that the solution develops a Type-II singularity and converges to the Bryant soliton after scaling if g0 has no minimal hyperspheres and is asymptotic to a cylinder. This proves a conjecture by Chow and Tian about Perelman’s standard solutions. Conversely, we show that if g0 has no minimal hyperspheres but its curvature decays at infinity, then the solution is immortal. In Chapter 2 we study the Ricci flow on R 4 starting at an SU(2)-cohomogeneity one metric g0 whose restriction to any hypersphere is a Berger metric. We prove that if g0 has no necks and is bounded by a cylinder, then the solution develops a global Type-II singularity and converges to the Bryant soliton when suitably dilated at the origin. This is the first example in dimension n > 3 of a non-SO(n)-invariant Type-II flow converging to an SO(n)-invariant singularity model. We also give conditions for the flow to be immortal and prove that if the solution is Type-I and controlled at spatial infinity, then there exist minimal 3-spheres for times close to the maximal time. In Chapter 3 we focus the analysis on the class of immortal Ricci flows derived in Chapter 2. We prove that if the initial metric has bounded Hopf-fiber, curvature controlled by the size of the orbits and opens faster than a paraboloid in the directions orthogonal to the Hopf-fiber, then the flow converges to the Taub-NUT metric in the Cheeger-Gromov sense in infinite time. We also obtain a uniqueness result for Taub-NUT in a class of collapsed ancient solutions

Convergence of Ricci flow solutions to Taub-NUT

We study the Ricci flow starting at an SU(2) cohomogeneity-1 metric g0 on R4 with monotone warping coefficients and whose restriction to any hypersphere is a Berger metric. If g0 has bounded Hopf-fiber, curvature controlled by the size of the orbits and opens faster than a paraboloid in the directions orthogonal to the Hopf-fiber, then the flow converges to the Taub-NUT metric gTNUT in the Cheeger-Gromov sense in infinite time. We also classify the long-time behaviour when g0 is asymptotically flat. In order to identify infinite-time singularity models we obtain a uniqueness result for gTNUT

Policy Uncertainty, Symbiosis, and the Optimal Fiscal and Monetary Conservativeness

This paper extends a well-known macroeconomic stabilization game between monetary and fiscal authorities introduced by Dixit and Lambertini (American Economic Review, 93: 1522-1542) to multiplicative (policy) uncertainty. We find that even if fiscal and monetary authorities share a common output and inflation target (i.e. the symbiosis assumption), the achievement of the common targets is no longer guaranteed; under multiplicative uncertainty, in fact, a time consistency problem arises unless policymakers� output target is equal to the natural level.Monetary-fiscal policy interactions, uncertainty, symbiosis.

Policy Uncertainty, Symbiosis, and the Optimal Fiscal and Monetary Conservativeness

This paper extends the stabilization game between monetary and fiscal authorities to the case of multiplicative (model) uncertainty. In this context, the “symbiosis assumption”, i.e. fiscal and monetary policy share the same ideal targets, no longer guarantees the achievement of ideal output and inflation, unless the ideal output is equal to its natural level. A time consistency problem arises.Monetary-fiscal policy interactions, uncertainty, symbiosis.