Finding Ricci-flat (Calabi-Yau) metrics is a long standing problem in geometry with deep implications for string theory and phenomenology.A new attack on this problem uses neural networks to engineer approximations to the Calabi-Yau metric within a given Kähler class.In this paper we investigate numerical Ricci-flat metrics over smooth and singular K3 surfaces and Calabi-Yau threefolds.Using these Ricci-flat metric approximations for the Cefalú family of quartic twofolds and the Dwork family of quintic threefolds, we study characteristic forms on these geometries.We observe that the numerical stability of the numerically computed topological characteristic is heavily influenced by the choice of the neural network model, in particular, we briefly discuss a different neural network model, namely spectral networks, which correctly approximate the topological characteristic of a Calabi-Yau.Using persistent homology, we show that high curvature regions of the manifolds form clusters near the singular points.For our neural network approximations, we observe a Bogomolov-Yau type inequality 3c 2 ≥ c 2 1 and observe an identity when our geometries have isolated A 1 type singularities.We sketch a proof that χ(X ∖ Sing X) + 2 |Sing X| = 24 also holds for our numerical approximations.

We consider the formation of trapped surfaces in the evolution of the Einstein-scalar field system without symmetries. To this end, we follow An's strategy to analyse the formation of trapped surfaces in vacuum and for the Einstein-Maxwell system. Accordingly, we set up a characteristic initial value problem (CIVP) for the Einstein-Scalar system with initial data given on two intersecting null hypersurfaces such that on the incoming slince the data is Minkowskian whereas on the outgoing side no symmetries are assumed. We obtain a scale-critical semi-global existence result by assigning a signature for decay rates to both the geometric quantities and the scalar field. The analysis makes use of a gauge due to J. Stewart and an adjustment of the Geroch-Held-Penrose (GHP) formalism, the T-weight formalism, which renders the connection between the Newman-Penrose (NP) quantities and the PDE analysis systematic and transparent.

We propose a generalized finiteness principle for physical theories, in terms of the concept of tameness in mathematical logic. A tame function or space can only have a finite amount of structure, in a precise sense which we explain. Tameness generalizes the notion of an analytic function to include certain non-analytic limits, and we show that this includes many limits which are known to arise in physics. For renormalizable quantum field theories, we give a general proof that amplitudes at each order in the loop expansion are tame functions of the external momenta and the couplings. We then consider a variety of exact non-perturbative results and show that they are tame but only given constraints on the UV definition of the theory. This provides further evidence for the recent conjecture of the second author that all effective theories that can be coupled to quantum gravity are tame. We also discuss whether renormalization group flow is tame, and comment on the applicability of our results to effective theories.

We consider generalizations of equivariant volumes of abelian GIT quotients obtained as partition functions of 1d, 2d, and 3d supersymmetric GLSM on S 1 , D 2 and D 2 × S 1 , respectively. We define these objects and study their dependence on equivariant parameters for non-compact toric Kähler quotients. We generalize the finite-difference equations (shift equations) obeyed by equivariant volumes to these partition functions. The partition functions are annihilated by differential/difference operators that represent equivariant quantum cohomology/K-theory relations of the target and the appearance of compact divisors in these relations plays a crucial role in the analysis of the nonequivariant limit. We show that the expansion in equivariant parameters contains information about genus-zero Gromov-Witten invariants of the target.

Topological defect lines (TDLs) are extended line operators which act on the Hilbert space of two-dimensional CFTs and satisfy non-trivial fusion algebras when forming junctions. Among the most interesting fusion algebras are the so-called Tambara-Yamagami (TY) fusion categories which are realized in (bosonized) Parafermionic CFTs. The corresponding TY[Z k ]-categories have been explicitly realized for the cases k = 2, k = 3, and k = 4 together with the action of the defect lines on the Hilbert space of the corresponding CFTs. For each of the cases, different methods have been used in the previous literature. In the current paper, we present a unified framework for finding the TDLs in bosonized Parafermionic CFTs. Our approach relies on generalizing several previously used methods by introducing the notion of an extended S matrix. We apply the method to the cases k = 2 to k = 5 to extract corresponding TDL fusion algebras.

We derive the framing anomaly of four-dimensional holomorphic-topological Chern-Simons theory formulated on the product of a topological surface and the complex plane. We show that the presence of this anomaly allows one to couple four-dimensional Chern-Simons theory to holomorphic field theories with Kac-Moody symmetry, where the Kac-Moody level k is critical k = −h ∨ . Applying this result to a holomorphic sigma model into a complex coadjoint orbit, we derive that four-dimensional Chern-Simons theory admits holomorphic monodromy defects.