Quantum non-local charges are central to the quantum integrability of a sigma model. In this paper we study the quantum consistency and UV finiteness of non-local charges of string theory in $$. We use the pure spinor formalism. We develop the near-flat space expansion of the transfer matrix and calculate the one-loop divergences. We find that the logarithmic divergences cancel at the level of one loop. This gives strong support to the quantum integrability of the full string theory. We develop a calculational setup for the renormalization group analysis of Wilson line type of operators on the string worldsheet.

In this paper we classify all free actions of finite groups on Calabi–Yau complete intersection of four quadrics in $$, up to projective equivalence. We get some examples of smooth Calabi–Yau three-folds with large nonabelian fundamental groups. We also observe the relation between some of these examples and moduli of polarized abelian surfaces.

The notion of causal boundary $$ for a strongly causal spacetime $$ has been a controversial topic along last decades: on one hand, some attempted definitions were not fully consistent, on the other, there were simple examples where an open conformal embedding $$ could be defined, but the corresponding conformal boundary $$ disagreed drastically with the causal one. Nevertheless, the recent progress in this topic suggests that a final option for $$ is available in most cases. Our study has two parts:

(I) To give general arguments on a boundary in order to ensure that it is admissible as a causal boundary at the three natural levels, i.e., as a point set, as a chronological space and as a topological space. Then, the essential uniqueness of our choice is stressed, and the relatively few admissible alternatives are discussed.

(II) To analyze the role of the conformal boundary $$. We show that, in general, $$ may present a very undesirable structure. Nevertheless, it is well behaved under certain general assumptions, and its accessible part $$ agrees with the causal boundary.

This study justifies both boundaries. On one hand, the conformal boundary $$, which cannot be defined for a general spacetime but is easily computed in particular examples, appears now as a special case of the causal boundary. On the other, the new redefinition of the causal boundary not only is free of inconsistencies and applicable to any strongly causal spacetime, but also recovers the expected structure in the cases where a natural simple conformal boundary is available. The cases of globally hyperbolic spacetimes and asymptotically conformally flat ends are especially studied.

We generalize the BF theory action to the case of a general Lie crossed module $$, where $$ and $$ are non-abelian Lie groups. Our construction requires the existence of $$-invariant non-degenerate bilinear forms on the Lie algebras of $$ and $$ and we show that there are many examples of such Lie crossed modules by using the construction of crossed modules provided by short chain complexes of vector spaces. We also generalize this construction to an arbitrary chain complex of vector spaces, of finite type. We construct two gauge-invariant actions for 2-flat and fake-flat 2-connections with auxiliary fields. The first action is of the same type as the BFCG action introduced by Girelli, Pfeiffer and Popescu for a special class of Lie crossed modules, where $$ is abelian. The second action is an extended BFCG action which contains an additional auxiliary field. However, these two actions are related by a field redefinition. We also construct a three-parameter deformation of the extended BFCG action, which we believe to be relevant for the construction of non-trivial invariants of knotted surfaces embedded in the four-sphere.

We prove that for spacetimes solving the Einstein–Maxwell (EM) equations, the electromagnetic field contributes at highest order to the nonlinear memory effect of gravitational waves. In Nonlinear nature of gravitation and gravitational-wave experiments, Christodoulou showed that gravitational waves have a nonlinear memory. He discussed how this effect can be measured as a permanent displacement of test masses in a laser interferometer gravitational-wave detector. Christodoulou derived a precise formula for this permanent displacement in the Einstein vacuum (EV) case. We prove in Theorem 2.6 that for the EM equations this permanent displacement exhibits a term coming from the electromagnetic field. This term is at the same highest order as the purely gravitational term that governs the EV situation. On the other hand, in Section 3, we show that to leading order, the presence of the electromagnetic field does not change the instantaneous displacement of the test masses. Following the method introduced by Christodoulou in Nonlinear nature of gravitation and gravitational-wave experiments, and asymptotics derived by Zipser in The global nonlinear stability of the trivial solution of the Einstein–Maxwell equations and Extensions of the stability theorem of the Minkowski space in general relativity: Solutions of the Einstein–Maxwell equations, we investigate gravitational radiation at null infinity in spacetimes solving the EM equations. We study the Bondi mass loss formula at null infinity derived in Extensions of the stability theorem of the Minkowski space in general relativity: Solutions of the Einstein–Maxwell equations. We show that the mass loss formula from Extensions of the stability theorem of the Minkowski space in general relativity: Solutions of the Einstein–Maxwell equations is compatible with the one in Bondi coordinates obtained in Gravitational waves in general relativity X: Asymptotic expansions for the Einstein–Maxwell field. And we observe that the presence of the electromagnetic field increases the total energy radiated to infinity up to leading order. Moreover, we compute the limit of the area radius at null infinity in Theorem 2.7.

We consider plane-symmetric spacetimes satisfying Einstein’s field equations with positive cosmological constant, when the matter is a fluid whose pressure is equal to its mass-energy density (i.e., a so-called stiff fluid). We study the initial-value problem for the associated Einstein equations and establish a global existence result. The late-time asymptotics of solutions is also rigorously derived, and we conclude that the spacetime approaches the de Sitter spacetime while the matter disperses asymptotically. A technical difficulty dealt with here lies in the fact that solutions may contain vacuum states as well as velocities approaching the speed of light, both possibilities leading to singular behavior in the evolution equations.

In this paper we discuss quantization of the Fayet–Iliopoulos parameter in supergravity theories with altered nonperturbative sectors, which were recently used to argue a fractional quantization condition. Nonlinear sigma models with altered nonperturbative sectors are the same as nonlinear sigma models on special stacks known as gerbes. After reviewing the existing results on such theories in two dimensions, we discuss examples of gerby moduli “spaces” appearing in four-dimensional field theory and string compactifications, and the effect of various dualities. We discuss global topological defects arising when a field or string theory moduli space has a gerbe structure. We also outline how to generalize the results of Bagger–Witten and more recent authors on quantization issues in supergravities from smooth moduli spaces to smooth moduli stacks, focusing particular attention on stacks that have gerbe structures.