Using the celebrated Witten–Kontsevich theorem, we prove a recursive formula of the $n$-point functions for intersection numbers on moduli spaces of curves. It has been used to prove the Faber intersection number conjecture and motivated us to find some conjectural vanishing identities for Gromov–Witten invariants. The latter has been proved recently by Liu and Pandharipande. We also give a combinatorial interpretation of $n$-point functions in terms of summation over binary trees.

Despite much recent progress in model building with $D$-branes, it has been problematic to find a completely convincing explanation of gauge coupling unification. We extend the class of models by considering $F$-theory compactifications, which may incorporate unification more naturally. We explain how to derive the charged chiral spectrum and Yukawa couplings in $N = 1$ compactifications of $F$-theory with $G$-flux. In a class of models which admit perturbative heterotic duals, we show that the $F$-theory and heterotic computations match.

The worldsheet sigma-model of the superstring in $AdS_5 × S^5$ has a one-parameter family of flat connections parametrized by the spectral parameter. The corresponding Wilson line is not BRST invariant for an open contour, because the BRST transformation leads to boundary terms. These boundary terms define a cohomological complex associated to the endpoint of the contour. We study the cohomology of this complex for Wilson lines in some infinite-dimensional representations. We find that for these representations the cohomology is nontrivial at the ghost number 2. This implies that it is possible to define a BRST invariant open Wilson line. The central point in the construction is the existence of massless vertex operators transforming exactly covariantly under the action of the global symmetry group. In flat space massless vertices transform covariantly only up to adding BRST exact terms. But in AdS we show that it is possible to define vertices so that they transform exactly covariantly.

Starting from the four normed division algebras — the real numbers, complex numbers, quaternions and octonions — a systematic procedure gives a 3-cocycle on the Poincaré Lie superalgebra in dimensions 3, 4, 6 and 10. A related procedure gives a 4-cocycle on the Poincaré Lie superalgebra in dimensions 4, 5, 7 and 11. In general, an$ (n + 1)$-cocycle on a Lie superalgebra yields a “Lie $n$-superalgebra”: that is, roughly speaking, an $n$-term chain complex equipped with a bracket satisfying the axioms of a Lie superalgebra up to chain homotopy. We thus obtain Lie 2-superalgebras extending the Poincaré superalgebra in dimensions 3, 4, 6 and 10, and Lie 3-superalgebras extending the Poincaré superalgebra in dimensions 4, 5, 7 and 11. As shown in Sati, Schreiber and Stasheff’s work on higher gauge theory, Lie 2-superalgebra connections describe the parallel transport of strings, while Lie 3-superalgebra connections describe the parallel transport of 2-branes. Moreover, in the octonionic case, these connections concisely summarize the fields appearing in 10- and 11-dimensional supergravity.

We discuss a natural form of Ricci-flow conjugation between two distinct general relativistic data sets given on a compact $n \ge 3$-dimensional manifold $\Sigma$. We establish the existence of the relevant entropy functionals for the matter and geometrical variables, their monotonicity properties, and the associated convergence in the appropriate sense. We show that in such a framework there is a natural mode expansion generated by the spectral resolution of the Ricci conjugate Hodge–DeRham operator. This mode expansion allows one to compare the two distinct data sets and gives rise to a computable heat kernel expansion of the fluctuations among the fields defining the data. In particular, this shows that Ricciflow conjugation entails a natural form of $L^2$ parabolic averaging of one data set with respect to the other with a number of desirable properties: (i) It preserves the dominant energy condition; (ii) It is localized by a heat kernel whose support sets the scale of averaging; (iii) It is characterized by a set of balance functionals, that allow the analysis of its entropic stability.

We construct infinite number of conserved nonlocal charges for type IIB superstring on the $\mathrm{AdS}_5×S^5$ space in the conformal gauge without assuming any $\kappa$ gauge fixing, and show that they satisfy the super Yangian algebra. The resultant algebra is the same as our previous work, where a special gauge was assumed in such a way that the Noether current satisfies a flatness condition. However the flatness condition for the Noether current of a superstring on the AdS space is broken in general. We show that the anomalous contribution is absorbed into the current where fermionic constraints play an essential role, and a resultant conserved nonlocal charge has different expression satisfying the same super Yangian algebra.

Given a smooth complex projective variety $X$ and a smooth divisor $D$ on $X$, we prove the existence of Hermitian–Einstein connections, with respect to a Poincaré-type metric on $X \setminus D$, on polystable parabolic principal Higgs bundles with parabolic structure over $D$, satisfying certain conditions on their restriction to $D$.