solutions of the Einstein vacuum equations toward a generic spacelike singularity. Starting from fundamental assumptions about the nature of generic spacelike singularities, we derive in a step-by-step manner the cosmological billiard conjecture: we show that the generic asymptotic dynamics of solutions is represented by (randomized) sequences of heteroclinic orbits on the “billiard attractor”. Our analysis rests on two pillars: (i) a dynamical systems formulation based on the conformal Hubblenormalized orthonormal frame approach expressed in an Iwasawa frame; (ii) stochastic methods and the interplay between genericity and stochasticity. Our work generalizes and improves the level of rigor of previous work by Belinskii, Khalatnikov, and Lifshitz; furthermore, we establish that our approach and the Hamiltonian approach to “cosmological billiards”, as elaborated by Damour, Hennaux, and Nicolai, can be viewed as yielding “dual” representations of the asymptotic dynamics.

We carry out an exact worldsheet computation of tree level threepoint correlators of chiral operators in type IIB string theory on AdS₃ × S³ × T⁴ with NS–NS flux. We present a simple representation for the string chiral operators in the coordinate basis of the dual boundary CFT. Striking cancelations occur between the three-point functions of the $\mathbb{H}_+^{3}$ and the SU(2) WZW models which result in a simple factorized form for the final correlators. We show, by fixing a single free parameter in the $\mathbb{H}_+^{3}$ WZW model, that the fusion rules and the structure constants of the N = 2 chiral ring in the bulk are in precise agreement with earlier computations in the boundary CFT of the symmetric product of T⁴ at the orbifold point in the large N limit.

We solve Bershadsky–Cecotti–Ooguri–Vafa (BCOV) holomorphic anomaly equation to determine the higher genus Gromov–Witten invariants ($g\geq 5$) of the derived equivalent Calabi–Yau 3-folds, which are of the appropriate codimensions in the Grassmannian Gr(2, 7) and the Pfaffian Pf(7).Let $q\geq n-p$

We describe a categorical framework for the classification of D-branes on non-commutative spaces using techniques from bivariant K-theory of C*-algebras. We present a new description of bivariant K-theory in terms of non-commutative correspondences which is nicely adapted to the study of T-duality in open string theory. We systematically use the diagram calculus for bivariant K-theory as detailed in our previous paper [12]. We explicitly work out our theory for a number of examples of noncommutative manifolds.

We explain the B-model origin of extended Picard–Fuchs equations satisfied by the D-brane superpotential on compact Calabi–Yau threefolds. The domainwall tension is identified with a Poincar´e normal function — a transversal holomorphic section of the Griffiths intermediate Jacobian — via the Abel–Jacobi map. Within this formalism, we derive the extended Picard–Fuchs equation associated with the mirror of the real quintic.