We propose a definition of mass for characteristic hypersurfaces in asymptotically vacuum space-times with non-vanishing cosmological constant Λ ∈ R ∗ , generalising the definition of Trautman and Bondi for Λ = 0. We show that our definition reduces to some standard definitions in several situations. We establish a balance formula linking the characteristic mass and a suitably defined renormalised volume of the null hypersurface, generalising the positivity identity of one of us (PTC) and Paetz proved when Λ = 0.

We show that specializations of the 4d =2 superconformal index labeled by an integer N is given by TrN where  is the Kontsevich-Soibelman monodromy operator for BPS states on the Coulomb branch. We provide evidence that the states enumerated by these limits of the index lead to a family of 2d chiral algebras N. This generalizes the recent results for the N=−1 case which corresponds to the Schur limit of the superconformal index. We show that this specialization of the index leads to the same integrand as that of the elliptic genus of compactification of the superconformal theory on S2×T2 where we turn on 12N units of U(1)r flux on S2.

We show that the semi-classical analysis of generic Euclidean path integrals necessarily requires complexification of the action and measure, and consideration of complex saddle solutions. We demonstrate that complex saddle points have a natural interpretation in terms of the Picard-Lefschetz theory. Motivated in part by the semi-classical expansion of QCD with adjoint matter on R3 ⇥ S1, we study quantum-mechanical systems with bosonic and fermionic (Grassmann) degrees of freedom with harmonic degenerate minima, as well as (related) purely bosonic systems with harmonic non-degenerate minima. We find exact finite action non-BPS bounce and bion solutions to the holomorphic Newton equations. We find not only real solutions, but also complex solution with non-trivial monodromy, and finally complex multi-valued and singular solutions. Complex bions are necessary for obtaining the correct non-perturbative structure of these models. In the supersymmetric limit the complex solutions govern the ground state properties, and their contribution to the semiclassical expansion is necessary to obtain consistency with the supersymmetry algebra. The multi-valuedness of the action is either related to the hidden topological angle or to the resurgent cancellation of ambiguities. We also show that in the approximate multi-instanton description the integration over the complex quasi-zero mode thimble produces the most salient features of the exact solutions. While exact complex saddles are more difficult to construct in quantum field theory, the relation to the approximate thimble construction suggests that such solutions may be underlying some remarkable features of approximate bion saddles in quantum field theories.

The present article surmmarizes our recent results about Calabi–Yau threefolds with infinite fundamental group. This class of Calabi–Yau manifolds is relatively simple yet rich enough to display the essential complexities of Calabi–Yau geometries, and thus it provides good testing-grounds for general theories and conjectures.

We study free scalar field theory on a graph, which gives rise to a modified version of discrete Green’s function on a graph studied in [8]. We show that this gives rise to a graph invariant, which is closely related to the 2-dimensional Weisfeiler-Lehman algorithm for graph isomorphism testing. We then consider the same theory over the integers, which leads to the consideration of certain quadratic forms over the integers as initiated in [14], associated to the graphs. The quadratic form represented by the combinatorial Laplacian respects a well-behaved wedge sum of graphs, and appears to capture important graph properties reg