Topological string theory with twistor space as the target makes visible some otherwise dificult to see properties of perturbative Yang–Mills theory. But left-right symmetry, which is obvious in the standard formalism, is highly unclear from this point of view. Here we prove that tree diagrams computed from connected D-instanton configurations are parity-symmetric. The main point in the proof also works for loop diagrams.

In [1], a new approach was suggested for quantising space-time, or space. This involved developing a procedure for quantising a system whose configuration space--or history-theory analogue--is the set of objects, Ob(Q), in a (small) category Q. The quantum states in this approach are cross-sections of a bundle A is in K[A] of Hilbert spaces over Ob(Q). The Hilbert spaces K[A], A are in Ob(Q)], depend strongly on the object A, and have to be chosen so as to get an irreducible, faithful, representation of the basic `category quantisation monoid'. In the present paper, we develop a different approach in which the state vectors are complex-valued functions on the set of arrows in Q. This throws a new light on the Hilbert bundle scheme: in particular, we recover the results of that approach in the, physically important, example when Q is a small category of finite sets.

In this paper we continue previous work on counting open string states between D-branes by considereing open strings between D-branes with nonzero Higgs vevs, and in particular, nilpotent Higgs vevs, as arise, for example, when studying D-branes in orbifolds. Ordinarily Higgs vevs can be interpreted as moving the D-brane, but nilpotent Higgs vevs have zero eigenvalues, and so their interpretation is more interesting - for example, they often correspond to nonreduced schemes, which furnishes an important link in understanding old results relating classical D-brane moduli spaces in orbifolds to Hilbert schemes, resolutions of quotient spaces, and the McKay correspondence. We give a sheaf-theoretic description of D-branes with Higgs vevs, including nilpotent Higgs vevs, and check that description by noting that Ext groups between the sheaves modelling the D-branes, do in fact correctly count open string states. In particular, our analysis expands the types of sheaves which admit on-shell physical interpretations, which is an important step for making derived categories useful for physics.

We discuss several aspects of the relation between asymptotically AdS and asymptotically dS spacetimes including: the continuation between these types of spaces, the global stability of asymptotically dS spaces and the structure of limits within this class, holographic renormalization, and the maximal mass conjecture of Balasubramanian-deBoer-Minic.

Recently, Witten showed that there is a natural action of the group SL(2, Z) on the space of 3 dimensional conformal field theories with U (1) global symmetry and a chosen coupling of the symmetry current to a background gauge field on a 3–fold N. He further argued that, for a class of conformal field theories, in the nearly Gaussian limit, this SL(2, Z) action may be viewed as a holographic image of the well–known SL(2, Z) Abelian duality of a pure U (1) gauge theory on AdS–like 4–folds M bounded by N, as dictated by the AdS/CFT correspondence. However, he showed that explicitly only for the generator T; for the generator S, instead, his analysis remained conjectural. In this paper, we propose a solution of this problem. We derive a general holographic formula for the nearly Gaussian generating functional of the correlators of the symmetry current and, using this, we show that Witten's conjecture is indeed correct when N = S³. We further identify a class of homology 3–spheres N for which Witten's conjecture takes a particular simple form.