We show that fractional flux from Wilson lines can stabilize the moduli of heterotic string compactifications on Calabi-Yau threefolds. We observe that the Wilson lines used in GUT symmetry breaking naturally induce a fractional flux. When combined with a hidden-sector gaugino condensate, this generates a potential for the complex structure moduli, Kähler moduli, and dilaton. This potential has a supersymmetric AdS minimum at moderately weak coupling and large volume. Notably, the necessary ingredients for this construction are often present in realistic models. We explore the type IIA dual phenomenon, which involves Wilson lines in D6-branes wrapping a three-cycle in a Calabi-Yau, and comment on the nature of the fractional instantons which change the Chern-Simons invariant. 1

In this paper we give an application of equivariant moduli spaces to the study of smooth group actions on certain 4-manifolds. A rich source of examples for such actions is the collection of algebraic surfaces (compact and nonsingular) together with their groups of algebraic automorphisms. From this collection, further examples of smooth but generally nonalgebraic actions can be constructed by an equivariant connected sum along an orbit of isolated points. Given a smooth oriented 4-manifold X which is diffeomorphic to a connected sum of algebraic surfaces, we can ask: (i) which (finite) groups can act smoothly on X preserving the orientation, and (ii) how closely does a smooth action on X resemble some equivariant connected sum of algebraic actions on the algebraic surface factors of X? For the purposes of this paper we will restrict our attention to the simplest case, namely X p2(C) #... # p2(C), a connected sum of n copies of the complex projective plane (arranged so that X is simply connected). Furthermore, ASSUMPTION. All actions will be assumed to induce the identity on H,(X, Z). In previous works [17], [18], [19], we considered problem (i) and a variant of problem (ii) when X p2(C). It turned out that the only finite groups which could act as above on p2(C) were the subgroups of PGL3(C) ([18] and [23] independently). For problem (ii) there are 2 interesting notions weaker than smooth equivalence. If (X, r) is a smooth action, then the isotropy group rx { 9 rclOx x}, x X, acts linearly on the tangent space TxX and we can ask the following. Question (iii) a. Given an action (X, n), is there an equivariant connected sum of actions on p2(C) with the same fixed point data and tangential isotropy representations? Question (iii) b. Given an action (X, re), is there an equivariant connected sum of actions on p2(C) which is n-homotopy equivalent or n-equivariantly homeomorphic to (X, n)? Partial results were obtained on these questions in [17] and [10]: if n acts smoothly on p2(C), inducing the identity on homology, and the action has an

We use the equivariant Yang–Mills moduli space to investigate the relation between the singular set, isotropy representations at fixed points, and permutation modules realized by the induced action on homology for smooth group actions on certain 4–manifolds.

There are several approaches to summarizing a mathematician’s research accomplishments, and each has its advantages and disadvantages. This article is based upon a talk given at Tulane that was aimed at a fairly general audience, including faculty members in other areas and graduate students who had taken the usual entry level courses. As such, it is meant to be relatively nontechnical and to emphasize qualitative rather than quantitative issues; in keeping with this aim, references will be given for some standard topological notions that are not normally treated in entry level graduate courses. Since this was an hour talk, it was also not feasible to describe every single piece of published mathematical work that Terry Lawson has ever written; in particular, some papers like [42] and [50] would require lengthy digressions that are not easily related to the central themes in his main lines of research. Instead, we shall focus on some ways in which Terry’s work relates to an important thread in geometric topology; namely, the passage from studying problems in a given dimension to studying problems in the next dimensions. Qualitatively speaking, there are fairly well-developed theories for very low dimensions and for all sufficiently large dimensions, but between these ranges there are some dimensions in which the answers to many fundamental