Related papers: Gopakumar-Vafa invariants associated to $cA_n$ sin…

Conformally variational Riemannian invariants (CVIs), such as the scalar curvature, are homogeneous scalar invariants which arise as the gradient of a Riemannian functional. We establish a wide range of stability and rigidity results…

This is the fourth in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global conformal invariants''; these are defined to be conformally invariant integrals of geometric scalars.…

We discuss some "folklore" results on categorical crepant resolutions for varieties with quotient singularities.

We consider the weak coupling limit of F-theory in the presence of non-Abelian gauge groups implemented using the traditional ansatz coming from Tate's algorithm. We classify the types of singularities that could appear in the weak coupling…

We give criteria for which a principal curvature becomes a bounded $C^\infty$-function at non-degenerate singular points of wave fronts by using geometric invariants. As applications, we study singularities of parallel surfaces and extended…

We formulate a Crepant Resolution Correspondence for open Gromov-Witten invariants (OCRC) of toric Calabi-Yau orbifolds by viewing the open theories as sections of Givental's symplectic vector space and the correspondence as a linear map of…

There is considerable current interest in applications of generalised Lie algebras graded by an abelian group $\Gamma$ with a commutative factor $\omega$. This calls for a systematic development of the theory of such algebraic structures.…

In previous articles we have studied the A_n tt*-Toda equations (topological-antitopological fusion equations of Toda type) of Cecotti and Vafa, giving details mainly for n=3. Here we give a proof of the existence and uniqueness of global…

Let $(F,J,\omega)$ be an almost K\"ahler manifold, $\alpha$ a $J$-holomorphic action of a compact Lie group $\hat K$ on $F$, and $K$ a closed normal subgroup of $\hat K$ which leaves $\omega$ invariant. We introduce gauge theoretical…

We construct a combinatorial invariant of 3-orbifolds with singular set a link that generalizes the Turaev torsion invariant of 3-manifolds. We give several gluing formulas from which we derive two consequences. The first is an…

Conifold geometries have recieved a lot of attention in string theory and string-inspired cosmology recently, in particular the Klebanov-Strassler background that is known as the "warped throat". It is our intention in this article to give…

This article studies the fundamental invariants of 2--nondegenerate CR geometries with simple models. We show that there are two sources of these invariants. The first source is the harmonic curvature of the parabolic geometry that appears…

We construct a covariant version of the Tolman-Oppenheimer-Volkoff equations in the case of isotropic sources. The new equations make evident the mathematical problems in the determination of interior solutions of relativistic stellar…

We survey briefly the definition of the Rozansky-Witten invariants, and review their relevance to the study of compact hyperkahler manifolds. We consider how various generalisations of the invariants might prove useful for the study of…

We consider the algebra of invariants of $d$-tuples of $n\times n$ matrices under the action of the orthogonal group by simultaneous conjugation over an infinite field of characteristic $p$ different from two. It is well-known that this…

We study solutions to conformally invariant equations with isolated singularties.

We define a Chern--Simons invariant of connections on stably trivial vector bundles over smooth manifolds, taking values in $3$-forms modulo closed forms with integral cohomology class. We show an additivity property of this invariant for…

In this dissertation we discuss various issues concerning application of the Dijkgraaf-Vafa (DV) conjecture to the study of supersymmetric gauge theories. The DV approach is very powerful in that it provides a systematic way of computing…

We offer a new construction of Lagrangian submanifolds for the Gopakumar-Vafa conjecture relating the Chern-Simons theory on the 3-sphere and the Gromov-Witten theory on the resolved conifold. Given a knot in the 3-sphere its conormal…

A new class of integrable two-dimensional dilaton gravity theories, in which scalar matter fields satisfy the Toda equations, is proposed. The simplest case of the Toda system is considered in some detail, and on this example we outline how…