Related papers: Lagrangians, SO(3)-instantons and the Atiyah-Floer…
Given a symplectic manifold M, we consider a category with objects finite ordered families of Lagrangian submanifolds of M (subject to certain additional constraints) and with morphisms Lagrangian cobordisms relating them. We construct a…
We present an alternate definition of the mod {\bf Z} component of the Atiyah-Patodi-Singer $\eta$ invariant associated to (not necessary unitary) flat vector bundles, which identifies explicitly its real and imaginary parts. This is done…
We discuss some physical issues related to the K-theoretic classification of D-brane charges, putting an emphasis on the role of D-brane instantons. The relation to D-instantons provides a physical interpretation to the mathematical…
We construct a family of points on the Lagrangian cone of a partial flag bundle associated to a (possibly non-split) vector bundle from any Weyl-invariant $I$-function of a prequotient. This result can be seen as the nonabelian analogue of…
This paper discusses the cobordism of Lagrangian submanifolds (in the monotone setting) and structures it as a category that is related in a functorial way to an appropriate (derived) Fukaya category. Are also discussed obstructions to…
We give a necessary and sufficient condition for lagrangians in a symplectic vector bundle to be deformed stably into transversal lagrangians. In the case of three lagrangians, we show that the associated Grothendieck group can be…
The construction of Atiyah, Drinfeld, Hitchin, and Manin [ADHM78] provided complete description of all instantons on Euclidean four-space. It was extended by Kronheimer and Nakajima to instantons on ALE spaces, resolutions of orbifolds…
We construct partial category-valued field theories in (2+1)-dimensions using Lagrangian Floer theory in moduli spaces of central-curvature unitary connections with fixed determinant of rank r and degree d where r,d are coprime positive…
We study the compact monotone Fukaya category of $T^*S^n$, for $n\geq 2$, and show that it is split-generated by two classes of objects: the zero-section $S^n$ (equipped with suitable bounding cochains) and a 1-parameter family of monotone…
In this note we give a short proof of Arnold's conjecture for the zero section of a cotangent bundle of a closed manifold. The proof is based on some basic properties of Lagrangian spectral invariants from Floer theory.
In a four-dimensional space, I shall construct all of the conformally invariant scalar-tensor field theories, which are flat space compatible; i.e., well-defined and differentiable when evaluated for a flat metric tensor and constant scalar…
We propose a single-step non-generational conjecture of all first class constraints,(involving only variables compatible with canonical Poisson brackets), for a realistic gauge singular field theory. We verify our proposal for the free…
Let G be a complex reductive group and let C be a smooth curve of genus at least one. We prove a converse to a theorem of Atiyah-Bott concerning the stratification of the space of holomorphic G-bundles on C. In case the genus of C is one,…
We state an improved version of the conjecture of Langlands and Rapoport, and we prove the conjecture for a large class of Shimura varieties. In particular, we obtain the first proof of the (original) conjecture for Shimura varieties of…
Jieao Song recently conjectured a formula for the class of a Lagrangian plane on a hyperk\"ahler variety of $K3^{[n]}$-type in terms of the class of a line on it. We give a proof of this conjecture if the line class is primitive.
(Generalizes theorem of Atiyah and Mumford.)
The recent proposal by Hawking and Turok for obtaining an open inflationary universe from singular instantons makes use of low-energy effective Lagrangians describing gravity coupled to scalars and non-propagating antisymmetric tensors. In…
Associated to each finite dimensional linear representation of a group G, there is a vector bundle over the classifying space BG. This construction was studied extensively for compact groups by Atiyah and Segal. We introduce a homotopy…
In this paper, which is a natural continuation of our previous paper math.DG/0504557, we describe some special Lagrangians of cohomogeneity one in the resolved conifold. Our main result gives a foliation of the resolved conifold by…
On the basis of Liouville theorem the generalization of the Nambu mechanics is considered. For three-dimensional phase space the concept of vector hamiltonian and vector lagrangian is entered.