On the geometry of isomonodromic deformations

This note examines the geometry behind the Hamiltonian structure of isomonodromy deformations of connections on vector bundles over Riemann surfaces. The main point is that one should think of an open set of the moduli of pairs (V,∇)$(V,\nabla )$ of vector bundles and connections as being obtained by “twists” supported over points of a fixed vector bundle _V0$V_0$ with a fixed connection _∇0${\nabla }_0$; this gives two deformations, one, isomonodromic, of (V,∇)$(V,\nabla )$, and another induced from the isomonodromic deformation of (_V0,_∇0)$(V_0,{\nabla }_0)$. The difference between the two will be Hamiltonian.     

The twistor geometry of the covariantly quantized Brink-Schwarz superparticle

We perform the Lorentz-covariant BFV-BRST quantization of the Brink-Schwarz (BS) superparticle by reducing it to a system whose constraints are all covariant, first class and independent (CFI).This procedure requires the presence of certain auxiliary pure gauge variables which are then recognized as (super-)twistors. In a particular covariant gauge, the system reduces to a purely (super-)twistor one.We present several extensions of this 4-dimensional construction to higher dimensions.     

The generation problem and the symplectic group

The generation structure of quarks and leptons is studied in a series of extended electroweak models. They are gauge models constructed from subgroups of left-right symmetric SU(4) × S_P^L(6) × S_P^R(6). In particular, SU_C(3) × S_P^L(6) × U_Y(1), as well as SU_C(3) × SU_L(3) × SU_R(3) × U_X(1), are investigated in greater detail. It is shown that models based on such subgroups yield massless first generation fermions at the tree level. They acquire masses from those of the third generation via two-loop radiative corrections. Moreover, since such corrections are induced by charged scalar fields, it is found that there is a natural correlation between the two apparently conflicting inequalities m_u < m_d and m_t > m_b.     

Projective geometry,Lagrangian subspaces,and twistor theory

The use of projective geometry for the characterization of Lagrangian subspaces and maps among them is of particular interest for the symplectic manifold that is twistor space. We raise some conjectures on how these should be interpreted on the space-time manifold by making use of the structure of projective twistor space.     

An interpretation of some Hitchin Hamiltonians in terms of isomonodromic deformation

This paper deals with moduli spaces of framed principal bundles with connections with irregular singularities over a compact Riemann surface. These spaces have been constructed by Boalch by means of an infinite-dimensional symplectic reduction. It is proved that the symplectic structure induced from the Atiyah–Bott form agrees with the one given in terms of hypercohomology. The main results of this paper adapt work of Krichever and of Hurtubise to give an interpretation of some Hitchin Hamiltonians as yielding Hamiltonian vector fields on moduli spaces of irregular connections that arise from differences of isomonodromic flows defined in two different ways. This relies on a realization of open sets in the moduli space of bundles as arising via Hecke modification of a fixed bundle.     

On the symplectic structure of general relativity

The relation between the symplectic structures on the canonical and radiative phase spaces of general relativity is exhibited.Alfred P. Sloan Research Fellow. Supported in part by the NSF contract PHY80-08155 and by a grant from the Syracuse University Research and Equipment FundSupported in part by crédits ministériels, tranche spéciale     

Three-dimensional topological field theory and symplectic algebraic geometry I

We study boundary conditions and defects in a three-dimensional topological sigma-model with a complex symplectic target space X (the Rozansky–Witten model). We show that boundary conditions correspond to complex Lagrangian submanifolds in X   equipped with complex fibrations. The set of boundary conditions has the structure of a 2-category; morphisms in this 2-category are interpreted physically as one-dimensional defect lines separating parts of the boundary with different boundary conditions. This 2-category is a categorification of the Z₂${\mathbb{Z}}_2$-graded derived category of X; it is also related to categories of matrix factorizations and a categorification of deformation quantization (quantization of symmetric monoidal categories). In Appendix B we describe a deformation of the B-model and the associated category of branes by forms of arbitrary even degree.     

BRST structures and symplectic geometry on a class of supermanifolds

By investigating the symplectic geometry and geometric quantization of a class of supermanifolds, we exhibit BRST structures for a certain kind of algebras. We discuss the undeformed andq-deformed cases in the classical as well as in the quantum cases.Alexander von Humboldt fellow. On leave from Institute of Physics, Chinese Academy of Sciences, Beijing.     

Integrable systems and invariant curve flows in centro-equiaffine symplectic geometry

In this paper, the theory for curves in centro-equiaffine symplectic geometry is established. Integrable systems satisfied by the curvatures of curves under inextensible motions in centro-equiaffine symplectic geometry are identified. It is shown that certain non-stretching invariant curve flows in centro-equiaffine symplectic geometry are closely related to the matrix KdV equations and their extension.     

Symplectic capacities and the geometry of uncertainty: The irruption of symplectic topology in classical and quantum mechanics

This paper aims at conducting an analysis of various uncertainty principles from a topological point of view where the notion of symplectic capacity plays a key role. The existence of symplectic capacities follows from a deep theorem of symplectic topology, Gromov’s non-squeezing theorem, which was discovered in the mid 1980’s, and which has led to numerous developments whose applications to Physics are not fully understood or exploited at the time of writing. We will show that the notion of symplectic non-squeezing contains, as a watermark, not only the Robertson–Schrödinger uncertainty relations (and a classical version thereof), but also Hardy’s uncertainty principle for a function and its Fourier transform. This observation will allow us to formulate the characterization of positivity for density matrices in a topological way. We also address some open questions and conjectures, whose solution cannot be given at the present time due to the lack of a sufficiently developed mathematical theory.     

The twistor connection and gauge invariance principle

It is shown that the twistor connection of the local twistor theory can be regarded as a gauge field whose Yang-Mills equations are equivalent to Bach equations of gravity.     

The development of ideas in twistor theory

This paper presents a review of the main concepts of twistor theory. The emphasis is on the evolution of the subject from the original motivating ideas to the more recent work. In particular the physical and philosophical reasoning behind the use of the various mathematical structures is discussed.     

On the cosmological constant problem and brane-world geometry

The cosmological constant problem is examined within the context of the covariant brane-world gravity, based on Nash’s embedding theorem for Riemannian geometries. We show that the vacuum structure of the brane-world is more complex than General Relativity’s because it involves extrinsic elements, in specific, the extrinsic curvature. In other words, the shape (or local curvature) of an object becomes a relative concept, instead of the “absolute shape” of General Relativity. We point out that the immediate consequence is that the cosmological constant and the energy density of the vacuum quantum fluctuations have different physical meanings: while the vacuum energy density remains confined to the four-dimensional brane-world, the cosmological constant is a property of the bulk’s gravitational field that leads to the conclusion that these quantities cannot be compared, as it is usually done in General Relativity. Instead, the vacuum energy density contributes to the extrinsic curvature, which in turn generates Nash’s perturbation of the gravitational field. On the other hand, the cosmological constant problem ceases to be in the brane-world geometry, reappearing only in the limit where the extrinsic curvature vanishes.     

The boost problem in general relativity

We show that any asymptotically flat initial data for the Einstein field equations have a development which includes complete spacelike surfaces boosted relative to the initial surface. Furthermore, the asymptotic fall off is preserved along these boosted surfaces and there exists a global system of harmonic coordinates on such a development. We also extend former results on global solutions of the constraint equations. By virtue of this extension, the constraint and evolution parts of the problem fit together exactly. Several theorems are given which concern the behaviour in the large of general classes of linear and quasilinear differential systems. This paper contains in addition a systematic exposition of the functional spaces employed.     

The curvature problem in general relativity

This essay investigates the relationships among the metric, the connection, the curvature, and the covariant curvature derivatives in general relativity. The extent to which the connection or the curvature together, possibly with certain curvature derivatives, determines the metric is considered, as well as other related problems. Some topological aspects of the problem are included and some applications to holonomy and symmetry groups are given.This article received honorable mention from the Gravity Research Foundation for 1987.     

Conformal symplectic geometry,deformations, rigidity and geometrical (KMS) conditions

Deformations admitting a unit element of a local associative algebra defined on the space of functions on a manifold. Definition and properties of the * ^f -products and conformal symplectic geometry. Deformations of a * ^f -products. A theorem of rigidity. Application to statistical mechanics (KMS conditions).     

The Knizhnik-Zamolodchikov system as a deformation of the isomonodromy problem

It is shown that the Knizhnik-Zamolodchikov (KZ) equation (and corresponding vector bundle) can be viewed as a quantization of the isomonodromy problem for differential equations with several singular points.