38 research outputs found
The symplectic and twistor geometry of the general isomonodromic deformation problem
Hitchin's twistor treatment of Schlesinger's equations is extended to the
general isomonodromic deformation problem. It is shown that a generic linear
system of ordinary differential equations with gauge group SL(n,C) on a Riemann
surface X can be obtained by embedding X in a twistor space Z on which sl(n,C)
acts. When a certain obstruction vanishes, the isomonodromic deformations are
given by deforming X in Z. This is related to a description of the deformations
in terms of Hamiltonian flows on a symplectic manifold constructed from affine
orbits in the dual Lie algebra of a loop group.Comment: 35 pages, LATE
BRST Extension of Geometric Quantization
Consider a physical system for which a mathematically rigorous geometric
quantization procedure exists. Now subject the system to a finite set of
irreducible first class (bosonic) constraints. It is shown that there is a
mathematically rigorous BRST quantization of the constrained system whose
cohomology at ghost number zero recovers the constrained quantum states.
Moreover this space of constrained states has a well-defined Hilbert space
structure inherited from that of the original system. Treatments of these ideas
in the Physics literature are more general but suffer from having states with
infinite or zero "norms" and thus are not admissible as states. Also the BRST
operator for many systems require regularization to be well-defined. In our
more restricted context we show that our treatment does not suffer from any of
these difficulties. This work was submitted for publication March 21,2006
The geometry of dual isomonodromic deformations
The JMMS equations are studied using the geometry of the spectral curve of a
pair of dual systems. It is shown that the equations can be represented as
time-independent Hamiltonian flows on a Jacobian bundle
A Brief Introduction to Poisson Sigma-Models
The theory of Poisson--models employs the mathematical notion of
Poisson manifolds to formulate and analyze a large class of topological and
almost topological two dimensional field theories. As special examples this
class of field theories includes pure Yang-Mills and gravity theories, and, to
some extent, the G/G gauged WZW-model. The aim of this contribution is to give
a pedagogical introduction, explaining many aspects of the general theory by
illustrative examples.Comment: 10 pages, LaTex. Based on two talks delivered in Schladming, March
199
Duality for the general isomonodromy problem
By an extension of Harnad's and Dubrovin's `duality' constructions, the
general isomonodromy problem studied by Jimbo, Miwa, and Ueno is equivalent to
one in which the linear system of differential equations has a regular
singularity at the origin and an irregular singularity at infinity (both
resonant). The paper looks at this dual formulation of the problem from two
points of view: the symplectic geometry of spaces associated with the loop
group of the general linear group, and a generalization of the self-dual
Yang-Mills equations
QUANTIZATION OF A CLASS OF PIECEWISE AFFINE TRANSFORMATIONS ON THE TORUS
We present a unified framework for the quantization of a family of discrete
dynamical systems of varying degrees of "chaoticity". The systems to be
quantized are piecewise affine maps on the two-torus, viewed as phase space,
and include the automorphisms, translations and skew translations. We then
treat some discontinuous transformations such as the Baker map and the
sawtooth-like maps. Our approach extends some ideas from geometric quantization
and it is both conceptually and calculationally simple.Comment: no. 28 pages in AMSTE
Generalized Kahler geometry and gerbes
We introduce and study the notion of a biholomorphic gerbe with connection.
The biholomorphic gerbe provides a natural geometrical framework for
generalized Kahler geometry in a manner analogous to the way a holomorphic line
bundle is related to Kahler geometry. The relation between the gerbe and the
generalized Kahler potential is discussed.Comment: 28 page
Supersymmetric Gauge Theories in Twistor Space
We construct a twistor space action for N=4 super Yang-Mills theory and show
that it is equivalent to its four dimensional spacetime counterpart at the
level of perturbation theory. We compare our partition function to the original
twistor-string proposal, showing that although our theory is closely related to
string theory, it is free from conformal supergravity. We also provide twistor
actions for gauge theories with N<4 supersymmetry, and show how matter
multiplets may be coupled to the gauge sector.Comment: 23 pages, no figure
Conformal proper times according to the Woodhouse causal axiomatics of relativistic spacetimes
On the basis of the Woodhouse causal axiomatics, we show that conformal
proper times and an extra variable in addition to those of space and time,
precisely and physically identified from experimental examples, together give a
physical justification for the `chronometric hypothesis' of general relativity.
Indeed, we show that, with a lack of these latter two ingredients, no clock
paradox solution exists in which the clock and message functions are solely at
the origin of the asymmetry. These proper times originate from a given
conformal structure of the spacetime when ascribing different compatible
projective structures to each Woodhouse particle, and then, each defines a
specific Weylian sheaf structure. In addition, the proper time
parameterizations, as two point functions, cannot be defined irrespective of
the processes in the relative changes of physical characteristics. These
processes are included via path-dependent conformal scale factors, which act
like sockets for any kind of physical interaction and also represent the values
of the variable associated with the extra dimension. As such, the differential
aging differs far beyond the first and second clock effects in Weyl geometries,
with the latter finally appearing to not be suitable.Comment: 25 pages, 2 figure
Dual giant gravitons in AdS Y (Sasaki-Einstein)
We consider BPS motion of dual giant gravitons on Ad where
represents a five-dimensional Sasaki-Einstein manifold. We find that the
phase space for the BPS dual giant gravitons is symplectically isomorphic to
the Calabi-Yau cone over , with the K\"{a}hler form identified with the
symplectic form. The quantization of the dual giants therefore coincides with
the K\"{a}hler quantization of the cone which leads to an explicit
correspondence between holomorphic wavefunctions of dual giants and
gauge-invariant operators of the boundary theory. We extend the discussion to
dual giants in where is a seven-dimensional
Sasaki-Einstein manifold; for special motions the phase space of the dual
giants is symplectically isomorphic to the eight-dimensional Calabi-Yau cone.Comment: 14 pages. (v2) typo's corrected; factors of AdS radius reinstated for
clarity; remarks about dual giant wavefunctions in T^{1,1} expanded and put
in a new subsectio