84,910 research outputs found
Numerical solution of gravitational dynamics in asymptotically anti-de Sitter spacetimes
A variety of gravitational dynamics problems in asymptotically anti-de Sitter
(AdS) spacetime are amenable to efficient numerical solution using a common
approach involving a null slicing of spacetime based on infalling geodesics,
convenient exploitation of the residual diffeomorphism freedom, and use of
spectral methods for discretizing and solving the resulting differential
equations. Relevant issues and choices leading to this approach are discussed
in detail. Three examples, motivated by applications to non-equilibrium
dynamics in strongly coupled gauge theories, are discussed as instructive test
cases. These are gravitational descriptions of homogeneous isotropization,
collisions of planar shocks, and turbulent fluid flows in two spatial
dimensions.Comment: 70 pages, 19 figures; v4: fixed minus sign typo in last term of eqn.
(3.47
Holography and colliding gravitational shock waves in asymptotically AdS_5 spacetime
Using holography, we study the collision of planar shock waves in strongly
coupled N=4 supersymmetric Yang-Mills theory. This requires the numerical
solution of a dual gravitational initial value problem in asymptotically
anti-de Sitter spacetime.Comment: 5 pages, 3 figure
An SO(3)-monopole cobordism formula relating Donaldson and Seiberg-Witten invariants
We prove an analogue of the Kotschick-Morgan conjecture in the context of
SO(3) monopoles, obtaining a formula relating the Donaldson and Seiberg-Witten
invariants of smooth four-manifolds using the SO(3)-monopole cobordism. The
main technical difficulty in the SO(3)-monopole program relating the
Seiberg-Witten and Donaldson invariants has been to compute intersection
pairings on links of strata of reducible SO(3) monopoles, namely the moduli
spaces of Seiberg-Witten monopoles lying in lower-level strata of the Uhlenbeck
compactification of the moduli space of SO(3) monopoles [arXiv:dg-ga/9710032].
In this monograph, we prove --- modulo a gluing theorem which is an extension
of our earlier work in [arXiv:math/9907107] --- that these intersection
pairings can be expressed in terms of topological data and Seiberg-Witten
invariants of the four-manifold. This conclusion is analogous to the
Kotschick-Morgan conjecture concerning the wall-crossing formula for Donaldson
invariants of a four-manifold with ; that wall-crossing formula and
the resulting structure of Donaldson invariants for four-manifolds with
were established, assuming the Kotschick-Morgan conjecture, by
Goettsche [arXiv:alg-geom/9506018] and Goettsche and Zagier
[arXiv:alg-geom/9612020]. In this monograph, we reduce the proof of the
Kotschick-Morgan conjecture to an extension of previously established gluing
theorems for anti-self-dual SO(3) connections (see [arXiv:math/9812060] and
references therein). Since the first version of our monograph was circulated,
applications of our results have appeared in the proof of Property P for knots
by Kronheimer and Mrowka [arXiv:math/0311489] and work of Sivek on Donaldson
invariants for symplectic four-manifolds [arXiv:1301.0377].Comment: x + 229 page
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