32 research outputs found
Melnikov's method in String Theory
Melnikov's method is an analytical way to show the existence of classical
chaos generated by a Smale horseshoe. It is a powerful technique, though its
applicability is somewhat limited. In this paper, we present a solution of type
IIB supergravity to which Melnikov's method is applicable. This is a brane-wave
type deformation of the AdSS background. By employing two
reduction ans\"atze, we study two types of coupled pendulum-oscillator systems.
Then the Melnikov function is computed for each of the systems by following the
standard way of Holmes and Marsden and the existence of chaos is shown
analytically.Comment: 37 pages, 5 figure
Towards Spinning Mellin Amplitudes
We construct the Mellin representation of four point conformal correlation
function with external primary operators with arbitrary integer spacetime
spins, and obtain a natural proposal for spinning Mellin amplitudes. By
restricting to the exchange of symmetric traceless primaries, we generalize the
Mellin transform for scalar case to introduce discrete Mellin variables for
incorporating spin degrees of freedom. Based on the structures about spinning
three and four point Witten diagrams, we also obtain a generalization of the
Mack polynomial which can be regarded as a natural kinematical polynomial basis
for computing spinning Mellin amplitudes using different choices of interaction
vertices.Comment: 32 pages, 2 figures, v2: typos corrected, clarification added,
references updated, to appear in NP
Anatomy of Geodesic Witten Diagrams
We revisit the so-called "Geodesic Witten Diagrams" (GWDs) \cite{ScalarGWD},
proposed to be the holographic dual configuration of scalar conformal partial
waves, from the perspectives of CFT operator product expansions. To this end,
we explicitly consider three point GWDs which are natural building blocks of
all possible four point GWDs, discuss their gluing procedure through
integration over spectral parameter, and this leads us to a direct
identification with the integral representation of CFT conformal partial waves.
As a main application of this general construction, we consider the holographic
dual of the conformal partial waves for external primary operators with spins.
Moreover, we consider the closely related "split representation" for the bulk
to bulk spinning propagator, to demonstrate how ordinary scalar Witten diagram
with arbitrary spin exchange, can be systematically decomposed into scalar
GWDs. We also discuss how to generalize to spinning cases.Comment: 40 pages, 4 figures, v2: typos corrected, references added, Appendix
E and a Mellin space discussion added, v3: typos correcte
Chaotic strings in a near Penrose limit of AdS
We study chaotic motions of a classical string in a near Penrose limit of
AdS. It is known that chaotic solutions appear on , depending on initial conditions. It may be interesting to ask whether
the chaos persists even in Penrose limits or not. In this paper, we show that
sub-leading corrections in a Penrose limit provide an unstable separatrix, so
that chaotic motions are generated as a consequence of collapsed
Kolmogorov-Arnold-Moser (KAM) tori. Our analysis is based on deriving a reduced
system composed of two degrees of freedom by supposing a winding string ansatz.
Then, we provide support for the existence of chaos by computing Poincare
sections. In comparison to the AdS case, we argue that no
chaos lives in a near Penrose limit of AdSS, as expected from the
classical integrability of the parent system.Comment: 19 pages, 9 figures, LaTeX, v2: typos corrected and some
clarifications adde
Lax pairs on Yang-Baxter deformed backgrounds
We explicitly derive Lax pairs for string theories on Yang-Baxter deformed
backgrounds, 1) gravity duals for noncommutative gauge theories, 2)
-deformations of S, 3) Schr\"odinger spacetimes and 4) abelian
twists of the global AdS\,. Then we can find out a concise derivation of
Lax pairs based on simple replacement rules. Furthermore, each of the above
deformations can be reinterpreted as a twisted periodic boundary conditions
with the undeformed background by using the rules. As another derivation, the
Lax pair for gravity duals for noncommutative gauge theories is reproduced from
the one for a -deformed AdSS by taking a scaling limit.Comment: 1+39 pages, v3: typos corrected and the reference [42] adde
Yang-Baxter sigma models and Lax pairs arising from -Poincar\'e -matrices
We study Yang-Baxter sigma models with deformed 4D Minkowski spacetimes
arising from classical -matrices associated with -deformations of
the Poincar\'e algebra. These classical -Poincar\'e -matrices
describe three kinds of deformations: 1) the standard deformation, 2) the
tachyonic deformation, and 3) the light-cone deformation. For each deformation,
the metric and two-form -field are computed from the associated -matrix.
The first two deformations, related to the modified classical Yang-Baxter
equation, lead to T-duals of dS and AdS\,, respectively. The third
deformation, associated with the homogeneous classical Yang-Baxter equation,
leads to a time-dependent pp-wave background. Finally, we construct a Lax pair
for the generalized -Poincar\'e -matrix that unifies the three kinds
of deformations mentioned above as special cases.Comment: 31 pages, v2: some clarifications and references added, published
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