(Non)-Integrability of Geodesics in D-brane Backgrounds

Abstract

Motivated by the search for new backgrounds with integrable string theories, we classify the D-brane geometries leading to integrable geodesics. Our analysis demonstrates that the Hamilton-Jacobi equation for massless geodesics can only separate in elliptic or spherical coordinates, and all known integrable backgrounds are covered by this separation. In particular, we identify the standard parameterization of AdS p ×S q with elliptic coordinates on a flat base. We also find new geometries admitting separation of the Hamilton-Jacobi equation in the elliptic coordinates. Since separability of this equation is a necessary condition for integrability of strings, our analysis gives severe restrictions on the potential candidates for integrable string theories.

Geodesics in supersymmetric microstate geometries

“…It has been shown in [16] that for the most symmetrical 2-charge microstate geometries the Hamilton-Jacobi equation for null geodesics separates in certain coordinates; we find that the same happens for the most symmetrical 3-charge microstate geometries. This separability of the Hamilton-Jacobi equation is due to the fact that these spacetimes have a 'hidden' symmetry related to a conformal Killing tensor which also allows the wave equation to separate in both cases [17,4].…”

“…The Hamilton-Jacobi equation for null geodesics (in 10d) is separable due to a conformal Killing tensor [16]. We can then separate the equation for geodesics in 6d, so in (2.8) let…”

Integrability and non-Integrability in N=2 SCFTs and their Holographic Backgrounds

“…This translates to the integrability of this particular geodesic motion. It seems plausible, following the ideas of [25], to study generically the integrability of geodesics in Gaiotto-Maldacena backgrounds.…”

“…Should this truncation display non-integrability then one can conclude that the parent string worldsheet theory is also non-integrable. This strategy was applied in the papers [8,[20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]. The method is to propose a string soliton with l-degrees of freedom, write its classical equations of motion, find simple solutions for (l − 1) of these equations and replace the solutions in a fluctuated version the last equation.…”

Toric data, Killing forms and complete integrability of geodesics in Sasaki-Einstein spaces $Y^{p,q}$

“…P φ is the third component of the SU (2) paper refers to geometries produced by D-branes on non-flat bases going towards the general goal of classifying all supersymmetric geometries with integrable geodesics. It is quite remarkable the fact that while the point-like strings (geodesic) equations are integrable in some backgrounds, the corresponding extended classical string motion is not integrable in general [3,4,24,7,8].…”