(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.

Maxwell’s equations in the Myers-Perry geometry

“…Such symmetries encoded in the Killing-(Yano) tensors have been explored in the past, both in higher-dimensional general relativity [42][43][44][45][46][47][48][49][50][51][52][53][54][55][56] and in string theory [61][62][63][64][65][66][67][68][69][70][71]. This paper will connect the properties of such tensors, in particular, their eigenvectors, to separation of the Maxwell's equations in an arbitrary number of dimensions.…”

Integrability of geodesics and action-angle variables in Sasaki–Einstein space $$T^{1,1}$$ T 1 , 1

“…Taking into account that the Hamiltonian (16) has no explicit time dependence, Hamilton's principal function can be written in the form [16] S(q, α, t) = W (q, α) − Et (20) where one of the constants of integration is equal to the constant value E of the Hamiltonian and W (q, α) is the Hamilton characteristic function…”

Chaos in the BMN matrix model

“…Recently, chaotic string solutions have been well studied in various ways [2][3][4][5][6][7][8][9][10][11][12]. Motivation lies on potential applications to the AdS/CFT correspondence [13][14][15] beyond integrability [16], but it has not been clarified yet what corresponds to chaotic strings in the gauge-theory side.…”