Abstract
We explore analytic integrability criteria for D1 branes probing 4D relativistic background with a null isometry direction. We use both the Kovacic’s algorithm of classical (non)integrability as well as the standard formulation of Lax connections to show the analytic integrability of the associated dynamical configuration. We further use the notion of double null reduction and obtain the world-volume action corresponding to a torsional Newton-Cartan (TNC) D0 brane probing a 3D torsional Newton-Cartan geometry. Moreover, following Kovacic’s method, we show the classical integrability of the TNC D0 brane configuration thus obtained. Finally, considering a trivial field redefinition for the D1 brane world-volume fields, we show the equivalence between two configurations in the presence of vanishing NS fluxes.
Highlights
Kovacic’s method: a reviewFor the sake of comprehensiveness, we briefly outline the essentials of Kovacic’s algorithm that was proposed originally in [23]
We explore analytic integrability criteria for D1 branes probing 4D relativistic background with a null isometry direction
We further use the notion of double null reduction and obtain the world-volume action corresponding to a torsional Newton-Cartan (TNC) D0 brane probing a 3D torsional Newton-Cartan geometry
Summary
For the sake of comprehensiveness, we briefly outline the essentials of Kovacic’s algorithm that was proposed originally in [23]. The steps are quite straightforward to follow: (1) choose an invariant plane in the dynamical phase space and (2) consider fluctuations normal to this plane. These fluctuations generally obey differential equations, a(τ )η(τ ) + b(τ )η(τ ) + c(τ )η(τ ) = 0. The algorithm sets rules to check whether NVE (2.1) admits Liouvillian solutions or not. To check this explicitly, it is customary first to note down an equivalent representation [23, 24] of (2.1), ξ = V (τ )ξ(τ ) ;. Polynomials goes under the name of Mobius transformations that generate the group of automorphisms of the Riemann sphere
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