Abstract
We study propagation of closed bosonic strings in torsional Newton-Cartan geometry based on a recently proposed Polyakov type action derived by dimensional reduction of the ordinary bosonic string along a null direction. We generalize the Polyakov action proposal to include matter, i.e. the 2-form and the 1-form that originates from the Kalb- Ramond field and the dilaton. We determine the conditions for Weyl invariance which we express as the beta-function equations on the worldsheet, in analogy with the usual case of strings propagating on a pseudo-Riemannian manifold. The critical dimension of the TNC space-time turns out to be 25. We find that Newton’s law of gravitation follows from the requirement of quantum Weyl invariance in the absence of torsion. Presence of the 1-form requires torsion to be non vanishing. Torsion has interesting consequences, in particular it yields a mass term and an advection term in the generalized Newton’s law. U(1) mass invariance of the theory is an important ingredient in deriving the beta functions.
Highlights
Background field quantizationThe quantum partition function that follows from the action (2.10) is defined by the Polyakov path integral.10 As for the bosonic strings [16], it will be very helpful to introduce the background field formalism to organize the perturbative ls2 expansion to study the quantum properties of the worldsheet sigma model
We study propagation of closed bosonic strings in torsional Newton-Cartan geometry based on a recently proposed Polyakov type action derived by dimensional reduction of the ordinary bosonic string along a null direction
We studied the non-linear sigma model for a bosonic string moving in torsional NewtonCartan geometry at one-loop
Summary
The geometric data of the TNC geometry in the absence of matter fields is encoded in a pair of vielbeins (τs, eis) and a U(1) connection ms collectively referred as the TNC metric complex. The vielbeins eis define a degenerate spatial metric through hmn = eimejnδij and it is possible to use the inverse of the square matrix (τm, ein), denoted as (−υm, eni ) with υmτm = −1 and τmemi = 0, to define an independent spatial inverse metric hmn = emi enj δij. The TNC geometry with this geometric data can be derived from a higher dimensional relativistic spacetime with an isometry in the extra null direction. It is possible to derive the world-sheet action for a string moving in the TNC geometry [15, 18] starting from the ordinary Polyakov action in the relativistic target space (2.1):. We will examine the quantum path integral defined by this Lagrangian in the rest of the paper, but we will first extend it to include Neveu-Schwarz matter, i.e. the Kalb-Ramond field and dilaton and discuss the symmetries of this generalized action both on the worldsheet and in the target space
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