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Abstract
Recently, using the assumption that the string theory effective action at the critical dimension is background independent, the classical on-shell effective action of the bosonic string theory at order alpha ' in a spacetime manifold without boundary has been reproduced, up to an overall parameter, by imposing the O(1, 1) symmetry when the background has a circle. In the presence of the boundary, we consider a background which has boundary and a circle such that the unit normal vector of the boundary is independent of the circle. Then the O(1, 1) symmetry can fix the bulk action without using the lowest order equation of motion. Moreover, the above constraints and the constraint from the principle of the least action in the presence of boundary can fix the boundary action, up to five boundary parameters. In the least action principle, we assume that not only the values of the massless fields but also the values of their first derivatives are arbitrary on the boundary. We have also observed that the cosmological reduction of the leading order action in the presence of the Hawking–Gibbons boundary term, produces zero cosmological boundary action. Imposing this as another constraint on the boundary couplings at order alpha ', we find the boundary action up to two parameters. For a specific value for these two parameters, the gravity couplings in the boundary become the Chern–Simons gravity plus another term which has the Laplacian of the extrinsic curvature.
Highlights
Jectured to be dual to a conformal field theory on the boundary [1,2]
We expect that the string theory classical effective action at the critical dimension which is a higher-derivative extension of the Einstein theory at the critical dimension, to be background independent too
In order to be able to extremize the effective action of the string theory at order α n, we propose that the massless fields and their derivatives up to order n should be arbitrary on the boundary, i.e., the massless fields are arbitrary on the boundary for the effective action at order α 0, the massless fields and their first derivatives are arbitrary on the boundary for the effective action at order α, and so on
Summary
The effective actions in the string theory have a double expansions. The genus-expansion which includes the classical tree-level and a tower of quantum loop-level corrections, and the stringy-expansion which is an expansion in terms of higher derivative couplings at each loop level. Imposing the O(1, 1)-symmetry on the most general gauge invariant couplings at order α 0, one finds that the effective actions of the bosonic string theory are fixed up to one extra parameter in the boundary action [25], i.e.,. In the bosonic string theory, there is no such symmetry One can fix this parameter by the principle of the least action as follows: Since the action is at two derivative order, only the massless fields are arbitrary on the boundary. We would like to impose the Z2-constraint (4), the constraint from the least action principle and the above constraint on the cosmological reduction of the boundary actions, to fix the effective actions of the bosonic string theory at order α when the spacetime has boundary.
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