Abstract
We perform the toroidal compactification of the full Bergshoeff-de Roo version of the Heterotic Superstring effective action to first order in α′. The dimensionally-reduced action is given in a manifestly-O(n, n)-invariant form which we use to derive a manifestly-O(n, n)-invariant Wald entropy formula which we then use to compute the entropy of α′-corrected, 4-dimensional, 4-charge, static, extremal, supersymmetric black holes.
Highlights
The dimensionally-reduced action is given in a manifestly-O(n, n)-invariant form which we use to derive a manifestlyO(n, n)-invariant Wald entropy formula which we use to compute the entropy of α -corrected, 4-dimensional, 4-charge, static, extremal, supersymmetric black holes
It is natural to try to extend those results to non-trivial toroidal compactifications, testing the O(n, n) invariance of the dimensionally-reduced action3 to first order in α and obtaining a manifestly O(n, n)-invariant Wald entropy formula that can be applied to more general black-hole solutions such as, for instance, the heterotic version of the 4dimensional, 4-charge, static, extremal black holes whose microscopic entropy was first computed in refs. [13, 14]
In this paper we have shown that the bosonic sector of the complete Heterotic Superstring effective action compactified on Tn is O(n, n) invariant to first-order in α
Summary
As a warm-up exercise (and because of the recursive definition of the action), we review the well-known dimensional reduction of the action at zeroth order in α using the ScherkSchwarz formalism [27]. Following Scherk and Schwarz, we consider the Lorentz components of the 3-form field strength, because they are automatically combinations of gauge-invariant objects These are given in terms of the world-indices components by. Rewrite the Kalb-Ramond field strength in the manifestly O(n, n)-invariant form As it is well-known, the zeroth-order action S(0) given in eq (2.15) is manifestly invariant under O(n, n) transformations which are understood as T-duality transformations from the 10-dimensional point of view. Because of the additional Yang-Mills Chern-Simons term in the Kalb-Ramond field strength, we do expect modifications in the definitions of the definitions of the (10−n)-fields that originate in the Kalb-Ramond 2-form, namely the (10 − n)-dimensional Kalb-Ramond 2-form Bμ(hν), the winding vectors B(h)m μ, with respect to their zeroth-order counterparts defined in eqs. Because of the additional Yang-Mills Chern-Simons term in the Kalb-Ramond field strength, we do expect modifications in the definitions of the definitions of the (10−n)-fields that originate in the Kalb-Ramond 2-form, namely the (10 − n)-dimensional Kalb-Ramond 2-form Bμ(hν), the winding vectors B(h)m μ, with respect to their zeroth-order counterparts defined in eqs. (2.11).
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