Heterotic Superstring Effective Action Research Articles

Non-extremal, α′-corrected black holes in 5-dimensional heterotic superstring theory

We compute the first-order α′ corrections of the non-extremal Strominger-Vafa black hole and its non-supersymmetric counterparts in the framework of the Bergshoeff-de Roo formulation of the heterotic superstring effective action. The solution passes several tests: its extremal limit is the one found in an earlier publication and the effect of a T duality transformation on it is another solution of the same form with T dual charges. We compute the Hawking temperature and Wald entropy showing that they are related by the first law and Smarr formula. On the other hand, these two contain additional terms in which the dimensionful parameter α′ plays the role of thermodynamical variable.

Non-supersymmetric black holes with α′ corrections

We study extremal 5- and 4-dimensional black hole solutions of the Heterotic Superstring effective action at first order in α′ with, respectively, 3 and 4 charges of arbitrary signs. For a particular choice of the relative signs of these charges the solutions are supersymmetric, but other signs give rise to non-supersymmetric extremal black holes. Although at zeroth order all these solutions are formally identical, we show that the α′ corrections are drastically different depending on how we break supersymmetry. We provide fully analytic mathcal{O} (α′) solutions and we compute their charges, masses and entropies and check that they are invariant under α′-corrected T-duality transformations. The masses of some of these black holes are corrected in a complicated way, but we show that the shift is always negative, in agreement with the Weak Gravity Conjecture. Our formula for the corrected entropy of the four-dimensional black holes generalizes previous results obtained from the entropy function method, as we consider a more general way in which supersymmetry can be broken.

The first law of heterotic stringy black hole mechanics at zeroth order in \u03b1\u2032

We prove the first law of black hole mechanics in the context of the Heterotic Superstring effective action compactified on a torus to leading order in α′, using Wald’s formalism, covariant Lie derivatives and momentum maps. The Kalb-Ramond field strength of this theory has Abelian Chern-Simons terms which induce Nicolai-Townsend transformations of the Kalb-Ramond field. We show how to deal with all these gauge symmetries deriving the first law in terms of manifestly gauge-invariant quantities. In presence of Chern-Simons terms, several definitions of the conserved charges exist, but the formalism picks up only one of them to play a role in the first law. We study explicitly a non-extremal, charged, black ring solution of pure mathcal{N} = 1, d = 5 supergravity embedded in the Heterotic Superstring effective field theory.This work is a first step towards the derivation of the first law at first order in α′ where, more complicated, non-Abelian, Lorentz (“gravitational”) and Yang-Mills Chern-Simons terms are included in the Kalb-Ramond field strength. The derivation of a first law is a necessary step towards the derivation of a manifestly gauge-invariant entropy formula which is still lacking in the literature. In its turn, this entropy formula is needed to compare unambiguously macroscopic and microscopic black hole entropies.

The first law and Wald entropy formula of heterotic stringy black holes at first order in \u03b1\u2032

We derive the first law of black hole mechanics in the context of the Heterotic Superstring effective action to first order in α′ using Wald’s formalism. We carefully take into account all the symmetries of the theory and, as a result, we obtain a manifestly gauge- and Lorentz-invariant entropy formula in which all the terms can be computed explicitly. An entropy formula with these properties allows unambiguous calculations of macroscopic black-hole entropies to first order in α′ that can be reliably used in a comparison with the microscopic ones. Such a formula was still lacking in the literature.In the proof we use momentum maps to define covariant variations and Lie derivatives and restricted generalized zeroth laws which state the closedness of certain differential forms on the bifurcation sphere and imply the constancy of the associated potentials on it.We study the relation between our entropy formula and other formulae that have been used in the literature.

O(n, n) invariance and Wald entropy formula in the Heterotic Superstring effective action at first order in \u03b1\u2032

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.

T duality and Wald entropy formula in the Heterotic Superstring effective action at first-order in \u03b1\u2032

We consider the compactification on a circle of the Heterotic Superstring effective action to first order in the Regge slope parameter α′ and re-derive the α′-corrected Buscher rules first found in ref. [42], proving the T duality invariance of the dimensionally-reduced action to that order in α′. We use Iyer and Wald’s prescription to derive an entropy formula that can be applied to black-hole solutions which can be obtained by a single non-trivial compactification on a circle and discuss its invariance under the α′-corrected T duality transformations. This formula has been successfully applied to α′-corrected 4-dimensional non-extremal Reissner-Nordström black holes in ref. [21] and we apply it here to a heterotic version of the Strominger-Vafa 5-dimensional extremal black hole.

On the supersymmetric solutions of the Heterotic Superstring effective action

We consider the effective action of the Heterotic Superstring to first order in α′ and derive the necessary and sufficient conditions that a field configuration has to satisfy in order to admit at least one Killing spinor using the spinor bilinear method in an arbitrary spinorial basis and corresponding arbitrary gamma matrices. As a previous step in this derivation, we compute the complete spinor bilinear algebra using the Fierz identities, obtaining as a by-product the algebra satisfied by the Spin(7) structure contained in the bilinears in an arbitrary basis. We find the off-shell relations existing between the bosonic equations of motion evaluated on supersymmetric field configurations using the Killing Spinor Identities instead of the (far more complicated) integrability conditions of the Killing Spinor Equations as it is common in the literature. We show how to include the Kalb-Ramond’s Bianchi identity in the Killing Spinor Identities.

Beyond the near-horizon limit: stringy corrections to heterotic black holes

We study the first-order in α′ corrections to 4-charge black holes (with the Reissner-Nordström black hole as a particular example) beyond the near-horizon limit in the Heterotic Superstring effective action framework. The higher-curvature terms behave as delocalized sources in the equations of motion and in the Bianchi identity of the 3-form. For some charges, this introduces a shift between their values measured at the horizon and asymptotically. Some of these corrections and their associated charge shifts, but not all of them, can be canceled using appropriate SU(2) instantons for the heterotic gauge fields. The entropy, computed using Wald’s formula, is in agreement with the result obtained via microstate counting when the delocalized sources are properly taken into account.

On a family of \u03b1\u2032-corrected solutions of the Heterotic Superstring effective action

We compute explicitly the first-order in α′ corrections to a family of solutions of the Heterotic Superstring effective action that describes fundamental strings with momentum along themselves, parallel to solitonic 5-branes with Kaluza-Klein monopoles (Gibbons-Hawking metrics) in their transverse space. These solutions correspond to 4-charge extremal black holes in 4 dimensions upon dimensional reduction on T6. We show that some of the α′ corrections can be cancelled by introducing solitonic SU(2) × SU(2) Yang-Mills fields, and that this family of α′-corrected solutions is invariant under α′-corrected T-duality transformations. We study in detail the mechanism that allows us to compute explicitly these α′ corrections for the ansatz considered here, based on a generalization of the ’t Hooft ansatz to hyperKähler spaces.

\u03b1\u2032-corrected black holes in String Theory

We consider the well-known solution of the Heterotic Superstring effective action to zeroth order in α′ that describes the intersection of a fundamental string with momentum and a solitonic 5-brane and which gives a 3-charge, static, extremal, supersymmetric black hole in 5 dimensions upon dimensional reduction on T5. We compute explicitly the first-order in α′ corrections to this solution, including SU(2) Yang-Mills fields which can be used to cancel some of these corrections and we study the main properties of this α′-corrected solution: supersymmetry, values of the near-horizon and asymptotic charges, behavior under α′-corrected T-duality, value of the entropy (using Wald formula directly in 10 dimensions), existence of small black holes etc. The value obtained for the entropy agrees, within the limits of approximation, with that obtained by microscopic methods. The α′ corrections coming from Wald’s formula prove crucial for this result.

α′-corrections to heterotic superstring effective action revisited

In this article we establish that the supersymmetric R2 effective action for the heterotic string, obtained from the supersymmetrisation of the Lorentz Chern-Simons term, is to order α' equivalent modulo field redefinitions to heterotic string effective actions computed by different methods.

Causality and supersymmetry in the superstring theory

Reduction of the ten-dimensional, heterotic-superstring effective action Ŝ, including quartic higher-derivative terms , yields a physical four-action S which is tachyon-free up to quadratic order ℛ2—the masses M0 and M2 of the spin-0 and spin-2 particles are both real—when the internal six-space is a Calabi–Yau metric ḡμν and N = 1 supersymmetry is preserved via the equation dH̄ ≡ tr(R̄ ∧ R̄) − Tr(F̄ ∧ F̄)/30 = 0, where H̄μνξ is the totally antisymmetric three-index field (while the bosonic string and type-II superstring may contain a spin-2 tachyon and a spin-0 tachyon, respectively). Here, we show that this crucial feature is independent of imposing supersymmetry, for if dH̄ ≠ 0, there is an additional contribution to ℛ2 which, however, produces no tachyons. Thus, the requirement of a stable, causal theory singles out the heterotic superstring irrespective of the requirement of N = 1 supersymmetry, which is essential at high energies rather for its roles in the vanishing of the cosmological constant and the maintenance of the gauge hierarchy.