We study black branes carrying both electric and magnetic charges in Einstein-Maxwell theory coupled to a dilaton-axion in asymptotically anti de Sitter space. After reviewing and extending earlier results for the case of electrically charged branes, we characterise the thermodynamics of magnetically charged branes. We then focus on dyonic branes in theories which enjoy an $SL(2,R)$ electric-magnetic duality. Using $SL(2,R)$, we are able to generate solutions with arbitrary charges starting with the electrically charged solution, and also calculate transport coefficients. These solutions all exhibit a Lifshitz-like near-horizon geometry. The system behaves as expected for a charged fluid in a magnetic field, with non-vanishing Hall conductance and vanishing DC longitudinal conductivity at low temperatures. Its response is characterised by a cyclotron resonance at a frequency proportional to the magnetic field, for small magnetic fields. Interestingly, the DC Hall conductance is related to the attractor value of the axion. We also study the attractor flows of the dilaton-axion, both in cases with and without an additional modular-invariant scalar potential. The flows exhibit intricate behaviour related to the duality symmetry. Finally, we briefly discuss attractor flows in more general dilaton-axion theories which do not enjoy $SL(2,R)$ symmetry.
We consider 5-dimensional Einstein-dilaton gravity with antisymmetric forms. Assuming staticity and a restriction on the dilaton coupling parameters, we derive 4-dimensional sigma-model with a target space $SL(2,R)/SO(1,1)\times SL(2,R)/SO(1,1)$. On this basis, using the symmetries of the target space, we develop a solution generating technique and employ it to construct new asymptotically flat and non-flat dyonic black rings solutions. The solutions are analyzed and the basic physical quantities are calculated.
We show that there are two classes of solutions that describe static spherically symmetric dyonic dilaton black holes with two nonsingular horizons. The first class includes only the already known solutions that exist for a few special values of the dilaton coupling constant. Solutions belonging to the second class have essentially different properties. They exist for continuously varying values of the dilaton coupling constant, but arise only for discrete values of the dilaton field at the horizon. For each given value of the dilaton coupling constant, there may exist several such solutions d…
We construct back-reacted asymptotically AdS_4 backgrounds with both electric and magnetic charge, at finite temperature and chemical potential. The system consists of a condensed bulk Dirac field hovering over a charged black brane. We give a detailed microscopic treatment of the bulk fermions, from which we derive the usual fluid approximations describing the condensate. In the context of holography, such a gravitational solution is dual to a (2+1)-dimensional CFT with a U(1) current, at finite density and strong coupling, subjected to a transverse magnetic field.
We study thermodynamic properties of dyonic black hole and its dual field theory. We observe that the phase diagram of a dyonic black hole in constant electric potential and magnetic charge ensemble is similar to that of a Van der Waals fluid with chemical potential. Phase transitions and other critical phenomena have been studied in presence of magnetic charge and chemical potential. We also analyse magnetic properties of dual conformal field theory and observe a ferromagnetic like behavior of boundary theory when the external magnetic field vanishes. Finally, we compute susceptibility of dif…
We report the discovered class of exact static solutions of several 4D Einstein-Maxwell-dilaton systems: string-induced, Liouville, trigonometric, polynomial, etc., for three basic topologies (spherical, hyperbolical and flat) being uniformly treated. In addition to the usual electric-magnetic duality this class obeys a certain extended duality between Maxwell-dilaton coupling and dilaton mass. Though major solutions we obtain are dyonic, the class also comprises interesting neutral models. As a by-product, we significantly succeded in resolving of the two important problems, one of which has …
We study dyonic solutions to the gravity-dilaton-antisymmetric form equations with the goal of identifying new $p$-brane solutions on the fluxed linear dilaton background. Starting with the generic solutions constructed by reducing the system to decoupled Liouville equations for certain values of parameters, we identify the most general solution whose singularities are hidden behind a regular event horizon, and then explore the admissible asymptotic behaviors. In addition to known asymptotically flat dyonic branes, we find two classes of asymptotically non-flat solutions which can be interpret…
We discuss dyonic black hole solutions in the case of $f(R)$-gravity coupled with a dilaton and two gauge bosons. The study of such a model is highly motivated from string theory. Our Black Hole solutions are extensions of the one firstly studied by Kallosh, Linde, Ortín, Peet and Van Proyen (KLOPV) in [arXiv:hep-th/9205027]. We will show that extreme solutions are unstable. In particular, these solutions have Bousso-Hawking-Nojiri-Odintsov (anti)evaporation instabilities.
We present the four-dimensional non-extremal dyonic black hole solution for Einstein-Maxwell-dilaton theory in absence of a scalar potential written in terms of integration constants only. These integration constants must satisfy a set of conditions imposed by the equations of motion. By defining a posteriori the mass $M$ of the black hole and the dilaton charge $Σ$, we show how to recover the dyonic black hole solution found by Kallosh et.al. In particular, our analysis show that there is a possibility in defining whether the dilaton charge or the mass of the black hole is an independent para…
In the present paper we investigate the Hawking radiation of asymptotically non-flat dyonic black holes in 4D Einstein-Maxwell-dilaton gravity in semi-classical approximation. We show that the problem allows an exact analytical treatment and we compute exactly the semi-classical radiation spectrum of both non-extremal and extremal black holes under consideration. In the high frequency regime we find that the Hawking temperature does not agree with the surface gravity when the magnetic charge is nonzero. Even more surprisingly the Hawking temperature is independent of the black hole intrinsic c…
New rotating dyonic dipole black ring solutions are derived in 5D Einstein-dilaton gravity with antisymmetric forms. The black rings are analyzed and their thermodynamics is discussed. New dyonic black string solutions are also presented.
We construct a new class of dyonic dilaton black hole solutions in the background of Anti-de Sitter (AdS) spacetime. In order to find an analytical solution which satisfy all the field equations, we should consider the string case where the dilaton coupling is $α=1$. The asymptotic behaviour of the solution ($r\rightarrow\infty$) is exactly AdS, where the dilaton field becomes zero and the metric function reduces to $f(r)\rightarrow -Λr^2/3$. In this spacetime, black hole horizons and cosmological horizons, can be a two-dimensional positive, zero or negative constant curvature surface. We stud…
We demonstrate that rotating dyonic black holes in Einstein-Maxwell-Dilaton-Axion (EMDA) theory is holographically dual with 2D conformal field theory. We first investigate the duality on dyonic Kerr-Sen black hole with non-vanishing dilaton and axion charges. The near-horizon geometry of extremal dyonic Kerr-Sen spacetime possesses the $SL(2,R) \times U (1)$ isometry allowing to employ the asymptotic symmetry group method to find the corresponding central charge. We find two different branches of the mass which correspond to CFT with two different central charges $c_L=12am_+$ and $c_L=12am_-$…
In this proceedings contribution we summarize and discuss results of Refs. \cite{Grumiller:2013swa, Grumiller:2015vaa} in the light of recent developments in $\textrm{AdS}_2$ holography \cite{Maldacena:2016upp, Jensen:2016pah, Engelsoy:2016xyb}.
We consider the D-dimensional massive dilaton gravity coupled to Maxwell and antisymmetric tensor fields (EMATD). We derive the full separability of this theory in static case. This discloses the core structure of the theory and yields the simple procedure of how to generate integrability classes. As an example we take a certain new class, obtain the two-parametric families of dyonic solutions. It turns out that at some conditions they tend to the D-dimensional dyonic Reissner-Nordström-deSitter solutions but with ``renormalized'' dyonic charge plus a small logarithmic correction. The latter h…
The properties of static spherically symmetric black holes, which are both electrically and magnetically charged, and which are coupled to the dilaton in the presence of a cosmological constant, Lambda, are considered. It is shown that apart from the Reissner-Nordstrom-de Sitter solution with constant dilaton, such solutions do not exist if Lambda > 0 (in arbitrary spacetime dimension >=4 ). However, asymptotically anti-de Sitter dyonic black hole solutions with a non-trivial dilaton do exist if Lambda < 0. Both these solutions and the asymptotically flat (Lambda = 0) solutions are studied num…
An exact solution of the low-energy string theory representing static, spherical symmetric dyonic black hole is found. The solution is labeled by their mass, electric charge, magnetic charge and asymptotic value of the scalar dilaton. Some interesting properties of the dyonic black holes are studied. In particular, the Hawking temperature of dyonic black holes depends on both the electric and magnetic charges, and the extremal ones, which have nonzero electric and magnetic charges, have zero temperature but nonzero entropy. These properties are quite different from those of electrically (or ma…
We investigate extremal dyon black holes in the Einstein-Maxwell-dilaton (EMD) theory with higher curvature corrections in the form of the Gauss-Bonnet density coupled to the dilaton. In the same theory without the Gauss-Bonnet term the extremal dyon solutions exist only for discrete values of the dilaton coupling constant $a$. We show that the Gauss-Bonnet term acts as a dyon hair tonic enlarging the allowed values of $a$ to continuous domains in the plane $(a, q_m)$ the second parameter being the magnetic charge. In the limit of the vanishing curvature coupling (a large magnetic charge) the …
$AdS_{5}$ with linear dilaton and non vanishing $B$-field is shown to be a solution of the non critical string beta function equations. A non critical $(D=5)$ solution interpolating between flat space-time and $AdS_{5}$, with asymptotic linear dilaton and non vanishing $B$-field is also presented. This solution is free of space-time singularities and has got the string coupling constant everywhere bounded. Both solutions admit holographic interpretation in terms of ${\cal N}=0$ field theories. Closed string tachyon stability is also discussed.
We show that previously known non-asymptotically flat static black hole solutions of Einstein-Maxwell-dilaton theory may be obtained as near-horizon limits of asymptotically flat black holes. Specializing to the case of the dilaton coupling constant $α^2 = 3$, we generate from the non-asymptotically flat magnetostatic or electrostatic black holes two classes of rotating dyonic black hole solutions. The rotating dyonic black holes of the ``magnetic'' class are dimensional reductions of the five-dimensional Myers-Perry black holes relative to one of the azimuthal angles, while those of the ``ele…
We consider the first subleading terms in the low-energy cross section for the absorption of dilaton partial waves by D3-branes. We demonstrate that these corrections, computed previously via supergravity, can be reproduced exactly in a worldvolume calculation using a deformation of N=4 SYM theory by a dimension eight chiral operator. The calculation does not depend on how the theory is regularized. This result provides another hint that holographic duality between the D3-brane worldvolume theory and the corresponding supergravity solution may be valid beyond the near horizon limit.
Asymptotic AdS Riemann space-times in five dimensions with a black brane (horizon) sourced by a fully back-reacted scalar field (dilaton) offer -- via the holographic dictionary -- various options for the thermodynamics of the flat four-dimensional boundary theory, uncovering Hawking-Page, first-order and second-order phase transitions up to a cross-over or featureless behavior. The relation of these phase structures to the dilaton potential is clarified and illustrating examples are presented. Having in mind applications to QCD we study probe vector mesons with the goal to figure out conditio…
We construct static and spherically symmetric particle-like and black hole solutions with magnetic and/or electric charge in the Einstein-Born-Infeld-dilaton-axion system, which is a generalization of the Einstein-Maxwell-dilaton-axion (EMDA) system and of the Einstein-Born-Infeld (EBI) system. They have remarkable properties which are not seen for the corresponding solutions in the EMDA and the EBI system.
Using the Sen's entropy function formalism, we compute the entropy for the extremal dyonic black hole solutions of theories in the presence of dilaton field coupled to the field strength and a dilaton potential. We solve the attractor equations analytically and determine the near horizon metric, the value of the scalar fields and the electric field on the horizon, and consequently the entropy of these black holes. The attractor mechanism plays a very important role for these systems, and after studying the simplest systems involving dilaton fields, we propose a general ansatz for the value of …
We present an interacting spin-2 gauge theory coupled to the two-dimensional dilaton-gravity in flat spacetime. The asymptotic symmetry group is enhanced to the central extension of Diff$(S^1)\ltimes C^\infty(S^1)\ltimes$Vec($S^1$) when the central element of the Heisenberg subgroup is zero (vanishing $U(1)$ level). Using the BF-formulation of the model we derive the corresponding boundary coadjoint action which is the spin-2 extension of the warped Schwarzian theory at vanishing $U(1)$ level. We also discuss the thermodynamics of black holes in this model.
In this note it is shown that near-extremal four dimensional dyonic black holes, where the dilaton is not constant, can be described by a microscopic model consisting of a one-dimensional gas of massless particles.
In the background of a charged AdS dilaton black hole, we investigate the movement of a self-graviting 3-brane and relevant holographic effects as the brane move close to the AdS boundary. The induced metric on brane corresponds to an exact FLRW geometry, while the evolution of brane is determined by Israel junction condition and the effective Einstein field equation on brane together. When the brane approaches the AdS boundary, AdS/CFT correspondence implies that a radiation dominated FLRW-universe ($P=\frac{1}{3}ρ$) should be given. According to the holographic renormalization procedure, we …
We discuss aspects of generic 2-dimensional dilaton gravity theories. The 2-dim geometry is in general conformal to $AdS_2$ and has IR curvature singularities at zero temperature: this can be regulated by a black hole. The on-shell action is divergent: we discuss the holographic energy-momentum tensor by adding appropriate counterterms. For theories obtained by dimensional reduction of the gravitational sector of higher dimensional theories, for instance higher dimensional $AdS$ gravity as a concrete example, the 2-dimensional description dovetails with the higher dimensional one. We also disc…
Classical solutions of the Yang-Mills-dilaton theory in Euclidean space-time are investigated. Our analytical and numerical results imply existence of infinite number of branches of dyonic type solutions labelled by the number of nodes of gauge field amplitude $W$. We find that the branches of solutions exist in finite region of parameter space and discuss this issue in detail in different dilaton field normalization.
We argue that vacua of string theory which asymptote at weak coupling to linear dilaton backgrounds are holographic. The full string theory in such vacua is ``dual'' to a theory without gravity in fewer dimensions. The dual theory is generically not a local quantum field theory. Excitations of the string vacuum, which can be studied in the weak coupling region using worldsheet methods, give rise to observables in the dual theory. An interesting example is string theory in the near-horizon background of parallel NS5-branes, the CHS model, which is dual to the decoupled NS5-brane theory (``littl…
Having in mind extensions of 2D holography beyond the Jackiw-Teitelboim model we propose holographic counterterms and asymptotic conditions for a family of asymptotically AdS$_2$ dilaton gravity models leading to a consistent variational problem and a finite on-shell action. We show the presence of asymptotic Virasoro symmetries in all these models. The Schwarzian action generates (a part) of the equations of motion governing the asymptotic degrees of freedom. We also analyse the applicability of various entropy formulae. By a dilaton-dependent conformal transformation our results are extended…
We investigate quantum phase transitions in a 2+1 dimensional gauge theory at finite chemical potential $χ$ and magnetic field $B$. The gravity dual is based on 4D $\mathcal{N}=2$ Fayet-Iliopoulos gauged supergravity and the solutions we consider---that are constructed analytically---are extremal, dyonic, asymptotically $AdS_4$ black-branes with a nontrivial radial profile for the scalar field. We discover a line of second order fixed points at $B=B_c(χ)$ between the dyonic black brane and an extremal "thermal gas" solution with a singularity of good-type, according to the acceptability criter…
We consider a system consisting of $5$ dimensional gravity with a negative cosmological constant coupled to a massless scalar, the dilaton. We construct a black brane solution which arises when the dilaton satisfies linearly varying boundary conditions in the asymptotically $AdS_5$ region. The geometry of this black brane breaks rotational symmetry while preserving translational invariance and corresponds to an anisotropic phase of the system. Close to extremality, where the anisotropy is big compared to the temperature, some components of the viscosity tensor become parametrically small compa…
In five dimensions, we consider a model described by the Einstein gravity with a source given by a scalar field and various Abelian gauge fields with dilatonic-like couplings. For this model, we are able to construct two dyonic black holes whose three-dimensional horizons are modeled by two nontrivial homogeneous Thurston's geometries. The dyonic solutions are of Lifshitz type with an arbitrary value of the dynamical exponent. In fact, the first gauge field ensures the anisotropy asymptotic while the remaining Abelian fields sustain the electric and magnetic charges. Using the Hamiltonian form…
In these lectures I review some recent progresses in counting the number of microstates of AdS supersymmetric black holes in dimensions equal or greater than four using holography. The counting is obtained by applying localization and matrix model techniques to the dual field theory. I cover in details the case of dyonic AdS$_4$ black holes, corresponding to a twisted compactification of the dual field theory, and I discuss the state of the art for rotating AdS$_5$ black holes.
Explicit dyonic dilaton black holes of the four-dimensional Einstein-Maxwell-dilaton theory are known only for two particular values of the dilaton coupling constant $a =1,\sqrt{3}$, while for other $a$ numerical evidence was presented earlier about existence of extremal dyons in theories with the discrete sequence of dilaton couplings $a=\sqrt{n(n+1)/2}$ with integer $n$. Apart from the lower members $n=1,\,2$, this family of theories does not have motivation from supersymmetry or higher dimensions, and so far the above quantization rule has not been derived analytically. We fill this gap sho…
Gauge theories living on stacks of D7-branes are holographically related to IIB gravitational backgrounds with a varying axion-dilaton field (F-theory). The axion-dilaton field is generated by D7, O7 and D-instanton sources and can be written in terms of the chiral correlators of the eight dimensional gauge theory living on the D7-branes. Using localization techniques, we prove that the same correlators determine the gauge coupling of the four-dimensional N=2 supersymmetric SU(2) gauge theories living on the elementary D3-brane which probes the F-theory geometries.
We consider a six-dimensional solitonic string solution described by a conformal chiral null model with non-trivial $N=4$ superconformal transverse part. It can be interpreted as a five-dimensional dyonic solitonic string wound around a compact fifth dimension. The conformal model is regular with the short-distance (`throat') region equivalent to a WZW theory. At distances larger than the compactification scale the solitonic string reduces to a dyonic static spherically-symmetric black hole of toroidally compactified heterotic string. The new four-dimensional solution is parameterised by five …
We show that a class of Einstein-Maxwell-Dilaton (EMD) theories are related to higher dimensional AdS-Maxwell gravity via a dimensional reduction over compact Einstein spaces combined with continuation in the dimension of the compact space to non-integral values (`generalized dimensional reduction'). This relates (fairly complicated) black hole solutions of EMD theories to simple black hole/brane solutions of AdS-Maxwell gravity and explains their properties. The generalized dimensional reduction is used to infer the holographic dictionary and the hydrodynamic behavior for this class of theori…
It is shown that the asymptotic growth of the microscopic degeneracy of BPS dyons in four-dimensional N=4 string theory captures the known corrections to the macroscopic entropy of four-dimensional extremal black holes. These corrections are subleading in the limit of large charges and originate both from the presence of interactions in the effective action quadratic in the Riemann tensor and from non-holomorphic terms. The presence of the non-holomorphic corrections and their contribution to the thermodynamic free energy is discussed. It is pointed out that the expression for the microscopic …
We discuss some aspects related to holography of Anti-de Sitter (AdS) dyonic hairy black holes, which break the conformal symetry of the boundary. We use counterterms for the scalar field that satisfies mixed boundary conditions to compute the Euclidean action and dual stress tensor. We apply these results to show that the Cardy-Verlinde formula is not satisfied. However, when the magnetic (or electric) charge vanishes, the conformal symmetry is preserved and the entropy of the black hole can be put in the Cardy-Verlinde form. In our analysis, there is no need of adding extra finite counterter…
We study extremal static dyonic black holes in four-dimensional Einstein-Maxwell-Dilaton theory, for general values of the constant $a$ in the exponential coupling $e^{aϕ}$ of the dilaton to the Maxwell kinetic term. Explicit solutions are known only for $a=0$, $a=1$ and $a=\sqrt3$, and for general $a$ when the electric and magnetic charges $Q$ and $P$ are equal. We obtain solutions as power series expansions around $Q=P$, in terms of a small parameter $ε= a^{-1}\, \log(Q/P)$. Using these, and also solutions constructed numerically, we test a relation between the mass and the charges that had …
We study several aspects of charged dilaton black holes with planar symmetry in $(d+2)$-dimensional spacetime, generalizing the four-dimensional results investigated in arXiv:0911.3586 [hep-th]. We revisit the exact solutions with both zero and finite temperature and discuss the thermodynamics of the near-extremal black holes. We calculate the AC conductivity in the zero-temperature background by solving the corresponding Schrödinger equation and find that the AC conductivity behaves like $ω^δ$, where the exponent $δ$ is determined by the dilaton coupling $α$ and the spacetime dimension parame…
Extremal black branes upon compactification in the near horizon throat region are known to give rise to $AdS_2$ dilaton-gravity-matter theories. Away from the throat region, the background has nontrivial profile. We interpret this as holographic renormalization group flow in the 2-dim dilaton-gravity-matter theories arising from dimensional reduction of the higher dimensional theories here. The null energy conditions allow us to formulate a holographic c-function in terms of the 2-dim dilaton for which we argue a c-theorem subject to appropriate boundary conditions which amount to restrictions…
We study the holographic non-relativistic fermions in the presence of bulk dipole coupling in charged dilatonic black hole background. We explore the nontrivial effects of the bulk dipole coupling, the fermion charge as well as the dilaton field on the flat band, the Fermi surface and the emergence of the gap by investigating the spectral function of the non-relativistic fermion system. In particular, we find that the presence of the flat band in the non-relativistic case will suppress the Fermi momentum. Besides, we observe that the effect of the dipole coupling in the dilaton gravity is more…
We study generic two-dimensional dilaton gravity with a Maxwell field and prove its triviality for constant dilaton boundary conditions, despite of the appearance of a Virasoro algebra with non-zero central charge. We do this by calculating the canonical boundary charges, which turn out to be trivial, and by calculating the quantum gravity partition function, which turns out to be unity. We show that none of the following modifications changes our conclusions: looser boundary conditions, non-linear interactions of the Maxwell field with the dilaton, inclusion of higher spin fields, inclusion o…
We show how D-instanton corrections modify the dilaton-axion profile emitted by an O7/D7 system turning it into the non-singular F-theory background which corresponds to the effective coupling on a D3 probe.
We study non perturbative superpotentials for N=1 super Yang Mills from the point of view of large $N$ dualities. Starting with open topological strings we work out the relation between the closed string sector dilaton tadpole, which appears in the annulus amplitude, and NSNS fluxes in the closed string dual on the resolved conifold. For the mirror closed string dual version on the deformed conifold we derive, for a non vanishing $G_{3}$ form, the $N$ supersymmetric vacua and the transformations of $G_{3}$ through domain walls. Finally, as an extension of Fischler Susskind mechanism we find a …
A class of four-dimensional static supersymmetric black hole solutions of effective supergravity Lagrangian of IIA superstring compactified on $T^6$ is constructed by explicitly solving Killing spinor equations (KSEs). These solutions are dyonic black holes parametrized by four charges, with dilaton and diagonal internal metric components as the only non-zero scalar fields, and preserve $1 \over 8$ of $N=8$ supersymmetry. The KSEs with only Neveu-Schwarz-Neveu-Schwarz charges relate spinors with opposite chirality from ten-dimensional view point, and have identical structures with KSEs of toro…
In this talk we present the latest results from our ongoing project on geometro-thermodynamics (also known as information geometry of thermodynamics or Ruppeiner geometry) of dilaton BHs in 4D in both Einstein and string frames and a dyonic dilaton BH and at the end we report very briefly results from this approach to the 2D dilaton BHs.
We present a revisitation of the Almheiri-Polchinski dilaton gravity model from a two-dimensional (2D) bulk perspective. We describe a peculiar feature of the model, namely the pattern of conformal symmetry breaking using bulk Killing vectors, a covariant definition of mass and the flow between different vacua of the theory. We show that the effect of the symmetry breaking is both the generation of an infrared scale (a mass gap) and to make local the Goldstone modes associated with the asymptotic symmetries of the 2D spacetime. In this way a non vanishing central charge is generated in the dua…
We develop models of 1+1 dimensional dilaton gravity describing flows to $AdS_2$ from higher dimensional $AdS$ and other spaces. We use these to study the effects of backreaction on holographic correlators. We show that this scales as a relevant effect at low energies, for compact transverse spaces. We also discuss effects of matter loops, as in the CGHS model.
We investigate bulk and holographic features of finite-temperature black brane solutions of 4D anti-de Sitter Einstein-Maxwell-dilaton-gravity (EMDG). We construct, numerically, black branes endowed with non trivial scalar hairs for broad classes of EMDG. We consider both exponential and power-law forms for the coupling functions, as well as several charge configurations: purely electric, purely magnetic and dyonic solutions. At finite temperature the field theory holographically dual to these black brane solutions has a rich and interesting phenomenology reminiscent of electron motion in meta…
We study the classical theory of a non-Abelian gauge field (gauge group $SU(2)$) coupled to a massive dilaton, massive axion and Einstein gravity. The theory is inspired by the bosonic part of the low-energy heterotic string action for a general Yang-Mills field, which we consider to leading order after compactification to $(3+1)$ dimensions. We impose the condition that spacetime be static and spherically symmetric, and we introduce masses via a dilaton-axion potential associated with supersymmetry (SUSY)-breaking by gaugino condensation in the hidden sector. In the course of describing the p…
