SLAC-PUB 7326

SNUTP-96/105

SU-ITP 96-44

hep-th/9610157

CLASSICAL AND QUANTUM ASPECTS OF BPS BLACK HOLES

IN HETEROTIC STRING COMPACTIFICATIONS^{1}^{1}1
Work supported in part by DOE DE-AC03-76SF00515, NSF Grant PHY-9219345,
NSF-KOSEF Bilateral Grant, KRF International Collaboration Grant and
Non-Directed Research Grant 81500-1341, KOSEF Purpose-Oriented Research
Grant 94-1400-04-01-3 and SRC Program, the Ministry of Education BSRI Grant
96-2418, and the Seoam Foundation Fellowship.

Soo-Jong Rey^{2}^{2}2email:
, ,

Stanford Linear Accelerator Center, Stanford University, Stanford CA 94309 USA

Department of Physics, Stanford University, Stanford CA 94305 USA

Physics Department, Seoul National University, Seoul 151-742 KOREA

ABSTRACT

We study classical and quantum aspects of BPS black holes for compactification of heterotic string vacua. We extend dynamical relaxation phenomena of moduli fields to background consisting of a BPS soliton or a black hole and provide a simpler but more general derivation of the Ferrara-Kallosh’s extremized black hole mass and entropy. We study quantum effects to the the BPS black hole mass spectra and to their dynamical relaxation. We show that, despite non-renormalizability of string effective supergravity, quantum effect modifies BPS mass spectra only through coupling constant and moduli field renormalizations. Based on target-space duality, we establish a perturbative non-renormalization theorem and obtain exact BPS black hole mass and entropy in terms of renormalized string loop-counting parameter and renormalized moduli fields. We show that similar conclusion holds, in the large limit, for leading non-perturbative correction. We finally discuss implications to type-I and type-IIA Calabi-Yau black holes.

Submitted to Nuclear Physics B

## 1 Introduction

In recent exciting development [2, 3, 4, 5], string theory has provided with a microscopic first–principle from which long–standing puzzle of black hole thermodynamics [6] can be understood. The development was made possible, on one hand, from better understanding of non-perturbative string theory including strong-weak coupling duality [7], various string-string dualities [8, 9, 10, 11, 12, 13] and D-brane soliton sector carrying Ramond-Ramond charges [14], and, on another, from deeper understanding of BPS states in string theory and their stability throughout weak to strong coupling regime.

While a definite relation between the statistical mechanics of microscopic stringy states and the macroscopic laws of the black hole thermodynamics has been established for various specific black holes, it has not yet provided a universal derivation of entropy–area relation for all classes of black holes. In particular, all specific examples explored so far have had large supersymmetries in four dimensions. Together with the fact that the scalar fields at black hole horizon were fixed [15, 16] so that the horizon area were independent of the scalar fields at infinity, large local supersymmetries were stringent enough to determine the macroscopic black hole entropy uniquely (up to constants). In view of this, extension of the previous studies to the less stringent yet controllable situations is posed as an interesting problem and might offer further insights to the black–hole physics. In this respect, black holes arising in supersymmetric theories are unique in that there exist controllable BPS states yet smaller supersymmetries renders underlying dynamics richer enough.

Recently, initiated by the pioneering work of Ferrara, Kallosh and Strominger [17], macroscopic aspects of BPS black holes have been studied extensively [18]–[22]. In these works, the special geometry [23] that governs interactions of supergravity with vector and hyper multiplets has played an important role. On the microscopic side, examples of D-brane configurations in compactifications have been found [24] and microscopic state counting has shown complete agreement with macroscopic entropy formula of corresponding black holes.

Particularly interesting subset of BPS black holes are are the ones having constant moduli everywhere outside black hole horizon [18, 19]. These, so-called double extreme black holes [20], are distinguished from other BPS black holes in that they have the lightest possible mass. With such special properties, one might expect that the double-extreme black holes play a special role among all BPS black holes and open up new understanding uncovered so far. In this respect, better understanding of the double-extreme black holes is desirable. The first motivation and contents of the present work is to address various aspects of them.

In all previous microscopic and macroscopic studies, however, only classical aspects of BPS black holes were considered. What distinguishes supersymmetry from ones is that nontrivial quantum effects are present generically at perturbative and non-perturbative levels [25]. In establishing entropy–area relation for black holes in supersymmetric theories, stability of BPS states against strong coupling extrapolation has served an integral part of underlying physics. Hence, it is of interest to what extent the nontrivial quantum effects are reflected in the black holes and their physical properties. The second motivation and contents of the present work is study quantum effects for BPS states in rigid and local theories.

In this paper, we study the above aspects for four–dimensional heterotic
string compactifications. For definiteness, we focus on rank-3, so-called
STU model, theories that arises from compactification on of a
heterotic string theory, which have been obtained from
by compactifying on with instanton numbers ^{3}^{3}3
This compactification is identical to other ones with different instanton
numbers or at least perturbatively. Non-perturbative
effects, however, may reveal differences among them[26, 27]
.. This theory is known to be dual either to the type IIA compactification
on the Calabi-Yau threefold [10]
or to the compactification of the type-I
orientifold on orbifold with
one tensor multiplet and completely Higgs gauge group [29].
Utilizing each duality map, we may learn otherwise inaccesible properties
of type IIA and type I black holes from heterotic STU black holes as well.

This paper is organized as follows. In Section 2, we first consider rigid theory and interpret BPS mass minimization as dynamical relaxation of scalar fields. We extend this to local theory and obtain what we call Kähler-BPS condition. This provides a simpler and more general derivation of the result by Ferrara and Kallosh [19]. In Section 3, we consider classical aspect of Kähler-BPS black holes for the heterotic STU model. In Section 4, we study quantum effects to Kähler-BPS configurations. After recalling the rigid supersymmetry non-renormalization theorems to BPS masses, we study quantum effects to BPS black holes of the heterotic STU model. We show that target-space duality symmetry provides a strong constraint to quantum corrections. We establish a perturbative non-renormalization theorem based on the symmetry and show that the BPS black holes continues to saturate the BPS bound in terms of renormalized string loop-counting parameter and moduli fields. In the large limit, we also derive leading non-perturbative corrections. In Section 5, we conclude with brief discussions on utilizing string-string duality maps to type–I and type–IIA string theory black holes.

## 2 Dynamical Relaxation of Bps Mass Spectra

### 2.1 BPS Mass and Dynamical Relaxatioin in Rigid Theory

#### 2.1.1 Rigid Special Geometry

Before we dwell into the technically more involved supergravity theory, we first study dynamical relaxation of the extremal BPS mass spectra for rigid supersymmetric Yang-Mills theory. The Lagrangian of the supersymmetric gauge theory is constructured out of vector multiplets. The vector multiplets are constrained chiral superfields and contain a complex scalar, doublet Majorana-Weyl gauginos and an abelian gauge field. We denote them as , for both left-handed and right-handed ’s. Associated with them, one introduces a holomorphic prepotential of the vector multiplets. Coupling of the scalar fields with the vector field strengths is then specified by a holomorphic tensor:

(2.1) |

To describe scalar self-interaction, we first combine and together, and construct a symplectic vector :

(2.2) |

Denoting symplectic inner product in terms of matrix multiplication

(2.3) |

Kähler potential defined by

(2.4) |

specifies a -dimensional Kähler manifold. By adopting so-called ‘rigid special coordinates’ , we find the Kähler metric

(2.5) |

The fact that couplings of scalar self-interactions and scalar-vector interactions are the same is nothing but a manifestation of the underlying supersymmetry. By the same reason, the scalar-gaugino interaction couplings should also be governed by . Indeed, expanding the Lagrangian defined by chiral -term of the prepotential into component fields, one obtains:

(2.6) |

Here, the ellipses denote non-minimal coupling terms including magnetic moment interactions and scalar potential.

Heuristically, one can draw an analogy to an electrodynamics in a macroscopic media [30]. The coupling matrix is then interpreted as a generalized electric permittivity and magnetic permeability tensors:

(2.7) |

The novelty of the analog macroscopic media is that the speed of light
remains unity as can be confirmed from the fact
.
This is a necessary condition to maintain the manifest Lorentz covariance
of the theory^{4}^{4}4 We note that effective supergravity theory of
the type-I open string provides an another interesting analog macroscopic
media with a unit speed of light,
for which the electric permittivity and the magnetic permeability
is self-field dependent:

(2.9) |

Symplectic vector of self-dual field strengths is defined by a complex conjugate relation of Eq.( 2.9). The field corresponds to the generalized electric and magnetic induction fields, and . Similarly, the field corresponds to the generalized electric displacement and magnetic fields, and . These two sets of field-strength sections are related each other

(2.10) |

This is a direct counterpart of the so-called ‘constitutive relation’ [30] in the electrodynamics of a macroscopic media, viz., a functioinal relation of in terms of .

In the presence of electric and magnetic four-currents , -component Maxwell’s equation is expressed compactly as:

(2.11) |

Integrating this equation over the space, we obtain a symplectic vector of microscopic electric and magnetic charges:

(2.12) |

Classically, the charges are continuous, real-valued in units of appropriate electric and magnetic coupling constants. Quantum mechanically, however, the charges should obey the Dirac-Schwinger-Zwanziger quantization condition, is an integer multiple of the Dirac unit . This in turn implies that the symplectic charge vector is covariant only under .

#### 2.1.2 Central Charge and Bps Spectra

One can derive the central charge directly from the supersymmetry algebra. The supersymmetry current and the supercharge of Eq.(2.6) are given by:

(2.13) |

From Eq.(2.6), one also derives anti-commutation relations for the gaugino fields:

(2.14) |

One then evaluates the supercharge anti-commutators:

(2.15) |

The central charge is defined from the second anti-commutator after integrating by parts and using the Maxwell’s equation Eq.(2.12):

(2.16) | |||||

Note the topological nature of the central charge as it is defined as the surface integral at spatial infinity. Diagonalizing the supersymmetry algebra, one finds the BPS inequality for the mass spectra:

(2.17) |

#### 2.1.3 Dynamical Relaxation of Bps Mass

We now introduce a notion of dynamical relaxation of BPS mass and free energy. Consider a single, isolated BPS state carrying electric and magnetic charges specified by the symplectic vector . The BPS mass defines a mass gap separating the BPS state from vacuum state, and is a function of the gauge coupling constants. In supersymmetric gauge theory, the gauge coupling matrix is not a constant but a function of the coordinates on Kähler manifold. These coordinates are dynamical fields in supersymmetric gauge theory, hence, can relax dynamically and minimize the mass gap . Since the BPS state is characterized by electric and magnetic charges it carries, the relaxation configuration of the special coordinates should be determined entirely in terms of the symplectic charge vector . Therefore, up to symplectic transformations, one can associate a one-to-one mapping from the -dimensional Kähler manifold parametrized by the scalar fields to the space of electric and magnetic charges of a given BPS state. Such a mapping is a harmonic one and the BPS mass can be taken as the positive-definite free energy associated with the harmonic mapping. Obviously, this notion can be extended to situations of multiple BPS states.

To exemplify the notion of dynamical minimization of BPS mass gap, consider a situation for the gauge group of rank one. The BPS mass spectra may be written as

(2.18) |

Here, denotes mass of heavy charged gauge boson and is related to vacuum expectation value of Higgs fields as . Using the Cauchy-Schwarz inequality [31],

(2.19) |

and the equality is saturated at .

It is straightforward to generalize the example to rank- () gauge group. For simplicity, we consider the minimal coupling , but the foregoing result can be generalized to non-minimal case straightforwardly . The gauge group is spontaneously broken to . Taking into account of the Witten effect [32] and introducing a notation for the quadratic form in the charge space, the BPS mass spectra is given by

(2.20) |

By introducing a new basis of electric and magnetic charges:

(2.21) |

one has

(2.22) |

Again, one finds that there exists a special configuration of the coupling constants at which BPS mass gap is minimized:

(2.23) | |||||

Here, we have used the Cauchy-Schwarz inequality and the Hölder’s inequalities [31] at the first and the second steps respectively. Each inequality then provides with separate conditions to the coupling constant and to the vacuum angle for which the BPS mass is minimized. First, the extremal vacuum angle is determined by saturating the Hölder’s inequality, viz., the second line in Eq.(2.23):

(2.24) |

Next, the extremal coupliing constant is determined by saturating the Cauchy-Schwarz inequality, viz., the first line in Eq.(2.23):

(2.25) |

To obtain the last expression, we have inserted the extremal vacuum angle Eq.(2.24). Alternatively, one may first tune the the vacuum angle to by a Peccei-Quinn transformation and determine the extremal gauge coupling constant by Cauchy-Schwarz inequality. It is given by:

(2.26) |

One then undo the Peccei-Quinn rotation of the vacuum angle . Recalling that the vacuum angle shift also introduces an electric charge by the Witten effect [32], one finds that the BPS mass minimized under the condition can be lowered further. Saturation of the new BPS mass then yields exactly the same result as the one based on the Hölder’s inequality in Eq.( 2.23). Hence, the vacuum angle relaxes to the extremal value Eq.( 2.24) and, in turn, the coupling constant further to the extremal value Eq.( 2.25). In either methods, one finally obtain the extremal BPS mass gap or free energy:

(2.27) |

The main idea of the BPS mass minimization is that the the gauge coupling constants parametrized by complex scalar fields on the Kähler manifold are actually not constants but can relax. Since the BPS mass-squared is a positive-definite quadratic form of electric and magnetic charges, it can be taken as a free energy that determines the relaxation configuration. In the analog electrodynamics of a macroscopic media, one allows the electric permittivity and the magnetic permeability to relax dynamically so that the screening of electric and magnetic monopole charges becomes as perfect as possible. Because of the Lorentz covariance relation , screening of electric and screening of magnetic charges compete each other. The above extremal configuration is where the competition is balanced. While it is rather artificial in non-supersymmetric theories, the notion of dynamical relaxation is quite natural in supersymmetric field theories, supergravity theories and superstring theory. In fact, the idea has been used repeatedly for minimizing vacuum energy and determine physical parameters dynamically for various situations [33]. The only novelty in the present situation is that the background under consideration is not a flat spacetime but a BPS soliton or, as we will extend later, a black hole carrying nonvanishing electric and magnetic charges.

### 2.2 Local Special Geometry and Supergravity

#### 2.2.1 Local Special Geometry

Consider the space of the complex scalar fields associated with vector multiplets in the supergravity. Locally this space form a Kähler-Hodge manifold, endowed with a Kähler potential and a Kähler metric . The local supergravity algebra constrains that the Riemann curvature tensor of the Kähler-Hodge manifold should obey so-called ‘special-geometry’ relation:

(2.28) |

To define the supergravity couplings, we start by defining symplectic sections of the Hodge bundle:

(2.29) |

They are covariantly holomorphic

(2.30) |

Projection of ’s to ’s is achieved by a gauge fixing. Demanding that the scalar and the graviton kinetic terms decouple, we find the choice:

(2.31) |

In addition to the section , one can construct new symplectic sections out of :

(2.32) |

Then the special-geometry constraint Eq.( 2.28) is solved by the above symplectic sections if they satisfy relations:

(2.33) |

Here, a symmetric matrix is solved by combining the sections together into matrix and inverting the relations:

(2.34) |

Note that the gauge fixing condition Eq.( 2.31) becomes

(2.35) |

It is straightforward to solve in terms of the symplectic sections and holomorphic matrix :

(2.36) |

One further finds that

(2.37) |

It is possible to fix an overall scale of the symplectic section as

(2.38) |

It follows that are holomorphic sections over a line bundle. All the above relations are then straightforwardly rewritten in terms of the holomorphic sections. In terms of , the Kähler potential is given by

(2.39) |

Under the Kähler transformation , . Therefore provides a homogeneous local coordinate system on the Kähler manifold. One possible choice of the coordinate system is so-called ‘special coordinates’:

(2.40) |

The electric and the magnetic charges provide with the source to the black hole mass. To manifest the symplectic structure, it is convenient to introduce anti-self-dual field strengths:

(2.41) |

The equations of motion and the Bianchi identities are then compactly expressed as

(2.42) |

As in the rigid theory, the electric and the magnetic charges combine to a symplectic vector:

(2.43) |

Again, quantum mechanically, the electric and the magnetic charges are required to obey the Dirac-Schwinger-Zwanziger quantization conditions. Therefore, the charge symplectic vector is covariant only under the transformations.

#### 2.2.2 Central Charge and Bps Mass Spectra

For a manifest supersymmetric multiplet formulation, it turns out convenient to reorganize the gauge field strengths into a new linearly independent combinations:

(2.44) |

That there are no other linearly independent field strength combinations is easy to understand from the two identities:

(2.45) |

The and are the gravi-photon of the supergravity multiplet and the gauge fields of vector multiplets respectively.

Associated to the new linearly independent combinations of the gauge field strengths are complex-valued, -component central charge vector:

(2.46) | |||||

(2.47) |

One notes that, under the Kähler transformation , the central charge transforms as holomorphic sections: , . These central charge vectors satisfy quadratic sum rules

(2.48) |

where

(2.49) |

and

(2.50) |

The matrix defines a symplectic transformation associated with the Witten effect [32]:

(2.51) |

Similarly, the matrices , and are defined by replacing in Eqs.( 2.49, 2.50) into respectively. In terms of factorized matrices, the quadratic sum rules Eq.(2.48) are given by:

(2.52) |

The two formula shows clearly that defines a negative-definite metric, while defines a metric of signature in the quadratic central charge sum rules.

Since the central charge and its derivatives are projections of the electric and the magnetic charges with respect to the symplectic sections, they are in general functions of moduli fields and complex-valued. The utility of the new linear combinations of the gauge field strengths and associated the central charges becomes transparent once one solves the condition for nontrivial BPS black holes to exist. We now turn to these conditions.

### 2.3 Supersymmetric Black Holes

Consider supergravity theory coupled to vector multiplets. Explicit construction of the supersymmetric black hole utilizing the special geometry was initiated by Ferrara, Kallosh and Strominger [17].

One obtains the metric, the gauge fields including the gravi-photon field and the scalar fields configurations by solving the supersymmetry Killing spinor conditions to the gravitino and the gaugino supersymmetry transformation rules:

(2.53) |

The classical, supersymmetric black hole configuration is obtained by demanding an existence of covariantly constant spinors, with an ansatz

(2.54) |

Here, and are arbitrary constant denoting the magnetic and electric charges measured from field at spatial infinity. Recall that, in an analogy with the electrodynamics of a macroscopic media, field corresponds to the generalized electric and magnetic induction fields, and . Therefore, one expects that the electric charge is not the microscopic charge but the total charge including the screening and the Witten effect [32]. Given the constitutive relations Eq.(2.42), one should then find a relation to the fundamental, microscopic charges . By a straightforward calculation, one finds that

(2.55) |

The afore-mentioned screening and Witten effects are manifest from the charge relations.

A particularly interesting class of the black hole configurations is the ones with a frozen special coordinate fields, constant [18, 19]. The gaugino supersymmetry transformation rules in Eq.(2.53) then imply that the gauge fields associated with the vector multiplets should vanish everywhere:

(2.56) |

Inferring Eq.(2.47) for the above gauge field configuration , one finds

(2.57) |

and concludes that the black holes with frozen special coordinate fields exhibit a special feature that central charge is covariantly constant with respect to the special coordinates. Being independent equations, Eq.(2.57) in turn determines uniquely the configuration of the special coordinates .

The fact that the constant moduli ansatz leads to the lowest BPS mass of the black hole may be understood as follows. The total energy of the black hole configuration may be expressed schematically as:

(2.58) |

That this expression is positive-definite is guaranteed by the Witten’s positive energy theorem [34], applied to the background of a black hole with nonvanishing electric and magnetic charges [35]. The second term represents a harmonic map to the Kähler manifold whose coordinates are represented by the complex scalar fields . Since the weight is manifestly positive definite, the lowest but nonzero BPS energy is achieved by a constant harmonic map: , viz. map the entire space of the black hole exterior (outside the horizon) to a single point in the Kähler manifold. Because of supersymmetry, this in turn requires that the the gauge field strengths paired with fields vanish as well. In what follows, we will call the constant harmonic map as Kähler-BPS limit, since it follows from the minimization of the Kähler sigma model contribution to the BPS black hole energy.

### 2.4 Dynamical Relaxation of the BPS Black Hole Spectra

#### 2.4.1 Kähler-Bps Scalar Fields

One now solves the Kähler-BPS condition and determines explicitly the constant scalar fields as a function of charges:

(2.59) |

In this case the two quadratic chrage sum rules Eq.(2.52) reduce down to the square of the central charge itself:

(2.60) | |||||

While and matrices are different in general, for the sum rules of the minimized central charge, quadratic forms formed out of either matrices are the same. In subsequent calculations, we will use exclusively the quadratic form using the holomorphic matrix mainly for calculational convenience. However, all the formulas we derive in this paper are straightforwardly generalizable to the representation using the coupling matrix by replacing wherever appears into .

The minimization condition Eq.( 2.59) determines vacuum expectation value of the moduli fields as a function of the electric and the magnetic charges. The condition can be inverted as follows. One first recalls the symplectic orthogonality relation Eq.( 2.37) of the symplectic covariant vector :

(2.61) |

Then the minimization condition Eq.( 2.59) is solved by a linear map between the symplectic section and the electric and the magnetic charge :

(2.62) |

where are complex-valued parameters to be determined. Consistency condition imposes the symplectic inner product of this solution with respectively:

(2.63) |

Hence, we finally get the Kähler-BPS condition:

(2.64) |

Note that the right hand side of Eq.(2.64) is invariant under the Kähler transformation. This is a necessary condition since the electric and the magnetic charges carry no Kähler weights.

In order to obtain an explicit form of the constant scalar fields, one needs to solve the Kähler BPS condition Eq.(2.64). Using the Kähler-BPS saturated quadratic sum rules of the central charges Eq.( 2.60), one finds the solution as:

(2.65) |

On the right-hand side, the first term denotes a particular solution that satisfies , while the second term is a homogeneous solution that satisfies the Kähler-BPS condition Eq.( 2.64). It is straightforward to check that the solution Eq.(2.65) satisfies the symplectic constraint . Expanding Eq.(2.65) in components,

(2.66) |

one finds an agreement with earlier result by Ferrara and Kallosh[19].

A comment is in order about the relation between the BPS black hole free energy and the topological free energy [36] which appears naturally in threshold corrections to string effective supergravity. In terms of holomorphic sections , the central charge is given:

(2.67) |

Here, is the so-called holomorphic mass. We have emphasized that the expression is for a fixed charge vector by denoting the dependence on it explicitly. The topological free-energy is given by

(2.68) |

where the determinant is over the fermionic mass matrix. Thus the topological free energy sums up contributions of all virtual BPS states, hence, is an infinite sum over the logarithm of the free-energy associated with a single BPS black hole background. In general, however, minima of the topological free energy is distinct from that for the free energy of a single BPS black. Minima of the BPS black hole free energy is given by the Kähler-BPS condition

(2.69) |

viz. condition for a Kähler covariantly constant holomorphic mass:

(2.70) |

On the other hand, the minima of topological free energy is determined by:

(2.71) |

In Eq.(2.71), while the summand equals to the condition Eq.(2.70), it does not necessarily require for each term in the summand to vanish. It is evident that minima of the topological free energy is generically different from that determined by the Kähler-BPS condition.

#### 2.4.2 Kähler-Bps Condition in the Shifted Basis

It is possible to simplify the solution given in Eq.( 2.65, 2.66) further by making a symplectic transformation that amounts to the Witten effect [32] and associated shift of the electric charge. One first recalls that the quadratic sum rule of the central charge becomes manifestly a positive-definite quadratic form once the Kähler-BPS condition is satisfied:

(2.72) |

Define the following new symplectic sections and symplectic
charges^{5}^{5}5As mentioned at the end of Section2.2, one
can define another symplectic transformed vectors and charges
in which the holomorphic matrix in the following equations
is replaced by the coupling matrix .:

(2.73) | |||||

(2.74) |

A few comments are in order for the electric and the magnetic charges
in the new shifted basis. First, while the fundamental electric and
magnetic charges of the symplectic charge vector are integer-valued, hence, independent of the special coordinates
, the shifted electric charge
after the symplectic transformation depends on the special coordinates,
hence, takes an arbitrary value. This is the manifestation of induced charge
effect both due to the screening from and due to the
Witten effect^{6}^{6}6 This difference is also reflected on the
Dirac-Schwinger-Zwanziger quantization condition of
charges:

One also notes that the constitutive relations for the new symplectic
sections is given by^{7}^{7}7
In deriving this relations we have used Eq.(2.30) and the fact that
, which can be checked directly
from Eq.(2.33).

(2.76) | |||||

For Kähler-BPS states, the quadratic form of the central charge reads in the new shifted basis as:

(2.77) | |||||

Since the Kähler-BPS solution Eq.(2.65) is symplectically covariant, one may simply replace the original sections and charges into the new
shifted ones so that the form of equation is unchanged^{8}^{8}8One can
derive this new, shifted Kähler-BPS solution directly from Eq.
(2.65) using Eq.(2.51). :

(2.78) |

It is easy to check that the matrix inside the bracket on the right hand side of Eq.( 2.78) has a rank only. This is as it should be since the and the are related each other for a given electric and magnetic charges. Therefore, it is enough to solve only half components of Eq.(2.78). Typically since the sections are not modified by the shift symplectic transformation, it is more convenient to solve them. In terms of the shifted holomorphic sections, this is easily seen:

(2.79) |

To keep the Einstein-Hilbert term in the supergravity Lagrangian, it is necessary to choose the gauge. This gauge choice then determines the Kähler-BPS central charge in terms of the Kähler potential once the electric and magnetic charges are specified:

(2.80) |

Therefore, the Kähler-BPS black hole mass and the macroscopic entropy is given: