1 Introduction

Quantum Kerr(Newman) degenerate stringy vacua in on a non-BPS brane

Sunita Singh,K. Priyabrat Pandey, Abhishek K. Singh and Supriya Kar1

Department of Physics & Astrophysics

University of Delhi, New Delhi 110 007, India

March 6, 2018

We investigate some of the quantum gravity effects on a vacuum created pair of -brane by a non-linear gauge theory on a -brane. In particular we obtain a four dimensional quantum Kerr(Newman) black hole in an effective torsion curvature formalism sourced by a two form dynamics in the world-volume of a -brane on . Interestingly the event horizon is found to be independent of a non-linear electric charge and the quantum black hole is shown to describe a degenerate vacua in string theory. We show that the quantum Kerr brane universe possesses its origin in a de Sitter vacuum. In a nearly -symmetric limit the Kerr geometries may seen to describe a Schwarzschild and Reissner-Nordstrom quantum black holes. It is argued that a quantum Reissner-Nordstrom tunnels to a large class of degenerate Schwazschild vacua. In a low energy limit the non-linear electric charge becomes significant at the expense of the degeneracies. In the limit the quantum geometries may identify with the semi-classical black holes established in Einstein gravity. Analysis reveals that a quantum geometry on a vacuum created -brane universe may be described by a low energy perturbative string vacuum in presence of a non-perturbative quantum correction.








1 Introduction

Conceptual ideas to explore a theory of quantum gravity have been in the folklore of theoretical physics for quite some time. There are a number of different techniques to investigate the quantum gravity effects to Einstein gravity [2, 3, 4, 5]. It is believed that a quantum gravity may be described by a background independent non-perturbation theory. Interestingly an eleven dimensional M-theory has been conjectured [6]. It attempts to unify five distinct string theories in ten dimensions. In a low energy limit, M-theory reduces to a supergravity theory. Furthermore M-theory on reduces to a type IIA superstring theory, where the even Dirichlet () branes are dynamical objects. Importantly a gravity theory in bulk and a gauge theory on its boundary have been established as an insightful tool using a holographic correspondence [7]. It explores a certain domain of quantum vacua via the gauge theoretic fluxes underlying a strong-weak coupling duality. On the other hand, there are theoretical attempts [8, 9, 10] to investigate an emergent quantum gravity underlying Einstein vacuum. An emergent gravity has also been conjectured to arise due to the statistical behavior of microscopic degrees encoded on a holographic screen [11].

In the past there have been attempts to construct extremal geometries on a BPS -brane in various dimensions [12]-[33]. They correspond to the near horizon black holes primarily described in a ten dimensional type IIA or type IIB superstring theory. Importantly a global NS two form in the string bulk couples to an electromagnetic field on a -brane to form a non-linear gauge invariant combination [12]. However a non-constant NS two form remains in the string bulk and hence does not play a role on a BPS D-brane. Nevertheless a closed string world-sheet conformal symmetry in presence of a NS two form of a two form lead to the vanishing of beta function equations and reassures a superstring effective action in ten dimensions. Interestingly a NS two form modifies the covariant derivative and is known to describe a torsion in an effective string theory [34, 35].

In the context an effective torsion curvature formalism underlying a two form in a gauge theory on a -brane was obtained by the the authors in the recent past [36, 37]. The five dimensional de sitter geometries obtained in the formalism were shown to be sourced by a geometric torsion. The formalism was exploited to describe a large number of tunneling vacua including de Sitter and anti de Sitter black holes in five dimensions on a vacuum created pair of -brane at the Big Bang. The gauge theory, underlying an effective curvature, allows a perturbative coupling of a global NS two form to a gauge theoretic torsion on a -brane. Interestingly some of the five dimensional torsion geometries on were shown to describe torsion free black holes on a vacuum created -brane within a pair [38]. Two of the brane universes have respectively been identified with a Reissner-Nordstrom (RN) and a Schwarzschild black hole in Einstein gravity. Subsequently the second order formalism was explored to obtain quantum Kerr geometries in five dimensions [39]. The torsion contribution was argued to be negligibly small in a low energy limit. Hence the torsion potential was ignored to obtain a Kerr black hole in a five dimensional brane universe. In the paper we investigate some of the plausible Kerr geometries on a vacuum created -brane universe within a pair using the formalism. More over the effective curvature scenario has been applied to various other interesting developments including a quintessence axion in string cosmology by the authors [40]-[46].

In particular a non-trivial space-time was argued to began on a vacuum created on a pair of -brane at an even horizon by a two form in a gauge theory on a -brane. A two form gauge theory is Poincare dual to an one form dynamics on a -brane. Thus the pair production idea is inspired by the established phenomenon of Hawking radiation at the event horizon of a background black hole by a photon in a gauge theory [2]. It may imply that a the gauge theory on a -brane may alternately be viewed through a vacuum created pair of -brane and an anti -brane in a five dimensional effective curvature formalism. An extra fifth dimension, transverse to a vacuum created pair of -brane, has been shown to play a significant role. It is believed take into account the perturbative closed string vacuum between a -brane and an anti -brane. In other words the formalism underlying a geometric torsion may seen to describe a low energy perturbative closed string vacuum in presence of a non-perturbative -brane and an anti -brane corrections [39, 44, 45, 46]. In the context a flat metric has been shown to be associated with a quantum correction. It indeed reassures the absence of closed string modes on a -brane. In particular a non-perturbative correction is shown to be sourced by a non-linear charge which possesses its origin in a geometric torsion. As a result a vacuum created brane universe may be approximated with the Einstein vacuum in presence of a non-perturbative quantum correction. The presence of a fifth dimension hidden to a 4D non-extremal quantum vacua may allow one to imagine the existence of another universe described by an anti-BPS brane. In fact a generic torsion curvature has been argued to began with a vacuum created pair of -instanton at the Big Bang by a two form [36]. Since the formalism evolves with an intrinsic torsion, a -brane is inevitable to nullify the torsion while establishing a correspondence with a known vacuum in Einstein gravity.

The underlying setup may be illustrated by considering an effective open string metric on an anti -brane underlying a global NS two form on its world-volume. The presence of a vacuum created anti -brane sourced by a geometric torsion in the formalism imply the existence of a -brane at a transverse distance. Thus the dynamics on a -brane, underlying the scenario of a pair of brane/anti-brane, may be described by a two form in presence of a background (open string) metric in a gauge theory. The non-linear quanta of a two form, or its Poincare dual one form, is argued to vacuum create a pair of -brane at the event horizon of an effective black hole established on an anti -brane. Needless to mention that a BPS brane and an anti BPS brane together breaks the supersymmetry and describes a non-BPS brane configuration. It may be noted that the formalism evolves with two independent two forms. One of them is a dynamical two form which is Poincare dual to an one form on a -brane. The other one is a global NS two form on an anti -brane which is known to describe an effective open string metric and hence an effective gravity may seen to emerge on an anti -brane. In other words a two form dynamics in a gauge theory on a -brane is explored in presence of a background black hole. Three local degrees of a two form on is described by an axionic scalar on an anti -brane and a non-linear one form on a -brane.

In the paper we obtain a number of quantum vacua on a created pair of -brane primarily sourced by a two form on a -brane. We construct the quantum Kerr-Newman geometries leading to a quantum black hole in four dimensions on a non-BPS brane in a superstring theory. The brane geometries in a certain regime are analyzed for their mass, angular velocity and charge if any. They are shown to map to Einstein vacua in string theory. A gauge choice for a two form leading to a vanishing torison in a generalized space-time curvature theory ensures Riemannian geometry underlying a vacuum . It is argued that the background fluctuations in on a -brane may have their origin in a dynamical two form described by a ten dimensional effective closed string action. In particular a five dimensional quantum Kerr black hole with two angular momenta have been obtained by the authors using a generalized curvature [39]. A quantum Kerr obtained in the formalism is analyzed in a low energy limit to describe a typical Kerr black hole in a five dimensional Einstein gravity [47]. Interestingly a number of rotating black holes in various dimensions are obtained in the folklore of Einstein gravity [48]-[54].

We observe that an event horizon in a 4D quantum Kerr(Newman) black hole is independent of its charge sourced by an one form. The charge independence identifies a magnetically charged quantum vacuum with an electrically charged one. In fact the quantum Kerr(Newman) black hole is argued to describe a degenerate Kerr vacua. A low energy limit removes the degeneracy in the Kerr and relates to a classical Kerr black hole in Einstein vacuum. Interestingly a (semi) classical Kerr black hole formally retains the causal patches of the quantum Kerr. It may imply that an exact geometry in perturbation theory validates a non-perturbative construction realized through a geometric torsion on a non-BPS brane. We show that a magnetic non-linear charge can be absorbed by a renormalized mass to describe a quantum Schwarzschild black hole for an -restoring geometry. In a low energy limit the quantum black hole may be identified with the Schwarzschild black hole in Einstein vacuum. On the other hand an electric non-linear charge underlying the degeneracies in a quantum Kerr, for an -restoring geometry, is shown to describe a RN quantum black hole in four dimensions. Its low energy vacuum is shown to describe a RN-black hole in Einstein-Maxwell theory. The renormalization of the mass in a RN quantum black hole has been invoked to argue for a plausible tunneling to a Schwarzschild quantum black hole.

We plan the paper as follows. A non-perturbative geometric torsion leading to an effective curvature underlying a non-linear theory on a -brane is revisited in section 2. We work out the quantum Kerr(Newman) geometries on a vacuum created pair of -brane in section 3. The degenerate Kerr(Newman) vacua are analyzed in a limit to obtain Reissner-Nordstrom and Schwarzschild quantum black holes in section 4. We explore a low energy limit leading to some of the semi-classical Einstein vacua in section 5. We conclude by summarizing the results obtained in an effective five dimensional curvature formalism underlying the quantum effects and outlining some of the future perspectives in section 6.

2 A non-perturbative setup sourced by a two form

2.1 Geometric torsion on a -brane

A BPS brane carries an appropriate RR-charge and is established as a non-perturbative dynamical object in a ten dimensional type IIA or IIB superstring theories. In particular, a -brane is governed by a supersymmetric gauge theory on its five dimensional world-volume. However, we restrict to the bosonic sector and begin with the gauge dynamics in presence of a constant background metric on a -brane. A linear one form dynamics is given by

(1)

where denotes the gauge coupling. Remarkably a non-linear gauge symmetry is known to be preserved in an one form theory in presence of a constant two form on a -brane. In the past there were several attempts to approximate a non-linear gauge dynamics by Dirac-Born-Infeld action coupled to Chern-Simmons on a BPS D-brane [13, 14]. The BPS brane dynamics is known to describe an extremal black hole which corresponds to a near horizon geometry in a string theory.

In the context a non-linear gauge dynamics may also be re-expressed in terms of a two form alone on a -brane which is Poincare dual to an one form. The duality allows one to address the one form non-linear gauge dynamics in presence of a constant two form on a -brane. A Poincare duality interchanges the metric signature between the original and the dual. The two form gauge theory on a -brane may be given by

(2)

where denotes a gauge coupling. The local degrees in two form on a -brane have been exploited to construct an effective space-time curvature scalar in a second order formalism [40]-[39]. The curvature is primarily sourced by a two form gauge theory on a -brane. Generically an irreducible scalar governs a geometric torsion which is primarily described by a gauge theoretic torsion on a -brane. Unlike the extremal brane geometries, the dynamics of a geometric torsion on an effective curvature formalism addresses some of the non-extremal quantum vacua in string theory. In fact the emergent black holes are described by a pair of brane and anti-brane separated by a transverse dimension. The -pair breaks the supersymmetry and may describe a non-BPS brane in string theory.

A priori the required modification to incorporate a geometric notion may be viewed via a modified covariant derivative defined with a completely antisymmetric gauge connection: . The appropriate derivative may be given by

(3)

Under an iteration the geometric torsion in a second order formalism may be defined with all order corrections in in a gauge theory. Formally a geometric torsion may be expressed in terms of gauge theoretic torsion and its coupling to two form. It is given by

(4)

An exact covariant derivative in a perturbative gauge theory may seen to define a non-perturbative covariant derivative in a second order formalism. Thus a geometric torsion constructed with a modified covariant derivative (3) may equivalently be described by an appropriate curvature tensor which has been worked out by the authors [36]. Explicitly the effective curvature tensors are given by

(5)

The fourth order tensor is antisymmetric within a pair of indices, and , which retains a property of Riemann tensor . However the effective curvature do not satisfy the symmetric property, under an interchange of a pair of indices, as in Riemann tensor. Nevertheless, for a constant torsion the generic tensor: . As a result, the effective curvature constructed in a non-perturbative formalism may be viewed as a generalized curvature tensor. It describes the propagation of a geometric torsion in a second order formalism.

2.2 Emergent metric fluctuations

A geometric torsion in a second order formalism may seen to break the gauge invariance of a two form in the underlying gauge theory. Nevertheless an emergent notion of metric fluctuation, sourced by a two form local degrees, restores gauge invariance in a generalized irreducible space-time curvature . The non-perturbative fluctuations, underlying a gauge invariance, turn out to be governed by the fluxes and are given by

(6)

where is an arbitrary constant and . The generalized curvature tensor may also be viewed though a geometric field strength which is Poincare dual to on a -brane. Then, a geometric may be given by

(7)

where is defined with a global NS on a -brane. It signifies a non-linear electromagnetic field and is gauge invariant under a non-linear transformations [12]. Apparently a non-zero seems to break the gauge invariance. Nevertheless, an action defined with a lorentz scalar may seen to retain the gauge invariance with the help of an emerging notion of metric fluctuations in the formalism. Then, the fluctuations (6) in its dual description may be given by

(8)

where is an arbitrary constant. The dynamical fluctuations in eqs.(6) and (8) modify the constant metric on a -brane. Then, the emergent metric on a -brane becomes

(9)

The fluxes in a bilinear combination are gauge invariant. The emergent metric sourced by the fluxes a priori seems to be unique. However an analysis reveals that the emergent metric may not be unique due to the coupling of -potential to underlying a geometric torsion . In other words -fluctuations do play a significant role to define the emergent geometries on a vacuum created brane universe. They lead to a generalized notion of metric on a -brane. It may be given by

(10)

The background fluctuations arising out of the non-dynamical components in may seen to deform the brane geometries significantly. They may lead to a large number of vacua and may correspond to the landscape quantum geometries in the formalism. The background fluctuations in two form may have their origin in a higher dimensional gauge theoretic torsion . They may couple to an electro-magnetic field in higher dimensions to define a gauge invariant non-linear .

2.3 Non-perturbative space-time curvature

The gauge dynamics on a -brane in presence of gauge connections, may be approximated by an irreducible generalized curvature theory in a second order formalism. A priori the effective curvature may seen to describe a geometric torsion dynamics on an effective -brane [36]. A geometric construction of a torsion in a non-perturbative formalism is inspiring and may provoke thought to unfold certain aspects of quantum gravity. Generically the action may be given by

(11)

where is a constant and . The cosmological constant , in the geometric action is sourced by a global NS two form in the theory. With a generalized curvature dynamics on , underlying an effective -brane [38], may appropriately be given by

(12)

The curvature scalar is sourced by a dynamical two form in a non-linear gauge theory. The field strength is governed by the equations of motion of a two form in presence of a flat background metric in a gauge theory. Explicitly the field equations of motion are given by

(13)

The field strength is appropriately modified to describe a propagating torsion in four dimensions underlying a second order formalism. The fact that a torsion is dual to an axion on an effective -brane ensures one degree of freedom. In addition describes a geometric one form field with two local degrees on an effective -brane. A precise match among the (three) local degrees of torsion in on with that in and reassure the absence of a dynamical dilaton field in the frame-work. The result is consistent with the fact that a two form on does not generate a dilaton field. The equation of motion for the one form is defined with an appropriate covariant derivative. It is given by

(14)

where and . The energy-momentum tensor is computed in a gauge choice:

(15)

Interestingly the in a non-linear gauge theory on a -brane incorporates an emergent metric in a generalized curvature theory, . The gauge choice ensures that the sources a nontrivial emergent geometry underlying a non-linear gauge theory. The covariant derivative satisfies . Thus, an emergent metric in the framework uniquely fixes the covariant derivative. It implies , which in turn incorporates a conserved charge in the formalism. The effective curvature theory may formally be viewed as a non-linear gauge theory. The energy-momentum tensor may be given by

(16)

The trace of energy-momentum tensor on a -brane becomes

(17)

It ensures that a vacuum, , may be defined in a gauge choice: ( and ) or (). We consider a gauge choice:

(18)

Then a in the gauge choice sources a generic ( and ) emergent metric on an arbitrary dimensional -brane underlying an effective curvature . It is given by

(19)

Thus the in a gauge theory sources the dynamics of a torsion in a generalized curvature theory. A higher dimensional can source a lower dimensional background fluctuations in two form on a brane. A gauge choice (18) in eq.(15) ensures and hence the metric fluctuation becomes significant on a -brane. Thus the non-trivial flux in a gauge choice leads to a dynamical geometric torsion underlying a generalized curvature on a -brane.

3 -deformed quantum vacua on a -brane

In this section, we obtain the quantum Kerr geometries constructed on a non BPS brane in four dimensions in presence of a non-linear charge . Interestingly a quantum Kerr-Newman black hole, on a non-BPS brane in a superstring theory, may seen to describe a number of vacua in various lower energy scales to Planckian energy. Arguably, the vacuum geometries tunnel among themselves at an intermediate energy scale. The brane geometries obtained in a non-perturbative formalism are analyzed in a low energy limit to realize, a Kerr-Newman, a Kerr, a Reissner-Nordstrom and a Schwarzschild, black holes in Einstein vacuum. In fact, we address some of the rotating quantum black holes obtained on an effective -brane underlying a generalized curvature on .

3.1 Flat metric

We use Boyer-Lindquist coordinates on a -brane underlying a Minkowski space-time. The range: and completely specify the flat space for the line-element. Now, we set in the paper. The cartesian coordinates may be defined by a spheroidal coordinate system. They are:

(20)

where is an arbitrary constant and shall be identified with a symmetry breaking background parameter. The coodinates ensure a circle in -plane with a varying -coordinate for . It is given by: and . Nevertheless, for and , the coordinates ensure a circle in -plane and is given by: and . Similarly for and , a circle is defined in -plane: and . Generically, all three circle equations satisfy:

(21)

The equations in -, - and -planes on the equator, respectively, define a ring of radius for , and . The Minkowski vacuum on a -brane defined with a flat metric in Boyer-Lindquist coordinates is given by

(22)

In a limit , the first expression assures a flat -symmetric vacuum on a -brane. It may seen to simplify the quantum geometry without changing its characteristic properties. Generically the limit may as well describes the geometry at poles for . The second and third expressions confirm that the -symmetry in a vacuum is broken by a perturbation parameter ’’. Contrary to a forbidden limit for the effective radius on a -brane [39], the limit turns out to be an allowed on the equator for on a -brane. In the limit, the metric possesses a coordinate ring singularity on a equatorial plane.

3.2 Gauge field ansatz

A two form on a -brane satisfies the equation of motion (13) in a non-linear gauge theory. A two form ansatz leading to a family of Kerr vacua in Einstein gravity is worked out. In presence of a spherical symmetry breaking perturbation parameter , the two form ansatz may be expressed as:

(23)

where is an arbitrary constant and shall be identified with a mass with a lower cutoff in quantum gravity.

On the other hand, the one form ansatz may be given by

(24)

where an arbitrary constant signifies a non-linear electric charge presumably with a lower cutoff . Generically a lower bound in and are enforced by their non-linearity, which can not be gauged away completely in a gauge theory. It shows that the component is sourced by the in presence of the background parameter . Thus, a magnetic field is generated via the spherical symmetry breaking parameter ’’ from an electric field. We shall see that the intermingle, of an electric with a magnetic field, phenomenon may well be described with a subtlety. Presumably, the parameter incorporates an euclidean notion of time within an ergo sphere of an emergent black hole. The non-zero components of field strength are worked out to yield:

(25)

In a limit the electromagnetic field in the geometric framework may only be described by a non-linear electric field. In the gauge choice the gauge theoretic torsion and hence the geometric torsion . The gauge ansatz freezes the local degrees of torsion on a -brane within a vacuum created pair of brane/anti-brane. In the case the brane dynamics is solely contributed by an one form in four dimensions. A nontrivial geometric field strength reduces in the gauge choice, , and is defined with a non-linear charge . The non-linearity in charge is due to a global NS two form [12]. A non-vanishing implies a non-trivial energy-momentum tensor in the gauge theory on a -brane. However a vanishing trace of energy momentum tensor in 4D may hint for a vacuum solution in Einstein gravity. Thus a conserved non-linear charge is defined in absence of a torsion on a -brane. The gauge field equations of motion in the case leads to for an appropriate conserved current and defines a conserved charge in the formalism.

3.3 Causal geometric patches

The -fluctuations in presence of may seen to source a non-trivial emergent metric on an effective -brane. For the emergent metric in the gauge choice on a -brane reduces to yield:

(26)

Two geometries for an emergent metric tensor is a choice keeping the generality in mind. Primarily they dictate the quantum geometric corrections and incorporate local degrees to a background metric on a -brane. The non-trivial metric components sourced by the gauge fields on a -brane are worked out to yield:

(27)

The emergent metric components on a -brane may be obtained under . For simplicity we analyze the geometries in a special limit on a brane. It reduces to yield:

(28)

The emergent quantum geometries for and on a brane may be re-expressed by a Kerr black hole in presence of geometric corrections coupled to a charge in a non-linear perturbation theory. They may be given by

(29)

where , signifies an -deformed in presence of the background parameter ’’. Interestingly, the quantum (geometric) corrections to possess their origin in a flat line-element (22) in a limit . Then the causal patches in eq(29) associated with an electric charge are obtained in a limit . It is given by

(30)

It reconfirms a zero curvature in the geometric patches associated with a quantum gravity correction on a pair of -brane in a type IIB superstring theory. We shall see that the quantum correction is indeed non-perturbative. A flat metric underlying a quantum correction to an emergent Kerr geometry is remarkable. It may turn out to be a potential tool to explore some of the unresolved issues in quantum gravity. On the other hand, the Kerr line-element in eq(29) may explicitly be given by

(31)

The causal patches characterize a 4D quantum Kerr black hole on a pair of -brane in presence of a fifth dimension. In fact the scale of an extra dimension distinguishes a low energy vacuum from its quantum vacuum. The Kerr vacuum is sourced only by a two form in a non-linear gauge theory. Under an interchange of angular coordinates with their appropriate normalizations, the angular velocity becomes significant at the expense of and leads to a precise Kerr vacuum in Einstein gravity. It reconfirms a Schwarzschild vacuum for in a global scenario underlying a pair of -brane. Presumably an extra dimension transverse to -brane and an anti -brane in a global scenario does not allow an annihilation of a pair of branes to a BPS brane. The irreversibility of pair creation process may also be argued from the non-linearity in a two form quanta in a superstring theory.

In fact an emergent Kerr geometry (31) on a -brane when comes in contact with that on a -brane, does not reduce to a flat line-element (22) on a BPS -brane. However for an angular velocity on a -brane nullifies that on a -brane and hence reduce to a quantum Schwarzschild black hole. Then the relation in eq(29) takes a form:

(32)

where a quantum Schwarzschild may be obtained from in eq(31) for . The quantum 4D Schwarzschild black hole obtained on a -brane hints at a small fifth dimension along with a deformed -geometry when compared with a a Schwarzschild black hole in Einstein vacuum. The extra dimension and the deformations are intrinsic to a geometric torsion on a -brane in the formalism. In other words the presence of a fifth dimension signifies a propagating torsion in quantum gravity.

Similarly the quantum geometric patches (28) in a limit may be rearranged in terms of their coupling to and a charge to yield:

(33)

where is defined in the limit. The is defined in eq(22) and all the remaining line-elements describe flat causal patches. Explicitly may be given by

A priori the fluxes Weyl scale the flat metric patches on a brane and may not lead to a black hole vacuum in absence of a -fluctuation. Thus a two form plays a significant role to describe an emergent graviton underlying a non-perturbative geometric torsion in the formalism. On the other hand an one form has been shown to incorporate a non-perturbative quantum correction underlying a flat geometry. Presumably it would describe a “graviton” within a gauge choice for a vanishing torsion in the formalism.

Furthermore eqs(30) and (33) confirm that an -deformation parameter ’’ incorporates geometric perturbations within a non-perturbative quantum correction. For the quantum geometries (33) reduce drastically on a pair of -brane in a pre-defined regime with and is obtained in eq.(32). Explicitly it is given by

(34)

Analysis reveals the significance of a two form over an one form in the formalism. It is in agreement with