Emergent gravity/Non-linear U(1) gauge theory correspondence

Kaluza-Klein gravity is revisted, with renewed interest, in a type IIB string theory on $S^1\times K3$. The irreducible curvature tensors are worked out in the, T-dual, emergent gravity in 4D to yield a non-linear U(1) gauge theory. Interestingly, the T-duality may be seen to describe an open/closed string duality at a self-dual string coupling. The obtained deformation in $AdS_5$ black hole is analyzed to introduce the notion of temperature in the emergent gravity underlying the recent idea of entropic force.


Introduction
The intrinsic notion of geometry, governing space-time curvature, is manifested beautifully in the success of the General Theory of Relativity (GTR). Importantly, the Einstein's field equations confirm the presence of non-linear matter and establishes a precise relation between the matter and geometry in GTR. With the help of an extra dimension to Einstein's theory, the matter/geometry relation may be explored to view as a gauge/gravity duality or in particular AdS 5 -gravity/gauge theory duality established in literatures [1,2,3].
Very recently, it has been conjectured that gravity emerges at a macroscopic level due to the entropic force described by the the matter field, which in turn results in a deformed geometry [4].
It is plausible that the deformation may be described by D-branes [5] contained within higher dimensional branes in string theory [6,7,8]. Similar to AdS/CFT, the emergent gravity is believed to be projected along a preferred holographic direction in space, which possibly governs a cut-off scale on the boundary. Since a flat space is not associated with any holographic direction, it is promising to begin with Newtonian potential to describe the emergent gravity phenomenon as a non-relativistic limit of GTR. As a result, the holographic direction may be seen to be governed by a gradient of the scalar potential, which in turn describes an accelerated frame and establishes the notion of temperature [9].
In the article, we attempt to focus on the notion of emergent gravity in 4D due to the two-form underlying a string theory in presence of D 3 -branes. In section 2.1, we elaborate on the idea of emergent gravity underlying "Kaluza-Klein" theory. In section 2.2, we re-investigate a deformed AdS 5 black hole geometry in the emergent gravity to emphasize the notion of temperature in the formalism. We obtain the dual actions in 4D by taking into account the irreducible gauge and gravity curvature tensors in section 3. We conclude the article in section 4.
2 Emergent gravity in 5D due to a two-form

"Kaluza-Klein" gravity
We begin with Kaluza-Klein gravity in presence of a cosmological constant Λ = 0. The action is given by where G N denotes the (appropriate) Newton's constant. We revisit the Einstein's geometric notion in GTR, by incorporating a covariantly constant dynamical two-form B mn along with a non-dynamical metric tensor g mn in the Kaluza-Klein gravity theory. Interestingly, the typical notion of gravity leading to nontrivial geometry may be seen to be prevailed with the source of two-form. In fact, the dynamical metric in Kaluza-Klein theory may alternately be obtained from a two-form using a relation: A priori, the "Kaluza-Klein" gravity due to the two-form source may be given by The cosmological constant becomes redundant in the two-form source action. However in the emergent gravity (2) description, Λ = 0 becomes significant. The irreducible scalar curvature K is obtained from the 4th order mixed curvature tensor. It is given by The new connections in the formalism may be expressed as: The covariant derivative is uniquely fixed by D p g mn = 0 and H mnp = (D m B np + cyclic) = 0 in the formalism. It is important to note that the generic curvature tensor K mnp q modifies the Riemann-Christoffel curvature R mnp q in Kaluza-Klein gravity. However for a constant torsion T s mn = −2Γ s mn , the geometric curvature K mnp q precisely reduces to R mnp q and Λ = 0. In other words, the Riemannian notion of geometry leading to a typical Kaluza-Klein gravity is restored for a constant torsion. As a result, a constant torsion geometry in the formalism is indistinguishable from the dynamical metric geometry. The field equations may be given by 2.2 Notion of temperature from deformed AdS 5 We consider the B-field ansatz in obtained recently in ref. [5]. They are given by: where 0 < θ ≤ π, 0 < ψ ≤ π, 0 ≤ φ ≤ 2π and (r 0 and q) are are arbitrary constants. The non-vanishing components of torsion become Γ θψφ = q 2 (sin θ sin 2 ψ) and Γ θ0φ = Γ θrφ = (qr 0 ) 2r 2 (sin θ sin 2 ψ) .
The geodesic describing an AdS 5 black hole may be obtained from the two-form using the relation (2) around an AdS vacua. The AdS 5 geometry is given by where f = q(cot θ sin ψ) and b denotes the AdS radius. The deformed S 3 geometry in AdS 5 black hole is a new feature in the formalism. Importantly, the deformation to the AdS 5 Schwarzschild black hole [10,11] in a typical Kaluza-Klein gravity is essentially due to the two-form in the formalism. In other words, the broken S 3 -symmetry is restored for f = 0, i.e. for a constant torsion and the AdS 5 geometry (9) reduces to the AdS 2 × S 3 Schwarzschild black hole geometry.
In fact, the S 3 deformation, by an S 2 , is independent of the torsion and is solely a characteristic of the non-vanishing two-form. Since a dynamical two-form is dual to a scalar potential, the notion of temperature (T ) may be seen to be governed by the deformation geometry in the formalism. For a constant torsion, the curvature is described by the Riemannian geometry and the deformation vanishes leading to T = 0. It is thought provoking to believe that the association of temperature is possibly defined naturally in any formalism defined with a non-vanishing twoform. For instance, the deformed black hole geometry obtained on a non-commutative D 3 -brane [12,13], is a consequence of the non-commutative parameter and has been argued to be the origin of non-zero temperature [14].

Dual geometries in 4D
The Kaluza-Klein compactification of the fifth dimension in the action (1) is worked out, a priori, to obtain the gravity and U (1) gauge curvatures. The irreducible curvatures, obtained by Kaluza-Klein compactification of action (3), give rise to where the geometric scalar curvature K (4) is precisely governed by the torsion in the formalism.
On the other hand, the torsion may be thought of as higher order derivative corrections to the electromagnetic field strength in a dual channel [5]. Alternately, one may obtain an U (1) gauge invariant field strength by incorporating the torsion. The non-linear U (1) field strength becomes The irreducible curvatures in the action (10) may be re-expressed in terms of non-linear gauge curvature in a dual channel. The action takes a simple form The non-linear U (1) gauge theory in 4D dual to "Kaluza-Klein" gravity is remarkable. Firstly, it re-assures Einstein's assertion that the space-time curvature is completely governed by the nonlinear matter field. Secondly, the underlying duality is accompanied with an a prior surprise due to its amazing correspondence between "Kaluza-Klein" gravity and the non-linear U (1) gauge theory in 4D. Interestingly, the gravity/gauge theory correspondence is along the line of holographic idea.
On the other hand, the apparent duality between the "Kaluza-Klein" gravity and gauge theory may be better understood in a closed string theory. For instance, we consider a D=10 type IIB string on S 1 × K3 obtained by one of the author in a collaboration [15]. The NS-NS sector in 5D may be worked out for a consistent truncation to yield "Kaluza-Klein" gravity (3). In particular, the geometric scalar curvature in the formalism may be identified with the gauge curvature in string theory, i.e. K ≡ −(dB) 2 , in a dual channel with a covariantly constant dynamical two-form.
In fact, the obtained AdS 5 geometry (9) may be viewed as an underlying AdS 5 × S 5 geometry in type IIB string theory. This in turn, corresponds to the open string boundary gauge theory on a D 3 -brane [1]. In other words, the non-linear U (1) gauge theory (12) may be identified with the D 3 -brane Born-Infeld dynamics in string theory [5]. As a result, the gravity/gauge theory correspondence may be viewed as closed/open string duality on the one hand and the T -duality (R → α ′ /R) in string theory on the other hand. The identification of gravity/gauge theory correspondence under two distinct (T -and S-) dualities in string theory is unique to the emergent gravity description due to the B-field source. This is essentially due to the fact that the underlying S-duality is defined at the self-dual string coupling, i.e. g s = 1, and the T -dual fifth dimension in closed string theory is transverse to the D 3 -braneworld [12].

Concluding remarks
The emergent Kaluza-Klein gravity due to the source of B-field is worked out in a type IIB string theory on S 1 × K3. Interestingly, the T -duality was identified with the S-duality at a self-dual string coupling in the formalism. The non-linear U (1) gauge theory obtained in the dual channel was identified with the Born-Infeld dynamics of D 3 -brane. In addition to the open/closed string duality, the formalism allows one to obtain a dual D 3 -brane description in 4D itself. On the one hand, the electro-magnetic field may be seen to govern the D 3 -brane dynamics underlying a fundamental open string theory (10). On the other hand, a non-linear electro-magnetic field governs a dual D 3 -brane dynamics underlying a NCOS theory (12). Interestingly, the D 3 -brane and its dual are manifestations of two distinct U (1) gauge symmetries established in literature [16].
The deformed AdS 5 black hole obtained in the emergent gravity description was re-investigated for the notion of temperature. It was shown that a two-form is solely responsible for the deformation geometry which in turn gives rise to a non-zero temperature in the formalism. Our analysis may be seen to be in conformity with the recent idea of entropic force underlying an emergent gravity [4]. We observe that the Einstein's geometric description of gravity is potentially a powerful tool and may help one to explore new physics in various other [17,18], a priori, unrelated branches in science.