Boosted Rotating Dyonic Strings in Salam-Sezgin Model

We show that the bosonic sector of the $N=(1,0),\, 6D$ Salam-Sezgin gauged supergravity model possesses a $T$-duality symmetry upon a circle reduction to $D=5$. We then construct a simple magnetic rotating string solution with two equal angular momenta. Applying the $T$-duality transformation to this solution, we obtain the general boosted rotating dyonic black string solutions whose global structures and thermodynamic quantities are also analyzed. Owing to the fact that the solutions are not asymptotically flat, we find that there are two distinct globally-different non-extremal solutions with two different sets of thermal dynamic variables, with both satisfying the thermodynamic first law and the corresponding Small relations. However, their BPS limit becomes the same and we show that it preserves one quarter of supersymmetry by directly solving the corresponding Killing spinor equations.


Introduction
Six-dimensional supergravities admit many different gaugings and diverse vaccuum solutions.
For instance, the gauged N = (2, 2) supergravity model was achieved in [1].Unlike maximum supergravities in four, five and seven dimensions, the six-dimensional gauged N = (2, 2) supergravity model does not admit a maximally supersymmetric anti-de Sitter (AdS) vacuum.On the other hand, the gauged N = (1, 1) supergravity constructed by Romans [2] does possess supersymmetric AdS 6 vacuum solutions, thus enjoying interesting applications in holography.
The 6D Romans' theory can be obtained from massive type IIA supergravity from consistent Pauli sphere reduction [3] and the corresponding AdS 6 vacuum can be interpreted as a D4/D8brane configuration [4].Gauged N = (1, 1) supergravity with general matter couplings was obtained in [5].
In this paper, we study the simplest 6D gauged supergravity model with (1,0) supersymmetry, namely the Salam-Sezgin model [6].From ungauged supergravity point of view, the minimum model contains a tensor and an abelian vector multiplet, in addition to the minimum supergravity multiplet.Extensions of this model by coupling to more matter multiplets in a way that is free of local anomalies were proposed in [7][8][9].(See also [10,11].)One intriguing feature of the Salam-Sezgin model is that it admits a half-supersymmetric Minkowski 4 × S 2 vacuum, where the S 2 is supported by the magnetic dipole charge carried by the U (1) vector field, together with the dilaton potential.It was later found that such a vacuum can also emerge in some variant N = (1, 1) gauged supergravities [12,13].Subsequently, a large class of gauged supergravities with Minkowski×sphere vacua were classified in [14].
Another intriguing feature of the Salam-Sezgin model is that for a vacuum to preserve supersymmetry, the supertransformation of the gaugino leads to (e which implies that preserving any amount of supersymmetry requires non-trivial U (1) flux F µν .
By contrast, in ungauged theory with g = 0, it is preferable to set F µν = 0 for the construction of BPS solutions.Indeed, 1  4 -BPS dyonic string solution in Salam-Sezgin model [15] involves a magnetic dipole charge of F µν .Its non-extremal generalization was recently constructed in [16] where the string charge lattice was also analyzed.
In this paper, we shall add angular momentum to the dyonic string and study their BPS limit, generalizing the results of [15,16].In six dimensions, a string solution has 4-dimensional transverse space so that the rotation group is SO(4) with two independent orthogonal rotations.
However, owing to the necessity of involving the magnetic dipole charge, the construction of the rotating solutions becomes more subtle.For the static solutions, the magnetic dipole charge has the effect of squashing U (1) fibre over the S 2 base of the 3-sphere in the transverse direction.
We therefore consider only "two equal" angular momenta J a = J b so that the rotation occurs only in the U (1) fibre direction, while the S 2 base space is preserved.This greatly simplify the construction, but it can be still rather complicated if the target solution carries both the electric and magnetic string charges associated with the 3-form field strength, magnetic dipole charge of the 2-form field strength as well as both angular and boosted linear momenta.
Our breakthrough comes from the observation that the scalar potential of Salam-Sezgin model takes exact the same form as the conformal anomaly term in non-critical strings [17].
It has the consequence that the T-duality symmetry at the level of supergravity Lagrangian is still preserved in the Salam-Sezgin model.We find that the five-dimensional theory from S 1 reduction of the Salam-Sezgin model has a nonlinearly realized SO(2, 1)/SO(2) coset symmetry, under which the three abelian vector fields, namely the Kaluza-Klein vector, winding vector from B µν and the vector descending directly from the 6D U (1) gauge field, form a triplet.We give explicit global symmetry transformations of the SO(2, 1) acting on various fields.With this T -duality symmetry, we are able to construct the boosted rotating dyonic string solutions.In a typical ungauged supergravity, any Ricci-flat metric is a vacuum solution.We can thus start with the known neutral Myers-Perry rotating black hole metric as a seed solution and perform the appropriate Kaluza-Klein reduction.The global symmetry of the reduced theory can be used to generate more solutions.Upon lifting back to the original dimension, one can obtain charged rotating black holes.However, Ricci-flat metrics are not solutions in the Salam-Sezgin model; therefore, the usual Myers-Perry metric cannot be used as a seed solution.Furthermore, the involvement of the magnetic dipole charges make the construction of a simpler seed solution even more complicated.Nevertheless, we overcome this problem and obtain the seed solution by direct construction.Another consequence of that Ricci-flat metrics are not solutions of the Salam-Sezgin model is that the black holes are not asymptotic to flat Minkowski spacetime.We therefore do not have a fiducial spacetime for fixing the scaling symmetry in the time direction.We find that this can lead to two globally-different solutions with two distinct sets of thermodynamic variables, but both satisfying the first law of thermodynamics and corresponding Small relations.
The paper is organized as follows.In section 2, we analyse the T-duality symmetry of Salam-Sezgin model reduced on S 1 .We obtain the symmetry transformation rule and manifestly SO(2, 1)-invariant form in both Einstein and string frames.In section 3, we construct magnetic rotating string solution as a seed solution and obtain the general boosted rotating dyonic solutions.We find two globally different non-extremal solutions with different set of thermodynamic rules.However, in section 4, we show that they give the same BPS limits and we obtain the Killing spinors.We conclude our paper in section 5.

T-duality symmetry of Salam-Sezgin model
The Salam-Sezgin model in six dimensions is the minimum N = (1, 0) gauged supergravity.
The bosonic sector consists of the metric and matter fields (B (2) , A (1) , ϕ).The Lagrangian is [6] For convenience we omit the universal factor √ −g throughout the paper.Note that the subscript (n) denotes the associated quantity is an n-form.This bosonic sector resembles the noncritical bosonic string theory with the conformal anomalous term that admits pseudosupersymmetry [18].
In fact, the bosonic Lagrangian (2), on its own, can also be obtained from the sevendimensional noncritical string theory via the S 1 Kaluza-Klein reduction, after setting the Kaluza-Klein and winding vectors equal to A We next derive the full set of symmetry transformation rules that is useful for our construction of the rotating dyonic solution.

Kaluza-Klein reduction to D = 5
The standard Kaluza-Klein reduction ansatz of the Salam-Sezgin model (2) on S 1 associated with x coordinate is For later purpose, it is advantageous to redefine the scalar field and the form fields The reduced five-dimensional theory in the Einstein frame becomes Here we define some shorthand notations Quantities with a superscript "0" denote close field strengths without Kaluza-Klein modifications, i.e.

Global symmetry
At the first sight, the two scalars (φ 2 , ψ) form a complex scalar describing the coset of SL(2, R)/SO(2) and ( F0 (2) , H0 ) form a doublet under the SL(2, R) global symmetry, but this does not fit the Kaluza-Klein modifications of the field strengths.Instead, we should treat the scalar pair as the coset structure of the isomorphic SO(2, 1)/SO(2), with the three vector fields forming a triplet under the SO(2, 1).To make this idea concrete, we introduce the 3×3 Cartan generator H, and the upper and lower triangular root generators They satisfy the algebra We can now parameterise the coset and define M = V T V.The kinetic term of (φ 2 , ψ) can be expressed in a standard way by M, We define a 1-form vector field triplet and its field strength The corresponding kinetic terms from an invariant bilinear construction are The 2-form potential and its 3-form field strength are singlets under the SO(2, 1) global symmetry.In order to see that its Kaluza-Klein modification is indeed a singlet, we define a matrix We can now express the three-form field strength manifestly as a singlet: With these, we can write the five-dimensional reduced theory (6) in a manifestly invariant form, i.e., It can be easily seen that the theory is invariant under the general SO(2, 1) global transfor- We can parameterize S by Specifically, we have In finite version, the transformation matrix S is given by We can now give the explicit transformation rules of scalar fields {φ, ψ} and form fields They are • S 2 : • S 3 : Note that the dilaton φ 1 generates a constant shift R symmetry, so that the total global symmetry is SO(2, 1)×R ∼ GL(2, R).Note that if we set the Kaluza-Klein and winding vectors equal, we can consistently truncate out the SO(2, 1)/SO(2) scalar coset.This is analogous the D = 7 to D = 6 reduction, where a vector multiplet can be consistently truncated out, commented above section 2.1.

String frame
It is also instructive to discuss the global symmetry in the string frame.Under the conformal transformation the Salam-Sezgin model ( 2) becomes We can now see clearly that the g 2 indeed appears as if it is the conformal anomaly term in a noncritical string.We consider the circle reduction and set ϕ = 1 2 φ − 2Φ.We obtain the five-dimensional theory in the string frame, namely It is manifestly invariant under SO(2, 1).Note that we obtained the above equations by appropriate conformal transformation to convert the calculations in section 2.2 in the Einstein frame to the string frame.We also verified that (28) can indeed be obtained directly from (26) on the reduction ansatz (27).Since many black hole thermodynamic expressions are developed in the Einstein frame, we find it is advantageous to construct and study the rotating black holes in the Einstein frame.

Rotating dyonic string solution
A direct construction of rotating solutions can be a formidable task in the Salam-Sezgin model, when all the fields will be necessarily turned on.In supergravities, one typically adopts the solution generating technique that utilizes the global symmetry in the Kaluza-Klein reduced theory.Notable examples of such construction include the Sen [19] and Cvetič-Youm solutions [20].Such global symmetries are typically broken in gauged supergravities, making the construction much more subtle, e.g.[21,22] Fortunately, as we have seen in the previous section, the T-duality survives in Salam-Sezgin model despite of the gauging.However, there is still an extra subtlety arising from the gauging.
For the ungauged theory, we can start with a neutral rotating black string, which is a direct product a line and five-dimensional Myers-Perry black hole, as the seed solution and generate the charged ones by the solution-generating technique.In the gauged theory, there is no Ricciflat vacuum.The only known solutions in literature are the static dyonic strings and their BPS limit [15,16].Furthermore, the "harmonic function" associated with the magnetic string charge takes the form H P ∼ P/ρ 2 .The BPS constraint requires that magnetic dipole charge k and the P are constrained by some algebraic relation.This implies that the minimum solution necessarily contains both (P, k) parameters.We therefore construct first the seed solution of rotating strings carrying both (P, k) parameters.

Magnetic seed solution
The 1 4 -BPS static dyonic string with magnetic dipole charge was constructed in [15].Its nonextremal generalization was obtained in [16].Turning off the electric charge, the two form fields are given by Here we parameterize the magnetic string charge P as P = 1 2 s 1 c 1 .In this paper, we denote (s i , c i ) as The metric takes the form with σ 3 = dψ + cos θdϕ and dΩ 2 2 = dθ 2 + sin 2 θdϕ 2 [15,16].Note that the level surfaces in the four-dimensional transverse is not the round 3-sphere, but the squashed one, described as a squashed U (1) bundle over S 2 .The rotation thus occurs in the fibre direction.We find that the full magnetic rotating string is where the functions (h 1 , h 2 , ∆ ρ , H 1 ) and the two constants (ξ 1 , ξ 2 ) are

Generating the electric string charges
In the effective theory of strings, there is a usual process of generating the electric string charge.One can perform a Lorenz boost along the string direction and then perform Kaluza-Klein circle reduction and obtain the electrically-charged black hole in one lower dimensions.
One can then use the T-duality to map the Kaluza-Klein charge to winding charge and lift the solution black to the original dimensions.We can follow the same procedure on the magnetic seed solution (32), since we have established that the T-duality is preserved in five dimensions even for the gauged theory.
We start by performing a Lorentz boost The solution (32) can be expressed in terms of the form that is ready for Kaluza-Klein reduction The reduced five-dimensional solution is therefore given by The scalar φ 1,2 combinations are It is clear that for the reduced solution, we have Consequently, the field strengths defined in (7) are For this reduced set of fields, the five-dimensional theory ( 6) is invariant under the transformation rule Under this transformation rule, we obtain a new solution in five dimensions where the electric charge is carried by the winding vector B′ 1 instead.We then lift the solution back to D = 6, and we obtain the dyonic string solution We can now remove the prime in (41) and perform a further boost We finally arrive at the rotating and boosted dyonic string in Salam-Sezgin model: where (h 1 , h 2 , ∆ ρ , H 1 , ξ 1 , ξ 2 ) are given by (33).

Black Hole thermodynamics
The boosted rotating dyonic string we constructed above is non-extremal, with a horizon located at the largest root ρ h of ∆ ρ = 0, where we can derive the temperature and entropy straightforwardly.The solution contains independent parameters (µ, a, δ 1 , δ 2 , δ 3 , k), giving rise to five independent conserved "charge" quantities: the mass M , the angular momentum J a = J b , electric Q e and magnetic Q m string charges associated with the 3-form field strength, the dipole charge Q D associated with the 2-form Maxwell field strength, and finally the boosted linear momentum P x along the string direction x.We find that the complete set thermodynamic variables are All the thermodynamic quantities, except for (M, Φ D ) are obtained in the standard way, with no particular subtlety.(See e.g.[24] for a pedagogical review on the subject.)Since the metric is not asymptotic to flat spacetime, we do not have an independent way of computing the mass.
We derive the formula of mass by requiring that the first law exists.Another subtlety is that, as explained earlier, the magnetic string charge Q m should be treated as a thermodynamic constant, i.e., δQ m = 0. We further require that δQ D = 0, influenced by the BPS condition studied later.This allows to show, quite nontrivially, that the first law works with the mass derived.We then relax the condition δQ D = 0 and determine its potential Φ D .It is important to note that Φ D does not depend on (δ 2 , δ 3 ).It is now straightforward to verify that the first law and the Smarr relation (e.g.[24]) are with the understanding that δQ m = 0. Note that the thermodynamic pair (Q D , Φ D ) does not appear in the Smarr relation, consistent with the fact that Q D is dimensionless.

A globally different solution
The equations of motion of Salam-Sezgin model is invariant under the trombone-like global symmetry since it has a consequence of uniformly scaling the whole Lagrangian.We can thus make the Lagrangian invariant by scaling the Newton's constant appropriately.Specifically, we consider with the rest of fields and coordinates fixed.Use this symmetry, we can rewrite the seed solution (32) in a new way: The remainder of the functions (h 1 , ∆ ρ , H 1 , ξ 1 , ξ 2 ) are given by (33).After the same solutiongenerating process, we have a new boosted rotating dyonic string Following the same strategy, we obtain the thermodynamical variables: The first law (with δQ m = 0) and the Smarr relation are We can see that in this new solution, the parameters (δ 1 , δ 2 , δ 3 ) enter the thermodynamic variables in a more symmetric manner, while in the old solution, the parameter δ 1 stands out from (δ 2 , δ 3 ) parameters.
It should be pointed out that the symmetry (46) at the level equation of motion exists in ungauged supergravity; however, we do not apply this transformation on the asymptoticallyflat string solutions since it can alter the asymptotic structure.In the gauged theory, the solutions are not asymptotically flat, and we do not have a fiducial spacetime.However, not all the scaling choices lead to a sensible description of black hole thermodynamics.We only find one such alternate globally-different dyonic string that satisfies the first law of thermodynamics.

Non-BPS and BPS extremal limits
We have constructed two globally-different non-extremal boosted dyonic string solutions.We shall call (43) as solution A, and (49) as solution B. The horizon of both solution are determined as the largest root of the same function ∆ ρ .The extremal limit corresponds to taking the temperature to zero.There are two ways of achieving this.One is simply set ρ h = √ 2a.In this case, µ = 2a 2 and ∆ ρ has double roots.This is a rather standard and straightforward extremal limit, and we shall not discuss this further.
The other is the BPS limit where the parameters δ i are sent to infinity while (µ, a) are sent to zeros such that charges and angular momentum remains finite and nonzero.We shall discuss this BPS extremal limit in more detail.Specifically, the limit is achieved by taking and then sending ε to zero while keeping the following parameters finite Solution A : In this limit, we find that both non-extremal dyonic solutions become the same and it is given by with Solution A : H V = 1 + q 3 ρ 2 ; Solution B : It is clear that H V is associated with the PP-wave component of the solution, arriving from the BPS limit of the Lorentz boost.Although the two H V 's appear to be different, but the difference is trivial; a coordinate transformation t → t + q 3 /(2q 1 ) renders them the same.In terms of the new variables, the charges become Note that the horizon of the extremal solution is located at ρ = 0, giving rise to the nearhorizon geometry as a direct product of the boosted AdS 3 (or BTZ) metric and squashed S 3 .
From the volume of the squashed S 3 , we obtain the entropy in the extremal limit: This entropy formula is analogous to that of the extremal rotating black hole in five-dimensional ungauged STU model [20,23].The extra contribution from the magnetic dipole charge enters to the entropy formula as an overall factor via ξ 1 .
The BPS limit on (44) and (50) leads to the following same thermodynamic potentials: The mass and Φ D obtained from (44) and (50) are somewhat different.The BPS limit of (44) gives The limit of (50) gives However, the first law at this zero temperature, with δQ m = 0 is satisfied for both set of thermodynamic quantities, namely The linear relation between the mass and charge suggests that the limits are BPS.In order to verify this, we need to construct Killing spinors, which satisfy three Killing spinor equations [6] ( We write the dyonic string metric in the following vielbein basis where σ 3 was given earlier and σ 1 and σ 2 are Note that the round S 2 metric is dΩ We have chosen the chiral condition on the spinors to be We further find that the existence of Killing spinors requires the condition that relates the magnetic string and dipole charges, namely In other words, the existence of Killing spinors require that the Q m and Q D charges are algebraically related, thereby reproducing the relation first given in [15].This particularly implies that Q D should not be treated as a thermodynamic variable in the BPS limit also, since we always imposed δQ m = 0. The Killing spinor is then given by ǫ = (q 3 + ρ 2 ) where ǫ 0 is a constant spinor, satifying the projection condition Note that the first projection arises from the magnetic dipole charge and the second projection arises from the magnetic string charge.Adding an electric string charge of appropriate sign does not affect the Killing spinor.The PP-wave component would add a projection of (γ 01 +1)ǫ = 0, which is automatically satisfied under the chiral projection.Thus the whole BPS solution preserves 1/4 of supersymmetry and the rotation has no effect on the Killing spinors.

Conclusions
In this paper, we showed that despite the U (1) R symmetry gauging, the Salam-Sezgin model still has a global T -duality symmetry upon a circle reduction.The symmetry acts nonlinearly on the SO(2, 1)/SO(2) scalar coset, but linearly on the three abelian vectors which form an SO(2, 1) triplet.The antisymmetric tensor is the singlet of SO(2, 1).
In our construction of string solutions, we first constructed a simpler rotating magnetic string as a seed solution, and derived the general boosted rotating dyonic strings by applying the T -duality symmetry.The general solution is characterized by the mass M , electric and magnetic string charges (Q e , Q m ), a magnetic dipole charge Q D and two equal angular momenta J a = J b , together with a linear momentum P x along the string direction.We analyzed the global structure and established the first law of thermodynamics.Owing to the fact that the solution are not asymptotically flat, we do not have a fiducial Minkowski spacetime.This allows us to use trombone symmetry to obtain a globally different dyonic string solution that satisfy a different version of the first law.Under the BPS limit we found that both non-extremal solutions reduces to the same solution.The existence of Killing spinor enforces an algebraic constraint on the magnetic string and the dipole charges, obtained in [15], but for the general non-BPS solutions, all charges are completely independent.It is worth pointing out that the seed solution carrying magnetic string and dipole charges preserves only 1/4 in the BPS limit, i.e. the minimum amount of supersymmetry.The proper inclusion of the electric string charge, the PP-wave momentum, or the angular momentum does not break the supersymmetry any further.
The reason we consider the two equal angular momentum case, i.e.J a = J b is that in the static case, the 3-sphere in the transverse space is already squashed to be a U (1) bundle over S 2 .It is not clear whether squashed 3-sphere allows to have a more general J a = J b rotations, but it certainly deserves further investigation.The Salam-Sezgin model by itself suffers from local anomalies.Its anomaly-free extensions [7][8][9] based on the Green-Schwarz mechanism necessarily introduce higher derivative terms in the effective action [10], analogous to the wellstudied heterotic supergravity in ten dimensions [25][26][27][28][29][30].It is thus interesting to see how these higher order interactions will modify the thermodynamic quantities of the solutions.This is more challenging than the case of heterotic supergravity compactified on K3, where the higher derivative corrections to the onshell action of black string [31] can be obtained using the trick of [32] without actually solving the field equations.With the U (1) R symmetry gauging, the solution is no longer asymptotic to Minkowski space nor to AdS, and therefore it is unclear whether the trick, or its AdS improved versions [33,34], still applies.

( 1 )
, which allows one to truncate out consistently one combination of the two dilatonic scalars.Since the conformal anomaly in string theory preserves the T-duality, the seven-dimensional noncritical string theory reduced on two torus will have SO(2, 2) T-duality global symmetry.Setting one pair of Kaluza-Klein and winding modes equal reduces the global symmetry to the diagonal SO(2, 1) ∼ SL(2, R).Thus the global symmetry of the Salam-Sezgin model reduced on S 1 should have SO(2, 1) global symmetry.