Poisson-sigma model for 2D gravity with non-metricity
aa r X i v : . [ h e p - t h ] J un MIT-CTP 3842 arXiv:0706.4070
Poisson-sigma model for 2D gravity withnon-metricity
M. Adak
Department of Physics, Faculty of Arts and Sciences, Pamukkale University,20017 Denizli, Turkey
D. Grumiller
Center for Theoretical Physics, Massachusetts Institute of Technology,77 Massachusetts Ave., Cambridge, MA 02139E-mail: [email protected] , [email protected] Abstract.
We present a Poisson-sigma model describing general 2D dilaton gravitywith non-metricity, torsion and curvature. It involves three arbitrary functions of thedilaton field, two of which are well-known from metric compatible theories, while thethird one characterizes the local strength of non-metricity. As an example we showthat α ′ corrections in 2D string theory can generate (target space) non-metricity.PACS numbers: 04.60.Kz, 02.30.Ik, 04.70.-s
1. Introduction
Dilaton gravity in two dimensions has found many classical, semi-classical and quantumapplications [1]. A very efficient way to describe it and to exhibit integrability isby means of a first order formulation [2] in terms of Cartan variables, dyad e aµ andconnection ω ab µ . It is closely related to a specific type of non-linear gauge theory [3],a Poisson-sigma model (PSM) [4]. However, it needs some extra structure beyond theone provided by the PSM, namely the specification of a tangent space metric η ab . Themetric then follows uniquely from this specification, g µν = e aµ e bν η ab , where the fields e aµ are determined from the dynamics of the PSM. In [5] it was suggested to introducethe tangent space metric as an external tensor, whose compatibility with the PSMstructure was analyzed carefully. All relevant examples presented there introduce η ab as a fixed metric depending on certain 0-form fields. Thus η ab is not considered as anindependent field. It would be nice if there was a pure PSM formulation that allowedto introduce the tangent space metric as an independent field. This is the first problemwe would like to address. A related drawback of the standard PSM formulation is thedifficulty to incorporate non-metricity. So far only isolated second order models are oisson-sigma model for 2D gravity with non-metricity U , V and W are some functions defining the model) I = k Z M h X d ω + X a (cid:0) δ ba d+ ǫ ab ω (cid:1) ∧ e b + ρ ab ∧ (cid:0) d η ab − W ( X ) η ab X c ǫ cd e d (cid:1) + 12 ( V ( X ) η ac + U ( X ) X a X c ) ǫ cb e b ∧ e a i . (1)We shall prove in Section 2 that (1) is a specific PSM. Our notation is explained thereas well. Section 3 provides a derivation of all classical solutions descending from (1).The final Section 4 contains a discussion, an application and a comparison with existingspecial cases of two-dimensional dilaton gravity models with (or without) non-metricity.
2. Action
Let us consider a PSM [4] (we drop all boundary terms in this work) I = k Z M h A I d X I − P IJ A J ∧ A I i = k Z M h X I d A I − P IJ A J ∧ A I i , (2)with the field content X I = ( X, X a , η ab ) and A I = ( ω, e a , ρ ab ). The former are thetarget space coordinates of a Poisson manifold with the Poisson tensor P IJ dependingon them, while the latter are gauge field 1-forms, e.g. e a = e a µ d x µ . Here the index a runs over two values, which we denote by + , − , while the index ab runs over threevalues, which we denote by ++ , + − , −− . For convenience we also introduce the value − + for ab and assume η ab = η ba and ρ ab = ρ ba . This allows us to consider the index ab as an index pair. The indices I, J, . . . always run over the full set X, + , − , ++ , + − , −− .The index position was chosen for sake of similarity with the PSM notation, but onecould equally define target space coordinates of the form X a , η ab and gauge fields e a , ρ ab .The construction of a metric g = e a e b η ab or g = e a e b η ab requires that X a and η ab havethe same index positions. A priori there is no device to manipulate indices, besides theLevi-Civita and Kronecker symbols. The real coupling constant k is irrelevant for ourdiscussion. The suggestive notation is chosen for sake of clarity, but we emphasize thatat this point the fields do not have some geometric interpretation in terms of “dilatonfield”, “Zweibeine” or “connection”. Neither do the indices ± necessarily imply lightcone variables, the use of which is very convenient in two-dimensional gravity [9, 10].We employ the Einstein summation convention and introduce the abbreviation ǫ ab := ˜ ǫ ac η cb (3)with the Levi-Civita symbol ˜ ǫ ab = − ˜ ǫ ba , where ˜ ǫ − + = 1. From these definitions we derivethe useful relations ǫ aa = 0 and ǫ ac ǫ cb = ( − det η ) δ ba , where det η := η ++ η −− − η + − η + − oisson-sigma model for 2D gravity with non-metricity δ ba is the Kronecker symbol. For the Poisson tensor P IJ = − P JI we choose P X,a = X b ǫ ba (4) P a,b = − V ( X ) (cid:0) η ac ǫ cb − η bc ǫ ca (cid:1) − U ( X ) (cid:0) X a X c ǫ cb − X b X c ǫ ca (cid:1) (5) P X,ab = 0 (6) P ab,cd = 0 (7) P a,bc = W ( X ) X d ǫ da η bc (8)We shall comment on the specific form of P IJ in Section 4 and confine ourselves toa couple of immediate remarks. The choices (4)-(5) are standard [1, 4], while (6)-(7)were chosen for simplicity. With (4)-(7) as Ansatz the non-trivial entry (8) emerges assolution of the non-linear Jacobi identities J IJK := P IL ∂ L P JK + P JL ∂ L P KI + P KL ∂ L P IJ = 0 . (9)Only if they are fulfilled then P IJ as defined in (4)-(8) really is a Poisson tensor andthe action (2) is a PSM [or, equivalently up to a surface term, the action (1)].We check now the validity of (9). To reduce clutter we shall immediately dropall terms containing P X,ab or P ab,cd or derivatives thereof. Decomposing the genericindices I, J, . . . into
X, a, ab the identities we have to check split into J X,a,b = J X,a,bc = J X,ab,cd = J a,b,cd = J a,bc,de = J ab,cd,ef = 0. Let us start with the first one: Without lossof generality we can simplify J X,a,b = 0 to J X, + , − = 0 and get P X,c ∂ c P + , − + P + , − ∂ a P a,X + P + ,cd ∂ cd P − ,X + P − ,cd ∂ cd P X, + = 0 . (10)The first term P X,c ∂ c P + , − ∝ X b ǫ bc ∂ c ( X − X d ǫ d + − X + X d ǫ d − ) = 0 vanishes by itself. Thesecond term vanishes because ∂ a P a,X = 0. Therefore we obtain the condition X − ( P + , + − − P − , ++ ) = X + ( P + , −− − P − , + − ) , (11)which is fulfilled for (8). The second identity J X,a,bc = 0, P X,d ∂ d P a,bc = P d,bc ∂ d P X,a , (12)holds because (8) has a structure very similar to (4). The third identity J X,ab,cd = 0holds identically because P X,ab and P ab,cd vanish. The fourth identity J a,b,cd = 0 splitsinto three parts, P a,I ∂ I P b,cd − P b,I ∂ I P a,cd = P e,cd ∂ e P a,b . (13)The terms on the left hand side actually cancel each other, so the right hand side mustvanish by itself, which indeed is the case: P e,cd ∂ e P a,b = − / U W η cd X f ǫ f e ∂ e ( X a X g ǫ gb − X b X g ǫ ga ) = 0. The fifth identity J a,bc,de = 0 simplifies to P f,bc ∂ f P a,de = P f,de ∂ f P a,bc . (14)Evidently this relation holds for (8). The sixth and final identity, J ab,cd,ef = 0, is triviallyfulfilled because of the choice (7). Thus we conclude that (4)-(8) is a valid Poisson tensorbecause all Jacobi identities (9) hold. For transparency we introduce a comma between first and second index. oisson-sigma model for 2D gravity with non-metricity
3. Classical solutions
The gauge symmetries [3] of the PSM action (2), δ ε X I = P IJ ε J , δ ε A I = − d ε I − (cid:0) ∂ I P JK (cid:1) ε K A J , (15)render the model a topological one in the sense that there are no propagating physicaldegrees of freedom [4]. The integration of the equations of motion,d X I + P IJ A J = 0 , d A I − (cid:0) ∂ I P JK (cid:1) A K ∧ A J = 0 , (16)leads to four conserved quantities in the present case. This can be deduced from theform of the Poisson-tensor (4)-(8): its dimension is six, but its rank equals to two, sothere is a four-dimensional kernel corresponding to the four conserved quantities, alsoknown as Casimir functions. We demonstrate now explicitly how to obtain them usingthe equations of motion (16). We shall need only the first set of equations ( ˆ X a := X b ǫ ba ),d X = − ˆ X a e a , (17)d X a = ˆ X a ω − P a,b e b − W ( X ) ˆ X a η bc ρ bc , (18)d η ab = W ( X ) ˆ X c e c η ab . (19)The first and third equation lead to three conserved quantities η ab (0) ,d (cid:0) e P ( X ) η ab (cid:1) = 0 ⇒ η ab = η ab (0) e − P ( X ) , (20)with P ( X ) := Z X d y W ( y ) . (21)It is now obvious that the function W is responsible for non-metricity because d η ab vanishes if W vanishes. Introducing Y := X a ˆ X b ˜ ǫ ba = X a X b ǫ bc ˜ ǫ ca (22)and manipulating (18) [inserting repeatedly (17) and exploiting (19) to simplify d ˆ X a = ǫ ba d X b + W ˆ X a ˆ X b e b ] yieldsd Y = − X a ˜ ǫ ab P b,c e c − W Y d X = (cid:2) V det η − ( U + W ) Y (cid:3) d X . (23)The last equation together with (20) establishes the fourth conserved quantity M ,d (cid:0) e Q ( X )+ P ( X ) Y + w ( X ) (cid:1) = 0 ⇒ e Q ( X )+ P ( X ) Y + w ( X ) = M . (24)Here we have defined Q ( X ) := Z X d y U ( y ) , w ( X ) := − η (0) Z X d y e Q ( y ) − P ( y ) V ( y ) . (25) oisson-sigma model for 2D gravity with non-metricity P is of no relevance and can be absorbed intoa redefinition of η ab (0) . A constant shift of Q corresponds to a rescaling of the physicalunits for length and mass, a well-known feature of the metric-compatible case where thesame issue arises [1, 11]. A constant shift of w can be absorbed into a redefinition of M .Much like in the metric-compatible case [1], M has an interpretation as “mass”.We show this by constructing the metric g µν = e a µ e b ν η ab (0) e − P ( X ) . (26)To simplify the discussion we shall assume η ±∓ (0) = 1 and η ±± (0) = 0, which establishes ǫ ++ = − ǫ −− = − e − P and Y = 2 e − P X + X − . A simple class of solutions is obtainedfor X ± = 0, which by virtue of (17) leads to constant X . Equation (18) then requiresthat X be a solution of V ( X ) = 0 and (19) establishes metricity, d η ab = 0. Thesesolutions, so-called “constant dilaton vacua”, are therefore the same as in the metriccompatible case and allow only for constant curvature solutions, cf. e.g. [11]. They arenon-generic because V ( X ) = 0 need not have any solution in the range of definitionof X . We shall now discuss the generic class of solutions, which requires X = const . Following [1, 12] we make the Ansatz e − = X + Z , where Z is a 1-form. This Ansatz isvalid in a patch where X + = 0. Then (17) yields e + = e P d X/X + + X − Z , while (18)yields ω − W η ab ρ ab = − e P d X + /X + − e P P + , − Z . Because of redundancy only one of theright equations (16),d e − + e − P ( ω − W η ab ρ ab ) ∧ e − = ∂ − P + , − e − ∧ e + , (27)is needed. Defining Z = e Q ˆ Z and using the previous equations simplifies (27) tod ˆ Z = 0. Since ˆ Z is closed, locally it is exact, ˆ Z = d u . This yields e − = X + e Q d u and e + = e P d X/X + + X − e Q d u . Using X and u as coordinates we obtain from (24)and (26) the line-elementd s = g µν d x µ d x ν = e Q ( X ) (cid:2) X d u + e − P ( X ) ( M − w ( X )) d u (cid:3) (28)in Eddington-Finkelstein gauge. Besides the “constant dilaton vacua” above, this is themost general classical solution for the line-element in an Eddington-Finkelstein patch.It is parameterized by a single constant of motion M and exhibits a Killing vector ∂ u .For P = 0 equation (28) agrees with results of the metric-compatible case [1, 12].On a sidenote we mention that it is possible to obtain the same classical solutions(28) also from a Riemannian second order action, I ∝ Z M d x √− g h ˜ XR + ˜ U ( ˜ X )( ∇ ˜ X ) − V ( ˜ X ) i , (29)where d ˜ X = e P ( X ) d X and˜ U ( ˜ X ) = ( U ( X ) − W ( X )) e − P ( X ) , ˜ V ( ˜ X ) = V ( X ) e − P ( X ) . (30) Other choices for η ab (0) would lead to a similar discussion provided det η (0) = 0. oisson-sigma model for 2D gravity with non-metricity X , butrather to ˜ X .
4. Discussion
We demonstrated that (4)-(8) is a valid Poisson tensor for a PSM (2) which allowsan interpretation as a first-order gravity system with non-metricity (1). We thenconstructed all classical solutions for the line-element in a basic Eddington-Finkelsteinpatch (28). We did not address global properties, but such a discussion can be performedin analogy to [1, 12]. For vanishing non-metricity potential, W = 0, we recovered well-known results.Actually the last point can be seen already at the level of the action. Consider (1)with W = 0. Then the η, ρ sector decouples and can be integrated out trivially, leadingto three Casimirs η ab = η ab (0) . Let us choose them as η ±± = 0 and η ±∓ = 1. Then (3)simplifies to ǫ ∓∓ = ± I = k Z M h X d ω + X ± (d ∓ ω ) ∧ e ± + (cid:0) V ( X ) + X + X − U ( X ) (cid:1) e − ∧ e + i . (31)This coincides with the PSM action for ordinary dilaton gravity, parameterized by thepotentials U and V . For instance, with k = − e ± = e ∓ the result (31) coincideswith Eq. (2.2) in [11]. So the simple case W = 0 is well understood. The conceptualdifference to previous approaches is that the tangent space metric η ab here is not anexternal input but rather emerges from the integration of (19). By choosing differentvalues for the constants of motion η ab (0) we can obtain either signature of the tangentspace metric. Thus, the choice of signature in our approach happens only at the levelof the equations of motion and not at the level of the action.We discuss now in a bit more detail the specific form (4)-(8) of the Poisson tensorand the geometric interpretation of our fields. The entry (4) is basically fixed byrequiring that the torsion 2-form T a := d e a + ω ab ∧ e b = d e a + ǫ ab ω ∧ e b (32)appears in the actions (1) and (2). This interpretation, however, requires anti-symmetryof the connection ω ab . We achieve this via (6) which eliminates any coupling of theconnection different from (32). Since this does not seem to be the usual way of It would be interesting to see the effect of W = 0 on global properties. Singularities of the non-metricity potential W typically do not change the number and types of Killing horizons, but theycan be of relevance for the asymptotic structure of space-time or geodesic (in)completeness properties.Moreover, the fact that we doubled the number of target space coordinates as compared to previousPSM approaches may have an impact on global considerations. Because we demand compatibility with the PSM index structure quantities like torsion and curvaturehave non-standard index positions of the tangent space indices. oisson-sigma model for 2D gravity with non-metricity Q ab := d η ab + Λ ab + Λ ba (33)requires a symmetric contribution to the full connection, Λ ab = − Λ ba , because normallythe term d η ab vanishes since the tangent space metric is assumed to be constant.However, rather than shifting the burden of non-metricity to the symmetric part of thefull connection one can achieve non-metricity also with an anti-symmetric connection ω ab by choosing a tangent space metric which is not constant. A reasonable non-trivialchoice for η ab in physical applications could be an ( A ) dS metric, for instance. Thisre-interpretation of non-metricity is not tied to our two-dimensional discussion butgeneralizes to higher dimensions. For sake of completeness we mention that the fullcurvature R ab = d ω ab = ǫ ab d ω − ˜ ǫ ac ω ∧ d η cb (34)is in general inequivalent to the Riemannian curvature derived from our solution forthe line-element (28). Solely for U = W = 0 both notions of curvature agree witheach other, because then torsion and non-metricity vanish. After this digression wereturn to the discussion of the Poisson tensor. The choice (5) appears to be the mostgeneral expression compatible with the index structure, anti-symmetry and the Jacobi-identities. A novel feature is that (5) contains terms at most quadratic in X a , whereas inthe traditional approach arbitrary coupling to X a X b η ab is allowed since η ab is introducedthere as an external structure. This is interesting by itself because supergravity imposesa similar restriction to quadratic coupling [15], but we shall not pursue this issue anyfurther here. As mentioned before the choice (7) was made for simplicity. However,the Jacobi-identities (9) are very restrictive concerning contributions to (7) and it couldwell be they imply P ab,cd = 0. The main purpose of our Poisson tensor is to producenon-metricity, d η bc = e a f abc , (35)where f abc = f acb . This is the generic form of non-metricity because we can decomposeany 1-form into basis 1-forms e a . The equations of motion (16) imply f abc = P a,bc if P X,ab = P ab,cd = 0. Thus, it is natural to introduce a non-vanishing P a,bc component,but no P ab,cd component is required to generate non-metricity. Our choice (8) containsone free function parameterizing the strength of non-metricity and appears to be thegeneric solution of the Jacobi-identities (9) once the Ansatz (4)-(7) is taken for granted.It is evident from (19) that we have only a trace part of non-metricity. So either a PSMformulation does not allow for a tracefree contribution to non-metricity or it requiresa different choice of the Poisson tensor. Therefore it would be interesting to check inwhat sense our choice (4)-(8) is generic, possibly by adapting the discussion in [5].It is worthwhile mentioning that the symmetries (15) are non-linear because thePoisson tensor (4)-(8) is at least quadratic in the target space coordinates. Hence thequantities ∂ I P JK become structure functions rather than structure constants. Evenfor the metric-compatible case W = 0 this differs from the simpler situation in (31): oisson-sigma model for 2D gravity with non-metricity V ∝ X , U = 0 [16] simplifies to a SO (1 ,
2) gaugetheory. The reason for this difference comes from our treatment of η ab as target spacecoordinates.Our main result derived from the action is the line-element (28), which depends onone additional function P ( X ) as compared to standard results. This additional freedomcan be useful in “reverse-engineering procedures” where one attempts to construct anaction for a given family of classical solutions for dilaton field and line-element. Forinstance, it provides a new possibility to evade the no-go argument of [17] and toconstruct a PSM action for the exact string black hole [18] which differs from the oneconstructed in [19]. A possible choice for the exact string black hole potentials is U = − XX + 1 , V = − λX √ X + 1 √ X + 1 , W = U √ X + 1 . (36)In this way the α ′ corrections contained in the exact string black hole are encoded innon-metricity. In the weak coupling limit ( X → ∞ ) non-metricity becomes irrelevant( P → /X ) and the model asymptotes to the Witten black hole [20].Finally, we would like to comment on the relation to previous approaches. Theaction (1) contains as special cases all models with non-metricity constructed so far. Inparticular, the results of [8] are recovered for U = − e − − W ( z ) (cid:20) A W ( z ) + B (1 + W ( z )) (cid:21) , V = 2 βk e − − W ( z ) (1 + W ( z )) ,W = − k e − − W ( z ) (1 + W ( z )) , z := k X e − e . (37)The real parameters A = ( k + p ) k/ k / B = k ( α + µk + ν ( p − q )) /
16 arerelated to parameters defined in that work ( k, p, q, α, β, µ, ν ). For brevity we have setan additional parameter to zero, l = 0, but also the case l = 0 allows a comparisonand leads to somewhat lengthy expressions for U, V and W . The function W denotesthe principal branch of the Lambert-W function [21]. It is convenient to choose the freeintegration constant in P such that P = − − W ( z ) (since X typically is non-negative z ≥ − /e and W ( z ) ≥ −
1, so that P ≤ p = q = ν = 0. The results of [7] are recovered for U ( X ) = B ˜ X , V ( X ) = C ˜ X + D ˜ X + E ˜ X , W ( X ) = A . (38)The real parameters A = a /a , B = a /a , C = λ/a + D/a , D = a (4 a − / (2 ab )and E = − a / (4 ab ) are related to parameters defined in that work ( a, a , a , b, λ ). Thedilaton X is determined from ˜ X by the relation between the equations (29) and (30),which integrates to AX = ln ( A ˜ X ). Our action (1) not only encompasses all these specialcases, but generalizes them while maintaining integrability. It could be interesting tocouple matter to the system, to supersymmetrize it and/or to quantize it. This should bepossible by analogy to the metric compatible case [1], even though there will be technicaland conceptual differences because the tangent-space metric η ab now is dynamical. oisson-sigma model for 2D gravity with non-metricity Acknowledgments
We thank Tekin Dereli, Roman Jackiw, Yuri Obukhov and especially Thomas Strobl fordiscussions. This work is supported in part by funds provided by the U.S. Departmentof Energy (DOE) under the cooperative research agreement DEFG02-05ER41360. DGhas been supported by the Marie Curie Fellowship MC-OIF 021421 of the EuropeanCommission under the Sixth EU Framework Programme for Research and TechnologicalDevelopment (FP6). DG would like to thank the Erwin-Schr¨odinger InternationalInstitute for Mathematical Physics (ESI) for the hospitality during the final preparationsof this manuscript.
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