BTZ entropy from topological M-theory

By determining the relation between topological M-theory and the Chern-Simons actions for a gauge field constructed from the Lie algebra of either SL(2,R)x SL(2,R) or SL(2,C)x SL(2,C), depending on the sign of the space-time curvature, we show that the standard and exotic actions of 3-dimensional gravity can be recovered from topological M-theory. With this result, we provide a concrete realisation of a conjecture by Dijkgraaf et al. stating that the partition function of topological M-theory is equivalent to the partition function of a black hole in a related theory. We do this for the standard and exotic BTZ black holes in 3-dimensional gravity.

By determining the relation between topological M-theory and the Chern-Simons actions for a gauge field constructed from the Lie algebra of either SL(2, R) × SL(2, R) or SL(2, C) × SL(2, C), depending on the sign of the space-time curvature, we show that the standard and exotic actions of 3-dimensional gravity can be recovered from topological M-theory. With this result, we provide a concrete realisation of a conjecture by Dijkgraaf et al. stating that the partition function of topological M-theory is equivalent to the partition function of a black hole in a related theory. We do this for the standard and exotic BTZ black holes in 3-dimensional gravity.
One of the most useful tools to understand the gravitational interaction, is three dimensional gravity. Not only has 2+1 gravity been quantized, it has other remarkable features that are of great value as a guide to understand the foundations of gravity. Some of these features can be easily derived from the fact that it can be written as a Chern-Simons action [1]. And although 2+1 gravity is topological and therefore might seem physically unrealistic (it lacks propagating degrees of freedom), there is a black hole solution, known as the BTZ black hole [2]. The BTZ solution is asymptotically anti-de Sitter and has no singularity, but it has many of the features of the Kerr black hole, it has an event horizon and an inner horizon for the rotating case and thermodynamic properties analogous to 4 dimensional black holes. Interestingly, the BTZ solution solves any 2+1 gravity model that admits anti-de Sitter vacuum, and the mass and angular momentum to some linear combination of the parameters. When the role of mass and angular momentum is reversed, the resulting black hole is known as an exotic BTZ black hole. The entropy of the BTZ black hole is in agreement with Hawking-Bekenstein entropy, but for the exotic case the entropy is related to the inner horizon. This appalling contradiction was resolved in [3] by considering that the BTZ is a solution to the standard action and the exotic BTZ is a solution to the exotic action, and therefore the entropy must be given by, The standard and exotic actions, are the two independent actions that are derived in the Chern-Simons formulation of 2+1 gravity [1,4]. The description of the gravitational field in terms of gauge fields or p-forms has been continuously developed. In these theories the metric does not appear explicitly but it is reconstructed from the dynamical fields under * javier.chagoya@fisica.uaz.edu.mx † grace@cimat.mx ‡ msabido@fisica.ugto.mx consideration. These descriptions are referred to as form theories of gravity. Some of these form theories, including Chern-Simons (CS) three dimensional (3d) gravity and the A and B models of topological strings can be unified in a seven dimensional space-time, X, through the topological M-theory (TMT) proposed by Dijkgraaf et al. [5]. Essential in this theory is the volume form, V , constructed from an invariant p-form whose existence is characteristic of special holonomy manifolds. The study of manifolds admitting stable non-degenerate forms is an interesting topic by itself, see for example [6] for a classification of all stable forms on R n . In particular, for seven dimensions, there are two non-trivial p-forms invariant under the holonomy group G 2 , one of which is a 3-form and the other a 4-form. The same is true for the stable p-forms invariant under the dual groupG 2 . Using the 3-form Φ with holonomy in G 2 , Dijkgraaf et al.
showed that the equations of motion for 2+1 gravity are recovered under a convenient partition of X. However, it is known that for non-vanishing cosmological constant, λ, there are two classically equivalent actions to describe gravity in 2+1 dimensions, known as standard and exotic actions [1]. In [7], the authors obtain these actions from TMT. To write down the standard and exotic actions they first show how to obtain the CS actions for 2+1 gravity. This result opens up the possibility to apply the formalism and ideas of TMT to several models of 2+1 gravity that are built in terms of the CS actions. Using this result we present a concrete realisation of a conjecture that states that the partition function of a theory with an action defined by the Hitchin volume functional, is related to the partition function of a BPS black hole in the gravitational theory allowed by the p-forms used to construct the volume functional [5]. Since we deal with 3d gravity, the relevant black hole solution will be the BTZ space-time [2]. As a side result we shed some light on the proposal in [3] for the entropy of BTZ black holes.
In [8], it is conjectured that the partition function of a 4d BPS black hole is related to the topological string partition function by Furthermore, it is pointed out that the topological par-tition function can be interpreted as a wave function, this interpretation comes from [9]. Thus, the conjecture above becomes A similar proposal exists in TMT [5], where the partition function Z H of a 6d theory (contained within TMT and constructed from a volume form) is associated to a Wigner function arising from the B model of topological strings.
Here we present a realization of these ideas, but in the context of 3d gravity. As shown in [5] at the level of the equations of motion and in [7] at the level of the action, 3d gravity is contained in TMT as a particular splitting of the 7d manifold. In order to give a concrete example of the relation between Z H and the black hole entropy we consider an extremal BTZ black hole, compute its volume form in terms of the 2+1 dimensional standard and exotic actions for gravity, then we obtain Z H , and finally we compare it to the norm of the wave function for the same black hole [10].
The organization of this work is as follows. First, we review and formalize the derivation of the standard and exotic actions for 2+1 gravity from TMT and construct the topological partition function. Then, we also review the BTZ black hole solutions and its partition function obtained from canonical quantization. Finally, we show how these results are related.

I. STABLE FORMS IN 7D
In this section we study the relation between invariant stable forms and structures on a 7d Riemannian manifold, R 7 . To understand the geometric structures defined by stable forms, we need to study the isotropy subgroup of such forms under the action of the general linear group GL(7). We start by recalling the structure on R 7 . Later we use such construction to understand the case of a manifold X.
Let V be a real 7d vector space with basis {e i } and consider the space where e ijk = e i ∧ e j ∧ e k and {e i } is a basis for V * . Consider the group G = GL(7) of automorphisms of V . There is a natural action G ∧ 3 V * and it is known that there are two distinguished orbits given by this action, namely where ω i is the form defined as To each form corresponds an isotropy group, the Lie group It is proved in [11] that G 2 is compact, connected, simple, simply connected, 14-dimensional and it fixes the Euclidean metric G 2 also preserves the orientation of the forms ω 1 and * ω 1 with respect to g 1 , and G 2 is isomorphic to the group of automorphisms of the octonians. There are analogous results for the groupG 2 , this group preserves ω 2 , * ω 2 , the metric induced by and it is the non-compact dual of G 2 . It is also connected, of dimension 14 and simple.
In this case the natural identification is in fact a diffeomorphism. Since dim(G) = 49 and dim(G 2 ) = dim(G 2 ) = 14 then the dimension of these orbits dim(G · ω i ) = 49 − 14 = 35 coincides with the dimension of the ambient space dim(∧ 3 V * ) = 35 and we conclude as in [11] that both orbits are open and the forms ω 1 and ω 2 are stable. In [6], the authors show that the forms ω 1 , ω 2 are essentially the unique stable forms, in the sense that any stable form ω ∈ ∧ 3 V * is either in the orbit of ω 1 or ω 2 . The scenario we study in this paper is the case when X is a complete 7d Riemannian manifold, x ∈ X is a point and V = T x X. A stable form induces a G ω i -structure on X, as follows (see [12]) : Consider the fiber bundle ∧ 3 T * X and the open subbundle P i (X) with fiber where in the last definition f is an oriented isomorphism. From the previous discussion P 3 x ∼ = G · ω i . Fix a form ω over X such that ω| p ∈ P i x = g · ω i and consider the frame bundle F of X with fiber Let Q be the principal subbundle of F whose fiber consists in isomorphisms preserving ω. Hence the fiber is Q x ∼ = G ω i and ω determines Q which defines a G ω istructure on X, preserving the metric g ω induced by the inner product There is a converse for this construction: given an oriented G ω i -structure we can define a metric g, a 3-form ω and * ω requiring that the corresponding metric is preserved by the action of G ω i .
Let X be a Riemannian 7d manifold with a G 2structure (ω, g) and denote as ∇ g the Levi-Civita connection associated to g. Let ∇ g ω be the torsion of this G 2structure. We say that (ω, g) is torsion-free if ∇ g ω = 0. Finally define a G 2 -manifold as a triplet (X, ω, g) such that (ω, g) is torsion-free.
Consider a G 2 -manifold X. The existence of a G 2 holonomy metric is equivalent to the existence of a 3form Φ satisfying as in [5], A stable 3-form can be written in terms of a 7d vielbein as where Ψ ijk are the structure constants of the imaginary octonions. There are analogous constructions for stable forms on aG 2 -manifold, since the orbits of ω 1 , ω 2 correspond with the holonomy groups G 2 andG 2 respectively. In order to define a volume on a G ω i -manifold X consider a 3-form Φ on X as before, invariant by the corresponding holonomy group and define a volume as As above since in the 7d case there are only two open orbits of maximal dimension hence is natural to consider only forms in these orbits to get a notion of genericity as in [5].

II. 3D GRAVITY FROM TOPOLOGICAL M-THEORY
In [5], Dijkgraaf et al. introduced a notion for TMT in 7d with the property that it seems to unify several lower dimensional topological models. In particular, they find a dimensional reduction that recovers the equations of motion of 2+1 gravity from the volume of the 7d manifold X discussed in the previous section. A similar construction was given by Bryant et al. [11], where starting from a rank-4 spin bundle S over a 3d space of constant curvature (space form), a 3-form Φ satisfying dΦ = d * Φ Φ = 0 is constructed by making use of the structure equations for a manifold with constant sectional curvature κ ≡ 4Λ, i.e., where {e 1 , e 2 , e 3 } is a basis of the tangent space at a point of the 3-manifold, and A is a Levi-Civita connection 1form. As [5,11] point out, a 3-form that generalizes ω 1 (7) can satisfy the conditions dΦ = d * Φ Φ = 0 in some special cases. In order to write down this 3-form ψ it is convenient to introduce first a set of local coordinates on the 4d fibre. Let y i be those coordinates, we define r = y i y i . Notice that this is SO(4)-invariant. With the following 2-forms, we can write the 3-form Φ that satisfies dΦ = d * Φ Φ = 0 as Since f and g depend only on r, Φ preserves the SO(4) invariance of ω 1 . Remembering that SO(4) is a subgroup of G 2 , and by the discussion of the previous section, the fact that Φ is SO(4)-invariant is a good indicator that it can define a G 2 structure -thus satisfying the required equations. The local coordinates y i are also used to define a basis of 1-forms in the fibre direction as The four components of α are identified as a local basis on the fibre, α i = e i , i = 4, 5, 6, 7. As a consequence of eqs. (13), these 1-forms satisfy Using Eqs. (13), (17), and * Φ Φ = − in [13] it is showed that the equations Conversely, the authors of [5] start with dΦ = d * Φ Φ = 0 and verify that the above assumptions for f (r) and g(r) lead to the structure equations, (13), i.e., in their interpretation, the equations of motion for 3d gravity arise from the equations for a 3-form with G 2 -holonomy. If these equations of motion are recovered from such a 3form Φ, it is natural to look for a Lagrangian for Φ that encompasses the main points of the derivations above and reduces to the known Lagrangians for 3d gravity. This Lagrangian is given precisely in terms of the volume form discussed around Eq. (12). In order to convert Eq. (12) into an expression that we can recognise as the action for 2+1 gravity we perform the following steps. First, we rewrite the integrand Φ ∧ * Φ Φ using the antisymmetry of the wedge product and of the Levi-Civita tensor, obtaining Now, let Σ be the curvature of a connection α, i.e., and cyclically for the others. Later on we will relate this α to the connection 1-form A. Notice that this is compatible with the equations (14) that express Σ i in a local orthonormal basis [14]. Using again the properties of the wedge product, and noticing that as a consequence of the structure equations (13) we have d(e 567 ) = 0 [13], the volume V 7 can be written as The argument of the differential does not depend on r, therefore, by an appropriate choice of coordinates, its prefactor can be integrated out so that it becomes a global factor of a 6d integral. We can further reduce these dimensions by using Stokes theorem, obtaining 1 Finally, since the argument of the integral only depends on quantities defined over the 3-manifold M with basis {e 5 , e 6 , e 7 }, the volume can be expressed as Expanding the wedge product in components, relabeling the internal indice as (a, b, c) and using (i, j, k) for the spacetime indices, we get 1 We have to be careful with the notation: all p-forms are integrated over p-dimensional manifolds. If the dimensions of the integral and the order of the p-form obtained by counting wedge products does not match, this means that one of the differentials dx i has been integrated out, and we have to remember this when writing the form in component notation. This is the Chern-Simons action. At this point it is convenient to notice that the 2-forms Σ are anti-self-dual, i.e., * Σ i = −Σ i . For this reason, we rename it as − Σ i , with associated connection − α i , and we also rename the form Φ given in eq. (15) as − Φ. Now we are ready to see the relevance of the discussion of the previous section. The form − Φ is constructed out of the stable form ω 2 presented in eq. (8). However, we have seen that the volume form can also be constructed in terms of ω 1 , eq. (7). Furthermore, these two possibilites, ω 1 and ω 2 are unique in the sense discussed in the previous section. With these considerations in mind, we construct a volume form for each of the 3-forms where + Σ m are the self-dual 2-forms + Σ 5 = e 12 + e 34 , + Σ 6 = e 13 + e 42 , and r is defined in the same way as described before. When f (r) = g(r) = 1, − Φ, + Φ are equivalent to ω 2 and ω 1 , respectively. The 4-forms associated to − Φ and + Φ are * Φ We can use either of ± Φ to construct the volume of the 7-manifold X, By the same steps of the previous section, V 7 can be written as where + α i is the connection associated to + Σ i . Thus, we have found two Chern-Simons actions derivable from the volume of a 7-manifold that admits two special stable forms. Now we want to understand how these two actions are related to 2+1 gravity. From the results of [5,13], we know that the equations of motion arising from the volume of − Φ are those of 2+1 gravity with a cosmological constant. Since V ( + Φ) describes the same volume as V ( − Φ), the 3d equations of motion derived from both actions have to coincide. This is remarkably similar, and consistent, with the results of [1], where it is shown that there are two 3d actions, named standard and exotic, that lead to the same equations of motion that we are interested in. Furthermore, they show that these actions can be written precisely in terms of the Chern-Simons actions (31) by setting where A i and e i are the fields introduced around Eq. (13). The combinations where ± I are the integrals in Eq. (31), give respectively the standard and exotic actions. Now we can reinterpret the standard and exotic actions in terms of the volume functional as where h ± are the inverses of the proportionality factors in Eq. (31). In this way, we can see the standard and exotic actions as two different combinations of pieces of the volume of the 7-manifold X. Applications of the ideas developed so far to the Immirzi ambiguity in 3d gravity have been presented in [7]. In the next section we explore the entropy of the BTZ black hole from the point of view of TMT and we discuss the relation of our results to the conjecture Z BH = |Z top | 2 .

III. BTZ BLACK HOLE: PARTITION FUNCTION
Using the results described above we can provide evidence that the conjecture discussed around Eq. (2) also applies for G 2 -manifolds and 3d black holes, i.e., that in general, the partition function of a theory with action defined by a Hitchin functional is related to the partition function for a BPS black hole in the gravitational theory allowed by the p -forms used to construct the Hitchin functional. The possibility that the relation between BPS objects and form theories of gravity extends to G2-manifolds was hinted in [5]; however, it was only studied for 4d and 5d black holes embedded in a 6d SU (3)-manifold. In this work we show explicitly that the partition function of the BTZ black hole is recovered from the partition function associated to the volume V 7 . Given the different ways of writing down V 7 either in terms of V + , V − or both, one could think that the result only applies to the extremal case, which turns out to be associated to the situation were we demand that the linear combinations of V + and V − -for instance I st and I ex -preserve a given multiple of V 7 ; but as we argue below, the partition function obtained from TMT correctly gives the BH partition function even away from the extremal case.
In the case of TMT, the total space X is 7d and as we shown in the previous sections, its volume can be constructed with either of the 3-forms + Φ and − Φ. A certain combination of these volumes, Eq. (33), results in the standard action for 3d gravity. In this theory, a black hole solution is given by the BTZ space-time [2], whose metric can be written as where the lapse N and shift N φ are The integration constants M and J are interpreted respectively as the mass and angular momentum of the black hole, and ℓ is related to the cosmological constant of the theory by ℓ −2 = Λ/3. The lapse function vanishes at two distinct values of r, defining two coordinate singularities, r ± , When J = 0 only r + is different from zero, and in the extremal case J = M ℓ the two horizons coincide. The entropy of the BTZ black hole can be computed by different methods, for example, by Euclidean path integral or by Noether charges [see e.g. 15], and it is given by These computations do not depend only on the metric but also on the action, that is usually taken to be the standard action, hence the superscript st. Originally, this result comes from geometrical considerations on the standard action of 2+1 gravity, and then deriving the entropy from the grand canonical partition function in the classical approximation [16] Z = exp(I st ) .
Since the standard action is recovered from TMT, the entropy of a BTZ black hole described by such an action is recovered as well.
The same techniques that lead to Eq. (40) have been applied to the exotic action, finding an entropy proportional to the inner BTZ horizon, r − . The fact that the entropy is proportional to the inner horizon raised doubts about the validity of black hole thermodynamics. However, it has been shown that these laws hold [3]. Indeed, the result is even more general: an entropy of the form is in agreement with black hole thermodynamics. Eq. (41) arises naturally in the context we are studying in this work. Hitchin's partition function is defined in terms of the volume functional, Thus, when we write TMT as a theory of a 4d vector bundle over a 3d base space such that the 7d manifold X has a G 2 -structure, we can separate V 7 in terms of the volume functionals V ± , for some coefficients λ, β ± . Notice that, so far, all the properties that hold for a theory based on V 7 hold for a theory based on a multiple λ of V 7 . In addition, V ± are proportional to the Chern-Simons actions, Eq. (31), with proportionality constants 1/h ± . Putting all together, we write Hitchin's partition function as As before, the basis of the 7d manifold can be decomposed into a 3d base space and a 4d bundle. The coefficients β ± can be chosen in such a way that the linear combination of ± I in the argument of the exponential reproduces either the standard or the exotic action, or a combination of both. For the choice that leads to the standard action, by the discussion above we confirm that Hitchin's entropy is related to the BTZ entropy, On the other hand, for a different choice of parameters we can have i.e., the Hitchin partition function for the exotic action is also related to a black hole partition function, only that in this case Z BH corresponds to the exotic BTZ black hole. The extremal case, r + = r − , admits an interpretation from the point of view of TMT. Suppose we fix λ, e.g. λ = 1. This imposes a constraint on the linear combinations in Eq. (43), such that any choice of β ± leads to a fixed V 7 and the same Z H (Φ). Therefore, all combinations lead to the same black hole entropy, and this is only possible if r + = r − , i.e., the extremal case corresponds to a constraint on the parameters β ± .

IV. DISCUSSION
3d gravity can be embedded in a 7-manifold with G 2holonomy. The volume form of this manifold is constructed in terms of a stable (generic, in the sense of [5]) form. Indeed, there are essentially two unique such forms and by using these two stable forms, we split the volume of the 7-manifold into contributions from the distinct orbits. Using the structure equations appropriated for our geometrical set-up, we find that these two contributions can be rephrased as Chern-Simons actions, one for a self-dual curvature and one for an anti-self-dual curvature. This observation allows us to recover the two classically equivalent known actions of 3d gravity, i.e., Witten's standard and exotic actions, thus completing the picture shown in [5,13].
In a context that is more general than the theory that we study here, it has been conjectured that topological and black hole partition functions are related. Our results give a concrete realisation of this conjecture: by writing the action of TMT in terms of the contributions from the two unique stable forms, we can tune the theory so that it reproduces the partition function of the standard action of 3d gravity, thus agreeing with the result for the BTZ black hole; or we can choose to reproduce the exotic action, obtaining the correct entropy for the exotic BTZ black hole. It is worth noticing that a combined standard/exotic entropy is in agreement with black hole thermodynamics [3], and our results provide a scenario where such combined models can be embedded.
The topological partition function is also conjectured to be related to a wave function. The wave function for a static BTZ black hole in the region outside the horizon has been computed within a canonical quantization scheme [10]. When evaluated at the horizon, their result takes the form (more details in the Appendix): whereμ is a quantized number related to the energy levels of the system. This result indeed reassembles the Euclidean partition function for the BTZ black hole. It would be interesting to explore the quantization of a nonstatic BTZ black hole, so that the relation between the wave function and the black hole partition function can be explored for the extremal case, i.e., the case that would correspond to the conjectures in [8]. This is left for future work.