Color decorations of Jackiw-Teitelboim gravity

We introduce the colored version of Jackiw-Teitelboim (JT) gravity which is the two-dimensional dilaton gravity model with matrix-valued fields. It is straightforwardly formulated in terms of BF action with su(N, N) gauge algebra so that the standard JT gravity is embedded as su(1, 1) ⊂ su(N, N) subsector. We also elaborate on the respective metric formulation which is shown to involve the JT fields plus su(N) non-Abelian fields as well as su(N)-matrix valued metric and dilaton fields. Their interactions are governed by minimal couplings and potential terms of cubic and quartic orders involving derivatives.

Note that the original JT gravity can be extended by adding more fields following two different tracks using either metric or frame (BF) formulations. Introducing a metric explicitly can be more useful in the context of finding exact solutions in dilaton gravity models in the presence matter or gauge fields, see e.g. [15][16][17][18][19]. Within the BF formulation, which is essentially the Cartan approach to gravity, all extensions of the original theory are basically boiled down to extending the gauge algebra. A direct product sl(2, R) × K, where K is some Lie algebra provides Yang-Mills type extensions, see e.g. [17]. Another way is to embed sl(2, R) ⊂ H, where H is some (in)finite-dimensional Lie algebra. For example, the extension of the gauge algebra sl(2, R) to higher-rank gauge algebras sl(M, R) reveals higher-spin JT gravity models with finite spectra of higher-rank fields [17,20,21]. Moreover, there are higher-spin JT models with infinite number of fields [22][23][24][25]  based on the infinite-dimensional extension of the gauge algebra sl (2, R), known as the algebra sl [λ] or hs [λ], with a real parameter λ [26][27][28]. 1 In this paper we take the second route and introduce a colored version of JT gravity which can be obtained by promoting the algebra sl(2, R) ∼ = su (1,1) to the higher-rank algebra su (N, N ). The corresponding BF theory naturally extends JT gravity by adding matrix-valued fields and can be viewed as the dilaton gravity carrying Chan-Paton color charges, see figure 1. Such a choice of the gauge algebra is naturally inherited from 3d colored AdS (higher-spin) gravity [35][36][37] and 3d colored Poincare gravity [38] as well as their non-relativistic limits [39].
The paper is organized as follows. In section 2 we shortly review both BF and metric formulations of JT gravity. Section 3 introduces the colored JT gravity as BF theory with su(N, N ) gauge algebra. Here, we address the gauging procedure and describe the resulting spectrum of the theory. In section 4 we develop the metric formulation by solving the constraints for the singlet auxiliary fields arising in the JT sector of the theory. However, the resulting action still contains matrix auxiliary fields subject to complicated matrix constraints that can be solved only perturbatively. Nonetheless, it is still possible to find the general form of the second order colored gravity action that can be split into manifest JT gravity and su(N ) BF actions along with colored field contributions defined by implicit second order derivative terms and explicit algebraic terms. Conclusions and future perspectives including preliminary notes on the colored AdS backgrounds and the color symmetry breaking are discussed in section 5. Appendix A contains various technical details.

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are given by 2 [ The fields are differential 0-form Φ = Φ A J A and 1-form Ω = dx µ Ω A µ J A taking values in the o(2, 1) adjoint representation. The BF formulation of JT gravity on a two-dimensional manifold M 2 is given by the following action [40][41][42] where κ is a dimensionless coupling constant and 2-form curvature reads The equations of motion that follow from the BF action (2.2) have the form of the covariant constancy condition and the zero-curvature condition, Now the o(2, 1) basis elements J A are split as translations P a = (J 0 , J 1 ) and Lorentz rotation L = J 0 with the commutation relations: [L, P a ] = 2 ab P b , and [P a , P b ] = −2 ab L. This allows representing the fields and the curvature in the Lorentz basis as Ω A = (ω, e a ), where e a and ω are the frame field and the Lorentz spin connection, and φ is the dilaton field. Fields ω and φ a are auxiliary expressed in terms of e a and ϕ through their equations of motion which are given by respective components of the general equations (2.3), Introducing g µν = η ab e a µ e b ν and e = det e a µ = √ −g, the above constraints can be solved as 3 (2.5) Further, we introduce the standard scalar curvature R so that the 2-form curvature R = R µν dx µ dx ν is cast into the form R = 1 2 (R + 4) Then, expressing the scalar curvature in terms of the metric R = R(g) we recover the second-order JT action [7,8] which therefore describes the coupled system of the metric field g µν and the dilaton field φ. 2 The o(2, 1)-covariant Levi-Civita symbol ABC is defined by 0 01 = 1.
. The Levi-Civita symbol µν and Lorentz tensor ηµν with the world indices µ, ν, . . . = 0, 1 are defined by the same relations. We omit the wedge product ∧ symbol in exterior products: 3 Keeping o(2, 1) basis elements dimensionless one can introduce a length scale in the theory by redefining e a µ → e a µ / AdS , where AdS is the AdS radius.

Colored JT gravity as su(N, N ) BF theory
In this section we formulate the colored JT gravity in the BF form and derive the spectrum of fields.

Matrix realization
The gauge algebra su(N, N ) can be decomposed as that makes manifest that the spectrum of gauge fields can be naturally divided in three subsets (see below). Here, I 2 and I N are respective identity elements. In the sequel, we make use of the following matrix realization of the gauge algebra where A, B, and C are N × N matrix blocks with complex entries satisfying the anti-Hermitian conditions A † = −A, C † = −C, and the trace condition Tr (A + C) = 0, the block B is arbitrary, and † is the Hermitian matrix conjugation. Solving the above constraints and using real parameterization we can express an arbitrary matrix M ∈ su(N, N ) as where s A , t α , p A,α , with indices A = 0 , 0, 1 and α = 1, . . . , N 2 − 1, are real parameters (in total, there are 4N 2 − 1 real parameters that equals dim(su(N, N )), while (I 2 , J A ) are u(1, 1) basis elements, (3.4) and Hermitian N × N matrices (I N , T α ) are u(N ) basis elements (for more details see, e.g. [36]). The basis elements in (3.3) are identified with matrices J A ⊗ I N , I 2 ⊗ T α , J A ⊗ T α which are Kronecker products naturally inherited from the block form (3.2). The basis elements are known to satisfy 4 where g αβγ and f αβγ are respectively totally symmetric and totally anti-symmetric structure constants. 5 The commutation relations of su(1, 1) and su(N ) subalgebras read It directly follows from that any complex N × N matrix A can be decomposed as A = B + i C, where B and C are Hermitian matrices. Since A ∈ Mat(N, C) then the product of any two Hermitian matrices can be expressed via the structure constants of the matrix algebra. 5 Note that totally symmetric structure constants vanish at N = 2. In this case, identifying Tα ∼ σa we obtain the standard Pauli matrix relation σaσ b = δ ab I +i abc σc underlying the su(2) commutation relations. Changing from su (2) to su(1, 1) we see that the foregoing relation reproduces the first relation in (3.5).

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Following (3.3) we denote the basis elements in the su(N, N ) algebra as Their commutation relations are given by 6 Finally, let us list the matrix traces of bilinear combinations of the basis elements. Since then using (3.5) and (3.12) we obtain From the footnote 6 and (3.12), (3.13), we find non-vanishing diagonal traces of bilinear combinations of (3.7) to be while all off-diagonal ones are zero, From the commutation relations (3.9) it follows that the linear space of the A = su(N, N ) algebra decomposes as where [·, ·] and {·, ·} are the standard matrix (anti)commutators.

Field content
The fields are 1-form A = dx µ A µ (x) and 0-form B = B(x) taking values in the adjoint of su(N, N ) algebra, i.e. in matrices (3.2). In the basis (3.3) they can be represented as The component expansion (3.17) defines the spectrum of 2d colored JT gravity: • are 1-form and 0-form fields taking values in the adjoint of so(2, 1) ≈ su(1, 1) ⊂ su(N, N ) subalgebra. This color singlet subsector describes the JT dilaton gravity. In total, we have the JT graviton and dilaton, (N 2 − 1) colored gravitons and (N 2 − 1) colored dilatons, su(N ) gauge fields and matter fields in the su(N ) adjoint multiplet. At N = 1 both the colored dilaton gravity sector and su(N ) sector trivialize so that only the standard JT gravity is left.

First-order action
An action for the colored JT gravity can be naturally given in the standard BF form where M 2 is a two-dimensional manifold, tr is defined by (3.14)-(3.15), and F = dA + AA is the 2-form curvature associated with 1-form A, the κ is a dimensionless coupling constant. In the basis (3.17) the curvature decomposes as where the expansion coefficients are given by (A.1). It follows that the action (3.18) can be cast into the component form The action is invariant with respect to the gauge transformations

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where following (3.17) the 0-form gauge parameter is given by The respective equations of motion take the form of zero-curvature and covariant constancy conditions The explicit component form of (3.22) can be found in appendix A. There are no local physical degrees of freedom (PdoF) in the colored gravity. However, there are finitely many global PDoF arising as integration constants in the sector of 0-forms B. Indeed, the last equation being the covariance constancy condition is solved in terms of constant B taken in a fixed point x = x 0 . Therefore, #PDoF= 4N 2 − 1 which is the dimension of the su(N, N ) adjoint.

Action for matrix-valued fields
Below, when considering the total action with the separated JT gravity term we explicitly evaluate partial trace related to the basis elements (I 2 , J A ), leaving dependence on the basis elements (I N , T α ) implicit. To this end, it is convenient to change the notation by singling out the u(1, 1) basis elements (3.4) in a general A-valued p-form X as follows . Then, recalling the basic definition (3.17) the BF fields can be cast into the form Using the algebra decomposition A = B + C (3.16) we represent A-valued fields as X = X B + X C , where the B-valued part corresponds to the first two terms in (3.23), while the C-valued part is given by the third term. Then, Using (3.25) in the original BF action (3.18) one gets the factorized form of the action where ∇ B is the B-covariant derivative defined on A-valued p-forms as follows The respective equations of motion also provides a systemic view of the component equations (A.1), (A.2). Taking A-valued p-form in round brackets (X (p) ) B,C means projecting onto subspaces B, C ⊂ A. Equations (3.28) and (3.29) admit the obvious solution with all C-valued fields set to zero. This leaves only B-valued fields so that the resulting truncated theory is given by JT gravity plus su(N ) BF theory. The subalgebra B = su(1, 1) ⊕ su(N ) can be further restricted to one of its parts yielding respective solutions of the original equations. In particular, isolating the su(1, 1) ⊂ B sector according to (3.24) we can introduce The derivative ∇ B acting on the C-valued part of a p-form X (3.23) is actually reduced Using this formula along with (3.24) and partially evaluating the trace we can give an alternative representation of the action (3.26) in terms of the matrix-valued fields as where the trace is now given only by the second relation in (3.13). Note that the coloursinglet gravity field enters the action through the JT term plus minimal coupling to the colored fields. The interacting part contains at most cubic terms. The respective equations of motion are given by, in the singlet sector, in the su(N ) sector, in the non-singlet sector,

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Similar to the representation (3.24) one can introduce the 0-form gauge parameter (3.38) and using the relation (3.8) rewrite the gauge transformations (3.21) in the matrix basis as In particular, one observes that the singlet gravity and dilaton gauge transformations receive corrections from the matrix-valued fields.

Towards the second-order formulation of colored JT gravity
In order to obtain the colored gravity action in the second-order form one needs to identify a set of auxiliary fields and solve the corresponding equations in terms of dynamical fields. The resulting action is given by the standard JT action (2.6) extended by a number of terms encoding the colored field dynamics.

4)
7 Let us note that other sets of auxiliary fields are possible which however yield second-order formulations for different dynamical fields. In general, this leads to the so-called dual formulations of the same theory with the first-order action playing the role of a parent action. In d-dimensional higher-spin (linearized) gravities this effect of choosing different sets of dynamical and auxiliary fields was investigated in [43,44]. Our choice of dynamical and auxiliary fields is standard and is given by the solution (2.5). It is rationalized by providing the JT action as a part of the resulting colored gravity action. Dual forms of 2d higher-spin dynamics are also possible and some preliminary discussion can be found in [20,22].

and in the 0-form sector [Lorentz scalar components of (3.34) and (3.37)]
dφ + 2 ab e a ϕ b + 2 N ab tr (W a Ψ b ) = 0 , (4.5) Note that equations (4.3), (4.4) and (4.5), (4.6) are obtained by variation of the action (3.32) with respect to the auxiliary 0-forms (ϕ a , Ψ a ) and 1-forms (ω, W), respectively. Altogether they can be traded as generalized torsion-free constraints of the colored dilaton gravity in the first-order formulation. The auxiliary equation system is linear in all auxiliary fields that allows expressing the auxiliary fields. The auxiliary equations can be processed in two stages. Firstly, we express the singlet auxiliary fields (ω, ϕ a ) thereby singling out the JT action. Indeed, solving the first (singlet) equations in each sector in (ω, ϕ a ) one can substitute the resulting expressions back into the second (matrix) equations still having a linear system with a number of equations exactly matching a number of the matrix auxiliary field variables (W, Ψ a ). At the second stage, (W, Ψ a ) can be expressed in terms of all dynamical fields that leads to the final second-order formulation. All in all, the auxiliary fields can be expressed as m , φ, W b , Ψ, A, B) .   o(1, 1) components. Note also that the decomposition can be done in manifestly o(2, 1)-covariant manner by introducing the compensator field [45]. 9 See appendix A for a summary of various types of covariant derivatives introduced in this paper.

Generalized (singlet) torsion-free constraints
Let us consider first the pair of the singlet auxiliary fields (ω, ϕ a ). Their equations are given by (4.3) and (4.5), Obviously, both equations are algebraic with respect to ω and ϕ a and their number is sufficient (four in total) to express two components of 1-form ω and two components if 0-form ϕ a in terms of the dilaton-gravity fields e a , ϕ and the colored fields W, W a , Ψ b . For future convenience we make the following redefinition where new fields ω and ϕ a satisfy the constraints with the colored graviton terms being neglected, which therefore reproduce the constraints in the pure BF JT theory (2.4). 10 Splitting off the Lorentz connection ω is not necessary, however, it allows to manifestly control the diffeomorphism covariance. On the other hand, splitting ϕ a helps to distinguish between derivative and algebraic contributions in the interaction part of the resulting action. Indeed, the original ϕ a enters the action (4.9) algebraically while the final expression for ϕ a (see (4.16) below) is given by derivative part (ϕ a ) and algebraic part (φ a ). Then, it follows that fieldsω andφ a satisfy the following constraints (4.14) Converting the world indices of p-forms as X µ 1 Here, W m and Ψ m are in turn the matrix auxiliary fields subject to the remaining (unsolved) constraints (4.4) and (4.6).
To summarize, the singlet constraints (4.11) and (4.12) are solved by Here, the leading terms are given by (2.5) which therefore explicitly manifest the contribution of the JT sector in the full colored gravity theory. Note that the solution for the singlet 10 Indeed, the constraints are algebraic linear inhomogeneous equations that can be schematically represented as Ax = B provided that det A = 0 which encodes det eµ a = 0. Splitting B = B1 + B2 and x = x1 + x2, where x1 satisfies Ax1 = B1 we obviously find that Ax2 = B2 and the solution is given by . In our case, B1 stands for ∂e and ∂φ, while B2 stands for the colored terms.

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auxiliary fields is linear in the matrix auxiliary fields. Keeping the Lorentz spin connection ω implicit refers to the so-called 1.5th-order formulation that allows one to simplify finding a second-order action if it derives from the original first-order action.
Using the explicit expressions forω and ϕ a ,φ a one obtains the following form of the colored dilaton gravity action (4.9) with partially eliminated auxiliary fields where JT and su(N ) BF parts (see section 2) are given by the standard expressions (modulo the overall normalization κN ) while all cross-terms are assembled into S X which contains both derivative and potential contributions where Here, all world indices in 1-forms have been converted to tangent ones by means of the frame e m = e m µ dx µ as 11 Also, from the standard torsion-free constraint (2.4) we have De m = 0 that allows to represent 2-form terms as DX = −e m e n D m X n thereby producing the action integral measure by means of e m e n = d 2 x √ −g mn . The base manifold M 2 (4.18)-(4.20) is now endowed with the metric which form is fixed by the JT equations of motion for g µν sourced by colored fields. Note that the potential V contains both cubic and quartic terms given by single and double trace combinations. Apart from the colored fields it explicitly depends on the singlet dilaton φ and su(N ) BF scalar B.

Generalized (matrix) torsion-free constraints
The remaining matrix auxiliary fields are subject to constraints (4.4) and (4.6) with the singlet auxiliary fields redefined according to (4.13): whereω andφ are given by (4.15). These constraints now follow from the action (4.18) provided all world indices are converted to tangent ones according to (4.23).
Though the above constraints are linear algebraic equations with respect to the matrix auxiliary fields they are hard to solve. However, we can still analyze the general structure of the action (4.18) with all auxiliary fields eliminated and derive a precise form of algebraic contributions to the potential term. To this end, let us denote the matrix auxiliary fields as and the relevant parts of the dynamical fields as Then, the derivative (4.21) and potential (4.22) terms can be schematically represented as It follows that contrary to other fields the auxiliary fields M aux contribute to the total action (4.18) only quadratically. The respective equations of motion δS cJT /δM aux = 0 can be represented in the following schematic form which encodes the constraints (4.24) and (4.25) with the matrix auxiliary fields explicitly isolated. Here, we assume that the inverse operator on the left-hand side does exist in a given context. The schematic equation (4.30) makes it clear that substituting the solution M aux into K + V results in reshuffling terms so that one produces K + V with K containing second order derivative terms in the dynamical fields, where the dots stand for terms with derivatives coming from V, while the potential V is obtained by setting M aux = 0 in V (4.29), i.e.

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Thus, the residual (at most cubic) potential is given by Finally, the second-order action for the colored dilaton gravity can be represented in the form with implicit K and explicit V defined as above. The above form of the action manifests that the color generalization of JT gravity corresponds to the system consisting of the usual JT gravity, the SU(N ) BF gauge theory and the color charged "spin 2" fields. Besides the gravitational and gauge interaction, the latter fields have non-trivial self-interactions through the potential V.

Discussion
In this paper we have introduced a colored version of JT gravity. It has been formulated as su(N, N ) BF theory. The theory describes the standard (singlet) dilaton and metric fields interacting with su(N ) vector and dilaton fields along with adjoint su(N ) multiplet of colored dilaton and metric fields. We developed both the first-order (BF) and the second-order (JT type) formulations though explicit form of the second-order colored gravity action involves potential terms with derivatives which were obtained only implicitly. The reason lies in complicated matrix constraints on auxiliary fields to be solved. This is a characteristic feature of the transition between the first-order and second-order formulations which therefore allows for the perturbative consideration only. 12 Note that the colored JT gravity is a topological theory that is manifest in the BF form. Local degrees of freedom can be introduced in the standard fashion by imposing appropriate boundary conditions so that a dynamics is restricted to the boundary. The respective Schwarzian-type one-dimensional boundary theory is considered in the paper [48].
A few comments are in order. Having in mind the isomorphisms sl ( [17,20,21] and the su(N, N ) series in the present paper (see figure 1). A related comment concerns a higher-spin extension of the su (N, N ) BF theory. Taking a formal limit lim N →∞ su(N, N ) one finds a higher-spin algebra hs[λ] for some particular value of λ [27].
Lastly, the colored JT gravity admits various "colored AdS vacua" like the colored 3d gravity [35]. The mechanism allowing for these vacua is simple: we take the ansatz A and JHEP08(2022)286 for solutions of the equations (3.33)-(3.37). Here, X is a constant traceless N × N matrix belonging to su(N ), and Ω A = (ω, e a ) and Φ A = (φ, ϕ a ) are the fields of the color-singlet gravity sector. This configuration corresponds to the su(N, N ) 1-and 0-form fields, where Y = I N + X. The equations are Using the first and second properties, we can diagonalize Y as where p is the number of the negative eigenvalues. The last condition determines α in terms of p as We find that the net cosmological constant increases in its absolute value as the number of the color background p increases. For p = 0, the background breaks the color symmetry from su(N ) to su(p) ⊕ su(N − p) ⊕ u(1). Let us briefly discuss the Higgs mechanism resulting from the symmetry breaking at the level of the linearized field equations for the gauge 1-form. For simplicity, we choose a fluctuation

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The linearized flatness condition around the backgroundĀ (5.2) reads dη + [Ā, η] = 0 , (5.12) and its component decomposition gives Note that the roles of the spin connection and the spin-one mode are exchanged in the symmetry broken part. Otherwise, ( a b ψ b , b) satisfy exactly the same equations as those satisfied by (h a , a) , and hence they also describe a massless spin-two mode. This is in contrast with the 3d colored gravity [35] where the symmetry breaking generates spin-two Goldstone modes which combine with CS gauge fields to become partially-massless spin-two fields.
Let us emphasize that the basic idea behind our study is to probe various extensions of JT gravity in the BF form with a gauge algebra necessarily containing sl(2, R) subalgebra. This latter requirement is naturally related to that the low-energy SYK model has a conformal symmetry [1][2][3]6]. As is noted in the Introduction, the bulk dual of the SYK model is presently unknown and, therefore, it seems that BF theory with extended gauge symmetries (containing conformal sub-sector, or, equivalently, a gravitational JT sub-sector) is a natural option to investigate the dual bulk dynamics.
The color-extended gauge symmetry in our model implies that a dual SYK-like model would have the extended emergent reparametrization symmetry. Like SYK-dual of higherspin version of JT gravity, it will be of great interest to find a microscopic Hamiltonian for SYK-like model which has extended nearly-conformal symmetry. Furthermore, the "colored Euclidean wormhole" and the genus expansion of the colored JT gravity will be a tantalizing future work. This will lead us to investigate how the color extension can alter the ensemble of random matrix theory [49] of the colored JT gravity and its dual SYK-like model.
In the present paper we considered the SU(N, N ) BF model which can be interpreted as a colored JT gravity. However, it is to be understood that in two dimensions general gauge and matter fields can carry only scalar/spinor degrees of freedom (i.e. can be reduced to Klein-Gordon or Dirac equations) otherwise they are topological. Therefore, in the case of pure topological BF theory any direct dynamical interpretation can be ambiguous. This is true both for colored and higher-spin versions of JT gravity. The only thing that matters here is the choice of a particular gauge algebra.
On the other hand, both sl(2N ) and SU(N, N ) are two real forms of the same complex algebra which means precisely that the corresponding BF theories in spite of different space-time interpretations (see figure 1) extend JT gravity in the same conceptual direction. Therefore, both higher-spin and colored JT gravities could be equally relevant in finding AdS dual of SYK according to higher-spin proposals of refs. [10,17,23].

A Technical details of su(N, N ) BF formulation
Component form of equations. The component form of the equations of motion in the basis (3.17) take the following form: • 1-form sector: (A.1) • 0-form sector: Using (3.5) one can show that these equations are equivalent to the matrix equations (3.33)- (3.36). Note that the equations of motions of pure JT and su(N ) BF theories are now sourced by terms with colored fields.
Covariant derivatives. Throughout the paper we introduced a number of covariant derivatives associated to various subalgebras of the original gauge algebra A = su (N, N ). It is useful to bring them together: • ∇ A is a covariant derivative (3.22); • D is a o(1, 2) covariant derivative; • ∇ B is a covariant derivative (3.27) of the subalgebra B ⊂ A (3.16); • ∇ is a tensor product part of ∇ B acting on C-valued fields (3.31); • D is o(1, 1) ⊕ su(N ) covariant derivative (4.10); • ∇ L is the Lorentz o(1, 1) covariant derivative being a part of D (4.10).

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