Flat space (higher spin) gravity with chemical potentials

We introduce flat space spin-3 gravity in the presence of chemical potentials and discuss some applications to flat space cosmology solutions, their entropy, free energy and flat space orbifold singularity resolution. Our results include flat space Einstein gravity with chemical potentials as special case. We discover novel types of phase transitions between flat space cosmologies with spin-3 hair and show that the branch that continuously connects to spin-2 gravity becomes thermodynamically unstable for sufficiently large temperature or spin-3 chemical potential.


Introduction
Chemical potentials are ubiquitous in physics (and chemistry) ever since their introduction by Gibbs. In gauge theories chemical potentials µ are usually introduced by giving the 0-component of the gauge connection a vacuum expectation value (see e.g. [1]).
In AdS 3 /CFT 2 it is rewarding to study Bañados-Teitelboim-Zanelli (BTZ) black holes [73,74]. The flat space analogue of these objects are flat space cosmologies [75,76]. Much like it is possible to consider BTZ black holes or their higher spin versions with chemical potentials switched on, it is plausible that there should be a flat space counterpart thereof, both in flat space gravity and in flat space higher spin gravity.
Flat space higher spin gravity in three dimensions combines all these research avenues and may serve to gain a better and deeper understanding of higher spin gravity, flat space holography, microscopic aspects of flat space cosmologies, string theory in the tensionless limit and, more broadly, quantum gravity and the holographic principle itself.
In the present work we consider specifically spin-3 gravity in flat space. Our main goal is to introduce chemical potentials for the spin-2 and spin-3 field in flat space, and to address some of their consequences, in particular the entropy and free energy of flat space cosmologies with spin-3 charges. Technically, we do this by working in the Chern-Simons formulation of spin-3 gravity and introducing chemical potentials as in (1.1), i.e., by deforming the zero-component of the gauge connection, analog to [29].
One of the most surprising results that we find is that the "physical" branch that connects continuously to spin-2 physics becomes thermodynamically unstable at large temperature or large spin-3 chemical potential. The phase transition can be of first or zeroth order, which differs qualitatively from the situation in AdS, where the corresponding phase transitions discovered so far were of zeroth order [77][78][79].
This paper is organized as follows. In section 2 we review aspects of flat space spin-2 and spin-3 gravity. In section 3 we include chemical potentials and present the main results. In section 4 we display our results applied to flat space Einstein gravity with chemical potentials. In section 5 we discuss some applications to flat space cosmologies, calculate their entropy and free energy, discover novel types of phase transitions, remark on flat space orbifold singularity resolution in spin-3 gravity and mention further developments and open issues.

Flat space higher spin gravity
In this section we review results of flat space higher spin gravity and present them in a way that is considerably simpler than in the original publications. In section 2.1 we recall the Chern-Simons formulation in terms of an isl(3) connection. In section 2.2 we display a recent representation of this connection reminiscent of similar representations in the AdS case. In section 2.3 we present the canonical charges and their algebra. In section 2.4 we provide simple formulas for metric and spin-3 field by means of a twisted trace.

Chern-Simons formulation
Like in the AdS case, it is very convenient to use the Chern-Simons formulation of the theory. The Chern-Simons action contains a coupling constant k (the Chern-Simons level; k = 1/(4G N ), where G N is Newton's constant) and the Chern-Simons 3-form The bilinear form · , · will be specified below. In the present work the connection will always be isl (3) The L n generate (Lorentz-)rotations, M n generate translations, and U n , V n generate associated spin-3 transformations. The factor σ fixes the overall normalization of the spin-3 generators U n and V n . In the present work we choose 2 It is also noteworthy that one can equip the algebra (2.4) naturally with a Z 2 grading so that the generators L n , U n are even and M n , V n are odd. Then even with even gives even, even with odd gives odd and odd with odd vanishes. If one constructs isl(3) as aṅ Inönü-Wigner contraction from so (2,2) it is possible to do so introducing a Grassmann parameter , and the contraction then consists in dropping all expressions quadratic in ; the -independent generators are then the even generators and the generators linear in are the odd generators [66].
We exploit this Grassmann trick in appendix A.1 to define the generators. Moreover, we use it to define a (twisted) trace over a product of k isl(3) generators G n i (with i = 1 . . . k) that is useful to construct the spin-2 and spin-3 fields from the isl(3) connection.
The right hand side contains the usual matrix trace and involves the matrix (A.4). Here are some relevant properties of the twisted trace: • Oddness. The twisted trace vanishes identically if at least one of the generators G n i is even, i.e., one of the generators L n or U n . Therefore, the only non-vanishing twisted traces involve exclusively the odd generators M n and V n .
• Relation to matrix trace. If k is even, then all factors of γ * cancel and the twisted trace is equivalent to the ordinary trace, upon taking into account the oddness property and up to a factor 1 2 . If k is odd, then one factor of γ * remains, which ensures that the twisted trace does not vanish identically for all odd numbers of odd generators (the ordinary trace, however, does vanish for all odd numbers of odd generators, essentially due to the vanishing trace of the Pauli matrix σ 3 ).
• Relation to sl(3) trace. If we consider just the sl(3) block of the generators then the twisted trace is equivalently defined as the matrix trace over the products of the corresponding sl(3) blocks, again upon taking into account the oddness property. For the sake of this property we introduced the factor 1 2 in the definition (2.6). We shall employ the twisted trace (2.6) to define the spin-2 and spin-3 fields.
With respect to the above generators the (degenerate) bilinear form is given by Here η mn given by η = antidiag (1, − 1 2 , 1) is proportional to the sl(2) Killing form and the sl(3) part is given by K = antidiag (12, −3, 2, −3, 12), both of which have non-zero entries only on the anti-diagonal. The bilinear form can be represented as a trace as follows [again using the matrix (A.4)]: We shall refer to this trace as "hatted trace" to discriminate it from the twisted trace (2.6) and the ordinary matrix trace.

Spin-3 flat space connection
Explicit expressions for isl (3) connections that obey asymptotically flat boundary conditions were established independently in [41] and [42] (see also [80]). However, we shall not use either of these expressions, but use instead the one introduced in [70] since it is considerably simpler. Namely, we represent the connection A as gauge transformation of another connection a with very simple properties and simple gauge group element b: The form (2.9) is reminiscent of the similarly useful form of the AdS connection in spin-2 gravity (see e.g. [81,82]) and higher spin gravity [83,84]. We use coordinates u, r and ϕ ∼ ϕ+2π adapted to flat space in (outgoing) Eddington-Finkelstein coordinates. As we shall see below, the background line-element in the absence of chemical potentials is then given by (2.12) The manifold is topologically a filled cylinder. The asymptotic boundary cylinder (corresponding to null infinity) is reached in the limit where the radial coordinate r tends to infinity.
Note that any connection A with the properties (2.9)-(2.13) automatically solves the Chern-Simons field equations F = dA + [A, A] = 0 (2.14) provided the state-dependent functions in (2.13) are constrained as follows.
Dots (primes) denote derivatives with respect to retarded time u (angular coordinate ϕ).
The constraints (2.15) are solved in terms of four arbitrary functions of the angular coordinate ϕ, all of which appear in the canonical charges [41].

Canonical charges and their algebra
The canonical charges of Brown-Henneaux type [85] are constructed in the usual way [83,84]. We use here the notation of [37]. The first step is to identify all gauge transformations that preserve the flat space boundary conditions. This was done already in [41,42]. We rephrase now these results in terms of the new representation of the connection (2.9)-(2.11). To this end, we similarly define the gauge parameter as with the same group element (2.10) as before. The results of [41,42] for the boundary condition preserving gauge transformations then translate into the following expression: 3 The functions , σ, χ and ρ depend on ϕ only, and we have the relations τ = σ + u and κ = ρ + uχ . When acting with such a gauge transformation on an isl(3) connection A with the properties (2.9)-(2.16) the gauge transformed connection A = A + δ ε A also has the properties (2.9)-(2.16), in general with some shifted values for the state dependent functions, M = M + δ ε M, and similarly for N , V and Z. The canonical charges also follow the general prescription of (non-)AdS holography summarized in [37]. Their field variation, also known as the canonical currents, is given by Note the appearance of the hatted trace (2.8). 3 There are three differences to the results in [41], whose conventions we use: 1. due to the convenient representation (2.17) with (2.10) we do not have any r-dependent terms, which are automatically generated through the Baker-Campbell-Hausdorff formula, 2. we have corrected three numerical coefficients, which all differ by factor of −3 from the expressions given in [41], namely the coefficients of the V-and Z-terms in the components L− and M−, and 3. we have rescaled τ by a factor of 2 to make the results look more symmetric. We note finally that we use in two ways in this paper, as Grassmann parameter and as function in the boundary condition preserving gauge transformations (2.18), but we believe that the meaning should always be clear from the context. It is now evident that the canonical currents can be integrated in field space to canonical boundary charges The canonical charges are integrable, finite and conserved in (retarded) time, ∂ u Q = 0. The algebra of the canonical charges was derived classically [41,42] and quantummechanically [41] starting from the Poisson-bracket algebra of the canonical charges (2.22) and then expanding in Fourier modes, e.g. (2.23) and similarly for the other three state-dependent functions appearing in (2.22). After a suitable shift of the zero mode M 0 → M 0 + k 2 and converting Poisson-brackets into commutators one obtains finally the asymptotic symmetry algebra as a commutator algebra of the modes L n , M n , U n and V n . It is anİnönü-Wigner contraction of two copies of the W 3 algebra, with the following non-vanishing commutators.
We used the definitions of bi-linears in the generators where normal ordering is defined by : L n M m : = L n M m if n < −1 and : L n M m : = M m L n otherwise. It is interesting to note that the algebra (2.24), with some standard assumptions, does not have unitary highest weight representations for non-vanishing k [49].

Metric and spin-3 field
The metric in AdS higher spin gravity is usually defined as trace over the zuvielbein [83,84]. In flat space higher spin gravity the line-element takes the form which for the connection (2.9)-(2.16) simplifies to This is the same result as in Einstein gravity [60]. Exploiting the Grassmann-structure there is a neat way to define the metric again as a trace. Namely, take the matrix representation of the generators L n , M n , U n and V n (see appendix A.1) and the twisted trace definition (2.6). Then the metric is equivalently defined by Only bilinear expressions in the odd generators contribute to the line-element, which is precisely the statement of (2.25) or (2.27). The spin-3 field is similarly defined from the cubic sl(3)-Casimir or, equivalently, by using again the twisted trace which for the connection (2.9)-(2.16) simplifies to Only expressions trilinear in the odd generators contribute to the spin-3 field.

Flat space higher spin gravity with chemical potentials
In this section we generalize the discussion to flat space spin-3 gravity with chemical potentials µ M , µ L , µ V , µ U for the spin-2 and spin-3 fields. We start by stating our main result in section 3.1 and perform consistency checks in section 3.2. In section 3.3 we discuss the canonical charges and variational principle in the presence of chemical potentials. Finally, we display results for the metric and the spin-3 field in section 3.4.

Statement of the main result
To include chemical potentials we solve the equations of motion (2.14) assuming the representation of the connection as in (2.9)-(2.11). Following the procedure of [29] we also assume that the form of a ϕ remains unchanged by chemical potentials, in order to maintain the structure of the canonical charges. We obtain ϕ being the connection (2.13) in the absence of chemical potentials and where the subscript M → L denotes that in the corresponding quantity all odd generators and chemical potentials are replaced by corresponding even ones, As before, dots (primes) denote derivatives with respect to retarded time u (angular coordinate ϕ). The equations of motion (2.14) impose the conditionṡ with the inverse substitution rules to above, viz.
The chemical potentials µ M , µ L , µ V and µ U are arbitrary functions of the angular coordinate ϕ and the retarded time u. In many applications they are constant so that many formulas simplify.
In the next subsection we provide several checks on the correctness of the results presented above and discuss in a bit more detail how we obtained them.

Checks
Note first that in the absence of chemical potentials, µ M = µ L = µ V = µ U = 0, corresponding results from section 2 are recovered. In particular, the on-shell conditions (3.3) simplify to (2.15). In the presence of chemical potentials the on-shell conditions (3.3) contain information about the asymptotic symmetry algebra (2.24). For example, the µ L -terms in (3.3a) are an infinitesimal Schwarzian derivative, while the µ U -terms exhibit transformation behavior of a spin-3 field. Since any solution to the field equations must be locally pure gauge, and any solution that obeys our boundary conditions can be generated by the boundary condition preserving gauge transformations (2.18), it should be possible to obtain (3.2) directly from a gauge transformation. Indeed, comparing the expressions for (2.18) with the expressions in (3.2) we see that they coincide upon identifying → µ L , τ → µ M , κ → µ V and χ → µ U . This comparison provides an independent check on the correctness of our results.
It is possible to derive the results of section 3.1 in various ways. For instance, one can start from equation (3.7)-(3.12) in [29] and use the Grassmann-approach of [66] to derive the flat space connection with chemical potentials, dropping in the end all terms quadratic in the Grassmann-parameter. This is the procedure we have used. The map that leads from (3.7)-(3.12) in [29] (left hand side) to the results presented in section 3.1 (right hand side) is given by coordinates: spin-2 generators: spin-3 generators: After using the map (3.4) one is supposed to drop all terms quadratic (or higher power) in the Grassmann parameter . Note that no inverse powers of appear anywhere in the connection, despite of their appearance in various expressions above. Equivalently, one can do a straightforwardİnönü-Wigner contraction, sending the AdS radius to infinity. Alternatively, one could directly solve the flat space field equations (2.14) with the condition that a ϕ remains unchanged as given in (2.13) and only a u obtains contributions from chemical potentials. All these procedures lead to the same results displayed above in section 3.1.

Canonical charges with chemical potentials
Since the canonical currents (2.20) only depend on a ϕ , which has not changed by introducing chemical potentials, the results for the canonical charges remain unchanged and all expressions displayed in section 2.3 also apply to the case of non-vanishing µ M , µ L , µ V and µ U . In fact, this property was the very reason why we allowed only a deformation of a u . In particular, from (2.22) we have the following four zero-mode charges.
They can be interpreted, respectively, as mass, angular momentum, odd and even spin-3 charges.
The canonical charges will be important for our later discussion of entropy in section 5.1 below. They also feature prominently in the variational principle. To determine the boundary term required for a well-defined variational principle we vary first the bulk action (2.1).
Evaluating the boundary term explicitly yields ( denotes equality up to total ϕ-derivative terms, which vanish upon integration over the ϕ-cycle) This confirms the result [41] that the bulk action (2.1) has a well-defined variational principle in the absence of spin-3 chemical potentials. In their presence, however, the last two terms are incompatible with a well-defined variational principle. Therefore, we subtract a boundary counterterm to restore a well-defined variational principle for this case, In total we get (Q N is Q L with L replaced by N , and similarly for Q Z with Q U ) In conclusion, the action (3.8) has a well-defined variational principle, in the sense that the first variation of the full action vanishes on-shell for arbitrary (but fixed) chemical potentials. As expected, the response functions (3.10) are determined by the canonical charges, and the chemical potentials act as sources.

Metric and spin-3 field in presence of chemical potentials
Plugging the results for the connection with chemical potentials, (3.1)-(3.3) with (2.9)-(2.11), into the definitions for the metric (2.27) yields and the contributions g Note that for zero-mode solutions with constant chemical potentials, M = N = µ M = µ L = µ V = µ U = 0, all the expressions in appendix B vanish and thus the spin-2 and spin-3 fields simplify considerably in this case (see also appendix C).

Flat space Einstein gravity with chemical potentials
If we set to zero the spin-3 charges and spin-3 chemical potentials, V = Z = µ V = µ U = 0, we recover flat space Einstein gravity with chemical potentials µ M and µ L . While this is merely a special case of the more general results of section 3, it seems convenient for future applications to collect these results separately and to elaborate on them. This is what we do in this section.
In section 4.1 we present the general solution for the isl(2) gauge connection and the metric with arbitrary spin-2 chemical potentials. In section 4.2 we focus on zero mode solutions with constant chemical potentials and provide a canonical interpretation of the latter. In section 4.3 we linearize the solutions in the chemical potentials, which is useful for some applications, like the holographic dictionary, which we address in section 4.4.

General solution
The connection is given by (2.9), (2.10), (2.11) with The corresponding line-element reads with the on-shell conditionṡ

Zero mode solutions with constant chemical potentials
We consider now zero mode solutions, M = N = 0, with constant spin-2 chemical potential, µ M = µ L = 0. Then the results above simplify further. The line-element reads with the on-shell conditionsṀ =Ṅ = 0.
If we set to zero the even chemical potential, µ L = 0, then the line-element (4.4) simplifies to the vacuum solution (2.12), but with u replaced byũ = (1 + µ M )u. Therefore, a constant odd chemical potential µ M effectively rescales the retarded time coordinate. In canonical general relativity language the odd chemical potential µ M rescales the lapse function.
If instead we set to zero the odd chemical potential, µ M = 0, then the line-element (4.4) simplifies to Comparing this result with the vacuum solution (2.12) in ADM-like form, we see that the even chemical potential µ L changes only the last term. In canonical general relativity language the even chemical potential µ L shifts the shift vector.

Perturbative solutions linearized in chemical potentials
A different kind of simplification arises when linearizing in the chemical potentials. Expanding the metric (3.11) in the chemical potentials, with the background line-elementḡ µν dx µ dx ν given by the right hand side of (2.26), yields for the linear terms The terms in the second line vanish for constant chemical potentials.

Comparison with holographic dictionary
From a holographic perspective, the first two terms in the linearized solution (4.8) show the typical coupling between sources (chemical potentials) and vacuum expectation values (canonical charges). The r 2 µ L du dϕ term and the µ M dr du term correspond to the essential terms in the two towers of non-normalizable 4 solutions to the linearized equations of motion.
In the holographic dictionary, these non-normalizable contributions should correspond to sources of the corresponding operators in the dual field theory. Indeed, this is what happens as shown in [68]. Note, however, that [68] worked in Euclidean signature, restricted to zero mode solutions and imposed axial gauge for the non-normalizable solutions to the linearized Einstein equations on a flat space background, so a direct comparison is not straightforward. Exploiting our interpretation of constant chemical potentials as modifications of lapse and shift (see section 4.2) we can interpret the results of [68] as follows (see their section 3.4): their quantity δξ J corresponds precisely to the (linearized) even chemical potential δξ J ∼ µ L , and their quantity δξ M corresponds to twice the (linearized) odd chemical potential, δξ M ∼ 2µ M . This identification is perfectly consistent with the holographic interpretation summarized above.

Applications
In this section we address some applications, without claiming to be exhaustive.
In section 5.1 we calculate the entropy of flat space cosmologies with spin-3 charges by solving all holonomy conditions. In section 5.2 we determine the free energy and discover novel types of phase transitions. In section 5.3 we conclude with some remarks on the recent spin-3 singularity resolution of flat space orbifolds. In section 5.4 we provide an outlook to some further possible applications.
As supplements, in appendix C we discuss zero-mode solutions with constant spin-3 and vanishing spin-2 chemical potentials and in appendix D we consider more general solutions to the field equations, dubbed "chemically odd", by restricting to odd chemical potentials only and by allowing specific deformations of a ϕ .

Entropy
To discuss thermodynamical aspects we restrict ourselves to zero mode solutions with constant chemical potentials. The main quantity of interest is the entropy of solutions like flat space cosmologies with spin-3 charges switched on. As we shall demonstrate by solving holonomy conditions, entropy is given by a hatted trace, The quantity β L is not necessarily the inverse temperature, but rather the length of the relevant cycle appearing in the holonomy condition below. The zero mode charges Q i are displayed in (3.5). We start by proposing the holonomy condition that we want to solve.
exp iβ L a u = 1l (5.2) This condition is completely analogous to corresponding holonomy conditions for higher spin black holes in AdS [24]. To solve the holonomy condition (5.2) we exploit the representation summarized in appendix A.2 in terms of 9 × 9 matrices. By a similarity transformation we can diagonalize the ad-part of a generic matrix of the form (A.5).
A matrix of this form is easily exponentiated. Assuming that ad has zero as eigenvalue with geometric and algebraic multiplicity n and denoting v = A −1 odd yields The first set of relations (5.5) is precisely the same as in AdS spin-3 gravity for one chiral half. Therefore, we must be able to represent these conditions in the same way as it was done in AdS. In fact, a plausible guess for the two holonomy conditions that follow from the first set of relations (5.5) is given by (compare with corresponding conditions in the AdS case, particularly Eqs. (3.32) and (3.33) We prove now that this is indeed the correct result. Since the matrix A −1 adA is diagonal, it must lie in the Cartan subalgebra of sl (3); diagonalizing simultaneously L 0 and U 0 we find with some functions f L , f U of the charges and chemical potentials that can be determined by explicitly calculating the characteristic polynomial of the matrix iβ L a u for the eigenvalues λ as derived from the solution (3.2) (with constant charges and chemical potentials) and comparing it with the characteristic polynomial that follows from (5.8). The first set of relations (5.5) yields the conditions Thus, the first half of the holonomy conditions leads to a discrete family of solutions parametrized by two integers n and m. For the choice m = 2 and n = 1 these conditions reproduce precisely the guess (5.6) and (5.7). This choice is unique by requiring that in the absence of spin-3 chemical potentials and spin-3 charges the holonomy conditions reduce to the ones for flat space cosmologies. We will therefore always make this choice in the present work. So far we have obtained and solved only half of the holonomy conditions. The other half emerges from imposing the second set of relations (5.5). After a straightforward calculation 5 we find that one of these conditions is linear in the charges and chemical potentials, while the other is quadratic in the charges and linear in the chemical potentials These results are considerably simpler than the corresponding holonomy conditions in AdS, which are at least quadratic in chemical potentials and charges. The linear holonomy condition (5.10) simplifies entropy (5.1) to For the special case µ U = 0 entropy (5.12) depends only on spin-2 charges and chemical potentials (see appendices C and D). Moreover, the solution to the four holonomy conditions (5.6), (5.7), (5.10), (5.11) becomes elementary.
For that case entropy is given by the Bekenstein-Hawking area We included absolute values to ensure that entropy is positive regardless of the sign of the charge L. Inverse temperature then coincides with the spin-2 result (see e.g. [59]; note that in their conventions M = r 2 + and |L| = |r 0 r + |).

T = 1 2π
There are numerous different ways to obtain these results, but it is not always easy to extract the simple conditions (5.10) and (5.11). For instance, one can contract the AdS holonomy conditions using the map (3.4), but this leads naturally to non-linear relations between charges and chemical potentials. Two combinations of these relations immediately provide the holonomy conditions (5.6) and (5.7), but it takes a bit of work to extract the other two conditions in their simplest form. Alternatively, one can explicitly construct the matrix A in (5.3) that diagonalizes the sl(3) part of the generators and then determine the two eigenvectors associated with the two zero eigenvalues. This approach makes it clear from the start that the remaining two holonomy conditions must be linear in the chemical potentials. The procedure we used is a simpler version thereof that avoids complete diagonalization, but merely puts the generators into block form with a 2 × 2 block of zeros, since the remaining two holonomy conditions are restricted to the subspace associated with the zero eigenvalues.
The minus sign in the definition (5.15) is reminiscent of the inner horizon first law of black hole mechanics [86][87][88][89][90] as explained in [59]. From the corresponding first law − dQ M = T dS + Ω dQ L (5.17) we deduce the angular potential which again coincides with the spin-2 result [59].
In the general case µ U = 0 not all holonomy conditions are linear. Instead, we have to solve one quadratic and one cubic equation, similar to the AdS case. Defining µ = µ L µ U and η = µ L /µ U + 1 9 M 2 /V the holonomy conditions (5.6), (5.7) simplify to Solving the cubic equation ( Heneceforth, we shall always assume the inequality In other words, we consider from now on exclusively the case of positive discriminant, D > 0. In this case there are three real solutions for η. The resulting entropy is real for all three branches. However, only one branch recovers the same entropy (5.12) as for the spin-2 case in the limit V → 0. Therefore, we take that branch. On this particular branch, there is a neat way to express all results in terms of the charges M, L, U and a new parameter R that depends on the ratio of spin-3 and spin-2 charges V 2 /M 3 , just like in the AdS case [24]: The restriction to R > 3 guarantees that we sit on the correct branch. The chemical potentials then read while entropy is given by The expression for entropy (5.29) is the main result of this section. The pre-factor containing the spin-2 charges M, L coincides with the spin-2 result (5.14). The spin-3 correction depends non-linearly on one of the combinations of spin-3 charges, R, and linearly on the other, P.
For some purposes it can be useful to have a simpler perturbative result for entropy in the limit of small spin-3 charge V (large R), which we present below.
We close the entropy discussion by addressing sign issues. We have assumed that the mass is positive, M > 0, motivated by the necessity of this condition in the spin-2 case. The sign of L does not matter, which is why we included absolute values in the final result for entropy (5.29). Here is our argument. Suppose that L > 0 (L < 0). Then we exploit the sign ambiguity in the definitions of µ L , µ U by choosing µ L > 0 (µ L < 0) so that the first term in (5.12) is always positive and thus entropy is positive in the limit of vanishing spin-3 fields. The sign of V is taken care of by the definition (5.24), which ensures positive R regardless of the sign of V. Thus, the only remaining signs of potential relevance are the signs of the spin-3 charge U and the corresponding chemical potential µ U . The latter is fixed through the sign choice of µ L explained above, but the former is free to change, and this change is physically relevant. This implies that the quantity P defined in (5.30) can have either sign, so that the last term in the entropy (5.29) can have either sign. Demanding positivity of entropy then establishes an upper bound on U.

Grand canonical free energy and phase transitions
In the previous section we found that there are three branches of solutions of all the holonomy conditions, and we simply took the branch that connects continuously to the spin-2 results in the limit of vanishing spin-3 charges. However, it is not guaranteed that this procedure picks out the correct branch from a thermodynamical perspective in the whole parameter space. What we should do is to compare the free energies of all branches for given values of the chemical potentials and check which of the branches leads to the lowest free energy. This is precisely the aim of this subsection.
We start by writing the general result for the (grand canonical) free energy, regardless of the specific branch (we set k = 1 in this subsection). We already have a thermodynamic potential, namely entropy in terms of extensive quantities (charges), so all we need to do is to Legendre transform with respect to all pairs charge/chemical potential. 6 The zero mode charges are given by (3.5) and the intensive quantities by the chemical potentials.
In order to express free energy in terms of intensive variables we have to invert the holonomy conditions and solve for the charges in terms of chemical potentials. Before doing so, it is instructive to consider free energy expressed in terms of charges in certain limits. In the large R limit (weak contribution from spin-3 charges) we recover the spin-2 result In the R → 3 limit (strong contribution from spin-3 charges) we obtain Thus, we have a universal ratio F weak F strong = 3 . Performing the Legendre transformation (5.32) with the entropy (5.12) yields In order to obtain free energy as function of intensive variable we have to solve the nonlinear holonomy conditions (5.6), (5.7) for the charges in terms of the chemical potentials. Solving (5.6) for V allows us to express free energy in terms of the mass M and of chemical potentials.
Plugging the solution for the spin-3 charge V in terms of the mass M into the other holonomy condition (5.7) establishes a quartic equation for the mass M, which leads to four branches of solutions for free energy. The discriminant of that equation is positive, provided the spin-3 chemical potential obeys the bound Another way to read the inequality (5.42) is that it provides an upper bound on the temperature for given spin-3 chemical potential Ω U . The maximal temperature is given by In the limit of small Ω U it turns out that only one of the branches has finite free energy. This is the branch that continuously connects with spin-2 results, on which free energy yields The term before the parentheses reproduces the spin-2 result for free energy. The term in the parentheses depends only on two linear combinations of the chemical potentials [on t and v introduced in (5.46) below]. As in the spin-2 case [59] there will be a phase transition between flat space cosmologies and hot flat space at some critical temperature. A novel feature of the spin-3 case is that there are additional phase transitions between the various flat space cosmology branches. To see this we consider the difference between the free energies of two branches.
There are two zeros in the difference (5.45), an obvious one when the masses of the two branches coincide, M 1 = M 2 , and a non-obvious one when the expression in the last parentheses in (5.45) vanishes. We focus in the following on the difference between the branch that continuously connects to spin-2 results (branch 1) and the other branch that ceases to exist if the bound (5.42) is violated (branch 2). The other two branches are then branch 3 and 4; they will play only minor roles.
To reduce clutter we assume from now on that temperature and the chemical potentials are non-negative. Moreover, we introduce dimensionless combinations of chemical potentials The quantity t is a dimensionless temperature, while v is essentially a ratio of odd over even spin-3 chemical potential. Expressing the difference of free energies (5.45) between branches 1 and 2 as function of these two combinations, up to a non-negative overall constant, yields The corresponding free energy differences near these temperatures read, respectively We arrive therefore at the following picture, depending on the value of the parameter v: 7 • 0 < v < 3 2 : Branch 1 is thermodynamically unstable for all temperatures.
• v = 3 2 : Branch 1 degenerates with branch 2 at vanishing temperature and is thermodynamically unstable for all positive temperatures.
• 3 2 < v < 2: Branch 1 degenerates with branch 2 at some positive temperature. Below that temperature branch 1 is thermodynamically unstable. At that temperature there is a phase transition from branch 2 to branch 1. Above that temperature branch 1 is stable (modulo the phase transition to hot flat space [59]). 7 Positivity of entropy imposes additional constraints on the existence of branches; we checked that the existence of the first order phase transition between branches 1 and 2 that we describe below is not influenced by such constraints. • v = 2: Branch 1 degenerates with branch 2 at the maximal temperature (5.43) and is thermodynamically stable for all temperatures (again modulo the phase transition to hot flat space).
• v > 2: Branch 1 is thermodynamically stable for all temperatures (with the same caveat as above).
To illustrate the results above we show an example in figure 1. In all six graphs the thick line depicts free energy for branch 1 and the dashed line for branch 2 (the other two branches are not essential for this discussion; if visible they are plotted as dotted lines). The three upper plots show explicitly the phase transition between branches 1 and 2, depending on the choice of v. The three lower plots show that there are further phase transitions involving the branches 3 and 4, if branch 1 is unstable for all values of temperature. In addition to all these new phase transitions there is the 'usual' phase transition to hot flat space [59], which in the present case can be of zeroth, first or second order. Since there are several phase transitions possible there exist also multi-critical points where three or four phases co-exist.
The most striking difference between the AdS results by David, Ferlaino and Kumar [77] and our flat space results is that we observe the possibility of first order phase transitions between various branches (see the right upper and middle lower plot in figure 1). By contrast, in AdS the only phase transitions (other than Hawking-Page like) arise because two of the branches end, at which point the free energy jumps (we also recover these zeroth order phase transitions in flat space, see e.g. the left lower plot in figure 1).

Remarks on flat space singularity resolutions
String theory is believed to resolve (some of) the singularities that arise in classical gravity, see e.g. [92] and references therein. If this is true and if higher spin gravity can be thought of as emerging from string theory in the tensionsless limit, then it is suggestive that also higher spin gravity could resolve (some of) the singularities that arise in Einstein gravity. Regardless of how plausible this line of reasoning appears, it is certainly of interest to investigate the issue of singularity resolution in (three-dimensional) higher spin gravity.
Indeed, Castro et al. discovered corresponding singularity resolutions for black holes [93] and conical surpluses [94] in three-dimensional AdS higher spin gravity. More recently, Krishnan et al. considered the singularity resolutions of the Milne universe [65] and the null orbifold [95] in three-dimensional flat space higher spin gravity. We discuss now some of their findings from the perspective developed in the present work, starting with the second example, the null orbifold singularity resolution.
In our conventions the null orbifold is a configuration with This configuration leads to the null orbifold line-element with vanishing spin-3 field. The null orbifold exhibits a singularity at r = 0, see e.g. [96][97][98][99].
Comparing the original null orbifold configuration (5.53) with the resolved one (5.55) we see that the difference is in the a ϕ component, not the a u component. Therefore, we cannot interpret the additional terms proportional to p as coming from a chemical potential as introduced in section 3.
We check now whether the (spin-3) transformation that maps (5.53) to (5.55) is a small gauge transformation. If p is a state-dependent function then the term proportional to p in (5.55) leads to a contribution to the canonical currents of the form (see appendix D) δQ ∼ dϕ (χ − Mχ)δp (5.58) which is finite and conserved, but not integrable in field space unless p = p(M). However, if p is not a state-dependent parameter but merely some (gauge-)parameter then its fieldvariation vanishes, δp = 0, and the canonical charges remain unchanged. Therefore, we conclude that the spin-3 singularity resolution discussed in [95] is based on a small gauge transformation, i.e., it neither changes the canonical charges nor the chemical potentials. Our results thus support their conclusions.
The same remarks apply to the spin-3 singularity resolution of the Milne universe [65], which manifestly uses a gauge transformation of the type discussed in appendix D, with state-independent constant gauge parameters v 0 = 0 = v 2 .

Further applications, developments and generalizations
Above we have presented some applications of flat space (spin-3) gravity with chemical potentials. Below we mention several other possible applications and generalizations that we leave for future work.
• Flat space higher spin Cardy formula. The usual Cardy formula [100,101] was generalized in (at least) two ways: 1. by including higher spin fields [102][103][104] and 2. by taking the flat space limit [57,58,71,72]. It seems both natural and interesting to combine these two generalizations and to derive a Cardy-like formula for the entropy of spin-3 flat space cosmologies that (hopefully) matches our result (5.1).
• Flat space family of solutions beyond flat space cosmologies. It could be rewarding to study in detail solutions of the holonomy conditions (5.9) for integers m = 2 and n = 1. The ensuing family of solutions could play an analogous role for flat space (higher spin) gravity as the SL(2, Z) family of Euclidean saddlepoints in AdS spin-2 gravity.
• Flat space spin-3 holographic dictionary. Following our discussion in section 4 it would be interesting to continue the flat space holographic dictionary, in particular by identifying the sources (or non-normalizable modes) for the spin-3 field. To this end we linearize the result (3.13) in µ V and µ U .
For simplicity, we set to zero the spin-2 chemical potentials and charges, as well as the spin-3 charges, and assume that all the spin-3 chemical potentials are constant. The background solutionΦ µνλ dx µ dx ν dx λ is given by the right hand side of (2.29). With these assumptions, the linear piece in the chemical potentials yields By analogy to the discussion after (4.8), we conjecture that the two terms in (5.60) should correspond to the essential pieces in the two towers of non-normalizable solutions to the linearized spin-3 equations of motion.
• isl(N ). Everything we have done in the present work should generalize straightforwardly to higher spin gravity theories based on an isl(N ) connection, with N > 3. In fact, for the principal embedding we expect that all our conclusions remain essentially unchanged. All flat space results should be obtainable from a suitableİnönü-Wigner contraction of corresponding AdS results based on an sl(N ) ⊕ sl(N ) connection. Similar generalizations in AdS were considered in [78,79,105].
• Non-principal embeddings. Whenever the corresponding AdS results are known, again we expect that all flat space results should be obtainable from a suitableİnönü-Wigner contraction. There could be interesting surprises for non-principal embeddings in the flat space limit, however, as the discussion in [49] shows.
• Gravitational anomalies. It is of interest to generalize the result (5.29) for entropy to theories which are obtained as flat spaceİnönü-Wigner contractions from AdS theories with gravitational anomalies so that c L = c−c = 0. In [71] such an expression was found, which correctly reproduces (5.29) (up to a different choice of normalization of L and U). Moreover, it also gives a prediction for the thermal entropy of flat space cosmology solutions in the presence of gravitational anomalies. This result can be obtained using the methods presented in section 5.1 upon replacing the hatted trace with (one quarter of the) trace and the level k with c L /24.
• First order phase transitions in AdS higher spin gravity. Some of our results resemble corresponding AdS results. For instance, the branch that continuously connects to spin-2 gravity also becomes unstable beyond a critical temperature in AdS [77]. Moreover, this temperature agrees quantitatively with our result (5.43), upon replacing our ratio Ω 2 /|Ω U | by their µ −1 . However, the first order phase transitions discovered in section 5.2 do not arise in AdS, despite of the fact that the main ingredient we used was to solve the non-linear holonomy conditions (5.6), (5.7) for the charges in terms of chemical potentials, and these holonomy conditions are identical to the ones in AdS higher spin gravity [29]. It could be interesting to make a scan through all possibilities in AdS higher spin gravity to see if some novel first order phase transitions can arise, and if not, to understand better why AdS and flat space behave so differently in this regard.
• Holographic entanglement entropy. Entanglement entropy of Galilean CFTs, the dual field theories that arise in flat space spin-2 gravity, was derived recently [106]. It would be very interesting to generalize the discussion to the spin-3 case (or even higher spins), both on the field theory and the higher spin gravity sides, along the lines of [107,108] or by suitably contracting the results of [109].
This work was supported by the START project Y 435-N16 of the Austrian Science Fund (FWF) and the FWF projects I 952-N16, I 1030-N27 and P 27182-N27. During the final stages MG was supported by the FWF project P 27396-N27.

A Matrix representations of isl(3) generators
A.1 6 × 6 representation In most of our work we use the following matrix representation of isl(3) generators in terms of 6 × 6 block-diagonal matrices. It is convenient to write them as a 3 × 3 block tensored by a simple diagonal 2 × 2 matrix. The block structure is a remnant of the decomposition of the AdS algebra so(2, 2) ∼ so(2, 1) ⊕ so(2, 1) before theİnönü-Wigner contraction.
Even spin-2 generators: Even spin-3 generators: All odd generators can be written as a product of corresponding even generators times a γ * -matrix, with a Grassmann-parameter, 2 = 0, and Equivalently, one can replace in the formulas (A.1), (A.2) everywhere the factor 1l 2×2 by the diagonal Pauli matrix σ 3 = diag(1, −1) times the Grassmann parameter in order to obtain the odd generators from the corresponding even ones.

A.2 8 + 1 representation
For deriving entropy and holonomy conditions we use the following matrix representation of isl(3) generators in terms of 8 + 1-dimensional matrices with a "tensor"-and a "vector"block. Generic generators G are written in the form where ad 8×8 is an 8 × 8 matrix that is an element of sl(3) in the adjoint representation and odd 8×1 is an 8 × 1 column vector. The even generators L n and U n have ad = O, odd = O; the odd generators M n and V n have ad = O, odd = O. In fact, we can (and will) use the odd generators as unit basis vectors, The ad-parts of the even generators compatible with the algebra (2.4) are then given by the following 8 × 8 matrices.

B Non-constant contributions to spin-2 and spin-3 fields
In this appendix we collect contributions that vanish identically for zero mode solutions with constant chemical potentials, M = N = µ M = µ L = µ V = µ U = 0. We start with expressions for the metric appearing in (3.12a).
We continue with the spin-3 field. The four coefficient-functions in Φ uuu contained in (3.14a) read explicitly with the quadratic part and the constant part

C Flat space cosmologies with spin-3 chemical potential
The general result for spin-2 and spin-3 fields, (3.11)-(3.14) together with the formulas from appendix B, is fairly lengthy. It is therefore useful to consider a simple non-trivial class of configurations for applications. In this appendix we achieve this by studying zero mode solutions with most (but not all) chemical potentials switched off. This analysis provides flat space cosmology solutions with spin-3 hair, which can be considered as the flat space analogue of BTZ black holes with spin-3 hair [31,33].
We consider now zero mode solutions, M = N = V = Z = 0, with vanishing spin-2 chemical potentials, µ M = µ L = 0, and constant spin-3 chemical potentials, µ V = µ U = 0. If we have µ U = 0 then g uu acquires a contribution quadratic in the radial coordinate r, see (3.12a). Since we want to consider solutions that in the spin-2 sector look like flat space cosmologies [75,76] we must not have such a contribution. Therefore, we switch off the even spin-3 chemical potential as well, µ U = 0. In this case entropy (5.29) simplifies to the spin-2 result (5.14).
The metric (3.11) simplifies to g µν dx µ dx ν = M + 24Vµ V + 4 3 M 2 µ 2 V du 2 − 2 dr du + L + 8Uµ V 2 du dϕ + r 2 dϕ 2 (C.1) and the spin-3 field (3.13) simplifies to 2) The metric thus receives a contribution from the spin-3 charges V and U, by contrast to what happens in the absence of a spin-3 chemical potential [41,42]. Switching on a spin-3 chemical potential therefore leads to deformed geometries, some of which can be interpreted as flat space cosmologies with spin-3 hair.
More specifically, flat space cosmologies with mass parameter m and angular momentum parameter j, ds 2 = m du 2 − 2 dr du + 2j du dϕ + r 2 dϕ 2 (C. 3) are obtained for the choices Note, however, that these solutions are singular in general, because the holonomy conditions in sections 5.1 require V = 0, which uniquely determines the mass parameter m for regular solutions as

D Chemically odd configurations
If we keep only the odd chemical potentials, µ M = 0 = µ V , and switch off the even ones, µ L = 0 = µ U , then the connection (3.1) simplifies considerably. In particular, the component a u now only contains odd generators. This feature permits us to consider a simple generalization where a ϕ is deformed. Namely, we replace the connection components (3.1) by The additional term a (ν) ϕ commutes with the group element b as defined in (2.10) and with all contributions to a u , since all commutators involve exclusively two odd generators. Moreover, the expression da (ν) vanishes since a (ν) has only a ϕ-component and all the functions therein depend on ϕ only. Therefore, the additional term (D.2) does not contribute to gauge curvature and the full connection (2.9)-(2.11) with (D.1), (2.13), (3.2), (3.3) and (D.2) solves the Chern-Simons field equations (2.14).
The asymptotic behavior of the metric and spin-3 field remain essentially unchanged, so that it may be tempting to consider these generalized flat space solutions of the equations of motion as legitimate field configurations. However, as we now show the canonical charges are in general not well-defined, unless there are some further restrictions on the functions m n and v n in (D.2).
The boundary condition preserving gauge transformations do acquire additional terms, ε (0) + ∆ε (0) , as compared to (2.18). where δQ[ε] is the previous contribution (2.21). The first two new terms, the whole second line and the first term in the last line are not integrable in general, which means that the canonical charges are not well-defined. The simplest way to obtain integrable canonical charges is to demand δm 1 = δv 2 = δv 1 = δv 0 = 0 . The coordinate transformation r + m 0 → r together with the redefinition of the function L + m 0 → L allows to eliminate the function m 0 from the line-element (D.10) and the canonical charges (D.11). A redefinition of the function L − 2m −1 → L eliminates the function m −1 from the line-element (D.10) and the canonical charges (D.11). Therefore, the functions m 0 and m −1 play no physical role. We expect that essentially the same is true for the quantities v −1 and v −2 , replacing diffeomorphisms by spin-3 gauge transformations. Note, however, that there are more complicated ways to obtain integrable charges than demanding (D.8). We do not study this issue exhaustively here, but just provide one non-trivial example. Choosing