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(If you run after two hares, you will catch neither.)二兎を追う者は一兎をも得ず。– Japanese Proverb
C
hemical potentials were first introduced by Josiah Willard Gibbs at the end of the 19thcentury and have played an important role in botch chemistry and physics since then. Mostly known from statistical physics one can also introduce chemical potentialsµin gauge theories by giving the0-component of the gauge connection a vacuum expectation value (see e.g. [158])A0 →A0+µ. (13.1)
Using holography to describe quantum field theories with non-zero (higher-spin) chemical potentials one principally has to think about how to implement them in a gravitational context. In three dimensions, where gravity can be reformulated as a Chern-Simons gauge theory this can be done in a very clear and straightforward way. Chemical potentials were introduced in spin-3 AdS gravity in the past few years, first in the form of new black hole solutions with spin-3 fields by Gutperle and Kraus [159] (see also [135]), next perturbatively in the spin-3 chemical potential [160], then to all orders by Compère, Jottar and Song [161] and independently by Henneaux, Perez, Tempo and Troncoso [162]. A comprehensive recent discussion of higher spin black holes with chemical potentials is provided in [163]. However, the discussion so far was focused mostly on AdS and holographic aspects thereof [32], see [IV,33–35] for reviews.
In this chapter1I will describe how to extend the AdS considerations to flat space. In a similar manner as it is rewarding to study Bañados–Teitelboim–Zanelli (BTZ) black holes [12,60] in AdS3/CFT2it is also rewarding to study flat space cosmologies with higher-spin charges and chemical potentials in order to get a better understanding of flat space holography.
In Chapter12I have reviewed how to describe flat space higher-spin gravity in terms of a Chern-Simons gauge theory, with a special focus on spin-3 gravity. In this chapter
1Since this chapter is based on the publication [VII] for which I collaborated with my co-authors Mirah Gary, Daniel Grumiller and Jan Rosseel several parts of this chapter coincide with the content found in [VII].
I will generalize this discussion to flat space spin-3 gravity with additional chemical potentialsµM,µL,µV,µU, one for each of the spin-2 and spin-3 fields, respectively.
13.1 Adding Chemical Potentials
The starting point is the connection (12.8), where following the procedure of [163] I also assume that the form ofaϕremains unchanged by chemical potentials, in order to maintain the structure of the asymptotic canonical boundary charges.
Thus, I will make a general ansatz forau with some arbitrary coefficients, which are then fixed by solving the equations of motionF = dA+A ∧ A = 0. Associating the coefficients of the highest weight components with the corresponding chemical potentials i.e. αM1→µMM1 one obtains
au =a(0)u +a(µu M)+a(µuL)+a(µu V)+a(µu U), aϕ =a(0)ϕ , (13.2) witha(0)u ,a(0)ϕ being theupart of the connection (12.8) and
a(µuM)=µMM1−µ0MM0+12 µ00M−12MµMM−1+12VµMV−2, (13.3a) a(µuL)=a(µuM)M→L−12NµLM−1+ZµLV−2, (13.3b) a(µuV)=µVV2−µ0VV1+12 µ00V− MµVV0+16 −µ000V +M0µV+52Mµ0VV−1
+ 241 µ0000V −4Mµ00V−72M0µ0V+32M2µV− M00µVV−2−4VµVM−1, (13.3c) a(µu U)=a(µuV)
M→L−8ZµUM−1− NµUV0+ 56Nµ0U+13N0µU
V−1
+ −13Nµ00U− 247N0µ0U−121N00µU+14MNµU
V−2, (13.3d)
where the subscriptM →Ldenotes that in the corresponding quantity all odd gen-erators and chemical potentials are replaced by corresponding even ones,Mn→Ln, Vn→Un,µM→µLandµV→µU, i.e.
a(µu M)
M→L=µLL1−µ0LL0+12 µ00L −12MµLL−1+12VµLU−2, (13.3e) a(µuV)
M→L=µUU2−µ0UU1+12 µ00U− MµUU0+16 −µ000U +M0µU+52Mµ0UU−1
+241 µ0000U −4Mµ00U−72M0µ0U+32M2µU− M00µU
U−2−4VµUL−1. (13.3f) Here, dots (primes) denote derivatives with respect to retarded time u (angular coordinateϕ).
The equations of motion (12.11) impose the additional conditions
M˙ =−2µ000L + 2Mµ0L+M0µL+ 24Vµ0U+ 16V0µU, (13.4a) N˙ = 12M˙
L→M + 2Nµ0L+N0µL+ 24Zµ0U+ 16Z0µU, (13.4b) V˙ = 121 µ00000U −125 Mµ000U −58M0µ00U −38M00µ0U+13M2µ0U
−121M000µU+13MM0µU+ 3Vµ0L+V0µL, (13.4c) Z˙ = 12V˙L→M −125 Nµ000U −58N0µ00U−38N00µ0U+23MNµ0U
−121 N000µU+13(MN)0µU+ 3Zµ0L+Z0µL, (13.4d) with
1
2M˙L→M =−µ000M +Mµ0M+ 12M0µM+ 12Vµ0V+ 8V0µV, (13.4e)
1
2V˙L→M = 241 µ00000V −245 Mµ000V −165 M0µ00V −163 M00µ0V+16M2µ0V
−241 M000µV+16MM0µV+ 32Vµ0M+12V0µM. (13.4f) The chemical potentialsµM,µL,µVandµUcan be in principal arbitrary functions of the angular coordinateϕand the retarded timeu. In many applications, however, they are constant, which simplifies most of the formulas considerably.
13.2 Consistency Checks
After having added the chemical potentials I will perform some consistency checks before proceeding.
In the absence of chemical potentials, µM = µL = µV = µU = 0 one should recover the results from Chapter12. This is indeed true. In particular, the on-shell conditions (13.4) simplify to (12.11).
In the presence of chemical potentials the on-shell conditions (13.4) should contain information about the asymptotic symmetry algebra (10.30). For example, theµL-terms in (13.4a) are an infinitesimal Schwarzian derivative, while theµU-terms exhibit transformation behavior of a spin-3 field.
Since any solution to the field equationsF = 0 must be locally pure gauge, and any solution that obeys the boundary conditions (12.8) can be generated by the boundary condition preserving gauge transformations (12.12), it should be possible to obtain (13.3) directly from a gauge transformation. Indeed, comparing the expressions for (12.12) with (13.3) one can see that they coincide upon identifying→µL,τ →µM,κ→µVandχ→µU.
It is possible to derive the results of section13.1also in a different way. One could start from equation (3.7)-(3.12) in [163] and use the Grassmann-approach of [70]
to derive the flat space connection with chemical potentials along similar lines as in Section10.1, dropping in the end all terms quadratic in the Grassmann-parameter . The map that leads from (3.7)-(3.12) in [163] (left hand side) to the results presented in section13.1(right hand side) is given by
Coordinates: x±= u±ϕ, (13.5a)
Connection 1-form: 2a±(x+, x−) = 1au(u, ϕ)±aϕ(u, ϕ), (13.5b)
Spin-2 Generators: 2L±n =Ln±1Mn, (13.5c)
Spin-3 Generators: 2Wn±=Un±1Vn, (13.5d)
Spin-2 Fields: 24
c±
L±(x±) =M(u, ϕ)±2N(u, ϕ), (13.5e)
Spin-3 Fields: − 3
c±
W±(x±) =V(u, ϕ)±2Z(u, ϕ), (13.5f) Spin-2 Chemical Potentials: 14ξ±(x+, x−) = 1 +µM(u, ϕ)±1µL(u, ϕ), (13.5g) Spin-3 Chemical Potentials: 14η±(x+, x−) =µV(u, ϕ)±1µU(u, ϕ). (13.5h) As expected, this procedure leads to the same results as displayed above in Sec-tion13.1.
13.3 Canonical Charges and Chemical Potentials
Since I have not changedaϕ the results for the canonical charges remain unchanged and all expressions displayed in Chapter12are also valid for non-vanishingµM,µL, µVandµU. In particular, from (12.15) one can read off the following four zero-mode charges
QM= k
2M, QL=kL, QV = 4kV, QU = 8kU, (13.6) which can be interpreted as mass, angular momentum, odd and even spin-3 charges, respectively. These zero-mode charges play a prominent role in the variational principle and the calculation of entropy of flat space cosmologies with higher-spin hair. But before proceeding in calculating physical observables one has to check whether or not introducing chemical potentials spoils the variational principle of the Chern-Simons action.
As shown in Chapter2, varying the Chern-Simons action (2.3) in general yields a boundary term of the form 4πk RhA ∧δAi. Evaluating this term explicitly for the connection (12.8) including chemical potentials as in (13.2) one finds
hAϕδAu−AuδAϕi ' MδµM+ 2NδµL+ 12VδµV+ 24ZδµU+ 4µVδV+ 8µUδZ. (13.7)
For vanishing spin-3 potentials this means that the bulk Chern-Simons action has a well defined variational principle. For non-vanishing spin-3 chemical potentials, however, the last two terms are incompatible with a well defined variational principle.
This can be fixed by adding an appropriate boundary term by hand. In this case this term has the form
Γ[A] =SCS[A]−SB[A], with SB[A] = k 4π
Z
dudϕhA¯uAϕi, (13.8) where
¯au =au−2(1 +µM)M1−2µLL1−2µVV2−2µUU2. (13.9) Thus, after adding this term one obtains
δΓEOM= k 4π
Z
dudϕ hAϕδAu−AuδAϕi −δhA¯uAϕi
= Z
du QMδµM+QN δµL+QVδµV+QZδµU
. (13.10)
This small modification is already sufficient for the action (13.8) to have 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 re-sponse functions (13.10) are determined by the canonical charges, and the chemical potentials act as sources.