It was mentioned in the previous section that the first class constraints φa0 and K¯a generate gauge transformations. In this section I will explicitly show how to construct the canonical charges which generate those gauge transformations by using Castellani’s algorithm [85]. In general one can construct such a gauge generator by G=ε(t)G0+ ˙ε(t)G1, (3.19) withε(t)˙ ≡ dε(t)dt and whereε(t)is an arbitrary function oft. The constraintsG0 and G1 have to fulfill the following relations
G1 =CPFC, (3.20a)
G0+{G1,HT}=CPFC, (3.20b) {G0,HT}=CPFC, (3.20c) whereCPFCdenotes a primary first class constraint. These relations are fulfilled for G0 = ¯KaandG1 =φa0 =πa0.
For the following considerations it will prove to be convenient to work with a smeared generator, which can be obtained by integrating over the spatial surfaceΣ as
G[ε] = Z
Σ
d2xD0εaπa0+εaK¯a. (3.21) One can show by a straightforward but tedious calculation that this smeared genera-tor generates the following gauge transformations viaδε•={•,G[ε]}
δεAa0=D0εa, (3.22a)
δεAaµ¯ =Dµ¯εa, (3.22b)
δεπa0=−fabcεbπc0, (3.22c) δεπaµ¯ = k
4πµ¯¯νhab∂ν¯εb−fabcεbπcµ¯, (3.22d) δεφaµ¯ =−fabcεbφcµ¯. (3.22e) The generatorG that has been constructed so far is only a preliminary result. The reason for this is that I am considering a Chern-Simons theory with a boundary which renders the generatorGnon-functionally differentiable.
In order to make this statement more precise I will first perform the full variation of the generator for a field independent gauge parameterεa
δG[ε] = Z
Σ
d2x(δ(D0εaπa0) +εaδK¯a) =
= Z
Σ
d2x
fabcεcπaµδAbµ+Dµεaδπaµ+ k
4πµ¯¯νhab∂µ¯εaδAbν¯−
∂µ¯ k
4πµ¯¯νhabεaδAbν¯+εaδπaµ¯
. (3.23)
The first three terms are regular bulk terms and thus do not spoil functional differen-tiability. The last term on the other hand is a boundary term that spoils functional differentiability. In order to fix this one has to add a suitable boundary term to the gauge generator in such a way that the variation of this additional boundary term cancels exactly the boundary term in (3.23) i.e.
δG[ε] =¯ δG[ε] +δQ[ε], (3.24)
This expression for the variation of the canonical boundary charge δQ[ε] can be further simplified by first setting the second class constraintsφaµ¯ ≈0strongly equal to zero and thus going to the reduced phase space. One can then use in addition Stoke’s theorem4, which simplifies the variation of the boundary charge even further to Whether or not this expression is functionally integrable or not depends on the specific form of the gauge parameterεaand thus differs from theory to theory. It is, however, important to note that the general expression for the variation of the canonical boundary charge (3.26) is independent of the theory under consideration.
One simple example where (3.26) is functionally integrable is when the gauge parameter is field independent. One then obtains the following canonical boundary charge
As a final step one now has to determine the Dirac brackets of the canonical bound-ary charge Q with itself in order to determine the asymptotic symmetry algebra generated by those charges5, which read in general
{Q[ε],Q[λ]}=Q[σ(ε, λ)] +Z[ε, λ], (3.28) whereZ[ε, λ]denotes possible central terms andσ(ε, λ)is a composite gauge param-eter consisting ofεandλ. At this point it is important to note that the appearance of those central terms is exactly the mechanism which is responsible for having non-trivial physics at the boundary, as these central terms reduce some of the first class constraints to second class constraints. Therefore, these transformations are not
4I assume here that the boundary of the surfaceΣis parametrized byϕ.
5To be more precise the starting point is again the Poisson bracket algebra of the improved canonical charges G¯which reduces to the Dirac bracket algebra of the canonical boundary charges after setting the second class constraints strongly to zero.
proper gauge transformations anymore but rather correspond to global symmetry transformations which change the state of the physical system.
This asymptotic symmetry algebra can of course be determined by brute force evalu-ation of the Dirac brackets and the basic relevalu-ations (3.17). As this is usually a rather tedious calculation I will use the following shortcut when determining asymptotic symmetry algebras in my thesis.
Given two functionsV,W and a canonical boundary chargeQ[ε] = Rdϕ ε(x)V(x) one can use the fact that this charge generates infinitesimal gauge transformations via δε• = {•,Q[ε]}. Thus, knowing for example how W transforms under an in-finitesimal gauge transformation with gauge parameterεone can determine the Dirac bracket6{V(ϕ),W( ¯ϕ)}using
δεW(y) =−{Q[ε],W(y)}=− Z
dϕ ε(x){V(x),W(y)}. (3.29) As an example let me consider the functionL(ϕ) which transforms under gauge transformations with gauge parameter(ϕ)as
δL= k
4π000+ 20L+L0, (3.30) where a prime denotes derivative with respect toϕ. Furthermore assume that the canonical boundary charge is given by
Q[] = Z
dϕ L. (3.31)
The Dirac bracket{L(ϕ),L( ¯ϕ)}then is calculated using δL( ¯ϕ) =−{Q(),L( ¯ϕ)}=−
Z
dϕ (ϕ){L(ϕ),L( ¯ϕ)}. (3.32) Equation (3.32) can be satisfied for
{L(ϕ),L( ¯ϕ)}= k
4πδ000(ϕ−ϕ) +¯ 2L( ¯ϕ)δ0(ϕ−ϕ)¯ − L0( ¯ϕ)δ(ϕ−ϕ)¯ , (3.33) withδ0(ϕ−ϕ) =¯ ∂ϕδ(ϕ−ϕ). This can also be written in terms of¯ δLas
{L(ϕ),L( ¯ϕ)}=−δL( ¯ϕ)
∂nϕ¯( ¯ϕ)=(−1)n∂nϕδ(ϕ−ϕ)¯ . (3.34) The expression (3.34) can be seen as a convenient shortcut that allows one to determine the Dirac bracket algebra of the asymptotic symmetries directly from the transformation behavior of the state dependent fields under infinitesimal gauge transformations.
6Please note that for the sake of compactness I will from now on omit the additional subscript that I used previously to distinguish Poisson and Dirac brackets i.e. from now on{·,·} ≡ {·,·}D.B
whenever I am using the term Dirac bracket.
Example: sl(2, R) ⊕ sl(2, R) Chern-Simons Theory
I want to close this chapter by reviewing asymptotically AdS3 boundary conditions [11] for asl(2,R)⊕sl(2,R)Chern-Simons theory and how to determine the asymp-totic symmetry algebra using the methods previously described in this chapter [86].
In [87] it was shown that the metric ds2 =`2
"
dρ2− 2π k
L(dx+)2+ ¯L(dx−)2− e2ρ+4π2
k2 LLe¯ −2ρ
!
dx+dx−
# , (3.35) is a solution of Einstein’s equations in 3d for any functionsL ≡ L(x+),L ≡¯ L(x¯ −), wherex±= t` ±ϕ,`is the AdS radius andkis given by (2.13).
For constantLandL¯one obtains the BTZ black hole [12] with massM and angular momentumJ via the identification
LBTZ=− 1
4π(M `−J), (3.36a)
L¯BTZ=− 1
4π(M `+J). (3.36b)
Global AdS3is obtained forJ = 0andM `=−k2 which corresponds to LAdS= ¯LAdS= k
8π, (3.37)
while Poincaré patch AdS is obtained for M = J = 0 as well as an additional decompactification of the boundary coordinateϕ.
Solutions with different (and in general non-constant)LandL¯can be related by global symmetry transformations which correspond to finite transformations of the form (2.14) that change the canonical boundary charges. I again want to emphasize the fact that this is purely due to the presence of a boundary in the theory, making it necessary to introduce the boundary canonical charge (3.27). This makes it possible that some of the first class constraints can be reduced to second class constraints.
This in turn changes some of the gauge symmetries in the bulk to global symmetries at the boundary.
In order to describe the metric (3.35) in a Chern-Simons formulation using two Chern-Simons gauge fieldsAandA¯I will first choose the following basis ofsl(2,R) generators
[Ln, Lm] = (n−m)Ln+m, (3.38) withn, m=±1,0whose invariant bilinear form in the fundamental representation is given by (A.2).
In order to simplify the canonical analysis one can also use some of the gauge freedom provided by Chern-Simons theories to fix the radial dependence as
A=b−1ha(x+, x−) + dib, (3.39a) A¯=bh¯a(x+, x−) + dib−1, (3.39b)
withb=b(ρ) =eρL0. One can then verify usinggµν = `22DAµ−A¯µ, Aν −A¯ν
correctly reproduce the line element (3.35).
The next step consists of determining the gauge transformationsε=b−1(0)(x+)b, ε¯=b¯(0)(x−)b−1 which preserve the structure of (3.40). Those transformations are given by where a prime denotes a derivative with respect to the argument of the function it is acting on i.e. f0(x±) =∂x±f(x±).
One can now also determine how the functions L and L¯ transform under those gauge transformations. This transformation behavior is given by
δL=L0+ 2L0+ k
The corresponding variations of the canonical boundary charges can be integrated and read
One can now use (3.34) in order to determine the Dirac bracket algebra7of these canonical boundary charges straightforwardly as
{L(ϕ),L( ¯ϕ)}= k
4πδ000(ϕ−ϕ) +¯ 2L( ¯ϕ)δ0(ϕ−ϕ)¯ − L0( ¯ϕ)δ(ϕ−ϕ)¯ , (3.44a) {L(ϕ),¯ L( ¯¯ ϕ)}= k
4πδ000(ϕ−ϕ) +¯ h2 ¯L( ¯ϕ)δ0(ϕ−ϕ)¯ −L¯0( ¯ϕ)δ(ϕ−ϕ)¯ i. (3.44b) For many purposes it is already sufficient to have the asymptotic symmetries available in the form of Dirac brackets as in (3.44). It is, however, often also useful to go one step further and represent the algebra (3.44) in terms of its Fourier modes. In order to cast the algebra into a possibly more familiar form one first has to suitable decompose the functionsL(ϕ)and L(ϕ)¯ in terms of Fourier modes and quantize
7Dirac brackets of the form {A(t±ϕ), B(¯t±ϕ)}¯ are evaluated att = ¯t, where A and B are some arbitrary functions of their repective argument. Thus, one can also equivalently write {A(t±ϕ), B(¯t±ϕ)}¯
t=¯t={A(ϕ), B( ¯ϕ)}.
the system, i.e. replacing i{·,·} →[·,·]. After doing so one obtains the following asymptotic symmetry algebra given by
[Ln, Lm] =(n−m)Ln+m+ c
12n(n2−1)δn+m,0, (3.45a) [ ¯Ln,L¯m] =(n−m) ¯Ln+m+ ¯c
12n(n2−1)δn+m,0, (3.45b) where
c= ¯c= 6k= 3`
2GN
. (3.46)
This is the famous result Brown and Henneaux obtained in [11] which showed that there are boundary conditions for AdS3that can be chosen in such a way that the asymptotic symmetries are given by two copies of the Virasoro algebra as in (3.45), which in turn gave rise to the idea that the holographic dual of AdS3 is a two dimensional conformal field theory.