CFT(4) Partition Functions and the Heat Kernel on AdS(5)

0.17 MB·English

by  Shailesh Lal

ICTS/2012/12
PreprinttypesetinJHEPstyle-HYPERVERSION
CFT(4) Partition Functions and the Heat
3
1
Kernel on AdS(5)
0
2

n
a
J

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1
Shailesh Lal∗
]
h
International Center for Theoretical Sciences – TIFR,
t
-
p TIFR Centre Building, Indian Institute of Science,
e
Bangalore, India 560012.
h
[

2
v
0 Abstract: We explicitly reorganise the partition function of an arbitrary CFT in
5
0 four spacetime dimensions into a heat kernel form for the dual string spectrum on
1
AdS(5). Onverygeneralgrounds, theheatkernel answer canbeexpressed intermsof
.
2
a convolution ofthe one-particlepartitionfunction ofthe four-dimensional CFT. Our
1
2 methods are general and would apply for arbitrary dimensions, which we comment
1
: on.
v
i
X
Keywords: String theory, AdS/CFT correspondence, Partition Functions, Heat
r
a Kernel methods.
∗
shailesh DOT lal AT icts DOT res DOT in
Contents
1. Introduction 1
2. The Heat Kernel for AdS(5): A Review 2
3. The Heat Kernel for Mixed Symmetry Fields 5
3.1 The Heat Kernel for Massive Fields 6
3.2 The Heat Kernel for Mixed Symmetry Massless Fields 7
4. From the CFT Partition Function to the AdS Heat Kernel 7
5. Conclusions 10
A. Incorporating Short Multiplets 10
B. A Consistency Check 12
1. Introduction
Heat kernel methods have of late played animportant role inextracting out quantum
effects in gravitational physics. Such applications include the extraction of leading
quantum corrections to black hole entropy [1]–[4], the asymptotic symmetries of
gravitational theories [5]–[9], precision tests of AdS/CFT [10], as well as the many
more applications extensively reviewed in [11]. This is essentially because the heat
kernel method powerfully captures the leading quantum properties of a given theory.
In many cases (especially quantum gravity) while the full quantum theory is poorly
understood, these leading properties are potentially tractable.
In this paper, we shall briefly describe some progress in bringing heat kernel
methods to bear on another significant arena, the string theory sigma model on AdS.
As is well known, this sigma model is presently quite intractable at the quantum
level. In this light, one potential starting point to gain a foothold on the sigma
model could be to use the AdS/CFT correspondence [12, 13, 14]. In particular, to
take the spectrum of the CFT dual to the AdS string and to reproduce the planar
– 1 –
CFT partition function in terms of quadratic fluctuations of the dual fields in AdS1.
Typically, these quadratic fluctuations would arrange themselves into determinants
of the Laplacian acting over fields of varying spin. This would essentially provide
us with a first-quantised description of the particles that form the string spectrum.
More ambitiously, one could attempt to reconstruct the full vacuum amplitude (the
torus string amplitude with no vertex operator insertions) in AdS by interpreting
the heat kernel proper time in terms of the modulus of the torus worldsheet [24]2.
In this paper, we shall perform the first of these two tasks. We shall organise
the CFT partition function in terms of quadratic fluctuations of the particles that
constitute the string spectrum on AdS. We shall do this in the specific instance of the
AdS /CFT duality. However, we make no assumptions about the matter content of
5 4
the CFT or about supersymmetry.
A brief overview of this paper is as follows. In Section 2 we will begin with a
brief review of the heat kernel methods of [30] and their subsequent applications in
[9]. We shall then present our ansatz for the extension of these results to (massive
or massless) fields of mixed symmetry in Section 3. Finally, in Section 4, we use
the above results to interpret the given CFT partition function in terms of the heat
kernel ofthedual AdS(5)spectrum. The equation(4.12)obtainedthereisthecentral
result of this paper. Subtleties that occur due to the appearance of short multiplets
in the CFT operator spectrum are relegated to the Appendix A as they do not affect
the final results.
2. The Heat Kernel for AdS(5): A Review
The essential ingredient that we employ to compute the one-loop partition function
in the bulk is its relation to the determinant of the Laplacian, which in turn may be
evaluated from the (traced, coincident) heat kernel. In this section, we shall review
the results of [30] for the heat kernel of the Laplacian acting over tensor fields on
1For a free planar CFT defined on the compact space S3 S1 this partition function is very
⊗
explicitly known by counting the spectrum of gauge invariant operators [15, 16], see also [17] and
the related papers [18]–[23].
2There are good reasons to expect this approach to be a fruitful one. In particular, it has
been proposed that the general relation between the heat kernel proper time and the closed string
moduli is closely connected to the phenomenon of gauge-string duality [25]–[27], see also [28] for
an interesting applicationof this approach. Encouragingly,this overallapproachbasedonthe heat
kernel method has also previously been successful for string theory in flat space [29]. We thank
Rajesh Gopakumar for helpful discussions and correspondence about these points.
– 2 –
AdS. In summary, for a spin-S particle moving on a spacetime manifold ,
M
∞ dt ∞ dt
ln (S) lndet 2 = Trln 2 = Tret∇2(S) K(S)(t),
Z ≃ −∇(S) −∇(S) − t ≡ − t
Z0 Z0
(cid:0) (cid:1) (cid:0) (cid:1) (2.1)
where the trace of the Laplacian is taken over both the spin and the spacetime in-
dices. Now for an arbitrary curved manifold , such expressions are prohibitively
M
hard to compute, but symmetric spaces such as spheres and Euclidean AdS (i.e.
hyperboloids) admit important simplifications due to an underlying group theoretic
structure–theycanberealisedascosetsofLiegroups(seeforexample[31]–[35],[24]).
The determinants of the Laplacian were explicitly evaluated for the symmetric,
transverse-traceless (STT) fields in euclidean AdS in [30] by the heat kernel method
(see [24] for a more detailed exploration of AdS ). We shall briefly recollect the main
3
results, specialising to the case of AdS .
5
Firstly, euclideanAdS isthesymmetricspaceSO(5,1)/SO(5). Thespinofafield
5
over AdS is given by the unitary irreducible representation (UIR) of the isotropy
5
group SO(5) that is carried by the field. UIRs of SO(5) are labelled by the array
S = (s ,s ) s.t. s s 0. (2.2)
1 2 1 2
≥ ≥
We shall also need UIRs of SO(5,1), which are labelled by the array
R = (iλ,m ,m ), m m . (2.3)
1 2 1 2
≥ | |
where R contains S if [36, 35]
s m s m . (2.4)
1 1 2 2
≥ ≥ ≥ | |
The m’s ands’s must all simultaneously be integers or half-integers. Further, them’s
defineanSO(4)UIRm~ = (m ,m ), anditsconjugaterepresentationm~ˇ = (m , m ).
1 2 1 2
−
With these ingredients, the heat kernel of a spin-S particle on (a quotient of) AdS
5
is given by
β ∞
K(S)(γ,t) = dλχλ,m~ γk etER(S), (2.5)
2π
k∈Z m~ Z0
XX (cid:0) (cid:1)
where χ is the Harish-Chandra character in the principal series of SO(5,1), which
λ,m~
has been evaluated [37] to be
e−iβλχSO(4)(φ ,φ )+eiβλχSO(4)(φ ,φ )
χ (β,φ ,φ ) = m~ 1 2 m~ˇ 1 2 , (2.6)
λ,m~ 1 2 e−2β 2 eβ eiφi 2
i=1| − |
Q
– 3 –
the eigenvalue of the Laplacian E(S) is a function of λ
R
E(S) = λ2 +C (S) C (m~ )+22 , (2.7)
R − 2 − 2
(cid:0) (cid:1)
and γ denotes the quotient of AdS which corresponds to turning on a temperature
5
β along with angular momentum chemical potentials φ ,φ along the SO(4) Cartans.
1 2
C denotes the quadratic Casimir of the appropriate SO group. The sum over m~ is
2
the sum over all values of m admitted by the branching rules (2.4). We refer the
reader to [30] for more details and explicit expressions.
There are many simplifications for the case of STT tensors. Firstly, as the tensor
is completely symmetric, S = (s,0), where s is the rank of the tensor. Secondly, the
branching rules (2.4) simplify to give
m = s, m = 0. (2.8)
1 2
The partition function of a massless spin-s particle was evaluated with these inputs
in [9]. In particular, it was found that the expression for the partition function as a
result ofevaluating thepathintegral atonelooparrangeditself interms oftransverse
traceless tensors, and finally (see Eq. (2.29) of [9])
∞ 1 e−mβ(s+2)
log (s) =
Z m 1 e−m(β−iφ1) 2 1 e−m(β−iφ2) 2×
m=1 | − | | − | (2.9)
X
χs (mα1)χs (mα2) χs−1 (mα1)χs−1 (mα2)e−mβ ,
× 2 2 − 2 2
h i
where we have expressed the SO(4) character χSO(4)(mφ ,mφ ) as a product of
(s,0) 1 2
SU(2) characters χs (mα1)χs (mα2), where α1 = φ1 +φ2, and α2 = φ1 φ2.
2 2 −
The reader will recognise (2.9) as the expression for the multiparticle partition
function in terms of the one-particle partition function , where is the SO(4,2)
Y Y
character evaluated over the short representation [s + 2, s, s] [38, 39]. Using the
2 2
AdS/CFT correspondence [12, 13, 14], if a CFT partition function contains a char-
acter of the representation [s+2, s, s], there must be bulk degrees of freedom giving
2 2
rise to one-loop determinants over STT fields, as reviewed above. For example, given
a long primary [∆, s, s] in the CFT, one can infer the presence of quadratic fluctua-
2 2
tions in AdS giving rise to a one-loop determinant of the operator 2+m2, where
−∇
m2 = (∆ 2)2 s 4. The corresponding heat kernel is given by
− − −
K[∆,2s,2s](γ,t) = β ∞ χs2 (kα1)χ2s (kα2) e−t(∆−2)2e−k24βt2
√πt e−2kβ ekβ eik(α1+α2) 2 ekβ eik(α2−α1) 2
k=1 2 2
X | − | | − |
(2.10)
– 4 –
for the dual bulk fluctuations. It may be verified by doing the t integral as in [30]
that this gives rise to the expected partition function. This forms the basis of the
analysis of Section 3.
3. The Heat Kernel for Mixed Symmetry Fields
In this section, we will compute the heat kernel for the AdS degrees of freedom
5
that correspond to primaries of mixed symmetry in the CFT. These correspond to
representations S of SO(5) where s = 0, i.e.
2
6
S = (s ,s ) s.t. s s > 0. (3.1)
1 2 1 2
≥
Representations of SO(5,1) R that contain S are determined by the branching rules
(2.4). The main ingredient of this calculation will be the tensors for which some of
the inequalities in the branching rules (2.4) get saturated. In particular, that
m = s , m = s . (3.2)
1 1 2 2
| |
We therefore have
R = (iλ,s , s ). (3.3)
1 2
±
The eigenvalues of the Laplacian, for such fields (which saturate the branching rules
as above) transforming in S = (s ,s ), are now given by
1 2
ES = λ2 +s +s +4 , R = (iλ,s , s ). (3.4)
R − 1 2 1 ± 2
(cid:0) (cid:1)
In what follows, we shall focus exclusively on such tensors in AdS. Using these
expressions, we will now obtain a formula for the heat kernel for the Laplacian in
AdS acting over such fields of mixed symmetry. This, by the above considerations,
is given by
β ∞
K(S)(γ,t) = 2π dλ χλ,(s1,s2) γk +χλ,(s1,−s2) γk eiERS, (3.5)
k∈ZZ0
X (cid:2) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3)
where we have used the degeneracy of eigenvalues (3.4) while carrying out the sum
over mˆ above. The sum
2cosβλ χSO(4) (φ ,φ )+χSO(4) (φ ,φ )
χ +χ (γ) = (s1,s2) 1 2 (s1,−s2) 1 2 . (3.6)
λ,(s1,s2) λ,(s1,−s2) he−2β eβ eiφ1 2 eβ eiφ2 2 i
| − | | − |
(cid:2) (cid:3)
Again, for later analysis it is more efficient to express the answer in terms of the
SU(2) SU(2) characters rather than the SO(4) ones. The precise dictionary is
⊗
χSO(4) (φ ,φ ) = χ (α )χ (α ), (3.7)
(s1,s2) 1 2 j1 1 j2 2
– 5 –
where
s +s
j = 1 2, α = φ φ ,
1 1 1 2
2 −
(3.8)
s s
j = 1 − 2, α = φ +φ .
2 2 1 2
2
With these ingredients, the heat kernel corresponding to a TT tensor field of spin
S = (s ,s ) (j ,j ) is given by
1 2 1 2
≡
β ∞ 2coskβλ(χ χ +χ χ )(kα ,kα )
K(j1,j2)(γ,t) = dλ j1 j2 j2 j1 1 2 e−t(λ2+2j1+4),
2π k∈ZZ0 e−2kβ ekβ eik(α1+2α2) 2 ekβ eik(α2−2α1) 2
X | − | | − |
(3.9)
Note that we cannot naively apply this expression to symmetric tensor fields by set-
ting j = j . Thisisbecause we have separately counted representations (iλ,s , s ).
2 1 1 2
±
If s = 0, or equivalently j = j , these representations would be the same and we
2 1 2
would count them only once. We will next apply these results to the case of massive
and massless fields, but before doing so, let us perform the λ integral to get an ex-
pression for the heat kernel purely in terms of the chemical potentials and the heat
kernel proper time. We use the identity
∞2coskβλe−tλ2dλ = πe−k24βt2 (3.10)
t
Z0 r
to write the heat kernel of 2 as
−∇
∞
K(j1,j2)(γ,t) = β (χj1χj2 +χj2χj1)(kα1,kα2) e−t(2j1+4)e−k24βt2,
√πt e−2kβ ekβ eik(α1+α2) 2 ekβ eik(α2−α1) 2
k=1 2 2
X | − | | − |
(3.11)
where
χ χ (kα ,kα ) χ (kα )χ (kα ). (3.12)
j1 j2 1 2 ≡ j1 1 j2 2
In the remainder of this section, we shall use (3.11) as a building block for the bulk
contributions that correspond to mixed symmetry primaries in the boundary CFT.
3.1 The Heat Kernel for Massive Fields
Consider now a long SO(4,2) representation of highest weight [∆,j ,j ] [∆,j ,j ].
1 2 1 2
⊕
This primary is dual to a massive field in the bulk. The corresponding heat kernel
is given by
∞
K[∆,j1,j2](γ,t) = β (χj1χj2 +χj2χj1)(kα1,kα2) e−t(∆−2)2e−k24βt2.
√πt e−2kβ ekβ eik(α1+α2) 2 ekβ eik(α2−α1) 2
k=1 2 2
X | − | | − |
(3.13)
– 6 –
This corresponds to the heat kernel ofthe operator 2+m2 evaluated onthetensor
−∇
fields (3.2), where m2 +2j +4 = (∆ 2)2. Then
1
−
1 1 ∞ dt
log = logdet 2 +m2 = K[∆,j1,j2](γ,t). (3.14)
Z −2 −∇ 2 t
Z0
(cid:0) (cid:1)
We carry out the t integral using equation 6.1 of [30]
1 ∞ dte−α4t2−β2t = 1e−αβ. (3.15)
2√π t3 α
Z0 2
We finally find that
∞
(χ χ +χ χ )(kα ,kα )
log = j1 j2 j2 j1 1 2 e−kβ∆. (3.16)
Z 1 e−kβeik(α1+α2) 2 1 e−kβeik(α2−α1) 2
k=1 2 2
X | − | | − |
To summarise, we have
1 ∞ dt
log = K[∆,j1,j2]. (3.17)
Z[∆,j1,j2]⊕[∆,j2,j1] 2 t
Z0
3.2 The Heat Kernel for Mixed Symmetry Massless Fields
Massless fields belong to short representations of the conformal group. The partition
function that corresponds to these fields is the character of a short representation
with highest weight [j +j +2,j ,j ], i.e.
1 2 1 2
∞
1
log = χ χ (kβ,kα ,kα ),
Z[j1+j2+2,j1,j2] k [j1+j2+2,j1,j2] − [j1+j2+3,j1−21,j2−21] 1 2
Xk=1 (cid:16) (cid:17)
(3.18)
where the characters on the right-hand side are the characters over long representa-
tions of SO(4,2). We can therefore write
1 ∞ dt
logZ[j1+j2+2,j1,j2] = 2 t K[j1+j2+2,j1,j2] −K[j1+j2+3,j1−21,j2−21] (β,α1,α2),
Z0 (cid:16) (cid:17) (3.19)
in the notation of (3.13).
4. From the CFT Partition Function to the AdS Heat Kernel
We now organise the full CFT partition function into the form of a heat kernel in
AdS for a theory which has only long representations in its spectrum. Remarkably,
5
however it turns out that the final answer (4.12) thus obtained is unchanged when
short multiplets [40, 41] are included. This is demonstrated in Appendix A.
– 7 –
Suppose the CFT has operators with quantum numbers [∆,j ,j ] appearing
1 2
N times. The one-particle partition function is a sum of SO(4,2) characters
[∆,j1,j2]
evaluated over the modules generated by these primaries.
q∆χ (a)χ (b)
(q,a,b) = N j1 j2 , (4.1)
Y [∆,j1,j2] 4 (1 qx )
∆X,j1,j2 i=1 − i
Q
The (multi-particle) partition function of the theory is then obtained by exponenti-
ating the one-particle partition function.
∞
1
log = N χ qk,ak,bk . (4.2)
Z k [∆,j1,j2] [∆,j1,j2]
Xk=1 ∆X,j1,j2 (cid:0) (cid:1)
where we have introduced notation
q = e−β, a = eiα1, b = eiα2. (4.3)
Shortly we will also define
x = √ab,x = a¯b,x = √a¯b,x = a¯¯b. (4.4)
1 2 3 4
p p
In what follows, it is useful to treat symmetric and mixed-symmetric tensors on a
different footing, i.e. sum up the j = j and j = j contributions separately. We
1 2 1 2
6
then have
∞ ∞
1 1 1 ′
log = N χ + N χ +χ . (4.5)
Z k [∆,j,j] [∆,j,j] k ·2 [∆,j1,j2] [∆,j1,j2] [∆,j2,j1]
Xk=1 X∆,j Xk=1 ∆X,j1,j2 (cid:0) (cid:1)
The prime over the second sum reminds us that in this sum, j = j . The factor
1 2
6
of half in the second term is from the fact that this sum counts each (j ,j ) pair
1 2
twice. The dependence on (qk,ak,bk) is implicit. We have imposed the condition
that N = N to club terms together in the second sum.
[∆,j1,j2] [∆,j2,j1]
We will now reinterpret, as per (3.13), each Verma module character above as
arising from a heat kernel in AdS . Using (3.17), we have
5
1 ∞ dt 1 ′
log = N K[∆,j,j]+ N K[∆,j1,j2] . (4.6)
Z 2 t [∆,j,j] 2 [∆,j1,j2]
Z0 X∆,j ∆X,j1,j2 !
We will now evaluate the sums over ∆,j ,j . To do so, the following identity is useful
1 2
e−t(∆−2)2 = 1 ∞ dye−y42t+iy(∆−2). (4.7)
4πt
r Z−∞
– 8 –
This follows from evaluating the Gaussian integral on the right-hand side. The heat
kernel formulae in our new notations are
K[∆,j,j](γ,t) = ∞ β ∞ dye−y2+4kt2β2e−2iy q2kχj(ak)χj(bk) ei∆y
2πt 4 1 qkxk
Xk=1 Z−∞ i=1 − i
K[∆,j1,j2](γ,t) = ∞ β ∞ dye−y2+4kt2β2e−2iyQq2k χ(cid:0)j1(ak)χj2((cid:1)bk)+χj2(ak)χj1(bk) ei∆y.
2πt 4 1 qkxk
Xk=1 Z−∞ (cid:0) i=1 − i (cid:1)
(4.8)
Q (cid:0) (cid:1)
We will now use these expressions to evaluate (4.6). As is apparent, most of (4.8)
does not depend on ∆,j ,j and factors out of the sum. The sum that we essentially
1 2
have to evaluate is
1 ′
N ei∆yχ (ak)χ (bk)+ N ei∆y χ (ak)χ (bk)+χ (ak)χ (bk)
[∆,j,j] j j 2 [∆,j1,j2] j1 j2 j2 j1
X∆,j ∆X,j1,j2 (cid:0) (cid:1)
= N ei∆yχ (ak)χ (bk).
[∆,j1,j2] j1 j2
∆X,j1,j2
(4.9)
We can now use the definition (4.1) (replacing q by eiy) to write this as
4
eiy,ak,bk 1 eiyxk . (4.10)
Y − i
i=1
(cid:0) (cid:1)Y(cid:0) (cid:1)
We therefore find that
log ∞ β ∞ dye−y2+4kt2β2e−2iyq2k eiy,ak,bk 4 1−eiyxki . (4.11)
Z ≃ 2πt Y 1 qkxk
k=1 Z−∞ i=1 (cid:0) − i (cid:1)
X (cid:0) (cid:1)Y
Including the integral over t explicitly, we find that (cid:0) (cid:1)
log = ∞ ∞dt β ∞ dye−y2+4kt2β2e−2iyq2k eiy,ak,bk 4 1−eiyxki .
Z 4πt2 Y 1 qkxk
k=1Z0 Z−∞ i=1 (cid:0) − i (cid:1)
X (cid:0) (cid:1)Y (4.12)
(cid:0) (cid:1)
This is an expression for the multi-particle partition function of the AdS theory in
5
terms of the heat kernel time t and the single-particle partition function of its dual
CFT. We remind the reader that the single-particle partition function has been
Y
very explicitly computed for a free, planar CFT by enumerating the gauge invariant
operators in the CFT spectrum [15, 16].3
3
Taking the free limit in the CFT side corresponds to the limit in which the string theory on
AdS becomes tensionless. This is of course a very non-trivial limit of string theory about which
much remains to be understood. However, from the AdS/CFT correspondence, the spectrum of
string theory should still match with the spectrum of conformal primaries of the CFT, and our
analysiswouldstillbe valid. Atgenericvaluesofthe coupling,the spectrumofstringtheoryisalso
implicitly known through TBA, see [42] for a review. We thank Arkady Tseytlin for discussions
regarding these points.
– 9 –