Structures on the Conformal Manifold in Six Dimensional Theories

0.32 MB·English

by  Hugh Osborn

DAMTP 2015/1
Structures on the Conformal Manifold
in Six Dimensional Theories
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Hugh Osborna and Andreas Stergioub
v
o
N
aDepartment of Applied Mathematics and Theoretical Physics, Wilberforce Road,

2
Cambridge CB3 0WA, England
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bDepartment of Physics, Yale University, New Haven, CT 06520 USA
]
h
t
-
p
e
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[ The tensors which may be defined on the conformal manifold for six dimensional CFTs with

3 exactly marginal operators are analysed by considering the response to a Weyl rescaling of
v
the metric in the presence of local couplings. It is shown that there are three symmetric two
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0
index tensors only one of which satisfies any positivity conditions. The general results are
3
1 specialised to the six dimensional conformal theory defined by free two-forms and also to the
0
. interacting scalar φ3 theory at two loops which preserves conformal invariance to this order.
1
0 All three two index tensor contributions are present.
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1
:
v
i
X
r
a
January 2015
[email protected], [email protected]
1. Introduction
If conformal field theories have exactly marginal operators there is a conformal manifold pa-
rameterised by the couplings for the marginal operators. In two and four dimensions CFTs
with associated conformal manifolds are not uncommon, at least with = 1 supersymme-
N
try [1]. The situation is much less clear in higher dimensions; whether any non trivial CFTs
with marginal operators exist in six dimensions remains doubtful but not inconceivable [2].
Here we aim to extend some results obtained in two and four dimensions to the significantly
more complicated case of six.
To this end we consider the response of a CFT extended to a curved space background
to a Weyl rescaling of the metric γ . In general, CFTs are invariant under Weyl rescalings
µν
of the background metric, γ e2σγ , up to a finite sum of local contributions formed
µν µν
→
from curvature tensors and σ, with coefficients commonly referred to as central charges. In
two dimensions there is just the Virasoro central charge c, so that the trace of the energy
momentum tensor is proportional to cR, with R the scalar curvature which is equal to the
two dimensional Euler density E . In four dimensions there are just two coefficients c,a, which
2
are related to the square of the Weyl tensor and the four dimensional Euler density E . These
4
results for CFTs on curved backgrounds may be used to construct effective field theories for
a dilaton τ, with terms O(τ2) in two dimensions, and O(τ3,τ4) in four dimensions, which
survive on reduction to flat space and are proportional to c,a respectively. By considering
dilaton scattering in four dimensions the crucial positivity constraints allowing arguments for
an irreversible RG flow between UV and IR fixed points have been obtained [3].
For CFTs with a conformal manifold it is convenient to allow the couplings gI for the
marginal operators to be local or x-dependent. The couplings can then be treated as sources
for the marginal operators. In that case there are additional local contributions under a Weyl
rescaling depending on derivatives of gI. Such terms are restricted by power counting. In two
dimensions this procedure generates a unique two index tensor g on the conformal manifold,
IJ
while in four dimensions a four index tensor is present also. In two dimensions g is identical
IJ
with the metric defined by Zamolodchikov [4] in terms of the two point functions for the scalar
operators coupled to gI and which for unitary theories is necessarily positive. A similar result
applies in the four dimensional case so the corresponding metric is again positive.
Away from a conformal critical point the response to Weyl rescalings with local couplings
must satisfy Wess–Zumino consistency conditions stemming from the fact that the Weyl group
is Abelian. The resulting equations relate the RG flow of the central charge c in two and a
in four dimensions to the corresponding g . For positive g the RG flow is irreversible
IJ IJ
[5–7]. In two dimensions this approach is equivalent to the Zamolodchikov c-theorem. In four
dimensions the metric is necessarily positive in the neighbourhood of a fixed point, but unlike
two dimensions there is no simple general non perturbative argument, although arguments
based on dilaton effective actions can be applied [7]. For renormalisable quantum field theories
1
in four dimensions the metric and related quantities may be calculated perturbatively in terms
of the vacuum amplitude, most directly with a curved space background and using local
couplings at two loops [8], but also just restricting to flat space at three loops [6,9].
It is natural to consider extensions to higher dimensions, in particular six. The dilaton
effective action was constructed in [10] and also [12]. The local RG approach was also extended
to six dimensions in [13]. The number of contributions which it is necessary to consider
increases significantly; in the approach followed in [13] there are O(100) different consistency
conditions to be analysed. Due to complications arising from the analytic structure of 3 3
→
amplitudes there is no derivation of irreversibility of RG flow along the same lines as that
applied in four dimensions [10], and recently a two loop calculation in six dimensional φ3
theory showed that the metric relevant for RG flow was not positive in this theory [14].
In this paper we endeavour to understand further the complications arising in six dimen-
sions by considering a six dimensional conformal field theory with exactly marginal operators.
The approach followed here, based on assuming local couplings for all marginal operators and
considering the response to Weyl rescalings of the metric, defines various tensors on any confor-
mal manifold. An infinitesimal Weyl rescaling determines the trace of the energy momentum
tensor. As is well known, in six dimensions on a curved background with fixed couplings and
neglecting scheme dependent contributions, this is expressible in terms of three scale dimen-
sion six Weyl invariants, with coefficients c ,c ,c , and the topological Euler density E , with
1 2 3 6
coefficient a [15,16]. Thus c ,c ,c ,a may be regarded as the central charges in six dimen-
1 2 3
sions, corresponding to the two dimensional c and four dimensional c,a. With local couplings
to marginal operators it is further possible to obtain three rank two symmetric tensors, as
well as rank four and rank six tensors. One rank two symmetric tensor can be related to
the two point function for marginal operators and is therefore positive. This may then be
taken as a metric for the conformal manifold. However, contrary to the case in two and four
dimensions, this is not the tensor that features in the equation for the RG flow of a. The
additional symmetric tensors present in six dimensions are constructed in terms of the Weyl
tensor and so are absent in any conformally flat space.
In the next section we review the response of a CFT containing exactly marginal operators
in four dimensions and then consider the extension to six. In six dimensions it is necessary to
consider Weyl transformations which are rather more involved than in four. Besides the Weyl
tensor the results can be expressed more simply in terms a basis involving the Cotton and
Bach tensors [19]. Their definitions and some basic properties are reviewed in appendix A. It
is also necessary to consider various conformally covariant differential operators which extend
the conformal Laplacian ∆ = 2 +ξR, where ξ = (d 2)/4(d 1) with d the spacetime
2
−∇ − −
dimension. In four dimensions the results involve ∆ , the conformal extension of ( 2)2, while
4
∇
in six dimensions it is necessary to consider the Branson operator ∆ [39] whose leading term
6
is ( 2)3.
− ∇
2
As an illustration of these results we consider in section 3 the conformal theory in six
dimensions which is obtained from the quantum field theory of free two-forms. In this case we
may introduce a local coupling in the action as 1/g2 which acts as a source for the dimension
six scalar operator formed by the gauge invariant classical Lagrangian density. After suitable
gauge fixing we determine the one loop anomalous contributions under a Weyl rescaling of the
metric, extending the results in [17] to include contributions involving derivatives of g. The
results fit the general structure determined in section 2.
In section 4 results obtained from calculations at two loops for φ3 theory on a curved
background with local couplings are also presented. This theory has non zero β-functions
and conformal invariance is broken but perturbative calculations should satisfy the constraints
obtained in section 2 to lowest order. We also present results for the central charges c ,c ,c ,a
1 2 3
to O(g2). To ensure that the results are compatible with the general analysis it is necessary
to ensure when using dimensional regularisation that the one loop counterterms are such as to
ensure the initial free theory is conformal away from d= 6. Although φ3 theory is problematic,
since it lacks a minimum energy ground state, we assume it may be stabilised by a small φ4
term and that it may then still be used to define an effective conformal theory, at least to
leading order.
We also consider in section 5 some positivity conditions which are obtained by relations to
two point functions. These serve as a check on the results for c which is related to the energy
3
momentum tensor two point function and also a two index tensor on the space of marginal
couplings which is related to the two point function for the exactly marginal dimension six
scalar operators. The coefficients c ,c as well as c determine the energy momentum tensor
1 2 3
three point function. This also satisfies positivity restrictions related to the energy flux at
infinity [18] and these are shown to be satisfied to lowest order beyond free theory by φ3
theory.
Various details are contained in four appendices. In appendix A we present a detailed
summary of results for conformal tensors, the Weyl, Cotton and Bach tensors, and also dif-
ferential operators which transform nicely under Weyl rescaling of the metric and are relevant
for our calculations. We also give an expression for the coincident limit of the Seeley–DeWitt
coefficient a , which determines the one loop results, in terms of the basis of conformal ten-
3
sors. In appendix B we describe briefly the six dimensional results obtained by integrating the
infinitesimal Weyl rescaling of the metric. Appendices C and D contain the detailed results
necessary to calculate the coincident limit of a for fermions and two-forms respectively.
3
3
2. Response to Weyl Rescalings for CFTs
In general the vacuum functional W, depending on the metric and couplings, for a CFT
responds to an infinitesimal Weyl rescaling, δ γ = 2σγ , in even d dimensions according to
σ µν µν
(4π)d2 δσW = ddx√ γ σLd, (2.1)
Z −
with L a local scalar of dimension d formed from the metric, the couplings and derivatives.
d
In general L is constrained by the integrability conditions following from (δ δ δ δ )W = 0.
d σ σ σ σ
′− ′
We initially consider solutions such that
δ L +dσL = (X µν∂ σ), X µν = X νµ. (2.2)
σ d d µ d ν d d
∇
We assume that in (2.1) L has the freedom
d
L L + Z µν, (2.3)
d d µ ν d
∼ ∇ ∇
since such contributions can in general be cancelled by local contributions to W. For variations
(2.3) compatible with (2.2) then
X µν X µν +2Z µν γµνZ λ if δ Z µν +dσZ µν = 0. (2.4)
d d d d λ σ d d
∼ −
Under a finite rescaling (2.1) extends to
(4π)d2 W e2σγµν W γµν = ddx√ γ d(σ), (2.5)
− Z − L
(cid:0) (cid:2) (cid:3) (cid:2) (cid:3)(cid:1)
where (σ) is obtained by a Taylor expansion,
d
L
(σ) = σL ∂ σ∂ σ 1 X µν + Jµ,
Ld d− µ ν r≥0(r+2)! d,r ∇µ
(2.6)
P
X µν = (δ +dσ)X µν, X µν = X µν,
d,r+1 σ d,r d,0 d
so that X µν = O(σr) and Jµ is arbitrary. The sum in (2.6) truncates after a finite number
d,r
of terms.
Before proceeding to the six dimensional case we recapitulate previous results obtained
in four dimensions [6]. The extra terms involving derivatives of the couplings depend on a
symmetric two index tensor g and also a four index tensor c . It is natural to express
IJ IJKL
the contributions to L using the Christoffel connection formed from g ,
4 IJ
ΓI = 1gIL(∂ g +∂ g ∂ g ), gIJ = (g−1)IJ . (2.7)
JK 2 J LK K LJ − L JK
We may also allow for a background gauge field A g coupled to conserved currents. If F
µ µν
∈
is the associated field strength, then
L = cWρµνλW aE 1κ F µνF
4 ρµνλ− 4− 4 ab a bµν
+ 1 g D2gID2gJ g ∂µgI(2P γ Rˆ)∂νgJ (2.8)
2 IJ − IJ µν − µν
+ 1 c ∂µgI∂ gJ ∂νgK∂ gL.
2 IJKL µ ν
4
Here,
D2gI = 2gI +ΓI ∂µgJ∂ gK, (2.9)
JK µ
∇
E is the Euler density, given by (A.9) for d = 4, and P and Rˆ are the Schouten tensor and its
4 µν
trace given by (A.1) for d = 4. In (2.8) clearly g = g , c = c = c and κ
IJ JI IJKL (IJ)(KL) KLIJ ab
is a symmetric invariant bilinear form in a convenient basis t for g so that, for any X g,
a
{ } ∈
X = X t . If g is simple then κ κδ . We may also extend ∂ gI ∂ gI+A (T g)I but
a a ab ab µ µ aµ a
→ →
for simplicity we neglect such contributions here.
It is straightforward to check that (2.8) satisfies (2.2) with
X µν = 8aG µν +g 2∂µgI∂νgJ γµν∂λgI∂ gJ , (2.10)
4 4 IJ λ
− −
(cid:0) (cid:1)
and G µν as in (A.13) with d= 4, so long as a is constant. From (2.6) it is easy to see that
4
X µν = 16a( µ∂νσ γµν 2σ), X µν = 16a(2∂µσ∂νσ+γµν∂λσ∂ σ). (2.11)
4,1 4,2 λ
∇ − ∇ −
Using (2.10), (2.11) in (2.6) gives
(σ) = σL + 1 g 2∂µgI∂νgJ γµν∂λgI∂ gJ ∂ σ∂ σ
L4 4 2 IJ − λ µ ν (2.12)
(cid:0) (cid:1)
+4a G µν∂ σ∂ σ+ 2σ∂µσ∂ σ+ 1 (∂µσ∂ σ)2 ,
4 µ ν ∇ µ 2 µ
(cid:0) (cid:1)
which reproduces the well known results for the four dimensional dilaton effective action and
the ∂g terms calculated in [6].
In six dimensions we follow a similar route by determining the general form for L satisfying
6
(2.2). There are various contributions which may be analysed independently. For any six
dimensional CFT in the absence of local couplings L is given by just
6
LR = c I +aE , (2.13)
6 i=1,2,3 i i 6
P
where an appropriate basis for the dimension six conformal scalars I , and also an explicit
i
expression for the Euler density E , are given in appendix A. I ,I are the two independent
6 1 2
scalars cubic in the Weyl tensor while I = Wρµνλ 2W + . Since δ I +6σI = 0 and
3 ρµνλ σ i i
∇ ···
δ E +6σE = 24 (G µν∂ σ), with G µν the six dimensional generalisation of the Einstein
σ 6 6 µ 6 ν 6
∇
tensor, it is easy to verify that (2.13) satisfies (2.2) with
XRµν = 24aG µν. (2.14)
6 6
There are also three potential dimension six conformal scalars formed from F for which
µν
we may take
LF = 1κ F µν( 2F ) 4RˆF µνF 2( +2P ) κ F µλF ν
6 − 4 ab a D µν b− a bµν − ∇µ∇ν µν ab a b λ (2.15)
(cid:0) (cid:1) (cid:0) (cid:1)
1κˆ W F µνF λρ+ 1f F µνF F λ ,
− 4 ab µνλρ a b 3 abc a bνλ c µ
with κ ,κˆ symmetric invariant tensors and f an antisymmetric invariant tensor.
ab ab abc
5
For free scalars, fermions and also two-form gauge fields the coefficients c ,a were calculated
i
in [17]. For scalars and fermions the results can be straightforwardly extended to include back-
ground gauge fields. For t the real antisymmetric or anti-hermitian generators determining
a
the gauge couplings to scalars or fermions, then, letting κ = κtr(t t ), κˆ = κˆtr(t t ),
ab a b ab a b
− −
f = ftr(t t t ), we have
abc [a b c]
−
7!c 7!c 7!c 7!a κ κˆ f
1 2 3
scalars 28 5 2 5 1 1 1
− 3 3 9 30 18 15 . (2.16)
fermions 896 32 40 191 16 8 52
− 3 − 9 15 9 15
two-forms 8008 2378 180 442
− 3 − 3
Our main motivation in this paper is to consider contributions depending on derivatives of
the couplings. We first consider terms which are the direct extension of the terms involving g
IJ
in (2.8). This can be constructed starting from a leading contribution involving six derivatives
S = 1 g DµD2gID D2gJ + 1 D2gI∂µgK∂ gLD2gJ , (2.17)
1 2 IJ µ 2RIKLJ µ
with D2gI defined as in (2.9) and
DµD2gI = ∂µD2gI +ΓI ∂µgKD2gL. (2.18)
KL
In (2.17) is the Riemann tensor defined as usual in terms of the Christoffel connection
IKLJ
R
ΓI in (2.7). The Weyl variation of S gives
KL 1
δ S +6σS = 8g D2gIDµ∂νgJ ∂ σ+3g D2gID2gJ 2σ
σ 1 1 IJ µ ν IJ
− ∇ ∇
+4g ∂µgI∂νgJ ∂ 2σ+8 (P ∂λσ)+2 (Rˆ∂ σ)
IJ µ ν µ νλ µ ν
∇ ∇ ∇ ∇
(cid:0) (cid:1)
2g ∂µgI∂ gJ 2 2σ+8 ν(P ∂λσ)+2 ν(Rˆ∂ σ)
IJ µ νλ ν
− ∇ ∇ ∇ ∇ (2.19)
(cid:0) (cid:1)
+ 4g D2gIDµ(∂νgJ∂ σ) 3g D2gID2gJ ∂µσ
µ IJ ν IJ
∇ −
(cid:0)
4g ∂µgI∂νgJ (∂ 2σ+8P ∂λσ+2Rˆ∂ σ)
IJ ν νλ ν
− ∇
+2g ∂λgI∂ gJ (∂µ 2σ+8Pµν∂ σ+2Rˆ∂µσ) .
IJ λ ν
∇
(cid:1)
If we then add the four derivative term
S = 4g (D2gIDµ∂νgJ +Dµ∂νgID2gJ)P +3g D2gID2gJRˆ
2 IJ µν IJ
− (2.20)
+ 4Pµλg ∂ gID2gJ 4g ∂µgI∂νgJ∂ Rˆ+2g ∂λgI∂ gJ∂µRˆ ,
µ IJ λ IJ ν IJ λ
∇ −
(cid:0) (cid:1)
we may obtain
δ (S +S )+6σ(S +S )= g ∂µgI∂νgJA + (Xgµν∂ σ), (2.21)
σ 1 2 1 2 IJ µν ∇µ 6 ν
for
A = 4 ∂ 2σ+32 λ(P ∂ σ) 16 λ(P ∂ σ)+16 (P ∂λσ)
µν µ ν λ(µ ν) µν λ (µ ν)λ
∇ ∇ ∇ − ∇ ∇
8 (Rˆ∂ σ) (2.22)
(µ ν)
− ∇
+γ 2 2 2σ 8 µ(P ∂νσ)+4 µ(Rˆ∂ σ)
µν µν µ
− ∇ ∇ − ∇ ∇
(cid:0) (cid:1)
6
and
Xgµν = 4g D2gIDµ∂νgJ 3γµνg D2gID2gJ
6 IJ − IJ
+8Pµνg ∂λgI∂ gJ 16 Pµλg ∂ gI∂νgJ +g ∂µgI∂ gJPλν (2.23)
IJ λ IJ λ IJ λ
−
(cid:0) (cid:1)
+16g ∂µgI∂νgJRˆ+γµν 16g ∂λgI∂ρgJP 8∂λgI∂ gJRˆ ,
IJ IJ λρ λ
−
(cid:0) (cid:1)
where P ,Rˆ are given by (A.1) with d = 6. The remaining A terms in (2.21) may be
µν µν
cancelled by taking
S = 4g ∂µgI∂νgJ B +6P P λ 4P Rˆ+ ∂ Rˆ
3 IJ µν µλ ν µν µ ν
− ∇ (2.24)
(cid:0) (cid:1)
2g ∂µgI∂ gJ 2P Pρλ 2Rˆ2+ 2Rˆ .
IJ µ ρλ
− − ∇
(cid:0) (cid:1)
Hence, we may satisfy (2.2) for d= 6 by taking
g
L = S S S
6 − 1− 2− 3 (2.25)
+g ∂µgI∂νgJ W W λρω +g ∂µgI∂ gJ W Wνλρω.
1,IJ µλρω ν 2,IJ µ νλρω
g
The terms involving g are a natural generalisation of the unique L , implicitly defined by
IJ 4
(2.8), and Lg = 1g ∂µgI∂ gJ. The sign is chosen so as to ensure later that g is positive
2 −2 IJ µ IJ
in unitary theories. In six dimensions there are further possibilities involving rank two tensors
which are formed in terms of the Weyl tensor, as included in (2.25). For these terms (2.2)
becomes essentially trivial.
Further contributions to L involve at least four g’s with derivatives. To construct these
6
we first consider
T = j 1 hIJµρ λhKL λhIJµν hKL
1 1,IJKL 2 ∇ρ ∇ µλ−∇ ∇λ µν
(cid:0) (cid:1)
+j 1∂λ(∂µgI∂ gJ)∂ (∂νgK∂ gL) (2.26)
2,IJKL 2 µ λ ν
+j µ∂νgI ∂ gJ 1 2gI 2gJ ∂ρgK∂ gL,
3,IJKL ∇ ∇µ ν − 4 ∇ ∇ ρ
(cid:0) (cid:1)
for hIJ symmetric and traceless,
µν
hIJ = ∂ gI∂ gJ 1γ ∂λgI∂ gJ , (2.27)
µν (µ ν) − 6 µν λ
and j = j = j . In this case
i,IJKL i,(IJ)(KL) i,KLIJ
δ T +6σT = ∂ σj 2 (hIJλνhKLµ )+∂λ(hIJµνhKL )
σ 1 1 λ 1,IJKL µ ν µν
∇ (2.28)
(cid:0) (cid:1)
∂ σ(j +j )∂λ ∂µgI∂ gJ ∂νgK∂ gL .
λ 2,IJKL 3,IJKL µ ν
−
(cid:0) (cid:1)
Terms involving two derivatives of σ may be cancelled by
T = j 2P hIJµλhKLν +RˆhIJµνhKL
2 1,IJKL µν λ µν
−
(cid:0) (cid:1)
+(j +j )Rˆ∂µgI∂ gJ ∂νgK∂ gL
2,IJKL 3,IJKL µ ν
(2.29)
∂ j hIJµλhKL ρ∂ gM 1 hIJµνhKL 2gM
− M 1,IJKL µρ∇ λ − 2 µν∇
(cid:0) (cid:1)
1 ∂ (j +j )∂µgI∂ gJ ∂νgK∂ gL 2gM .
− 4 M 2,IJKL 3,IJKL µ ν ∇
7
Hence
Lj = T +T +j W hIJµνhKLλρ (2.30)
6 1 2 4,IJKL µλνρ
satisfies (2.2) with
Xjµν = j 2hIJµλhKLν +γµνhIJλρhKL
6 1,IJKL λ λρ
(2.31)
(cid:0) (cid:1)
γµν(j +j )∂λgK∂ gJ ∂ρgK∂ gL.
2,IJKL 3,IJKL λ ρ
−
In (2.30) we have allowed for a possible trivial term involving the Weyl tensor. If in (2.26)
hKL D hKL and similarly ∂ (∂νgK∂ gL) D (∂νgK∂ gL), µ∂νgI Dµ∂νgI,
λ µν λ µν λ ν λ ν
∇ → → ∇ →
2gI D2gI, with D the covariant derivative including the the Christoffel connection (2.7),
λ
∇ →
then correspondingly in (2.29) ∂ j D j with D the covariant extension of
M i,IJKL M i,IJKL M
→
∂ .
M
The remaining potential contribution to L involves six g’s with derivatives,
6
Lk = 1k ∂µgI∂ gJ ∂νgK∂ gL∂ωgM∂ gN , (2.32)
6 2 IJKLMN µ ν ω
defining a rank six tensor with appropriate symmetries.
3. Two-Forms
In six dimensions there are three free conformal field theories. In four dimensions with abelian
gauge fields it is still possible to determine the leading one loop contribution to the metric on
the conformal manifold. Here we describe the analogous calculation in six dimensions following
the approach described in [21] and extending the six dimensional results in [17].
For a two-form B Ω(2), where Ω(n) is the space of n-forms comprised of antisymmetric
µν
∈
n-index tensors, the starting Lagrangian is just1
1
L = (dB)µνω(dB) . (3.1)
−12g2 µνω
This is invariant under gauge transformations B B +(dA) , A Ω(1). It is convenient
µν µν µν µ
→ ∈
here to add the covariant Feynman gauge fixing term,
g2
L = δ( 1 B) µ δ( 1 B) . (3.2)
g.f. − 2 g2 g2 µ
(cid:0) (cid:1) (cid:0) (cid:1)
Rescaling B gB the quantum theory is defined in terms of the functional determinants
µν µν
→
of the Laplacians
∆(n) = δ′d′+d′δ′ :Ω(n) Ω(n), d′ = 1 dg, δ′ = gδ 1 , (3.3)
→ g g
1The exterior derivative d : Ω(n) → Ω(n+1), is defined so that (dF)µ1...µn+1 = (n+1)∂[µ1Fµ2...µn+1] for
Fµ1...µn ∈ Ω(n) and is independent of the metric. The adjoint δ : Ω(n) → Ω(n−1) is correspondingly given
by (δF)µ1...µn−1 = −√1γγµ1ν1···γµn−1νn−1∂ω(√−γγωλγν1ρ1···γνn−1ρn−1Fλρ1...ρn−1) = −∇λFλµ1...µn−1. Of
course d2=δ2 =0. −
8
so that [17]
W = 1lnDet∆(2) +lnDet∆(1) 3lnDet∆(0). (3.4)
−2 − 2
∆(1) is related to a fermionic vector ghost and ∆(0) to a bosonic scalar ghost; the degrees of
freedom in d dimensions are then 1d(d 1) 2d+3= 1(d 2)(d 3).
2 − − 2 − −
Continuing to a Euclidean metric the functional determinant of an elliptic differential
operator ∆ may be defined in terms of the heat kernel by
1 ∞
lnDet∆ = ζ ′(0), ζ (s)= dτ τs−1Tr e−τ∆ . (3.5)
∆ ∆
− Γ(s) Z
0 (cid:0) (cid:1)
Under Weyl rescaling of the metric, for F an n-form, δ (dF) = 0, whereas
µ1...µn σ µ1...µn+1
δ (δF) = 2(d n+1)σ(δF) +2(d n)(δσF) . Hence, with d= 6,
σ µ1...µn−1 − − µ1...µn−1 − µ1...µn−1
δ ∆(2) = 2σ∆(2)+2σd′δ′+2d′δ′σ 4d′σδ′,
σ
− −
δ ∆(1) = 2σ∆(1)+2σd′δ′+4d′δ′σ 6d′σδ′+2δ′σd′ 2σδ′d′, (3.6)
σ
− − −
δ ∆(0) = 2σ∆(0) 4σδ′d′+4δ′σd′.
σ
− −
Using relations such as d′∆(1) = ∆(2)d′ we may obtain
δ Tr e−τ∆(2) 2Tr e−τ∆(1) +3Tr e−τ∆(0)
σ Ω(2) − Ω(1) Ω(0)
(cid:16) (cid:17)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (3.7)
d
= 2τ Tr σe−τ∆(2) 2Tr σe−τ∆(1) +3Tr σe−τ∆(0) ,
− dτ Ω(2) − Ω(1) Ω(0)
(cid:16) (cid:17)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
so that from (3.4) and (3.5)
δ W = Tr σe−τ∆(2) 2Tr σe−τ∆(1) +3Tr σe−τ∆(0) , (3.8)
σ (cid:16) Ω(2) − Ω(1) Ω(0) (cid:17)(cid:12)τ0
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:12)
with denoting the τ0 term in the Laurent expansion in τ. (cid:12)
τ0
|
In each case the Laplacians defined in (3.3) have the form
∆ = 2+2Rˆ1 +Y , (3.9)
V ∆
−D
for ∆ : V V and = +A with A an appropriate connection on V. For such elliptic
µ µ µ µ
→ D ∇
operators the associated heat kernel K (x,y;τ), corresponding to e−τ∆, has the well known
∆
expansion (4πτ)12dK∆(x,x;τ) ∼ n≥0a∆,n|(x)τn with a∆,n| the diagonal Seeley–DeWitt co-
efficients. Hence for d = 6 P
(4π)3Tr σe−τ∆ = d6x√γ σtr (a ), (3.10)
V τ0 Z V ∆,3|
(cid:0) (cid:1)(cid:12)
(cid:12)
with tr the matrix trace and
V
7! tr (a ) = dimV 5E 28 I + 5I +2I
V ∆,3| 9 6− 3 1 3 2 3
(cid:0) (cid:1)
+14 3tr (Iˆ)+5W tr (FµνFλρ) 8tr (FµνF Fλ )
V µνλρ V V νλ µ
− (3.11)
(cid:0) (cid:1)
7!tr 1Y 3+ 1 Y ∆ Y + 1 WρµνλW Y + 1 FµνF Y
− V 6 ∆ 12 ∆ 2 ∆ 180 ρµνλ ∆ 12 µν ∆
(cid:0) (cid:1)
+7! Z µν.
µ ν ∆
∇ ∇
9