Casimir energy on the sphere and 6D CFT trace anomaly
R Aros F Bugini DE Díaz C Núñez-Barra





^1 Departamento de Ciencias Fisicas, Universidad Andres Bello,
Sazie 2212, Piso 7, Santiago, Chile
^2 Departamento de Matemática y Física Aplicadas,
Universidad Católica de la Santísima Concepción,
Alonso de Ribera 2850, Concepción, Chile
^3 Departamento de Ciencias Fisicas, Universidad Andres Bello,
Autopista Concepcion-Talcahuano 7100, Talcahuano, Chile
^4 Facultad de Física, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, Santiago, Chile
raros@unab.cl,fbugini@ucsc.cl,danilodiaz@unab.cl,cnb@uc.cl

 We elucidate the dependence of the Casimir energy on the trace anomaly coefficients for a six-dimensional CFT on R× S^5. This extends the universal dependence on the central charge in 2D and the relation by Cappelli and Coste in 4D, unveiling the role of the trivial total derivatives in the anomaly that renders the Casimir energy scheme dependent. We obtain 
 
    E_o=-15/8 a_6 -5/12 (g_5+1/4 g_7+1/2 g_8-10  g_9+g_10),

 with a_6 being the type A central charge and the g's, the coefficients of five out of six terms that form a basis for trivial total derivatives. The derivation is based on the Polyakov formulas (conformal primitive) resulting from the integration of the trace anomaly. 
 
 Alternatively, on a 6D conformally flat background the above basis is redundant and one can simplify further to get, in terms of the Schouten scalar J and the Schouten tensor V, Branson's basis  for trivial total derivatives  ∇^2∇^2 J, ∇^2J^2 and ∇^2|V|^2+2∇·(V·∇ J) with coefficients γ_1, γ_2 and γ_3, respectively,    

    E_o=-15/8a_6-1/24(γ_1-γ_2 -1/8γ_3) .


 
 



§ INTRODUCTION

The Casimir effect <cit.> is a remarkable macroscopic quantum response of a system to external influences (e.g. boundary conditions or background fields). It has been extensively studied over the years and experimentally verified <cit.> (for a recent review, see <cit.>). One interesting route contemplates the extension of the original setting regarding the 4D electromagnetic field to a conformal field theory in a generic even dimension. 
Since conformal symmetry is quite restrictive, the Casimir energy of a 2D CFT on the cylinder is universal and given by the ubiquitous central charge c <cit.> of the trace anomaly,

    E_o=-c/12.

The aforementioned universality is partially lost in 4D,  yet the trace anomaly coefficients determine the vacuum energy[We stick to the conventions of <cit.>.] as shown by Cappelli and Coste <cit.>      

    E_o=3/4 a+3/8 g ,

where a is the universal type A trace anomaly coefficient and g is the regularization scheme dependent coefficient of the total derivative term 

    (4π)^2 ⟨ T⟩ = -a E_4 + c W^2 + g ∇^2 R .

The total derivative term ∇^2 R is a trivial anomaly since it comes from the conformal variation [By conformal we mean a local Weyl rescaling of the metric. We will use both terms, conformal and Weyl, indistinctly in what follows.] of a conformal primitive that happens to be a local curvature invariant with the appropriate scaling dimension[The other two candidates for a local curvature invariant Riem^2 and Ric^2 can be reduced to R^2 by exploiting the conformal invariance of the Euler density E_4 and of the quadratic contraction of the Weyl tensor W^2.], namely R^2, that can simply be added to the action.

There are good indications that in higher (necessarily even) dimensions the Casimir energy on the cylinder [Meaning that the spatial section is compactified to a sphere.] is still dictated by the trace anomaly of the CFT. The structure of the latter was asserted years ago by Deser and Schwimmer <cit.> to consist of [See also <cit.> for a cohomological approach. For the closely related question of the classification of global conformal invariants, see <cit.> on conformally flat manifolds and <cit.> in greatest generality.] a multiple of the Euler density (type A), a point-wise Weyl invariant (type B) and a trivial total derivative       

    (4π)^n/2·⟨ T ⟩   =    - (-)^n/2  a_n · E_n + {} +

Remarkably, a simple relation between the Casimir energy on the cylinder and the type A trace anomaly coefficient was found by Herzog and Huang in <cit.>

    E_o=(-1)^n/2· (n-1)!!/2^n/2· a_n ,

but only valid in a particular regularization scheme where the trivial total derivatives are absent. For example, in 6D one encounters discrepancies with the Casimir energy for the free (2,0) tensor multiplet as computed via standard ζ-function regularization (see, e.g. <cit.>). A 6D relation, like the Cappelli and Coste one, taking into account the role of the trivial total derivatives in the trace anomaly, is clearly still missing. This note aims to provide such an extension. 

Our results are based on the following rather simple observation: since the trivial total derivatives in the trace anomaly stem from a local conformal invariant in the action, it is enough to evaluate the latter on the geometry of the cylinder to obtain its contribution to the partition function and, in consequence, to the vacuum energy. 
For completeness, we will also re-derive the dependence on the type A central charge by the direct examination of the Polyakov formulas that provide the conformal primitive of the trace anomaly. We prefer Branson's construction where the Euler density is traded in for the Q-curvature, this reduces the answer to terms at most quadratic in the Weyl factor that relates the standard metric on the cylinder to that of a flat shell. Although our derivation is conceptually clear, in practice we face the complication that, unlike the Euler term, the number of independent trivial total derivatives increases significantly with the dimension; there are none in 2D, one in 4D, six in 6D, whereas 8D and beyond remain largely unexplored [There is, though, a basis of nine local conformal primitives on conformally flat backgrounds in 8D due to Branson and Peterson <cit.>.].

We start in Section 2 with a brief account of Polyakov formulas. We then consider the explicit cases of 2D, 4D, and 6D in Section 3 and test in Section 4 against data available in the literature. We then conclude and collect some useful relations in the appendix.  



§ POLYAKOV FORMULAS, Q-CURVATURE, AND LOCAL CURVATURE INVARIANTS


The (generalized) Polyakov formula provides a conformal primitive of the trace anomaly, as first shown by Polyakov in 2D <cit.>. In higher even dimensions, Branson asserted the form of the corresponding extension by making use of the suitable features of the critical Q-curvature (see, e.g. <cit.>). In a (closed and with Euclidean signature) conformally flat manifold the topological content of the trace anomaly is also captured by the Q-curvature and all point-wise Weyl invariants vanish so that the structure of the trace anomaly is significantly simplified            

    (4π)^n/2·⟨ T ⟩   =    - (-)^n/2  n· a_n · Q_n +

Here one exploits the linear dependence of the variation of the Q-curvature on the Weyl factor σ, that conformally relates two metrics ĝ=e^2 σg,  and integrates in an auxiliary parameter t along any interpolating path σ(t) such that σ(0)=0 and σ(1)=σ, to obtain for the partition functions

    Γ[g,ĝ]=-Γ[ĝ,g]   =   log Z[ĝ]-log Z[g] =-(-)^n/2 n· a_n/(4π)^n/2∫_M_n dv_g  σ·{ Q_n + 1/2 P_n σ}
    
          
        +{∫_M̂_n dv_ĝ ℱ̂-∫_M_n dv_g ℱ} .

The explicit dependence on σ is at most quadratic and the GJMS operator P_n can be integrated by parts to yield the so-called dilaton action for the Weyl factor, whereas higher order terms in σ are hidden in the finite conformal variation of the local conformal primitive ℱ.  

Now, considering the standard metric on the cylinder R× S^n-1 conformally related to flat space 

    ĝ=du^2+dΩ^2=r^-2{dr^2+r^2 dΩ^2}

via a Weyl factor σ=-ln r, we can readily read off the Casimir energy resulting from the compactification of the spatial section referred to the vanishing vacuum energy of flat space. Alternatively, one can consider a temperature circle with inverse temperature β and read off the dominant vacuum term as β goes to infinity on S^1_β× S^n-1.
  
In “flatland” (meaning flat space or an spherical shell for finite inverse temperature), the only contribution comes from the nth-order kinetic term |∂∂^2...∂^2σ|^2 or |∂^2...∂^2σ|^2 depending on the (even) dimension n, that produces the same answer ((n-2)!!)^2/r^n so that the radial integral results in ∫dr/r=∫ du or β. Performing the remaining volume integral on the sphere, the generic relation for the Casimir energy (modulo trivial total derivative contributions) becomes

    E_o=(-1)^n/2· n!!/2^n/2+1· a_n+... .

This ought to be compared with the result by Herzog and Huang in the Euler basis

    E_o=(-1)^n/2· (n-1)!!/2^n/2· a_n+... .


The apparent discrepancy above stems from the fact that each result is valid within a different regularization scheme where trivial total derivatives are discarded. However, evaluation of the local conformal primitive of the trivial total derivatives on the cylinder will complement the result and the agreement between the two schemes and a more general one will be achieved. The local conformal primitive vanishes on flat space, and must be a constant on the cylinder. This yields the scheme-dependent contribution to the Casimir energy foreseen above. 
The only technical difficulty in carrying out the above completion, as already pointed out, lies in the increasing complexity of the trivial total derivative terms and their conformal primitives with the dimension.



§ CASIMIR ENERGY AND TRACE ANOMALY: THE EXPLICIT RELATION


In what follows we will revisit the 2D and 4D cases and obtain, for the first time to our knowledge, the 6D extension of the Capelli-Coste relation. 



 §.§ 2D CFT

Let us start by reproducing the 2D relation [This was essentially obtained by Affleck <cit.>, except for the fact that his result applies to the modular transformed torus, i.e., to the Cardy or high temperature regime.] following Cappelli and Coste. There are no trivial anomalies in 2D, by convention Q_2=R/2 is the Gaussian curvature of the closed surface and P_2=-∇^2, that is integrated by parts to write down the answer in the well-known Liouville form 


    Γ[g,ĝ]=a_2/4π∫_flatland dv_g  {  |∇σ|^2 + σ· R  }
 
From this potential for the Weyl factor, also known in the conformal geometry literature as the Moser functional, evaluated on the fiducial flat metric with R=0 and for the conformal factor |∂σ|^2=1/r^2, one can readily obtain the leading contribution to the partition function as β→∞

    -β· E_o=a_2/4π·2π·β=a_2/2·β

This corresponds to the more familiar E_o=-c/12, when the central charge is tuned so that for the scalar Laplacian a_2=1/6 one gets c=1, i.e., a_2=c/6. In all, 

    E_o=-a_2/2.




 §.§ 4D CFT

Now we revisit the derivation by Cappelli and Coste in 4D, but in the light of Branson's insight that exploited the suitable transformation law of the Q-curvature under Weyl rescaling of the metric (see also Riegert's action <cit.> in local form). The trace anomaly at a conformally flat background becomes

    (4π)^2·⟨ T ⟩ = -a_4 · E_4 + g·∇^2 R = -4 a_4· Q_4 + g̃·∇^2 R ,

and its conformal primitive, referred to the fiducial flat metric, is then given by  
 
    Γ[g,ĝ]   =   -a_4/4π^2∫_flatland dv_g{σ· Q_4 + 1/2 σ· P_4 σ}+g̃/16π^2∫_S^1_β× S^3 dv_ĝ -1/12R̂^2
        
    
       =   -a_4/8π^2∫_flatland dv_g |∂^2σ|^2+g̃/16π^2∫_S^1_β× S^3 dv_ĝ -1/12R̂^2 .
 
We obtain contribution from the fourth-order kinetic term |∂^2σ|^2=4/r^4 and from the local curvature invariant -1/12R̂^2=-3, conformal primitive of the trivial total derivative, so that we end up with

    -β· E_o=-{a_4/2π^2+3g̃/16π^2}· 2π^2·β=-{a_4+3/8 g̃}·β .

Therefore, we obtain for the Casimir energy 

    E_o=a_4+3/8 g̃ .

To translate to the Euler basis, it is enough to remember that on conformally flat backgrounds 4Q_4=E_4-2/3∇^2 R, so that g̃=g-2/3a_4 and finally obtain the original result by Cappelli and Coste[Our convention follows <cit.> and slightly differs from that of the original Cappelli-Coste paper <cit.>.]

    E_o=3/4 (a_4+1/2  g) .

Notice that the precise shift g̃=g-2/3a_4 decorates the dilaton action (eq. <ref>), by using the conformal transformation of √(ĝ)R̂^2, as follows

    Γ[δ,δ̂]   =   -a_4/8π^2∫_flatland dv_δ |∂^2σ|^2-2/3a_4/16π^2∫_flatland dv_δ -1/12{-6∂^2σ-6|∂σ|^2}^2 +...
    
       =   a_4/8π^2∫_flatland dv_δ {2|∂σ|^2∂^2σ + (|∂σ|^2)^2 } ... ,

where the ellipsis stands for the contribution of the trivial anomaly g∇^2R. Evaluating the first part on the cylinder will again produce the dependence of the vacuum energy on the central charge a, while the contribution from the trivial anomaly can easily be reintroduced by evaluating the associated counterterm (conformal primitive) [ This shortcut route to the Cappelli-Coste relation was somehow already implicit in <cit.>(see appendix A.1 therein).].


 §.§ 6D CFT


It is at 6D that a subtle departure between total derivatives and trivial total derivatives occurs, stemming out from the difference between global conformal invariants and the trace anomaly of a CFT. The latter must comply with the Wess-Zumino consistency or integrability condition, requiring that the allowed total derivatives must be derived from the conformal variation of a local curvature invariant. In 6D, a basis for total derivatives is provided by the seven divergences C_1, C_2, C_3, C_4, C_5, C_6 and C_7 of <cit.>, however once they are expressed in the reduced basis for trivial total derivatives of M_5, M_6, M_7, M_8, M_9 and M_10 there remains an “offending term" ∇^2 |Ric|^2. The basis for trivial total derivatives was obtained in <cit.> by performing the conformal variation of all possible curvature invariants with the appropriate scaling dimension. There are seventeen elements K_1, ..., K_17, from which seven combinations are the total derivatives C_1, C_2, C_3, C_4, C_5, C_6, C_7 whose integrals vanish identically, another combination gives the Euler density E_6 which is a topological invariant, and  there are also three other combinations that produce three independent point-wise conformal invariants I_1, I_2 and I_3, so that in all there remain six independent combinations -conformal primitives that we denote F_5, F_6, F_7, F_8, F_9, F_10 -   whose conformal variations produce the trivial total derivatives M_5, M_6, M_7, M_8, M_9, M_10 (we collect explicit expressions in <ref>). 

Therefore, the 6D CFT trace anomaly at a conformally flat background becomes

    (4π)^3·⟨ T ⟩   =    a_6 · E_6 + g_5· M_5 + g_6· M_6 + g_7· M_7 + g_8· M_8 + g_9· M_9 + g_10· M_10
    
       =   6 a_6 · Q_6 + g̃_5· M_5 + g̃_6· M_6 + g̃_7· M_7 + g̃_8· M_8 + g̃_9· M_9 + g̃_10· M_10 .

The conformal primitive, referred to a fiducial flat metric where all curvature invariants vanish, is then given by  
 
    Γ[g,ĝ]   =   3 a_6/64π^3∫_flatland dv_g  |∂ ∂^2σ|^2
    
          +1/64π^3∫_S^1_β× S^5 dv_g{g̃_5·ℱ̂_5+g̃_6·ℱ̂_6+g̃_7·ℱ̂_7+g̃_8·ℱ̂_8+g̃_9·ℱ̂_9+g̃_10·ℱ̂_10}.
 
When evaluated at S^1_β× S^5 with the standard product metric, we obtain contributions from the sixth-order kinetic term |∇ ∇^2σ|^2=64/r^6 and from the local curvature invariants 

    ℱ̂_5=80/3  ,  ℱ̂_6=0  ,  ℱ̂_7=20/3  ,
    
    ℱ̂_8=40/3  ,  ℱ̂_9=-800/3  ,  ℱ̂_10=80/3  .

Collecting all contributions 

    -β· E_o   =   1/64π^3{3a_6·64 +g̃_5·80/3+g̃_7·20/3+g̃_8·40/3+g̃_9·-800/3+g̃_10·80/3}·π^3·β
       =   {3a_6+5/12 [g̃_5+1/4 g̃_7+1/2 g̃_8-10 g̃_9+g̃_10]}·β .

Therefore, we obtain for the Casimir energy 

    E_o=-3 a_6-5/12 [g̃_5+1/4 g̃_7+1/2 g̃_8-10 g̃_9+g̃_10] .

To translate to the Euler basis we have to express the 6D Q-curvature in terms of the Euler density and trivial total derivatives in a conformally flat background. The linearity under conformal transformations, a defining property of the Q-curvature, is shared by the whole family of pondered Euler densities Ẽ_6 introduced by Anselmi  in <cit.> (see <cit.> as well). In our conventions, the dictionary goes as follows

    E_6=Ẽ_6 +(67/10-5/2ζ)M_6+(5/2ζ-51/5)M_7+(1/16ζ-9/200)M_9-3/5M_10 ,

and results in the shift of the trivial anomaly coefficients

    g̃_5=g_5     ,     g̃_6=g_6+a_6(67/10-5/2ζ)  ,  g̃_7=g_7+a_6(5/2ζ-51/5)  ,
    g̃_8=g_8     ,     g̃_9=g_9+a_6(1/16ζ-9/200)  ,  g̃_10=g_10-3/5a_6 .

Substituting back in the Casimir energy relation (eqn.<ref>), we obtain the alternative relation that completes the partial result obtained by Herzog and Huang in 6D

    E_o=-15/8 a_6-5/12 [g_5+1/4 g_7+1/2 g_8-10 g_9+g_10] .

The above relation is robust in the sense that any construct with the same conformal transformation property as the critical Q-curvature Q_6 will produce the same result.
For completeness, let us also examine the two-parameter family of trivial total derivative terms that respects the linearity in a generic background [We are grateful to I.L. Shapiro for a useful discussion on this point.] found out by Hamada <cit.> (eqn.4.20 and 4.21 therein, see also <cit.>) at the expense of modifying Branson's operator P_6 :

    (i)   3/4∇^2W^2-3∇∇(WW) = -3/4C_5+3C_7 = 1/4M_5+1/2M_6-3/4M_7+M_8+9/160M_9
    
       (ii)   3/2∇^2W^2 = 3C_5 = -3M_6+3M_8+3/20M_9
 
It is reassuring to verify that none of the two combinations, ∇∇(WW) nor ∇^2 W^2, will produce a non-vanishing result in the relation for the Casimir energy. The above freedom agrees with the usual wisdom in Conformal Geometry that the pair of critical Q-curvature Q_n and critical GJMS operator P_n is “pure Ricci” , while any other construct that shares the same conformal properties is “contaminated” by explicit dependence on the Weyl tensor.



§ 6D CFT: EXAMPLES

There are few examples in 6D, worked out some years ago <cit.>, that can now be used to verify the correctness of the novel relation put forward above between Casimir energy and trace anomaly coefficients. We first write down the trace anomaly with the relevant coefficients, and in the table below we report the values for the Casimir energies computed with the relation we found out and the outcomes in all cases exactly match the known results in the literature (see, e.g., <cit.>).



  
Conformal scalar:
The first instance is of course the conformally coupled scalar field, for which we have

    (4π)^3·7! ⟨ T ⟩ = 5/9· E_6 -28/3· I_1+5/3· I_2+2· I_3+∇_i· J^i ,

with the trivial total derivative term given by 

    ∇_i· J^i    =   6/5· C_1 -2/5· C_2+4· C_3+12/5· C_4+17/5· C_6+12· C_7 ,
    
       =    M_5 - 4· M_7 + 7· M_8 +1/10· M_9 + 6/5· M_10 .




  
Massless Dirac squared:
The second instance of a conformally invariant free field corresponds to the (square of the) massless Dirac operator, for which we have

    (4π)^3·7! ⟨ T ⟩ = 191/9· E_6 -896/3· I_1-32· I_2+40· I_3+∇_i· J^i ,

with the trivial total derivative term given this time by 

    ∇_i· J^i    =    24· C_1 -148/15· C_2+136· C_3+48· C_4-168· C_5 +96· C_6+352· C_7 ,
    
       =   88/3· M_5 - 136· M_7 + 112/3· M_8 -5· M_9 + 24· M_10 .




  
Two-form:
The third instance of a conformally invariant free field corresponds to a two-form after a covariant gauge-fixing, for which we have

    (4π)^3·7! ⟨ T ⟩ = 442· E_6 -8008/3· I_1-2378/3· I_2+180· I_3+∇_i· J^i ,

and with the trivial total derivative term  

    ∇_i· J^i    =    -60· C_1 +2036/15· C_2-1152· C_3-120· C_4-504· C_5 - 646· C_6+856· C_7 ,
    
       =   214/3· M_5 + 312· M_7 - 14/3· M_8 +93· M_9 -60· M_10 .


In summary,


                                 

    conformal field      7!· a     7!· g_5     7!· g_6     7!· g_7     7!· g_8     7!· g_9     7!· g_10     E_o 
                                 
                                
  
   conformal scalar      5/9     1     0     -4      7     1/10     6/5      -31/60480
                                 
                                 

     massless Dirac squared     191/9     88/3     0     -136      112/3     -5     24      -367/24192 
                                 
                                 
   two-form     442     214/3     0     312     -14/3     93     -60      -191/2016 
                                 




As a verification of the above result for the conformal scalar in 6D, we can evaluate directly in the Polyakov formula obtained by Branson <cit.> (see also <cit.>) 


    -   3· 7! (4π)^3/2   logŶ/ Y=10∫_M_6 dv_g  σ·{ Q_6 + 1/2 P_6 σ} + 13{∫_M̂_6 dv_ĝ  |∇̂Ĵ|^2-∫_M_6 dv_g |∇J|^2}
    
    
                  +34{∫_M̂_6 dv_ĝ Ĵ^3-∫_M_6 dv_g J^3}-32{∫_M̂_6 dv_ĝ Ĵ |V̂|^2-∫_M_6 dv_g J |V|^2} ,

in terms of the Schouten tensor V and scalar J. Likewise, for the conformally related metrics we get

    -3· 7! (4π)^3 β· E_o   =   5∫_flatland dv_g  |∇ ∇^2 σ|^2 +34∫_S^1_β× S^5 dv_ĝ Ĵ^3 - 32∫_S^1_β× S^5 dv_ĝ Ĵ |V̂|^2
    
    
               =   β π^3{5·64 + 34· 8 - 32·3  } = 496 β π^3 ,

and we again obtain E_o=-31/60480.

For the Dirac operator, we start with

    -9· 7! (4π)^3/2 log∇̂^2/∇^2   =   -1146∫_M_6 dv_g  σ·{ Q_6 + 1/2 P_6 σ}
       -    507{∫_M̂_6 dv_ĝ  |∇̂Ĵ|^2-∫_M_6 dv_g |∇J|^2}
    
               -    1578{∫_M̂_6 dv_ĝ Ĵ^3-∫_M_6 dv_g J^3}
       +   1752{∫_M̂_6 dv_ĝ Ĵ |V̂|^2-∫_M_6 dv_g J |V|^2} ,

and thus for the conformally related metrics we obtain 

    9· 7! (4π)^3 β· E_o   =   -573∫_flatland dv_g  |∇ ∇^2 σ|^2 -1578∫_S^1_β× S^5 dv_ĝ Ĵ^3 + 1752∫_S^1_β× S^5 dv_ĝ Ĵ |V̂|^2
    
    
               =   β π^3{-573·64 - 1578· 8 + 1752·3  } = -44040 β π^3 ,

implying E_o=-367/24192.



§ SUMMARY AND OUTLOOK

In this work, the relation between the 6D CFT trace anomaly and the Casimir energy due to compactifying the spatial section on a five-sphere has been fully established. The resulting expression extends the 4D result of Cappelli and Coste to 6D by taking account of the trivial total derivatives in the trace anomaly. The scheme-dependence of the  Casimir energy is now controlled by the trivial total derivatives, or equivalently, by the finite counter-terms that can be added to the effective action. We expect this will allow a better understanding of the discrepancies in the vacuum energy in holographic scenarios. The present relation would also be well suited to further explore the connection, within ζ-function regularization, between Casimir energy and multiplicative anomaly <cit.> beyond conformal scalar fields [There is a misprint in eqn. 8.3 of <cit.>, the coefficient of γ_3 should be 1 and not 11. The total derivative ∇^2|V|^2 must also be completed to ∇^2|V|^2+2∇·(V·∇ J) to make it a  trivial total derivative.]. 

Let us conclude by mentioning that going beyond the conformally flat case seems to be quite a challenge (see e.g. <cit.>). One of the obvious difficulties consists in the choice of a fiducial metric, a suitable candidate could be the homogeneously squashed (or Berger), instead of the round, sphere where many explicit calculations are doable (see e.g. <cit.>).  

We thank F. Bastianelli, L. Casarin, J.S. Dowker, E. Friedman, L. Peterson and I.L. Shapiro for valuable conversations and comments. This work was partially funded through FONDECYT-Chile 1220335. We have benefited from the Cadabra software <cit.> to evaluate curvature invariants and the Mathematica package xAct <cit.> to verify their conformal transformations.




§ 6D BASIS OF CURVATURE INVARIANTS


Curvature invariants 

    K_1=R^3  ,   K_2=R  Ric^2  ,   K_3=R  Riem^2  ,   K_4=Ric^3  ,   K_5=-Riem Ric^2
    
          K_6=Ric Riem Riem  ,   K_7=Riem^3  ,   K_8=Riem'^3  ,   K_9=R ∇^2 R
    
    
          K_10=Ric ∇^2 Ric  ,   K_11=Riem ∇^2 Riem  ,   K_12=Ric ∇∇ R  ,   K_13=|∇ Ric|^2
    
    
          K_14=∇ Ric ∇ Ric  ,   K_15=|∇ Riem|^2  ,   K_16=∇^2R^2  ,   K_17=∇^4R .


Trivial anomalies:

    M_5   =    6K_6-3K_7+ 12K_8+K_10-7K_11-11K_13+ 12K_14-4K_15 ,
    
    M_6   =    -1/5 K_9  + K_10 + 2/5 K_12 + K_13 ,
    
    M_7    =    K_4 + K_5 - 3/20 K_9 + 4/5 K_12 + K_14 ,
    
    M_8    =    -1/5 K_9 + K_11 + 2/5 K_12 + K_15 ,
    
    M_9    =    K_16
    
    M_10   =    K_17.


Conformal primitives [These relations were reported in appendix C of <cit.>. We are indebted to F. Bastianelli for pointing this out to us.]:

    ℱ_5=1/30K_1-1/4K_2+K_6  ,  ℱ_6=1/100K_1-1/20K_2  ,
    ℱ_7=37/6000K_1-7/150K_2+1/75K_3-1/10K_5-1/15K_6  ,  ℱ_8=1/150K_1-1/20K_3  ,
    ℱ_9=-1/30K_1  ,  ℱ_10=1/300K_1-1/20K_9  .


Curvature invariants at S^1_β× S^5:

    K_1=R^3=8000  ,   K_2=R  Ric^2=1600  ,   K_3=R  Riem^2=800  ,
    
    
    K_5=-Riem Ric^2=-320  ,   K_6=Ric Riem^2=160  ,   K_9=R ∇^2 R=0  .




§ 6D BRANSON'S BASIS OF CONFORMAL PRIMITIVES

On conformally flat manifolds the general basis of six trivial total derivatives is reduced to only three independent terms, stemming from only three conformal primitives. One convenient choice of conformal primitives consists of J^3, |∇ J|^2 and J|V|^2.  
To compute the trivial total derivatives obtained by the conformal variation of these curvature invariants under g→ g_ω=e^2ωg,  it is enough to keep only the term linear in the conformal factor ω and discard boundary terms since the manifold is closed.

Let us start with J^3 

    ∫_M̂_6dv_ω J_ω^3=∫_M_6dv (J-∇^2ω)^3=∫_M_6dv (J^3-3J^2∇^2ω) → ∫_M_6dv ω (-3∇^2J^2) ,

so that J^3 turns out to be a conformal primitive for -3∇^2J^2.

In the case of |∇ J|^2, as conformal primitive it is equivalent to -J∇^2J since they add up to a boundary term 1/2∇^2J^2. A shortcut to compute the conformal variation is provided by considering the Yamabe operator Y=-∇^2 + 2J instead of the plain Laplacian, and exploit the conformal covariance of Y as follows

    ∫_M̂_6dv_ω J_ωY_ωJ_ω=∫_M_6dv (J-∇^2ω)   Y   (J-∇^2ω)=∫_M_6dv {JYJ-(∇^2ω)YJ-JY∇^2ω)}
        →    ∫_M_6dv ω (-2∇^2YJ) .

The integration by parts performed above is equivalent to the self-adjointness of both the Laplacian and the Yamabe operator. We end up with the conclusion that a conformal primitive for -2∇^2YJ is given by JYJ=J(-∇^2+2J)J or, equivalently, |∇ J|^2+2J^3. Combining this with the previous result for J^3, we obtain that |∇ J|^2 is a conformal primitive for 2∇^2∇^2J+2∇^2J^2.

A shortcut to compute the conformal variation of the remaining curvature invariant J|V|^2 involves the Paneitz operator P acting on the Schouten scalar 

    ∫_M̂_6dv_ω Q_4,ωJ_ω  =   ∫_M̂_6dv_ω P_ωJ_ω   =   ∫_M_6dv e^ωPe^-ω (J-∇^2ω) 
     = ∫_M_6dv {PJ-P∇^2ω+e^ωP^o(e^-ωJ)}   =   ∫_M_6dv {Q_4J-Q_4∇^2ω+JP^oω}
        →    ∫_M_6dv ω (-∇^2Q_4+P^oJ) .

We have kept only terms at most linear in ω and made use of the following properties of the Q-curvature Q_4 and Paneitz operator P in 6D

    P=P^o+Q_4 ,  Q_4=3J^2-2|V|^2-∇^2J ,  P_ω=e^-5ωPe^ω,

where P^o denotes the total derivative part of the Paneitz operator 

    P^o=∇^2∇^2+4∇·(V· - J)∇ .

We have therefore that Q_4J is a conformal primitive of -∇^2Q_4+P^oJ which equals 2∇^2∇^2J-5∇^2J^2+2∇^2|V|^2+4∇·(V·∇ J).
With all the instances gathered so far, we can finally disentangle the -J|V|^2 term in Q_4J, and obtain that it is a conformal primitive for ∇^2|V|^2+2∇·(V·∇ J)+∇^2J .
We summarize the above results, which have also been independently verified with the xAct package for Mathematica,  in the table below.



     

        trivial total derivative             conformal primitive     
    
    
  
   ∇^2∇^2J     1/2|∇ J|^2 +1/3J^3 
      
      
 ∇^2 J^2     -1/3J^3
      
    
  
   ∇^2|V|^2+2∇·(V·∇ J)     -J|V|^2+1/3J^3
      




Endowed with this correspondence, we can now determine the contribution to the partition function that originates in the trivial total derivatives of the trace anomaly on conformally flat backgrounds

    (4π)^3·⟨ T ⟩   =    -a_6 · E_6 + γ_1∇^2∇^2J + γ_2∇^2J^2+γ_3{∇^2|V|^2+2∇·(V·∇ J)}
    
       =   -6 a_6 · Q_6 + γ̃_1∇^2∇^2J + γ̃_2∇^2J^2+γ̃_3{∇^2|V|^2+2∇·(V·∇ J)} .

The contribution to the partition function from the trivial total derivatives reduces to the evaluation on the cylinder of their corresponding conformal primitives. Collecting similar terms, evaluating the conformal primitives J^3=8, |∇ J|^2=0 and J|V|^2=3, we arrive at volume times a particular combination of coefficients

    β π^3/64π^3{8γ_1/3-8γ_2/3+8γ_3/3 -3γ_3}=β/24{γ_1-γ_2-1/8γ_3 } ,

so that we obtain for the Casimir energy in the Euler basis 

    E_o=-15/8a_6-1/24(γ_1-γ_2 -1/8γ_3) ,

or, alternatively, in the Q-curvature basis

    E_o=-3a_6-1/24(γ̃_1-γ̃_2 -1/8γ̃_3) .

The degeneracy of the basis for total derivatives at conformally flat manifolds is summarized in the table below. 



     

        general basis of t.t.d.             reduced basis of t.t.d.     
    
    
  
   M_5     -24{∇^2|V|^2+2∇·(V·∇ J)}-7∇^2J^2 
      
    
  
   M_6     8{∇^2|V|^2+2∇·(V·∇ J)}-∇^2J^2 
      
      
 M_7     8{∇^2|V|^2+2∇·(V·∇ J)}-7/2∇^2J^2
      
    
  
   M_8     8{∇^2|V|^2+2∇·(V·∇ J)}-6∇^2J^2
      
      
 M_9     100∇^2J^2
      
    
  
   M_10     10∇^2∇^2J
      




This dictionary allows a direct verification of the vanishing on conformally flat manifolds of the freedom in Anselmi's pondered Euler density (eqn.<ref>) and the two freedoms found out by Hamada (eqns.<ref> and <ref>). It also allows to verify the relation for the Casimir energy in terms of the coefficients of the reduced basis, collecting their coefficients 

    E_o   =   -15/8a_6-1/24([10g_10]-[-7g_5-g_6-7/2g_7-6g_8+100g_9] -1/8[-24g_5+8g_6+8g_7+8g_8])
    
       =   -15/8a_6-5/12(g_5+1/4g_7+1/2g_8-10g_9+g_10) .






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