Supersymmetric Renyi Entropy and Anomalies in Six-Dimensional (1,0) Superconformal Theories
aa r X i v : . [ h e p - t h ] F e b Preprint typeset in JHEP style - HYPER VERSION
TAUP-3014/17
Supersymmetric R´enyi entropy andAnomalies in d ( , ) SCFTs
Shimon Yankielowicz , Yang Zhou
School of Physics and Astronomy, Tel-Aviv UniversityRamat-Aviv 69978, Israel
E-mails : [email protected], [email protected]
Abstract:
A closed formula of the universal part of supersymmetric R´enyi entropy S q for six-dimensional ( , ) superconformal theories is proposed. Within our argu-ments, S q across a spherical entangling surface is a cubic polynomial of ν = / q , with4 coefficients expressed as linear combinations of the ’t Hooft anomaly coefficients forthe R -symmetry and gravitational anomalies. As an application, we establish linearrelations between the c -type Weyl anomalies and the ’t Hooft anomaly coefficients.We make a conjecture relating the supersymmetric R´enyi entropy to an equivariantintegral of the anomaly polynomial in even dimensions and check it against knowndata in 4 d and 6 d . ontents
1. Introduction 1
2. Free d ( , ) multiplets 6
3. Interacting d ( , ) SCFTs 9
4. Relation with supersymmetric Casimir energy 125. Relation with anomaly polynomial 14
6. Discussion 17A. Perturbative expansion around q = A.1 S ′ q = and I ′′ q = S ′′ q = and I ′′′ q =
1. Introduction
Six-dimensional superconformal theories provide a framework to understand variousfeatures of lower-dimensional supersymmetric dynamics. By themselves, they aredifficult to study by traditional quantum field theory techniques. All known examplesof interacting CFTs in six dimensions are supersymmetric. The ( , ) theories shouldbe the simplest ones [1–3]. A large class of interacting ( , ) fixed points have beenconstructed in string theory or brane constructions [4–8]. Recently, F-theory providesa way to classify the known and new ( , ) fixed points [9–11].Since all the known interacting fixed points are supersymmetric, it is expectedthat supersymmetry constraints are important in computing their physical charac-teristic quantities, such as Weyl anomalies. Indeed, the a -anomaly in ( , ) super-conformal theories has been recently determined in terms of their ’t Hooft anomaly1oefficients [12,13] for the R -symmetry and gravitational anomalies [14] by analyzingsupersymmetric RG flows on the tensor branch [15] ¯ a = aa u ( ) = ( α − β + γ ) + δ , (1.1)where α , β , γ , δ are the coefficients appearing in the anomaly polynomial I = ( α c ( R ) + β c ( R ) p ( T ) + γ p ( T ) + δ p ( T )) . (1.2)Here c ( R ) is the second Chern class of the R -symmetry bundle and p , are thePontryagin classes of the tangent bundle. The relation (1.1) is analogues to the knownrelation [18] in four-dimensional N = a d = = k RRR − k R , where k RRR and k R are the Tr U ( ) R and Tr U ( ) R ’t Hooft anomalies. Although the anomalymultiplet in six dimensions has not yet been constructed, such linear relations arebelieved to follow from the anomaly supermultiplets which include ’t Hooft anomaliesas well as the anomalous trace of the stress tensor. The Weyl anomaly coefficientsin 6 d are defined from the latter [19–22] ⟨ T µµ ⟩ ∼ a E + ∑ i = c i I i , (1.3)where E is the Euler density and I i = , , are three Weyl invariants. In the presenceof ( , ) supersymmetry, c i = , , , satisfying a constraint c − c + c = a -anomaly determines both the uni-versal log divergence of the round-sphere partition function and the universal logdivergence in the vacuum state entanglement entropy associated with a ball in flatspace [31]. On the other hand, by the conformal Ward identities, the 2-point and 3-point functions of the stress tensor in the vacuum in flat space can be determined upto 3 coefficients [32, 33], which are linearly related to c -type Weyl anomalies c , , . Inthe presence of ( , ) supersymmetry, only two of them are independent as mentionedbefore. The subscript u ( ) means an Abelian ( , ) tensor multiplet. See [16] for the result in ( , ) theories and [17] for earlier investigation. The a -anomaly is proportional to the coefficient of the log divergence. q -deformed) sphere, which is directly related to the supersymmetric R´enyi entropy S q . Supersymmetric R´enyi entropy was first introduced in three dimensions [34–36], and later studied in four dimensions [37–39], in five dimensions [40, 41], in sixdimensions ( ( , ) theories) [42,43] and also in two dimensions ( ( , ) SCFTs) [44,45].By turning on certain R -symmetry background fields, one can calculate the partitionfunction Z q on a q -branched sphere S dq , and define the supersymmetric R´enyi entropyas S q = qI − I q − q , I q ∶ = − log Z q [ µ ( q )] , (1.4)which is a supersymmetric refinement of the ordinary R´enyi entropy (which is notsupersymmetric because of the conical singularity). The quantities defined in (1.4)are UV divergent in general but one can extract universal parts free of ambiguities.
The main result of this paper is the exact universal part of the supersymmetric R´enyientropy in 6 d ( , ) SCFTs. We show that, for theories characterized by the anomalypolynomial (1.2), it is given by a cubic polynomial of ν = / qS ( , ) ν = ∑ n = s n ( ν − ) n , (1.5)with four coefficients s = ( α − β + γ + δ ) ,s = ( α − β + γ + δ ) ,s = ( α − β + γ ) ,s = ( α − β + γ ) . (1.6)where α , β , γ , δ are the ’t Hooft anomaly coefficients defined in (1.2). The basicingredients in our arguments are the following: A branched sphere is a sphere with a conical singularity with the deformation parameter q − For CFTs, the R´enyi entropy (or the supersymmetric one) associated with a spherical entanglingsurface in flat space can be mapped to that on a sphere. Throughout this work we take the“regularized cone” boundary conditions. S ν of ( , ) free hyper multiplet and free tensor multiplet can be computed bythe heat kernel method closely following [43]. The results are given by S hν = ( ν − ) + ( ν − ) + ( ν − ) + , (1.7) S tν = ( ν − ) + ( ν − ) + ( ν − ) + . (1.8)These are the main results of Section 2.(B) S ν of A N − type ( , ) theories (which are of course ( , ) conformal theories)in the large N has been computed in [43]. The result is given by S ν [ A N → ∞ ] N = ( ν − ) + ( ν − ) + ( ν − ) + . (1.9)(C) Based on (A)(B) and (F) below, a reasonable assumption is that the generalform of S ν for ( , ) SCFTs is a cubic polynomial in ν −
1. However, so farwe do not have a sharp argument for this assumption. Furthermore, basedon (D)(E)(F) below, the four coefficients of the cubic polynomial are linearcombinations of α, β, γ, δ .(D) The value of S ν at ν = a -anomaly (1.1).(E) The first and second derivatives of S ν at ν = ν = c and c , ∂ ν S ν ∣ ν = = c − c , ∂ ν S ν ∣ ν = = c − c , (1.10)where c and c are believed to be given by linear combinations of ’t Hooftanomaly coefficients α, β, γ, δ .(F) The large ν behavior of S ν is controlled by the “supersymmetric Casimir en-ergy” [46]. This gives lim ν → ∞ S ν ν = ( α − β + γ ) . (1.11) We are interested only in the universal part, i.e. the coefficient of the UV log divergent part.This part should be given by a finite number of counter-terms, each of them an integral of localfunctions of the supersymmetric background including the metric (squashed sphere). Unfortunatelythe supersymmetric smooth squashed sphere in 6 d has not yet been constructed. ν expansion, the second Pontryagin class (with coefficient δ ) willnot contribute to the ν term (as we see from (F)) and the ν term. Becauseof the latter, one has ∂ δ ( ∂ ν S ν ∣ ν = ) = . (1.12)(H) For the conformal non-unitary ( , ) vector multiplet, a constraint for the c -type Weyl anomalies, c + c = , can be obtained by studying the higher-derivative operators on the Ricci flat background [26]. Together with (E),one has16 ( ∂ ν S “Vector” ν ∣ ν = ) − ( ∂ ν S “Vector” ν ∣ ν = ) = ( c + c )∣ “Vector” = . (1.13)From (A)(B)(C)(D)(E)(F)(G)(H), one can uniquely find the general expression ofthe supersymmetric R´enyi entropy given in (1.5)(1.6). We emphasize that among allthese ingredients (C) is an assumption, all the rest are derived results. The results(A),(D),(E),(F),(G) are new as far as we know. The precise agreement between(F) and (A)(B) can be considered as a nontrivial test of (F). Independently, weconjecture a relation between the supersymmetric R´enyi entropy and the anomalypolynomial in any even dimension, which perfectly agrees with (A)-(H). We considerthis precise agreement as a strong support of our result (1.5)(1.6). Note that (E)and (1.6) also establish the linear relations between c -type Weyl anomalies and the’t Hooft anomaly coefficients, c = − ( α − β + γ + δ ) ,c = − ( α − β + γ + δ ) ,c = ( α − β + γ + δ ) . (1.14)This paper is organized as follows. In Section 2 we employ heat kernel methodto study the supersymmetric R´enyi entropy of free ( , ) multiplets. In Section 3 wepropose a form of the universal supersymmetric R´enyi entropy with four non-trivialcoefficients, which works for general 6 d ( , ) SCFTs. We determine the coefficientsone by one. We study the relation between the supersymmetric R´enyi entropy andthe supersymmetric Casimir energy in Section 4, which is used to determine one of We thank Matteo Beccaria for explaining us this result first presented in [26]. The numerical coefficients for the c -anomalies here are different from those presented in [26],where an assumption concerning the structure of the linear combinations was made. We thankMatteo Beccaria for discussion on this issue. After our paper appeared on the arXiv the authorsof [26] clarified to us that the data they used did not allow them to fix c , , unambiguously. Therewas still a 1-parameter freedom consisting with our result. They fixed this freedom by anotherassumption/conjecture relating anomalies in 4 d and 6 d . d and 4 d . In Section 6, we discuss some openquestions, further applications of our results and some future directions of research.
2. Free d ( , ) multiplets We begin by studying the supersymmetric R´enyi entropy of free ( , ) multiplets,following [42]. For free fields, the R´enyi entropy associated with a spherical entanglingsurface in flat space can be computed by conformally mapping the conic space toa hyperbolic space S β × H and using the heat kernel method. A six-dimensional ( , ) hyper multiplet includes 4 real scalars, 1 Weyl fermion and a tensor multipletincludes 1 real scalar, 1 Weyl fermion and a 2-form field with self-dual strength.The 2-form field has a self-duality constraint which reduces the number of degreesof freedom by half. The partition function of free fields on S β = πq × H can be computed by the heatkernel log Z ( β ) = ∫ ∞ dtt K S β × H ( t ) , (2.1)where K S β × H ( t ) is the heat kernel of the associated conformal Laplacian. The kernelfactorizes because the spacetime is a direct product, K S β × H ( t ) = K S β ( t ) K H ( t ) . (2.2)On a circle, the kernel is given by K S β ( t ) = β √ πt ∑ n ≠ , ∈ Z e − β n t . (2.3)In the presence of a chemical potential µ , it will be twisted [47] ̃ K S β ( t ) = β √ πt ∑ n ≠ , ∈ Z e − β n t + i πnµ + iπnf , (2.4)where f controls the periodic/anti-periodic boundary conditions, namely f = f = H is homogeneous. Thus K H ( t ) can be writtenin terms of the equal-point kernel, K H ( t ) = ∫ d x √ g K H ( x, x, t ) = V K H ( , t ) . (2.5) In this section we use β = / T as the inverse temperature and hopefully this will not be confusingwith the anomaly coefficient β . For R´enyi entropy of free fields in other dimensions less than six, see for instance [48–51]. V = π log ( ℓ / ǫ ) , where ǫ is actually the UV cutoffin the flat space before the conformal mapping and ℓ is the curvature radius of H . For the K H ( , t ) of free fields with different spins we refer to [42] and referencesthere in.The R´enyi entropy of a hyper multiplet can be obtained by summing up thecontributions of 4 real scalars, 1 Weyl fermion and the R´enyi entropy of a tensormultiplet can be obtained by summing up the contributions of 1 real scalar, 1 Weylfermion and a self-dual 2-form, S hyperq = × S sq + S fq , (2.6) S tensorq = S sq + S fq + S vq . (2.7)where the R´enyi entropy for free fields with different spins can be computed by usingthe corresponding heat kernels. The final results for the R´enyi entropy of a 6 d complex scalar, a 6 d Weyl fermion and a 6 d S sq = ( q + ) ( q + ) ( q + ) q V π , (2.8) S fq = ( q + ) ( q + q + ) q V π , (2.9) S vq = ( q + ) ( q + ) + q + q q V π , (2.10)respectively. Note that, to obtain the correct R´enyi entropy for the two form field, wehave taken a q -independent constant shift which is associated with possible boundarycontributions [42]. Before moving on, let us represent S q in terms of S ν = π V S q , with ν = / q . The R´enyi entropy of free ( , ) multiplets are given by S hyperν = ( ν − ) + ( ν − ) + ( ν − ) + ( ν − ) + ν − + , (2.11) S tensorν = ( ν − ) + ( ν − ) + ( ν − ) + ( ν − ) + ( ν − ) + . (2.12)The reason why S ν is convenient is obvious, the series expansion near ν = S q near q = S ν instead of S q to express R´enyi entropy and supersymmetric R´enyi entropy from In the replica trick approach to compute the entanglement/R´enyi entropy, this is the q -foldspace with a conical singularity. For some relevant details of this computation we refer to [42]. q and the derivatives with respect to ν at q = / ν = ∂ ν S ν ∣ ν = = − ∂ q S q ∣ q = ⋅ π V , ∂ ν S ν ∣ ν = = ( ∂ q S q + ∂ q S q ) ∣ q = ⋅ π V , (2.13)which will be useful later. Finally, one can check that ∂ q = , ∂ q = and ∂ q = of both S hyperq and S tensorq are consistent with the previous results about the free ( , ) multiplets [23,29]. By “consistent”, we refer to the relations between the first and the secondderivatives of the R´enyi entropy at q = The supersymmetric R´enyi entropy of free multiplets can be computed by the twistedkernel (2.4) on the supersymmetric background. The R -symmetry group of 6 d ( , ) theories is SU ( ) R , which has a single U ( ) Cartan subgroup. Therefore one canturn on a single R -symmetry background gauge field (chemical potential) to twistthe boundary conditions for scalars and fermions along the replica circle S β [47]. The R -symmetry chemical potential can be solved by studying the Killing spinor equationon the conic space ( S q or S β = πq × H ), µ ( q ) ∶ = k A τ = q − , (2.14)with k being the R -charge of the Killing spinor under the Cartan U ( ) . We choose k = / A τ = ( q − ) . (2.15)For each component field in the free multiplets, one has to first figure out the associ-ated Cartan charge k i and then compute the chemical potential by k i A τ . After thatone can compute the free energy on S β × H using the twisted heat kernel with thechemical potential µ = k i A τ and obtain the supersymmetric R´enyi entropy.After summing up the component fields, the supersymmetric R´enyi entropy of afree ( , ) hyper multiplet and a free ( , ) tensor multiplet are S hν = ( ν − ) + ( ν − ) + ( ν − ) + , (2.16) S tν = ( ν − ) + ( ν − ) + ( ν − ) + , (2.17)respectively. See the appendix in [42]. . Interacting d ( , ) SCFTs
Having obtained the free multiplet results (2.16)(2.17), we will use them to rewrite S ( , ) ν in a general form which, we hope, works for general interacting 6 d ( , ) SCFTs, S ( , ) ν = A ( ν − ) + B ( ν − ) + C ( ν − ) + D , (3.1)where the coefficients
A , B , C , D will depend on the specific theories. Before determining
A , B , C , D for general ( , ) fixed points, let us summarizewhat we have learned so far for the existing examples. These are free ( , ) hypermultiplet, free ( , ) tensor multiplet, A N − type ( , ) theories in the large N limitand non-unitary conformal ( , ) vector multiplet [26,30]. We list A , B , C , D and therelevant anomaly data for them in Table 1. The anomaly data are from [14,21,23].
Table 1:
Supersymmetric R´enyi entropy and anomalies of known ( , ) fixed points A B C D α β γ δ c c c Hyper − − − Tensor
12 23240 − − − A N − N − −
13 19 “Vector” − − − − − − − − − − The coefficient D in (3.1) can be determined by using the fact that, the entan-glement entropy associated with a spherical entangling surface, which is nothing but S ν = , is proportional to the Weyl anomaly a . This is true for general CFTs in evendimensions as shown in [31]. Therefore S ( , ) ν = S ( , ) ν = = aa u ( ) = ∶ ¯ a . (3.2)By studying supersymmetric RG flows on the tensor branch, a / a u ( ) has been com-puted in [15], see (1.1). This allows us to fix D = S ( , ) ν = = ( ( α − β + γ ) + δ ) = ( α − β + γ ) + δ . (3.3)The coefficients C and B in (3.1) are the first and the second ν -derivatives of S ( , ) ν at ν =
1, respectively. The transformations between the ν -derivatives and the q -derivatives are given by (2.13). The relations between the q -derivatives and theintegrated correlators are given in Appendix A. Namely, the first q -derivative at This structure is not true for the ordinary (non-supersymmetric) R´enyi entropy [54]. We denote the conformal non-unitary vector multiplet by “Vector”. = ⟨ T T ⟩ and integrated ⟨ J J ⟩ in(A.23), S ′ q = = − V d − ⎛⎝ π d + Γ ( d )( d − )( d + ) ! C T − g π d + d − ( d − ) Γ ( d − ) C J ⎞⎠ . (3.4)This relation holds for general SCFTs with conserved R -symmetry in d -dimensions.Similarly the second q -derivative at q = R -current 3-point functionand some mixed 3-point functions. This is given explicitly in (A.27) S ′′ q = = I ′′′ q = = π [⟨ ˆ E ˆ E ˆ E ⟩ c − g ⟨ ˆ Q ˆ Q ˆ Q ⟩ c − g ⟨ ˆ E ˆ E ˆ Q ⟩ c + g ⟨ ˆ E ˆ Q ˆ Q ⟩ c ] S q = × H d − . (3.5)In 6 d ( , ) SCFTs, by the conformal Ward identities, the two- and three-pointfunctions of the stress tensor multiplet (including R -current) may be determined interms of two independent coefficients, which are linearly related to c and c . Becauseof this, C and B in (3.1) are also linear combinations of c and c . These relationscan be obtained by fitting to the free hyper multiplet and the free tensor multipletin Table 1, B = c − c , C = c − c . (3.6)Assuming B and C are linear combinations of α , β , γ , δ , we shall establish theexplicit relations. Because the second Pontryagin class p ( T ) does not contribute tothe ν term, we get ∂ δ B = . (3.7)To see that the ν term is independent of p ( T ) , let us consider the free energy on S q × H , which can be used to compute S q because S q × H is conformally equivalentto S q or S q × H . S q × H is similar to S q × S β → ∞ , but they are not the same dueto different boundary conditions on H and S β . The latter background preservingsupersymmetry is used to compute the supersymmetric Casimir energy in 6 d . Onecan formally define a supersymmetric R´enyi entropy on S q × S β → ∞ with the R´enyiparameter q by using the free energy βE c [ S q ] . As we will see in the next section, p ( T ) will not contribute to the 1 / q term in this supersymmetric R´enyi entropy,because p ( T ) contributes to E c in the following way (4.7) p ( T ) ω ω ω → ω ω ω ∑ i < j ω i ω j , ω = ω = , ω = / q . (3.8)The different boundary conditions on S q × H will not change the property that the1 / q term is independent of δ . We further confirm this fact by establishing a concreterelation between S q and the anomaly polynomial in Section 5.10ince B depends only on α , β , γ , it can be fixed by fitting to the three indepen-dent examples, the free hyper multiplet, the free tensor multiplet and the A N − typetheories in the large N , B = ( α − β + γ ) . (3.9)The same fitting method can be used to determine the α , β , γ , δ dependence of C ,but since C depends on all four of them, one free parameter is left. We fix theremaining free parameter by making use of the result of c + c for the conformalnon-unitary ( , ) vector multiplet in [26] (obtained by the heat kernel computationon the Ricci flat background) ( c + c )∣ “Vector” = . (3.10)Thus, the coefficient C as a linear combination of α , β , γ , δ is determined C = ( α − β + γ + δ ) . (3.11)(3.6)(3.9)(3.11) also establish the linear relations between c , , and α , β , γ , δc = − ( α − β + γ + δ ) ,c = − ( α − β + γ + δ ) ,c = − ( c − c ) = ( α − β + γ + δ ) . (3.12)The remaining coefficient A will be fixed as A = ( α − β + γ ) . (3.13)in the next section by studying the large ν behavior of the supersymmetric R´enyientropy. Obviously, the leading contribution in the limit ν → ∞ is determined onlyby A . As a summary, we can completely determine a closed formula for the universal partof supersymmetric R´enyi entropy for 6 d ( , ) SCFTs, S ( , ) ν = ( α − β + γ )( ν − ) + ( α − β + γ ) ( ν − ) + ( α − β + γ + δ )( ν − ) + ( α − β + γ + δ ) . (3.14)Given that ’t Hooft anomalies for general 6 d ( , ) SCFTs can be computed [13],the above formula tells us the universal supersymmetric R´enyi entropy for any ( , ) SCFT. 11or ( , ) theories labeled by a simply-laced Lie algebra g , (3.14) reduces to [43] S ( , ) ν = ( ¯ c − ¯ a ) H ν + ( a − c ) T ν , (3.15)where ¯ a and ¯ c are determined by the rank, dimension and dual Coxeter number of g , ¯ a = d g h ∨ g + r g , ¯ c = d g h ∨ g + r g . (3.16) T ν and H ν are the supersymmetric R´enyi entropy of the ( , ) tensor multiplet andthat of the ( , ) supergravity (large N ), respectively T ν = ( ν − ) + ( ν − ) + ( ν − ) + , (3.17) H ν = ( ν − ) + ( ν − ) + ( ν − ) + . (3.18)
4. Relation with supersymmetric Casimir energy
In this section we clarify the relation between the supersymmetric R´enyi entropy andthe supersymmetric Casimir energy in 6 d . Similar relation in 4 d has been obtainedin [38]. Recall that the partition function Z on M D − × S ̃ β is determined by theCasimir energy on the compact space M D − in the limit ̃ β → ∞ E c ∶ = − lim ̃ β → ∞ ∂ ̃ β log Z ( ̃ β ) , (4.1)which is equivalent to the statement lim ̃ β → ∞ log Z ( ̃ β ) = − ̃ βE c . (4.2)We consider the cases with supersymmetry. In even-dimensional superconformaltheories, the supersymmetric Casimir energy on S × S D − has been conjectured to beequal to the equivariant integral of the anomaly polynomial in [46], where the authorsprovided strong supports for this conjecture by examining a number of SCFTs in two,four and six dimensions. The equivariant integration is defined with respect to theCartan subalgebra of the global symmetries (that commute with a given supercharge)and one can write this as E D ( µ j ) = ∫ µ j I D + , (4.3)where the equivariant parameters µ j are the chemical potentials corresponding tothe Cartan generators. In equivariant cohomology, doing the integration (4.3) in 6 d is equivalent to the replacement rules (4.7). In this section we use ̃ β = / T for the inverse temperature in order to distinguish it from the ’tHooft anomaly β . For 6 d superconformal index, see [55–57]. See the appendix in [46] for details on the equivalence. d ( , ) SCFTs on S ̃ β × S ⃗ ω with squashing parameters ⃗ ω = ( ω , ω , ω ) . The squashing parameters are defined by coefficients appearing in theKilling vector K = ω ∂∂φ + ω ∂∂φ + ω ∂∂φ , (4.4)where φ , φ , φ are three circles representing the U ( ) isometries of the 5-sphere.The supersymmetric Casimir energy of superconformal ( , ) theories is given by theequivariant integral (4.3) E ( , ) ( µ j ) = − ∫ µ j I , (4.5)where the 8-form anomaly polynomial is I = ( α c ( R ) + β c ( R ) p ( T ) + γ p ( T ) + δ p ( T )) (4.6)as introduced in the introduction. The integration (4.5) is equivalent to the followingreplacement rules [46] c ( R ) → − σ , p ( T ) → ∑ i = ω i , p ( T ) → ∑ i < j ω i ω j , (4.7)where σ is the chemical potential for the R -symmetry Cartan and ω , , are thechemical potentials for the rotation generators (commuting with the supercharge).After the replacement, the result should be divided by the equivariant Euler class, e ( T ) = ω ω ω . (4.8)In the particular background of S q × S ̃ β , where S q is a q -deformed 5-sphere with themetric d s = ( sin θ + q cos θ ) d θ + q sin θ d τ + cos θ dΩ , (4.9)one should identify the shape parameters as ω = ω = , ω = q . (4.10)Note that there is a supersymmetric constraint for the chemical potentials, σ = ∑ j ω j . Evaluating (4.5) one obtains E ( , ) = − ω ω ω ⎛⎝ α σ − β σ ∑ j = ω j + γ ( ∑ j = ω j ) + δ ( ∑ i < j ω i ω j )⎞⎠ . (4.11)Therefore the free energy in the q → f [ S q → × S ̃ β → ∞ ] = ̃ βπ / ̃ βE c ∣ q → = − π α − β + γq , (4.12) We consider the minimal set of global symmetries without extra flavor symmetries. f ∶= IV , I ∶= − log Z . q -independent volume factor Vol [ D × S ̃ β ] = ̃ βπ / S q × H and S q × H , we have f [ S q → × S β → ∞ ] = f [ S q → × H ] = f [ S q → × H ] , (4.13)where the first equality follows from the background coincidence and the second onefollows from the conformal invariance of (supersymmetric) R´enyi entropy and S q → = − I q → , I q ∶ = − log Z q . (4.14)From (4.13) we obtain the asymptotic supersymmetric R´enyi entropy on S q × H S q → = − I q → = α − β + γq . (4.15)This fixes the undetermined coefficient A in (3.1) as A = ( α − β + γ ) . (4.16)Notice that this result perfectly agrees with the supersymmetric R´enyi entropy ofthe known ( , ) fixed points listed in Table 1.
5. Relation with anomaly polynomial
Inspired by the relation between the supersymmetric Casimir energy and the anomalypolynomial [46], we conjecture in this section a relation between the supersymmetricR´enyi entropy and the anomaly polynomial. Following this relation, the supersym-metric R´enyi entropy in even dimensions can be extracted directly from the anomalypolynomial of the theory. We conjecture that S q is determined by an equivariantintegral of the anomaly polynomial I D + with respect to the subalgebra formed bygenerators ( r, h j = ,...D / , h [ D + ] ) , where r is the R -symmetry Cartan generator and h j is the j -th orthogonal rotation generator in R D , while h [ D + ] generates an additional U ( ) rotation. We emphasize that we do not have yet a physical understanding ofthe extra U ( ) , but just employ it in the same way as the other rotational U ( ) ’s.We will check our conjecture against existing data in 6 d and 4 d . To simplify thenotation, we will use ̃ h = h [ D + ] from now on. The Cartan generators commutingwith a given supercharge Q have the corresponding chemical potentials denoted by σ, ⃗ ω, ̃ ω . Define an equivariant integral F ( σ, ⃗ ω, ̃ ω ) = ∫ ( σ, ⃗ ω, ̃ ω ) I D + (5.1) One can come up, for now, with some loose arguments that this equivariant integral gives thecoefficient of the universal log divergence in the free energy on a general D -dimensional squashedsphere. S q = V H qF − F q − q , F q = F ( σ, ⃗ ω, ̃ ω )∣ ⃗ ω = ⃗ , ̃ ω = / q . (5.2)Note that in the second equation in (5.2), the supersymmetric constraint for thechemical potentials was implicitly assumed. A volume V H = ( ℓ / ǫ ) was factorizedin S q because we work effectively on S D − q × H . We will test this conjecture for SCFTsin 4 d and 6 d in the following subsections. We have not been able, so far, to provethis conjecture. The fact that an equivariant integral appears in this conjecture mayhint towards some localization. In R , there is a U ( ) subalgebra in the rotation symmetries. The generatorscommuting with the supercharge have the corresponding chemical potentials, ω , , .The additional chemical potential is ̃ ω = ω . Consider superconformal theories with SU ( ) R R -symmetry. For the 8-form anomaly polynomial given in (4.6), the replace-ment rule in carrying out the equivariant integration (5.1) should be c ( R ) → − σ , p ( T ) → ∑ i = ω i , p ( T ) → ∑ i < j ω i ω j . (5.3)After these replacements in the anomaly polynomial, we divide it by ̃ e ( T ) = ω ω ω ω .The result is given by F ( σ, ω , , , ) = − ω ω ω ω ⎛⎝ α σ − β σ ∑ j = ω j + γ ( ∑ j = ω j ) + δ ( ∑ i < j ω i ω j )⎞⎠ . (5.4)Upon plugging in σ = ∑ i = ω i , ω = ω = ω = , ω = / q , (5.5)one obtains S q V H = qF − F q − q = α − β + γ q + α − β + γ q + α − β + γ + δ q + ( α − β + γ + δ ) . (5.6)The above result can be rewritten as S ν , S ν = ( ν − ) ( α − β + γ ) + ( ν − ) ( α − β + γ ) + ( ν − )( α − β + γ + δ ) + ( α − β + γ + δ ) . (5.7)15his agrees precisely with (3.14). Remarkably, a single conjectured formula by theequivariant integral (5.2) can give the a -anomaly, c , , -anomalies and also a certainpart of the supersymmetric Casimir energy simultaneously and precisely. We considerthese agreements as a strong support of both our results (1.5) and the conjectureitself. In R , there is a U ( ) subalgebra in the rotation symmetries. The generatorscommuting with the supercharge have the corresponding chemical potentials, ω , .The additional chemical potential is ̃ ω = ω . Consider superconformal theories with U ( ) R R -symmetry. The 6-form anomaly polynomial is I = ( k RRR c ( R ) + k R c ( R ) p ( T )) . (5.8)The supersymmetric Casimir energy is given by the equivariant integral of I [46] E = ∫ I = k RRR ω ω σ − k R ω ω ( ω + ω ) σ , (5.9)where the chemical potentials satisfy a supersymmetric constraint σ = ( ω + ω ) .Note that the relation between the conformal and the ’t Hooft anomalies in a 4 d N = k RRR = ( a − c ) , k R = ( a − c ) . (5.10)Plugging this in (5.9), one reproduces the familiar result [58, 59] E = ( a − c )( ω + ω ) + ( c − a ) ( ω + ω ) ω ω . (5.11)For our purpose, the equivariant parameters have been generalized to σ, ω , ω , ω .The equivariant integration (5.9) now becomes F ( σ, ω , , ) = k RRR ω ω ω σ − k R ω ω ω ( ω + ω + ω ) σ , (5.12)with a constraint σ = ( ω + ω + ω ) . Evaluating the supersymmetric R´enyi entropy(5.2), one obtains S ν = ( k R − k RRR ) + ( k R − k RRR ) ( ν − ) + ( k R − k RRR )( ν − ) , (5.13) = − a − c ( ν − ) − ( c − a ) ( ν − ) . (5.14)This is precisely the universal supersymmetric R´enyi entropy in 4 d N =
1. A fewremarks are in order. The leading coefficient in large ν , − ( c − a )/
27, preciselyagrees with the result in [38]. The first ν -derivative at ν = − c /
3, agrees with16A.23). The constant term, − a , agrees with the linear relation between the a -anomaly and the entanglement entropy in [31]. From (5.13) to (5.14), we have usedthe relations (5.10). Demanding the equivalence between (5.13) and (5.14), onecan reproduce the famous known relations between the conformal and the ’t Hooftanomalies.
6. Discussion
In this paper we proposed a closed formula for the universal log term of the six-dimensional supersymmetric R´enyi entropy and made a conjecture that the super-symmetric R´enyi entropy in even dimensions is equal to an equivariant integral ofthe anomaly polynomial. It remains a challenging problem to understand the extra U ( ) and to prove this conjecture. We leave it for future work.Let us mention a few other open question and further directions of research thatare related to this work.1. Proving our assumption that the expansion of the supersymmetric R´enyi en-tropy in 1 / q terminates (it is just a polynomial of 1 / q with degree 3 in 6 d ).For this we need the dependence of possible counter-terms on 1 / q . Hence, wehave to construct the six-dimensional supersymmetric curved background andin particular the smooth squashed six-sphere. The super-Weyl anomalies con-structed on this background will give the universal part of the supersymmetricR´enyi entropy. This approach will, hopefully, allow us to prove our assumption(C) in the introduction.2. A generalization of the discussion in appendix A implies that the third deriva-tive of the supersymmetric R´enyi entropy is related to a specific linear com-bination of 4-point functions of the stress tensor and other operators in itsmultiplet. On the other hand, according to our result (1.5)(1.6) it is related to s and hence via (1.1) and (1.14) to the Weyl anomalies. In 6 d , this is indeedconsistent with a long time expectation that the a -anomaly should determinesome specific term in the 4-point function of the stress tensor. This consistencybecomes manifest for ( , ) theories (3.15). It would be nice to demonstratethe relation between S ′′′ ν ∣ ν = and the integrated 4-point functions of operatorsin the stress tensor multiplet in a straight forward way.3. The supersymmetric R´enyi entropy has been proven to satisfy the four inequal-ities [60], ∂ q S q ≤ , ∂ q ( q − q S q ) ≥ , ∂ q (( q − ) S q ) ≥ , ∂ q (( q − ) S q ) ≤ . (6.1) In the sense that ∂ ν S ν = is a particular linear combination of C T and C J , therefore proportionalto c . d N = ≤ ac ≤ .Notice that the lower bound is not as tight as the 4 d N = d ( , ) superconformal theories, plugging (3.14) into (6.1) oneobtains P ∶ = α − ( β − γ ) ≥ , (6.2) P ∶ = α − β ≥ , (6.3)67 α − β + ( γ + δ ) ≥ , (6.4)9 α − ( β − γ − δ ) ≥ . (6.5)It is interesting to clarify the relations among different bounds in 6 d : theinformation theory bounds shown above, the unitary bound C T ∝ c ≥ α − β + γ + δ ≥ , (6.6)and the 6 d supersymmetric Hofman-Maldacena bounds (obtained by free-multipletestimaiton) in terms of α, β, γ, δ P ∶ = α − β + γ + δ ≥ P ∶ = α − β + γ − δ ≥ . (6.8)It is interesting to notice that, P is equal to s in (1.6) and, from P ≥ P ≥ C T ∝ c ∝ s ≥ s ≥
0. This indicates thatthe inequalities P ≥ , P ≥ , P ≥ , P ≥ s =
712 ¯ a = ( α − β + γ + δ ) = ( P + P + P + P ) ≥ , (6.9)which gives a proof of the positivity of the a -anomaly. We leave further inves-tigation on different bounds for future work. Acknowledgement
We are grateful for helpful discussions with Ofer Aharony, Matteo Beccaria, ClayCordova, Marcos Crichigno, Diego Hofman, Igor Klebanov, Zohar Komargodski, It would be interesting to understand whether these bounds (or some of them) can be saturatedby some specific theories. In terms of c and c , the Hofman-Maldacena bound reads from Table 1, 20 c ≤ c ≤ c . Wethank Clay Cordova for telling us about the free-multiplet estimation approach. A. Perturbative expansion around q = We review the perturbative expansion of supersymmetric R´enyi entropy (associatedwith spherical entangling surface) around q =
1. The great details have been givenin [43] and we will be brief. Although our main concern will be 6 d ( , ) SCFTs, wekeep the discussions valid for any SCFT with conserved R -symmetry in d -dimensions.Consider the supersymmetric partition function on S β = πq × H d − with R -symmetrybackground fields (chemical potentials), Z [ β, µ ] = Tr ( e − β ( ˆ E − µ ˆ Q ) ) . (A.1)which can be used to compute the supersymmetric R´enyi entropy associated witha spherical entangling surface in flat space. We work with the grand canonicalensemble. The state variables can be computed as follows E = ( ∂I∂β ) µ − µβ ( ∂I∂µ ) β , (A.2) S = β ( ∂I∂β ) µ − I , (A.3) Q = − β ( ∂I∂µ ) β , (A.4)where I ∶ = − log Z . The energy expectation value is given by (A.2) E = Tr ( ρ ˆ E ) Tr ( ρ ) , ρ = e − β ( ˆ E − µ ˆ Q ) , (A.5)and the charge expectation value is given by (A.4) Q = Tr ( ρ ˆ Q ) Tr ( ρ ) . (A.6)In the presence of supersymmetry, both β and µ are functions of a single variable q therefore I is considered as I q ∶ = I [ β ( q ) , µ ( q )] . (A.7)19he supersymmetric R´enyi entropy is defined as S q = qI − I q − q . (A.8)Consider the Taylor expansion around q =
1, with δq ∶ = q − S q = S EE + ∞ ∑ n = n ! ∂ n I q ∂q n ∣ q = δq n − . (A.9)The first q -derivative of I q is given by I ′ q = ( ∂I∂β ) µ β ′ ( q ) + ( ∂I∂µ ) β µ ′ ( q ) . (A.10)Using (A.2) and (A.4), one can rewrite it as I ′ q = ( E − µQ ) β ′ ( q ) − βQ µ ′ ( q ) . (A.11)Plugging in the supersymmetric background, β ( q ) = πq , µ ( q ) = g q − q , (A.12)one finally has I ′ q = π ( E − gQ ) . (A.13)Notice that µ ( q ) is solved from the Killing spinor equation. g is some numberdepending on the R -charge of the preserved Killing spinor. In general both E and Q are functions of q . Moreover, E and Q here are expectation values rather thanoperators. A.1 S ′ q = and I ′′ q = From (A.9) we see that S ′ q = = I ′′ q = . (A.14)Let us take one more q -derivative of (A.13) and make use of (A.5) and (A.6) I ′′ q = − π ⎛⎜⎝ Tr ( ρ ( ˆ E − g ˆ Q ) ) Tr ( ρ ) − [ Tr ( ρ ( ˆ E − g ˆ Q ))] [ Tr ( ρ )] ⎞⎟⎠ , (A.15)which can be simplified by using ρ = ρ ( µ = ) at q = S ′ q = = − π ⎛⎜⎝ Tr ( ρ ( ˆ E − g ˆ Q ) ) Tr ( ρ ) − [ Tr ( ρ ( ˆ E − g ˆ Q ))] [ Tr ( ρ )] ⎞⎟⎠ q = . (A.16)20A.16) can be written as connected correlators S ′ q = = − π [⟨ ˆ E ˆ E ⟩ c + g ⟨ ˆ Q ˆ Q ⟩ c − g ⟨ ˆ E ˆ Q ⟩ c ] S q = × H d − , (A.17)where we have used the fact that, ˆ Q is a conserved charge, [ ˆ E, ˆ Q ] =
0, to flip theorder of ˆ E and ˆ Q . Given that ⟨ ˆ E ˆ Q ⟩ c = ⟨ ˆ E ˆ E ⟩ c has been computed in [52], weget S ′ q = = − V d − π d / + Γ ( d / )( d − )( d + ) ! C T − π g ∫ H d − ∫ H d − ⟨ J τ ( x ) J τ ( y )⟩ cq = . (A.18) C T is defined through the flat space correlator ⟨ T ab ( x ) T cd ( )⟩ = C T x d I ab,cd ( x ) , (A.19)where I ab,cd ( x ) = ( I ac ( x ) I bd ( x ) + I ad ( x ) I bc ( x )) − d δ ab δ cd ,I ab ( x ) = δ ab − x a x b x . (A.20)Now the task is to compute the second term in (A.18). Following the way of com-puting ⟨ T T ⟩ on the hyperbolic space S q = × H d − , one can make use of the flat spacecorrelators in the CFT vacuum, ⟨ ˆ Q ˆ Q ⟩ c = − π d − V d − d − ( d − ) Γ ( d − ) C J , (A.21)where C J is defined through the R -current correlator in flat space ⟨ J a ( x ) J b ( )⟩ = C J x ( d − ) I ab ( x ) . (A.22)Our final result of S ′ q = becomes S ′ q = = − V d − ⎛⎝ π d + Γ ( d )( d − )( d + ) ! C T − g π d + d − ( d − ) Γ ( d − ) C J ⎞⎠ , (A.23)which shows that the first q -derivative of S q at q = C T and C J . This is intuitively expected because in the presence of supersymmetry,taking the derivative with respect to q is equivalent to taking the derivative withrespect to g ττ and A τ at the same time.In the particular case of 6 d ( , ) SCFTs, the 2-point function of the stress tensoris determined by the central charge c . Therefore the integrated 2-point function isproportional to c . Moreover, S ′ q = is also proportional to c , because the stress tensorand the R -current on the right hand side of (A.23) live in the same multiplet.21 .2 S ′′ q = and I ′′′ q = From (A.9) we see that S ′′ q = = I ′′′ q = . (A.24)It is straightforward to compute I ′′′ q by taking one more derivative on (A.15) I ′′′ q π = Tr ( ρ ( ˆ E − g ˆ Q ) ) Tr ( ρ ) − ( ρ ( ˆ E − g ˆ Q ) ) Tr ( ρ ( ˆ E − g ˆ Q ))[ Tr ( ρ )] + [ Tr ( ρ ( ˆ E − g ˆ Q ))] [ Tr ( ρ )] , (A.25)which may be simplified at q = µ = I ′′′ q = π = ( Tr ( ρ ( ˆ E − g ˆ Q ) ) Tr ( ρ ) − ( ρ ( ˆ E − g ˆ Q ) ) Tr ( ρ ( ˆ E − g ˆ Q ))[ Tr ( ρ )] + [ Tr ( ρ ( ˆ E − g ˆ Q ))] [ Tr ( ρ )] ) q = . (A.26)This can be further written in terms of connected correlation functions, S ′′ q = = I ′′′ q = = π [⟨ ˆ E ˆ E ˆ E ⟩ c − g ⟨ ˆ Q ˆ Q ˆ Q ⟩ c − g ⟨ ˆ E ˆ E ˆ Q ⟩ c + g ⟨ ˆ E ˆ Q ˆ Q ⟩ c ] S q = × H d − , (A.27)where we have used [ ˆ E, ˆ Q ] =
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