Supersymmetric Renyi Entropy and Anomalies in Six-Dimensional (1,0) Superconformal Theories

A closed formula of the universal part of supersymmetric R\'enyi entropy $S_q$ for six-dimensional $(1,0)$ superconformal theories is proposed. Within our arguments, $S_q$ across a spherical entangling surface is a cubic polynomial of $\nu=1/q$, with $4$ coefficients expressed as linear combinations of the 't Hooft anomaly coefficients for the $R$-symmetry and gravitational anomalies. As an application, we establish linear relations between the $c$-type Weyl anomalies and the 't Hooft anomaly coefficients. We make a conjecture relating the supersymmetric R\'enyi entropy to an equivariant integral of the anomaly polynomial in even dimensions and check it against known data in four dimensions and six dimensions.


Introduction
Six-dimensional superconformal theories provide a framework to understand various features of lower-dimensional supersymmetric dynamics.By themselves, they are difficult to study by traditional quantum field theory techniques.All known examples of interacting CFTs in six dimensions are supersymmetric.The (2, 0) theories should be the simplest ones [1][2][3].A large class of interacting (1, 0) fixed points have been constructed in string theory or brane constructions [4][5][6][7][8].Recently, F-theory provides a way to classify the known and new (1, 0) fixed points [9][10][11].Since all the known interacting fixed points are supersymmetric, it is expected that supersymmetry constraints are important in computing their physical characteristic quantities, such as Weyl anomalies.Indeed, the a-anomaly in (1, 0) superconformal theories has been recently determined in terms of their 't Hooft anomaly coefficients [12,13] for the R-symmetry and gravitational anomalies [14] by analyzing supersymmetric RG flows on the tensor branch [15] where α , β , γ , δ are the coefficients appearing in the anomaly polynomial Here c 2 (R) is the second Chern class of the R-symmetry bundle and p 1,2 are the Pontryagin classes of the tangent bundle.The relation (1.1) is analogues to the known relation [18] in four-dimensional N = 1 SCFTs, a d=4 = 9 32 k RRR − 3 32 k R , where k RRR and k R are the Tr U(1) 3  R and Tr U(1) R 't Hooft anomalies.Although the anomaly multiplet in six dimensions has not yet been constructed, such linear relations are believed to follow from the anomaly supermultiplets which include 't Hooft anomalies as well as the anomalous trace of the stress tensor.The Weyl anomaly coefficients in 6d are defined from the latter [19][20][21][22] where E 6 is the Euler density and I i=1,2,3 are three Weyl invariants.In the presence of (1, 0) supersymmetry, c i=1,2,3 , satisfying a constraint c 1 − 2c 2 + 6c 3 = 0 [23][24][25], are also believed to be linearly related the to 't Hooft anomaly coefficients [26][27][28].
Assuming that the linear relation indeed exist, one could determine its coefficients by considering the known values of the corresponding Weyl and 't Hooft anomalies in four independent examples.Unfortunately only three are known, i.e. the free hyper multiplet, the free tensor multiplet and supergravity [23,29].The naive vector multiplet is not conformal and the conformal version [30] involves higher derivatives.
Evaluating the anomalies via the heat kernel method will involve higher powers of the Laplacian operator.We will have, therefore, to consider another approach.
In even dimensions, it is known that the a-anomaly determines both the universal log divergence of the round-sphere partition function2 and the universal log divergence in the vacuum state entanglement entropy associated with a ball in flat space [31].On the other hand, by the conformal Ward identities, the 2-point and 3point functions of the stress tensor in the vacuum in flat space can be determined up to 3 coefficients [32,33], which are linearly related to c-type Weyl anomalies c 1,2,3 .In the presence of (1, 0) supersymmetry, only two of them are independent as mentioned before.
Because the round sphere is conformally flat, one expects that the nearly-round sphere partition function, which includes the response to a small deviation of the metric from the round sphere, is determined by the flat space stress tensor correlators.Due to these intrinsic relations and supersymmetric constraints, it is therefore tempting to ask whether one can fully determine the partition function on a branched (q-deformed) sphere, 3 which is directly related to the supersymmetric Rényi entropy S q .
Supersymmetric Rényi entropy was first introduced in three dimensions [34][35][36], and later studied in four dimensions [37][38][39], in five dimensions [40,41], in six dimensions ((2, 0) theories) [42,43] and also in two dimensions ((2, 2) SCFTs) [44,45].By turning on certain R-symmetry background fields, one can calculate the partition function Z q on a q-branched sphere S d q , and define the supersymmetric Rényi entropy as which is a supersymmetric refinement of the ordinary Rényi entropy (which is not supersymmetric because of the conical singularity). 4The quantities defined in (1.4) are UV divergent in general but one can extract universal parts free of ambiguities.

Summary of results
The main result of this paper is the exact universal part of the supersymmetric Rényi entropy in 6d (1, 0) SCFTs.We show that, for theories characterized by the anomaly polynomial (1.2), it is given by a cubic polynomial of ν = 1 q with four coefficients where α , β , γ , δ are the 't Hooft anomaly coefficients defined in (1.2).The basic ingredients in our arguments are the following: (A) S ν of (1, 0) free hyper multiplet and free tensor multiplet can be computed by the heat kernel method closely following [43].The results are given by These are the main results of Section 2.
(B) S ν of A N −1 type (2, 0) theories (which are of course (1, 0) conformal theories) in the large N has been computed in [43].The result is given by (1.9) (C) Based on (A)(B) and (F) below, a reasonable assumption is that the general form of S ν for (1, 0) SCFTs is a cubic polynomial in ν − 1.However, so far we do not have a sharp argument for this assumption. 5Furthermore, based on (D)(E)(F) below, the four coefficients of the cubic polynomial are linear combinations of α, β, γ, δ.
(D) The value of S ν at ν = 1 is the entanglement entropy associated with a spherical entangling surface, which is proportional to the a-anomaly (1.1).
(E) The first and second derivatives of S ν at ν = 1 can be written as linear combinations of integrated two-and three-point functions of operators in supersymmetric stress tensor multiplet.Because of this, one can relate the first and second derivatives at ν = 1 to c 1 and c 2 , where c 1 and c 2 are believed to be given by linear combinations of 't Hooft anomaly coefficients α, β, γ, δ.
(F) The large ν behavior of S ν is controlled by the "supersymmetric Casimir energy" [46].This gives (G) In the large ν expansion, the second Pontryagin class (with coefficient δ) will not contribute to the ν 3 term (as we see from (F)) and the ν 2 term.Because of the latter, one has (1.12) (H) For the conformal non-unitary (1, 0) vector multiplet, a constraint for the ctype Weyl anomalies, c 1 + 4c 2 = 62 45 , can be obtained by studying the higherderivative operators on the Ricci flat background [26]. 6Together with (E), one has , one can uniquely find the general expression of the supersymmetric Rényi entropy given in (1.5)(1.6).We emphasize that among all these 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, we conjecture a relation between the supersymmetric Rényi entropy and the anomaly polynomial in any even dimension, which perfectly agrees with (A)-(H).We consider this 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,7 This paper is organized as follows.In Section 2 we employ heat kernel method to study the supersymmetric Rényi entropy of free (1, 0) multiplets.In Section 3 we propose a form of the universal supersymmetric Rényi entropy with four non-trivial coefficients, which works for general 6d (1, 0) SCFTs.We determine the coefficients one by one.We study the relation between the supersymmetric Rényi entropy and the supersymmetric Casimir energy in Section 4, which is used to determine one of the coefficients in the previous section.In Section 5 we conjecture a relation between the supersymmetric Rényi entropy and the anomaly polynomial for SCFTs in even dimensions and test this conjecture in 6d and 4d.In Section 6, we discuss some open questions, further applications of our results and some future directions of research.

Free 6d (1, 0) multiplets
We begin by studying the supersymmetric Rényi entropy of free (1, 0) multiplets, following [42].For free fields, the Rényi entropy associated with a spherical entangling surface in flat space can be computed by conformally mapping the conic space to a hyperbolic space S 1 β × H 5 and using the heat kernel method. 8A six-dimensional (1, 0) hyper multiplet includes 4 real scalars, 1 Weyl fermion and a tensor multiplet includes 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 degrees of freedom by half.

Heat kernel and Rényi entropy
The partition function of free fields on S 1 β=2πq × H 5 can be computed by the heat kernel 9log where K S 1 β ×H 5 (t) is the heat kernel of the associated conformal Laplacian.The kernel factorizes because the spacetime is a direct product, On a circle, the kernel is given by In the presence of a chemical potential µ, it will be twisted [47] KS 1 where f controls the periodic/anti-periodic boundary conditions, namely f = 0 for bosons and f = 1 for fermions.The volume factor can be factorized in the kernels on the hyperbolic space, because H 5 is homogeneous.Thus K H 5 (t) can be written in terms of the equal-point kernel, The regularized volume is given by V 5 = π 2 log(ℓ ǫ), where ǫ is actually the UV cutoff in the flat space before the conformal mapping10 and ℓ is the curvature radius of H 5 .For the K H 5 (0, t) of free fields with different spins we refer to [42] and references there in.
The Rényi entropy of a hyper multiplet can be obtained by summing up the contributions of 4 real scalars, 1 Weyl fermion and the Rényi entropy of a tensor multiplet can be obtained by summing up the contributions of 1 real scalar, 1 Weyl fermion and a self-dual 2-form, where the Rényi entropy for free fields with different spins can be computed by using the corresponding heat kernels. 11The final results for the Rényi entropy of a 6d complex scalar, a 6d Weyl fermion and a 6d 2-form field are respectively.Note that, to obtain the correct Rényi entropy for the two form field, we have taken a q-independent constant shift which is associated with possible boundary contributions [42].Before moving on, let us represent S q in terms of The Rényi entropy of free (1, 0) multiplets are given by 12) The reason why S ν is convenient is obvious, the series expansion near ν = 1 has finite terms while the expansion of S q near q = 1 has infinite number of terms.We will use S ν instead of S q to express Rényi entropy and supersymmetric Rényi entropy from now on.It is worth to remember the relations between the derivatives with respect to q and the derivatives with respect to ν at q = 1 ν = 1, which will be useful later.Finally, one can check that ∂ 0 q=1 , ∂ 1 q=1 and ∂ 2 q=1 of both S hyper q and S tensor q are consistent with the previous results about the free (1, 0) multiplets [23,29].By "consistent", we refer to the relations between the first and the second derivatives of the Rényi entropy at q = 1 and the two-and three-point functions of the stress tensor derived in [52,53].

Supersymmetric Rényi entropy
The supersymmetric Rényi entropy of free multiplets can be computed by the twisted kernel (2.4) on the supersymmetric background.The R-symmetry group of 6d (1, 0) theories is SU(2) R , which has a single U(1) Cartan subgroup.Therefore one can turn on a single R-symmetry background gauge field (chemical potential) to twist the boundary conditions for scalars and fermions along the replica circle S 1 β [47].The R-symmetry chemical potential can be solved by studying the Killing spinor equation on the conic space (S 6 q or S 1 β=2πq × H 5 ),12 with k being the R-charge of the Killing spinor under the Cartan U(1).We choose k = 1 2 and the background field turns out to be For each component field in the free multiplets, one has to first figure out the associated Cartan charge k i and then compute the chemical potential by k i A τ .After that one can compute the free energy on S 1 β × H 5 using the twisted heat kernel with the chemical potential µ = k i A τ and obtain the supersymmetric Rényi entropy.
After summing up the component fields, the supersymmetric Rényi entropy of a free (1, 0) hyper multiplet and a free (1, 0) tensor multiplet are ) respectively.

− − −
The coefficient D in (3.1) can be determined by using the fact that, the entanglement entropy associated with a spherical entangling surface, which is nothing but S ν=1 , is proportional to the Weyl anomaly a.This is true for general CFTs in even dimensions as shown in [31].Therefore By studying supersymmetric RG flows on the tensor branch, a a u(1) has been computed in [15], see (1.1).This allows us to fix The coefficients C and B in (3.1) are the first and the second ν-derivatives of S (1,0) 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 the integrated correlators are given in Appendix A. Namely, the first q-derivative at q = 1 is given by a linear combinations of integrated ⟨T T ⟩ and integrated ⟨JJ⟩ in (A.23), This relation holds for general SCFTs with conserved R-symmetry in d-dimensions.
Similarly the second q-derivative at q = 1 is given by a linear combination of the integrated stress tensor 3-point function, the integrated R-current 3-point function and some mixed 3-point functions.This is given explicitly in (A.27) In 6d (1, 0) SCFTs, by the conformal Ward identities, the two-and three-point functions of the stress tensor multiplet (including R-current) may be determined in terms of two independent coefficients, which are linearly related to c 1 and c 2 .Because of this, C and B in (3.1) are also linear combinations of c 1 and c 2 .These relations can be obtained by fitting to the free hyper multiplet and the free tensor multiplet in Table 1, Assuming B and C are linear combinations of α , β , γ , δ, we shall establish the explicit relations.Because the second Pontryagin class p 2 (T ) does not contribute to the ν 2 term, we get To see that the ν 2 term is independent of p 2 (T ), let us consider the free energy on S 5 q × H 1 , which can be used to compute S q because S 5 q × H 1 is conformally equivalent to S 6 q or S 1 q × H 5 .S 5 q × H 1 is similar to S 5 q × S 1 β→∞ , but they are not the same due to different boundary conditions on H 1 and S 1 β .The latter background preserving supersymmetry is used to compute the supersymmetric Casimir energy in 6d.One can formally define a supersymmetric Rényi entropy on S 5 q × S 1 β→∞ with the Rényi parameter q by using the free energy βE c [S 5  q ].As we will see in the next section, p 2 (T ) will not contribute to the 1 q 2 term in this supersymmetric Rényi entropy, because p 2 (T ) contributes to E c in the following way (4.7) The different boundary conditions on S 5 q × H 1 will not change the property that the 1 q 2 term is independent of δ.We further confirm this fact by establishing a concrete relation between S q and the anomaly polynomial in Section 5.
Since B depends only on α , β , γ , it can be fixed by fitting to the three independent examples, the free hyper multiplet, the free tensor multiplet and the A N −1 type theories in the large N, B = 1 24 (2α − 5β + 8γ) . (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 the remaining free parameter by making use of the result of c 1 + 4c 2 for the conformal non-unitary (1, 0) vector multiplet in [26] (obtained by the heat kernel computation on the Ricci flat background) Thus, the coefficient C as a linear combination of α , β , γ , δ is determined (3.6)(3.9)(3.11)also establish the linear relations between c 1,2,3 and α , β , γ , δ (3.12) The remaining coefficient A will be fixed as in the next section by studying the large ν behavior of the supersymmetric Rényi entropy.Obviously, the leading contribution in the limit ν → ∞ is determined only by A.

A closed formula
As a summary, we can completely determine a closed formula for the universal part of supersymmetric Rényi entropy for 6d (1, 0) SCFTs, Given that 't Hooft anomalies for general 6d (1, 0) SCFTs can be computed [13], the above formula tells us the universal supersymmetric Rényi entropy for any (1, 0) SCFT.
For (2, 0) theories labeled by a simply-laced Lie algebra g, (3.14) reduces to [43] S where ā and c are determined by the rank, dimension and dual Coxeter number of T ν and H ν are the supersymmetric Rényi entropy of the (2, 0) tensor multiplet and that of the (2, 0) supergravity (large N), respectively

Relation with supersymmetric Casimir energy
In this section we clarify the relation between the supersymmetric Rényi entropy and the supersymmetric Casimir energy in 6d.Similar relation in 4d has been obtained in [38].Recall that the partition function Z on M D−1 × S 1 β is determined by the Casimir energy on the compact space M D−1 in the limit β → ∞ which is equivalent to the statement15 We consider the cases with supersymmetry.In even-dimensional superconformal theories, the supersymmetric Casimir energy on S 1 ×S D−1 has been conjectured to be equal to the equivariant integral of the anomaly polynomial in [46], where the authors provided strong supports for this conjecture by examining a number of SCFTs in two, four and six dimensions. 16The equivariant integration is defined with respect to the Cartan subalgebra of the global symmetries (that commute with a given supercharge) and one can write this as where the equivariant parameters µ j are the chemical potentials corresponding to the Cartan generators.In equivariant cohomology, doing the integration (4.3) in 6d is equivalent to the replacement rules (4.7).17  ⃗ ω with squashing parameters ⃗ ω = (ω 1 , ω 2 , ω 3 ).The squashing parameters are defined by coefficients appearing in the Killing vector where φ 1 , φ 2 , φ 3 are three circles representing the U(1) 3 isometries of the 5-sphere.
The supersymmetric Casimir energy of superconformal (1, 0) theories is given by the equivariant integral (4. 3) where the 8-form anomaly polynomial is18 as introduced in the introduction.The integration (4.5) is equivalent to the following replacement rules [46] where σ is the chemical potential for the R-symmetry Cartan and ω 1,2,3 are the chemical potentials for the rotation generators (commuting with the supercharge).
After the replacement, the result should be divided by the equivariant Euler class, In the particular background of S 5 q × S 1 β , where S 5 q is a q-deformed 5-sphere with the metric ds 2 = (sin 2 θ + q 2 cos 2 θ)dθ 2 + q 2 sin 2 θdτ 2 + cos 2 θdΩ 2 3 , ( one should identify the shape parameters as Note that there is a supersymmetric constraint for the chemical potentials, σ = 1 2 ∑ j ω j .Evaluating (4.5) one obtains Therefore the free energy in the q → 0 limit19 where we have divided by a q-independent volume factor Vol [D 4 × S 1 β ] = βπ 2 2. Because of the conformal equivalence between S 5 q × H 1 and S 1 q × H 5 , we have where the first equality follows from the background coincidence and the second one follows from the conformal invariance of (supersymmetric) Rényi entropy and From (4.13) we obtain the asymptotic supersymmetric Rényi entropy on S 1 q × H 5 This fixes the undetermined coefficient A in (3.1) as Notice that this result perfectly agrees with the supersymmetric Rényi entropy of the known (1, 0) fixed points listed in Table 1.

Relation with anomaly polynomial
Inspired by the relation between the supersymmetric Casimir energy and the anomaly polynomial [46], we conjecture in this section a relation between the supersymmetric Rényi entropy and the anomaly polynomial.Following this relation, the supersymmetric Rényi entropy in even dimensions can be extracted directly from the anomaly polynomial of the theory.We conjecture that S q is determined by an equivariant integral of the anomaly polynomial I D+2 with respect to the subalgebra formed by generators (r, ), where r is the R-symmetry Cartan generator and h j is the j-th orthogonal rotation generator in R D , while h [ D 2 +1] generates an additional U(1) rotation.We emphasize that we do not have yet a physical understanding of the extra U(1), but just employ it in the same way as the other rotational U(1)'s.We will check our conjecture against existing data in 6d and 4d.To simplify the notation, we will use ] from now on.The Cartan generators commuting with a given supercharge Q have the corresponding chemical potentials denoted by σ, ⃗ ω, ω.Define an equivariant integral20 with the corresponding chemical potentials as the equivariant parameters.The supersymmetric Rényi entropy can be determined as follow Note that in the second equation in (5.2), the supersymmetric constraint for the chemical potentials was implicitly assumed.A volume V H 1 = 2 log(ℓ ǫ) was factorized in S q because we work effectively on S D−1 q ×H 1 .We will test this conjecture for SCFTs in 4d and 6d in the following subsections.We have not been able, so far, to prove this conjecture.The fact that an equivariant integral appears in this conjecture may hint towards some localization.

Six dimensions
In R 6 , there is a U(1) 3 subalgebra in the rotation symmetries.The generators commuting with the supercharge have the corresponding chemical potentials, ω 1,2,3 .The additional chemical potential is ω = ω 4 .Consider superconformal theories with SU(2) R R-symmetry.For the 8-form anomaly polynomial given in (4.6), the replacement rule in carrying out the equivariant integration (5.1) should be After these replacements in the anomaly polynomial, we divide it by ẽ(T ) = ω 1 ω 2 ω 3 ω 4 .
The result is given by Upon plugging in one obtains The above result can be rewritten as S ν , This agrees precisely with (3.14).Remarkably, a single conjectured formula by the equivariant integral (5.2) can give the a-anomaly, c 1,2,3 -anomalies and also a certain part of the supersymmetric Casimir energy simultaneously and precisely.We consider these agreements as a strong support of both our results (1.5) and the conjecture itself.

Four dimensions
In R 4 , there is a U(1) 2 subalgebra in the rotation symmetries.The generators commuting with the supercharge have the corresponding chemical potentials, ω 1,2 .

Discussion
In this paper we proposed a closed formula for the universal log term of the sixdimensional supersymmetric Rényi entropy and made a conjecture that the supersymmetric Rényi entropy in even dimensions is equal to an equivariant integral of the anomaly polynomial.It remains a challenging problem to understand the extra U(1) and to prove this conjecture.We leave it for future work.
Let us mention a few other open question and further directions of research that are related to this work.
1. Proving our assumption that the expansion of the supersymmetric Rényi entropy in 1 q terminates (it is just a polynomial of 1 q with degree 3 in 6d).For this we need the dependence of possible counter-terms on 1 q.Hence, we have to construct the six-dimensional supersymmetric curved background and in particular the smooth squashed six-sphere.The super-Weyl anomalies constructed on this background will give the universal part of the supersymmetric Rényi 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 derivative of the supersymmetric Rényi entropy is related to a specific linear combination of 4-point functions of the stress tensor and other operators in its multiplet.On the other hand, according to our result (1.5)(1.6) it is related to s 3 and hence via (1.1) and (1.14) to the Weyl anomalies.In 6d, this is indeed consistent with a long time expectation that the a-anomaly should determine some specific term in the 4-point function of the stress tensor.This consistency becomes manifest for (2, 0) theories (3.15).It would be nice to demonstrate the relation between S ′′′ ν ν=1 and the integrated 4-point functions of operators in the stress tensor multiplet in a straight forward way.
(A.16) can be written as connected correlators where we have used the fact that, Q is a conserved charge, [ Ê, Q] = 0, to flip the order of Ê and Q.Given that ⟨ Ê Q⟩ c = 0 and ⟨ Ê Ê⟩ c has been computed in [52], we get where C J is defined through the R-current correlator in flat space ⟨J a (x)J b (0)⟩ = C J x 2(d−1) I ab (x) . (A.22) Our final result of S ′ q=1 becomes which shows that the first q-derivative of S q at q = 1 is given by a linear combination of 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 with respect to g τ τ and A τ at the same time.
In the particular case of 6d (1, 0) SCFTs, the 2-point function of the stress tensor is determined by the central charge c 3 .Therefore the integrated 2-point function is proportional to c 3 .Moreover, S ′ q=1 is also proportional to c 3 , because the stress tensor and the R-current on the right hand side of (A.23) live in the same multiplet.