Supersymmetric R\'enyi Entropy in Five Dimensions

We introduce supersymmetric R\'enyi entropies for $\mathcal{N}=1$ supersymmetric gauge theories in five dimensions. The matrix model representation is obtained using the localization method and the large-$N$ behavior is studied. The gravity dual is a supersymmetric charged topological AdS$_6$ black hole in the Romans $F(4)$ supergravity. The variation of the supersymmetric R\'enyi entropy due to the insertion of a BPS Wilson loop is computed. We find perfect agreements between the large-$N$ and the dual gravity computations both with and without the Wilson loop operator.


Introduction
An intriguing relation between entanglement entropy and partition functions holds in quantum field theories [1]. The Rényi entropy S n , one parameter generalization of entanglement entropy S = S 1 , can be written as where Z n is the partition function on the n-covering space M n around an entangling surface Σ. If quantum field theories in d dimensions have conformal symmetry and the entangling surface is a (d − 2)-dimensional hypersphere, Σ = S d−2 , M n becomes the n-covering space of a d-sphere [2]. Especially, the entanglement entropy S obtained in the n → 1 limit is equal to the free energy on S d , S = log Z 1 ≡ −F . This relation makes it easier to calculate the Rényi entropy for free fields [3], but cannot be applied to supersymmetric theories because the conical singularity at Σ breaks supersymmetries. To remedy the situation, a supersymmetric extension of the Rényi entropy was introduced in [4] by modifying (1.1) to where Z susy n is the supersymmetric partition function on the n-covering d-sphere with an R-symmetry background field to preserve supersymmetries. N = 2 superconformal field theories in d = 3 dimensions are considered in [4] and the matrix model representation of the supersymmetric Rényi entropies are derived by using the localization method [5][6][7][8][9]. The gravity dual of the supersymmetric Rényi entropy is constructed in [10,11] and turns out to be a supersymmetric AdS 4 charged topological black hole. The holographic computation precisely agrees with the large-N limit of the supersymmetric Rényi entropy both with and without Wilson loop operators. Similar constructions have been carried out recently in d = 4 dimensions by [12,13] extending the works [14,15] for N = 2 supersymmetric gauge theories on S 4 . The four-dimensional supersymmetric Rényi entropy has a logarithmic divergence with a coefficient determined by the a-anomaly of the Weyl symmetry.
The objective of this paper is to introduce the supersymmetric Rényi entropy (1.2) for N = 1 supersymmetric gauge theories in five dimensions and derive their matrix model representations by the localization technique. We will construct the theories by taking the rigid limit [16] of the N = 1 five-dimensional supergravity [17,18]. In some aspects, this can be regarded as a theoretical challenge to define field theories on a singular manifold with supersymmetry preserved. The partition functions of supersymmetric gauge theories are calculated on a round five-sphere in [19][20][21][22][23][24] and on a squashed five-sphere in [25][26][27][28]. More general five-dimensional manifolds admitting rigid supersymmetry are explored in [29][30][31][32][33][34][35][36][37]. We will show that the Killing spinor equations and additional equations for the Killing spinors can be solved on the resolved space of the n-covering five-sphere.
Using the supersymmetry generated by the solution of the Killing spinor equations, we perform the localization computation for the partition function on the resolved sphere that is the Hopf fibration over deformed CP 2 . There are three fixed points on CP 2 for U (1) 2 actions inside U (1) × SO(4) symmetry the resolved sphere has. We notice that the resolved five-sphere can be identified with the squashed five-sphere with the squashing parameters (ω 1 , ω 2 , ω 3 ) = (1/n, 1, 1) near the fixed points. Translating the results for the squashed sphere [25][26][27][28], we obtain the perturbative partition function on the n-covering five-sphere.
We evaluate the supersymmetric Rényi entropy (1.2) in the large-N limit of N = 1 U Sp(2N ) superconformal gauge theories and reveal the n-dependence that satisfies several inequalities the usual Rényi entropy does. We also consider an addition of a Wilson loop preserving supersymmetry, that can be physically interpreted as an insertion of a quark inside an entangling surface [38]. The variation of the supersymmetric Rényi entropy by the loop turns out to be independent of the parameter n.
Finally, we construct the gravity dual of five-dimensional supersymmetric Rényi entropy as a solution of the Romans F (4) supergravity [39]. It is a supersymmetric charged topological black hole in the Euclidean AdS 6 space whose boundary is S 1 × H 4 . 1 We compute the holographic free energy following [44,45] and the expectation value of the holographic Wilson loop. We find that the holographic supersymmetric Rényi entropy and its variation due to the loop perfectly agree with the results of the dual field theory in the large-N limit.
Note added: While this work was being completed, we were aware of the paper [46], which has a substantial overlap with this paper.

Rigid N = 1 supersymmetry in five dimensions
The N = 1 supergravity coupled to Yang-Mills and matter fields in five dimensions is constructed in [17]. This theory has an SU (2) R R-symmetry and the Weyl multiplet consists of the vielbein e a µ , the graviphoton A µ , the SU (2) R gauge field V IJ µ , the SU (2) R triplet scalar field t IJ , the dilaton α, the real anti-symmetric tensor v ab , the real scalar C, the SU (2) R -Majorana gravitino ψ I µ and SU (2) R -Majorana fermion χ I . Since the dilaton does not change under the supersymmetry transformation, we can fix the dilatational symmetry by α = 1. In addition, we only consider theories without central charge which allows us to turn off the graviphoton A µ = 0. This simplifies the construction of supersymmetric field theories on a curved space from the N = 1 supergravity as we will perform below.
Following [16], we take the rigid limit of the N = 1 supergravity by setting the gravitino ψ I µ and the fermion χ I and their variations to be zero while letting the space-time be curved: where ∇ µ is the covariant derivative with respect to the Lorentz index, and (2.2) In this limit, the supersymmetry transformations of the other background fields automatically vanish.

Supersymmetry algebra and multiplets
The N = 1 supersymmetry algebra in five dimensions is given by where D µ is the covariant derivative with respect to the gauge symmetry and translation, v µ , Θ ab , R IJ and γ are parameters for the translation, Lorentz rotation, SU (2) R -symmetry and gauge symmetry transformations 2 v µ = 2ξ 1 Γ µ ξ 2 , Vector multiplet. The vector multiplet contains the gauge field A µ , a real scalar σ, an SU (2) R triplet scalar Y IJ and an SU (2) R -Majorana fermion λ I . They transform under the supersymmetry as 3 (2.5) The covariant derivatives are (2.6) To put them into cohomological forms, we redefine the gaugino as [19] where Ψ µ = −2ξΓ µ λ , χ µν = ξΓ µν λ . (2.8) Then the supersymmetry transformation law (2.5) becomes (2.9) 2 We contract the SU (2)R indices from northwest to southeast direction when the indices are suppressed. 3 We can redefine the triplet scalar YIJ in [17,18,47,48] as DIJ = 2YIJ − 2tIJ σ to make contact with [20].
The supersymmetric action of the vector multiplet is where g is the gauge coupling constant. The Chern-Simons term can be added in five dimensions 4 whose supersymmetric completion is [19] where κ and Ψ are one-forms dual to the Killing vector κ ≡ v µ dx µ and Ψ ≡ Ψ µ dx µ , respectively.
Hypermultiplet. We can realize the supersymmetry algebra only for on-shell hypermultiplets. We, however, need one supercharge δ with unit norm ξξ = 1 for localization which satisfies where v µ = ξΓ µ ξ , with the modified supersymmetry transformation law by introducing an auxiliary field [20]. They satisfy The hypermultiplet consists of a scalar q I A , a fermion ψ A and an auxiliary field F A I with flavor indices A = 1, 2, · · · , 2r for r hypermultiplets. The index A is raised and lowered with a 2r × 2r antisymmetric matrix Ω AB as The reality conditions are imposed by The generators of the Lie group have one upper and one lower indices of the flavor symmetry, σ A B for example. Choosing the invariant tensor Ω of Sp(2r) to be the scalar field q I A satisfying the reality condition (2.17) is written as a two-component vector where φ + and φ − are in the fundamental and anti-fundamental representation of SU (r) respectively. The fermion ψ A is similarly decomposed to as a vector with spinors ψÂ in the fundamental representation of SU (r). The off-shell supersymmetry transformations realizing (2.13) are found to be where the checked parameterξ I satisfies [20] ξ I ξ I =ξ Iξ and the covariant derivatives are (2.23) We rewrite the transformation laws (2.21) with the fermionic variables [21] which satisfy the "chirality" conditions P + q A = q A with the projection operators P ± ≡ 1±v µ Γµ 2 . For a shifted auxiliary fieldF A ≡ξ IF I A (P −FA =F A ), the supersymmetry transformations of the fields (q A , ψ ± A ,F A ) are recast in the following form: (2.25) The matter lagrangian reads [18] where the flavour indices A, B are contracted from northeast to southwest.

Resolved space
We are interested in N = 1 supersymmetric field theories on a branched n-covering of fivesphere. To treat the conical singularity we replace it with a resolved space whose metric is given by where f (θ) is a smooth function behaving as for ǫ ≪ 1. ds 2 S 3 is the metric of a three-sphere The vielbein e i L in the left invariant frame and the spin connections satisfy (2.30) They can be parametrized by an element g of SU (2) group with the Pauli matrices σ i (i = 1, 2, 3). We choose the vielbein of the resolved space (2.27) as 32) and the spin connections are

Relation between resolved space and squashed five-sphere
It has U (1) 3 symmetry acting on the coordinates as The five-sphere has the Hopf fiber representation as the U (1) fibration over CP 2 . The translation along the U (1) fiber is described by the overall U (1) phase rotation a 1 = a 2 = a 3 in (2.34). There are three fixed points of U (1) 2 symmetry on the base CP 2 at (z 1 , z 2 , z 3 ) = (1, 0, 0), (0, 1, 0), (0, 0, 1). Let us introduce new coordinates by where the ranges of the angles are taken to be 0 ≤ θ < π/2, 0 ≤ τ < 2π, 0 ≤ φ < π, 0 ≤ χ < 4π and 0 ≤ ξ < 2π. The metric is locally that of a five-sphere, but globally the n-branched cover In this parametrization, the translation along the Hopf fiber is given by the shifts of the angles that is generated by a vector field The three fixed points are located at (2.39) The vielbein for S 3 are written as (2.40) Now we consider a deformation of a five-sphere satisfying |z 1 | 2 /n 2 + |z 2 | 2 + |z 3 | 2 = 1. This is a special case of the squashed five-sphere defined by with three squashing parameters chosen as One can parametrize the complex coordinates with the real angles by that gives where f n (θ) = 1/(n 2 cos 2 θ + sin 2 θ). This space may be regarded as a resolved space with f (θ) = f n (θ) if the partition function does not depend on the choice of f (θ). In appendix B, we show in detail that this identification is possible for evaluating the one-loop partition functions.

Killing spinor equations
We will solve the Killing spinor equations (2.1) on the resolved space (2.27). We let spinors ξ I be tensor products of spinors in two dimensions ζ I and spinors in three dimensions η I Correspondingly, the gamma matrices that are hermitian (Γ a ) † = Γ a can be written in tensor product forms: The charge conjugation matrix takes the form With the vielbein (2.32) and the spin connections (2.33), we find the background fields where ζ 1,2 are constant spinors in two dimensions and η ± are the Killing spinors on a unit three-sphere The SU (2)-Majorana condition leads the relations (2.52)

Localization
We will localize the infinite-dimensional path integral of the partition function to a finitedimensional matrix integral by adding a δ-exact term to the action I → I + tδV where the localizing term δV is taken to be positive semi-definite. Since the path integral does not depend on the δ-exact term, we let t be large so that the fixed points are given by δV = 0. After determining the fixed point loci for the gauge and matter sectors, we will read off the perturbative partition function on the resolved space by identifying it with the squashed five-sphere at the three fixed points on the base CP 2 .

Gauge sector
A localization term for the gauge sector is given similarly to [20] by whose supersymmetry variation yields where we let the hermitian conjugates of σ and Y IJ be σ † = −σ and Y † IJ = Y IJ . The saddle point of the localization term (3.2) is In the zero instanton sector, the gauge field is a flat connection and the saddle point becomes up to the gauge transformation.

Matter sector
We choose a localization term for the matter sector as whose supersymmetry variation yields (3.6) Then the path integral localizes to the following fixed locus Combining with (3.4), one finds 3.3 Partition function on the n-covering five-sphere As described in section 2.3, the resolved five-sphere (2.27) can be regarded as a squashed five-sphere with f (θ) = f n (θ) since they are locally equivalent near the fixed points of the four-dimensional base space in the Hopf fiber representation. Under this identification, the squashing parameters are (ω 1 , ω 2 , ω 3 ) = (1/n, 1, 1) and the perturbative partition function is [26,27] Z pert = dσ 0 e −I 0 α∈positive root where S 3 is the triple sine function (3.10) Here we take the mass of hypermultiplet as m, and I 0 is the classical contribution at the localization fixed point 6 where we used the volume of the n-covering five-sphere in the second equality. 7 There are also instanton contributions that we will neglect in the large-N limit.

Large-N limit of supersymmetric Rényi entropy
Consider N = 1 supersymmetric U Sp(2N ) gauge theories with N f flavors and a single hypermultiplet in the antisymmetric representation. In the large-N limit, the free energy F = − log Z on the n-covering five-sphere is approximated by the saddle point σ → −iN 1/2 x with the density [24,45] ρ(x) = 2 The free energy in the leading order of the large-N is Here, there is an imaginary unit in front of the Chern-Simons level k because of our anti-hermitian convention for Lie algebras. 7 Note that our convention leads to κ ∧ dκ ∧ dκ = 8vol(M5).
where F S 5 is for a round five-sphere Using the definition (1.2), the supersymmetric Rényi entropy in the large-N limit is 8 It is worth mentioning that the ratio of the supersymmetric Rényi entropy H n ≡ S susy n /S 1 in the large-N limit satisfies the inequalities that are satisfied by the usual Rényi entropy [49] and the supersymmetric Rényi entropy in three dimensions [4].

Adding Wilson loop
The supersymmetric Wilson loop in a representation R of the gauge group is This is invariant under the supersymmetry transformation if the contour of the loop is the same as the orbit of the Killing vector, namely,ẋ µ (s)/|ẋ(s)| = v µ . The variation of the entanglement entropy due to the Wilson loop is considered by [38] and those of the supersymmetric Rényi entropies are [11] S susy W,n = where · n stands for the expectation value taken on the n-covering sphere. Let us consider a Wilson loop in the fundamental representation wrapped on τ direction. This configuration is BPS for arbitrary real n. In the large-N limit, the expectation value can be approximated at the saddle point

19)
8 Note that S susy 1 = S1 because there is no R-symmetry flux in the n → 1 limit.

Gravity dual of supersymmetric Rényi entropy
The gravity dual of a squashed five-sphere with SU (3)× U (1) symmetry has been constructed in [44,45] using Romans F (4) supergravity in six dimensions [39]. We follow the conventions of [44,45] below. Instead of finding a solution dual to the resolved five-sphere, we map the n-covering of a five-sphere to the hyperbolic space S 1 × H 4 with the metric by a conformal transformation [2]. Hereτ ≡ nτ is the rescaled circle direction with the periodicityτ ∼τ + 2πn. We will look for an asymptotically Euclidean AdS 6 solution whose boundary is S 1 × H 4 in the Romans theory.

Romans F (4) supergravity
The bosonic part of the Romans F (4) supergravity includes the metric g µν , a dilaton φ, a one-form field A, a two-form field B and SU (2) gauge field A i (i = 1, 2, 3). The Euclidean action is 9 Since the gauge coupling g appears only in the overall factor, we set g = 1 in the following. The Hodge duality is defined by for p-forms α and β.
The equations of motion are where the covariant derivative D in the third line acts as Dω i = dω i − ǫ ijk A j ∧ ω k , and the Einstein equation is (4.5) The gauge symmetry A → A + 2λ/3, B → B − dλ allows us to set A = 0. Also, the second equation of (4.4) follows from the first one by taking the exterior derivative [45] .

Holographic free energy
The holographic free energy is the on-shell action on a manifold M 6 with boundary ∂M 6 I tot = I bulk + I bdy + I GH + I c.t. , (4.6) where I bulk and I bdy are the bulk and the boundary on-shell actions I GH is the Gibbons-Hawking term added at the boundary ∂M 6 with the induced metric h where * h is the Hodge dual with respect to h and K ≡ ∇ µ n µ is the trace of the extrinsic curvature with the unit normal vector n µ to ∂M 6 . The last one is the counterterm needed to make the action finite [44,45] (1 − X) 2 * h 1 . (4.9)

BPS charged topological AdS black hole
The reasonable ansatz for the metric that preserves the boundary symmetry SO(1, 4) × U (1) is Since the background fields break SU (2) R symmetry to the subgroup U (1) in the dual field theory, we correspondingly assume A 1 = A 2 = B = 0 and In [50], AdS black hole solutions were constructed with the fields given by

(4.12)
This solution becomes the supersymmetric AdS 6 topological black hole when µ = 0, but is no longer BPS for µ > 0. Actually, the BPS solution takes very close form to the above solution (4.13) One can check that the above solution satisfy the integrability equation of Killing spinor equation of the Euclidean Romans theory given in (A.5) of [45]. This BPS black hole is asymptotically AdS space with the AdS radius L AdS = 3 √ 2g and has a horizon at the largest root r = r H (γ) of f (r H (γ)) = 0 depending on the parameter γ r H (γ) = 2 cosh γ + 1 3 L AdS . (4.14) The temperature and the entropy of the black holes are Since the period of the circle in the boundary has 2πn periodicity, we take the parameter γ to be cosh γ = n + 1 2n . (4.18) Using the renormalized action in section 4.2, the holographic free energy becomes Comparing this result with the free energy on the n-covering five-sphere (3.13), they completely agree due to the relation [24,45]. Therefore, the holographic supersymmetric Rényi entropy also agrees with the large-N result (3.15).

Holographic Wilson loop
Consider a Wilson loop in the fundamental representation wrapped along τ direction at the origin of H 4 . It is dual to the fundamental string in the AdS 6 space [51,52] extending from the boundary r = ∞ and terminating at the horizon r = r H . The expectation value is given by where S string is the string world sheet action in a target space with the metric ds 2 string = G µν dx µ dx ν Uplifting the solution to the massive IIA supergravity, the scalar field X becomes the dilation field which distinguishes the string frame from the Einstein frame as ds 2 string = X −2 ds 2 Einstein . Taking into account this effect and choosing the world sheet coordinates to be ξ 1 = τ, ξ 2 = r, the string action becomes which is independent of the parameter n. If the solution is dual to N = 1 U Sp(2N ) superconformal theories, the AdS radius is fixed to be L 2 and (4.23) agrees with the field theory result (3.21).

A Convention
• The Greek indices µ, ν, · · · are for the space-time indices.
• The Roman indices a, b, · · · are for the local Lorentz frame. To translate them into the space-time indices, we use vielbein e µ a . For instance, v µν = v ab e µ a e ν b .
• The generators of the Lie algebra T A of a Lie group is anti-hermitian: with the structure constant f C AB and the Killing form • All spinors are Grassmann odd except Killing spinors that are treated as Grassmann even variables.
• The gauge transformation δ G for a gauge field A defined by and for a matter field Φ by

A.1 Gamma matrices and spinors
In the Euclidean five dimensions, the gamma matrices satisfying the Clifford algebra are 4 × 4 matrices. We use the gamma matrices with multiple indices where |σ| is the order of an element σ of the permutation group S n . With this notation, the product of the gamma matrices obeys the relation The charge conjugation matrix C satisfies We introduce a spinor ψ α with upper index α. Then the positions of indices of the matrices are We raise and lower the spinor index by the charge conjugation matrix and the contraction is always defined by Spinors have an SU (2) R index I (I = 1, 2) which can be raised or lowered by the skewsymmetric matrix ǫ IJ : We can impose the SU (2)-Majorana condition on spinors with the charge conjugation matrix If we denote spinors as ǫ = (ǫ 1 , ǫ 2 ) T , the condition yields The bilinear terms of spinors with SU (2) R indices behave under the exchange of the two spinors as ψ I Γ a 1 a 2 ···an χ J = t n χ J Γ a 1 a 2 ···an ψ I , t n = +1 , n = 0, 1, 4, 5 −1 , n = 2, 3 . (A.13) When the indices of SU (2) R are contracted, there appears a minus sign in the right hand side.

A.2 Spin connection and Lie derivatives
The spin connection for a given vielbein is defined by 15) and the Riemann tensor is given using the spin connection by The Ricci scalar is then This convention yields a negative Ricci curvature for a round sphere. The covariant derivatives for spinors are defined by
(B.6) ds 2 M 4 is the third line of this metric. The squashing parameters in (2.42) are read from the one-loop determinant in instanton configurations (3.3). For a while, we take the translation along ξ direction with charge δ into the translation of base M 4 generated by the Killing vector. This is expressed as a 2 − a 3 = δ = 0, and finally δ is taken to be zero 10 . Then the Killing vector which acts on M 4 is δ∂ ξ + ∂ τ ′′ . The fixed points of this translation are (z 1 , z 2 , z 3 ) = (n, 0, 0) , (0, 1, 0) , (0, 0, 1). In following, we determine the squashing parameters in (2.42) from the configurations around these fixed points. whose Omega deformation parameters are obtained from their charges.
From above results, we read a one-loop determinant Z one−loop = j w − 1 2 c j j around the saddle point configuration (3.3) from indD 10 = j c j e w j . Since the indices of vector multiplets in five-dimensional N = 1 theory are related to those of the self-dual complex indD SD [54], which are represented by the indices indD of the Dolbeault complex as indD SD = 1 2 (1 + e i(ǫ 1 +ǫ 2 ) ))indD . .
Euler class The Euler class for an oriented four-dimensional manifold M 4 is written in terms of their curvature: χ(M 4 ) = 1 32π 2 ǫ ijkl R ij R kl . In the case of R 4 ǫ i 1 ,ǫ i 2 , this is equal to Kähler-Hodge manifold When Kähler transformation defines a line bundle, its first Chern class is equivalent to its Kähler class: Ch(V ) = e c 1 (V ) = e tJ where J is the Kähler two-form.
One-loop determinant The Chern class of the base M 4 is derived from the zero-form values of the Chern class of R 4 ǫ i 1 ,ǫ i 2 which is identified with the local geometry around fixed points x i of M 4 . We assume that contributions from nonzero instanton sectors do not dominate in the large-N limit, and we are only interested in the perturbative part in the zero instanton sector. Under these assumptions, we can evaluate the contribution from the universal bundle as Ch(E) = Ch adj (e iλ ), and we can use the indices on R 4 × S 1 . The circumference of this S 1 , which is a period of τ ′ = n 2n+1 (τ + χ) = 2π 3n 2n+1 , is used for the normalization of the charge t of this S 1 rotation.
Putting all together, the index for vector multiplets is Each term has expansions in positive and negative power series [14], and the one-loop determinant are derived from their sum.