Supersymmetric Rényi entropy and defect operators

Journal of High Energy Physics, Nov 2017

We describe the defect operator interpretation of the supersymmetric Rényi entropies of superconformal field theories in three, four and five dimensions. The operators involved are supersymmetric codimension-two defects in an auxiliary \( {\mathbb{Z}}_n \) gauge theory coupled to n copies of the SCFT. We compute the exact expectation values of such operators using localization, and compare the results to the supersymmetric Rényi entropy. The agreement between the two implies a relationship between the partition function on a squashed sphere and the one on a round sphere in the presence of defects.

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Supersymmetric Rényi entropy and defect operators

HJE Supersymmetric Renyi entropy and defect operators Tatsuma Nishioka 0 1 3 4 Itamar Yaakov 0 1 2 4 0 Kashiwa , Chiba 277-8583 , Japan 1 Bunkyo-ku, Tokyo 113-0033 , Japan 2 Kavli IPMU (WPI), UTIAS, The University of Tokyo 3 Department of Physics, Faculty of Science, The University of Tokyo 4 induces a holonomy for A, around @ We describe the defect operator interpretation of the supersymmetric Renyi entropies of superconformal eld theories in three, four and tors involved are supersymmetric codimension-two defects in an auxiliary Zn gauge theory coupled to n copies of the SCFT. We compute the exact expectation values of such operators using localization, and compare the results to the supersymmetric Renyi entropy. The agreement between the two implies a relationship between the partition function on a squashed sphere and the one on a round sphere in the presence of defects. Nonperturbative E ects; Supersymmetric Gauge Theory; Supersymmetry - and Duality 1 Introduction Supersymmetric Renyi entropy and defects Renyi entropy and discrete gauge theories Supersymmetric Zn gauge theory Supersymmetric defects from supersymmetric Zn gauge theory Coupling defects to the n-copy theory Mass terms and vortices The matrix models The scalar moduli space and classical contributions Perturbative contributions Non-perturbative contributions | instantons and contact-instantons Sewing of instantons Classical and Chern-Simons contributions depending on instantons Fundamental hypermultiplets uctuations Adjoint hypermultiplet or vector multiplet uctuations The third point 3.7 n ! 1=n duality 4 Discussion A Conventions B Special functions C The instanton partition function { i { Introduction The non-local nature of quantum entanglement is one of the sharpest characteristics by which quantum physics di erentiates itself from classical physics. Entanglement occupies a central position in quantum information theory and, increasingly, in various branches of theoretical physics such as condensed matter and high energy physics. Entangled states are ubiquitous and of particular interest in many-body quantum systems. Renyi entropy is a re ned measure of the entanglement a given state possesses when the Hilbert space is split into states supported on a spatial region and those supported on its complement. In a local quantum amounts to eld theory, employing the replica trick [1], the nth Renyi entropy of a (Euclidean) manifold on which the theory is placed. The absolute value taken in the de nition (1.1) has no e ect in unitary theories, but is necessary to incorporate the case of a complex partition function which we will deal with when supersymmetry is implemented on a curved space. It follows from (1.1) that knowing the partition function Zn is more or less equivalent to calculating Renyi entropies, and there are a few situations where the exact values are known (see e.g. [1{3]). A common practice in handling the conical singularity around @ , present in the calculation of Zn for n > 1, is to smooth out the tip by introducing a regulator, calculate the partition function on the smoothed space, and take the singular limit [4]. This approach is highly advantageous as it reformulates the problem as a calculation in quantum eld theory on a curved space. Another complementary approach is to represent the partition function Zn as a product of correlation functions of twist operators that create the proper monodromies around [1, 5]. Twist operators are codimension-two (non-local in d > 2 dimensions) objects that specify the boundary conditions on the entangling surface, and the twisting is done for an n-fold copy of the original theory. In what follows, we will illustrate the interplay between the two approaches in a particular situation where the exact calculation of Zn is possible: the supersymmetric Renyi entropy computed using localization [6]. We will restrict ourselves to a spherical entangling surface @ = S d 2 in d dimensions, work in the vacuum state of the SCFT, and examine supersymmetric gauge theories of type 3d N = 2, 4d N = 2, and 5d N = 1. The motivation for doing so is twofold. First, it is interesting to compare the two di erent looking localization calculations and nd out how they match. Second, the microscopic de nition of the defect operators,1 presented in section 2, could be useful when examining dualities in which the SCFT participates. These could be dualities between di erent Lagrangian eld theories, or holographic dualities between an SCFT and string theory on an appropriate background. 1Twist operators are a subclass of defect operators. These terms will be used interchangeably in this paper. { 1 { The vacuum Renyi entropy is a non-local observable which can be de ned for any ddimensional quantum eld theory whose Hilbert space can be factorized in a local manner. For a conformal eld theory, and for integer Renyi parameter n, this observable is equivalent to two di erent objects, each of which can be de ned by a suitable Euclidean path integral: I. The partition function on an n-fold multiple covering of the d-sphere Sd, branched along @ , with appropriate boundary conditions at the branching loci [7]. II. The partition function of an n-fold copy of the theory - henceforth referred to as the n-copy theory - on Sd with codimension-one defect operators acting between copies (cf. [8]). The equivalence between these two objects is tautological. The object in II is simply a relabeling of the degrees of freedom, one for each of the n sheets of the branched sphere. The defect operators are de ned to reproduce the boundary conditions implied by the original geometry. One may introduce a linear eld rede nition, acting in the n-copy theory, to diagonalize the action of the defects, introducing defects which act on just one copy at a time. In the special case of a free theory, the action written using the rede ned elds does not couple the copies. The computation in this case is equivalent to a third object: III. The partition function of an n-fold copy of the theory on Sd with codimension-two defect operators acting on each copy - equivalently, the kth copy is coupled to a background connection with holonomy e2 i nk around @ , with k 2 f0; : : : ; n also [10{18] for further developments).3 It has been observed that the result in this case is equivalent to yet another object: IV. The partition function of a single copy of the theory on a squashed Sd, with a suitable supersymmetry preserving action. The squashing parameter, which determines the non-round metric, is related to the Renyi parameter n in a simple way, such that n = 1 corresponds to the round sphere. 2The holonomy prescription could be di erent if the eld in question is a fermion, or carries additional global symmetry charges [9]. We will make a speci c choice later on. 3Some of the 3d theories we consider are not superconformal. However, deformation invariance of the Renyi entropy of the SCFT to which the original theory ows [6]. means that the Euclidean action is invariant under some supersymmetry transformation . It is well known that the path integral in this case is insensitive to -exact deformations, either of the action or in the form of -exact insertions.4 The objects I and IV are related by such a deformation (c.f [19]). Despite their simplicity, there is something interesting to be said about forms III and IV of the supersymmetric Renyi entropy. The calculation of the path integrals representing either III or IV can be performed exactly using localization. This involves splitting the elds of the theory into an interacting part - the moduli - and a free part - the uctuations. The latter can be put into the form III. Since the two calculations look quite di erent, and in some contexts have di erent interpretations, it is interesting and potentially useful to determine exactly how they give the same result. This will be our primary goal. 1.2 Renyi entropy and discrete gauge theories The setup described above for calculating the Renyi entropy at integer Renyi parameter can be alternatively thought of as introducing a defect in a discrete gauge theory coupled to the n-copy theory.5 For our purposes, it is su cient to consider the gauge group Zn acting by cyclic permutation on the copies, although the n-copy theory is invariant under the full permutation group Sn. 6 If we choose to think of Zn as a gauge symmetry, we can reasonably treat the defects which implement the calculation of the Renyi entropy as codimension-two objects. Only operators charged under the Zn symmetry can detect the position of the codimension-one defect. Gauging Zn means that all such operators are projected out. We would now like to incorporate supersymmetry into the de nition of the Zn gauge theory and the defect. A simple way of doing so is to write down a version of the discrete gauge theory which is realized by higher form abelian gauge elds [20]. A BF type theory with one ordinary gauge eld (A) and one (d 2)-form eld (B) works nicely. The reduction to Zn gauge symmetry is implemented by using B as a Lagrange multiplier. The action for the gauge sector is in Z 2 SBF = F ^ B ; where F is the eld strength of A. SBF is invariant under gauge transformations for both A and B as long as n is an integer. The codimension-two defect could be realized by taking a at connection with prescribed holonomy and coupling it to the free elds of the n-copy theory. The way to do this in the BF theory is to insert the operator I exp ik = exp ik Z (1.2) (1.3) 4There are additional restrictions on the deformation. A -exact deformation to the action must be annihilated by and should have a positive semi-de nite real part. Speci cally, any deformation must be such that it does not alter the convergence properties of the path integral. 5The authors are grateful to Daniel Ja eris for suggesting a discrete gauge theory interpretation for our calculation. 6The symmetry which shifts the copies by one is known as the replica Zn symmetry. { 3 { where [@ ] is the Poincare dual to the cycle @ representing the entangling surface. This operator is invariant under the B gauge transformations as long as k is an integer. Integrating out B in the theory with the action (1.2), and in the presence of this operator, of strength e2 i nk . After the linear eld rede nition, A can serve as the at connection for the kth copy for the Renyi parameter n. One may then extend the elds A; B and the terms in the action to their supersymmetric versions to create a SUSY-BF theory [21].7 One possible obstruction to using ordinary gauge elds is that the original theory may have its own, possibly non-abelian, gauge elds. It may not be possible to couple the eld A to these non-abelian gauge elds in a manner consistent with gauge invariance. Note HJEP1(207) that the original description of the codimension-one defect already implies that any gauge group of the n-copy theory is broken to a diagonal subgroup, acting simultaneously and identically on the copy at either side, at the defect. This type of the reduction of the gauge invariance at the position of a defect is quite common (see e.g. [22]). Moreover, gauge elds often carry supersymmetric moduli which cannot be coupled to the codimension-two defect in the way described above. We will treat all of these problems in an ad hoc manner. The organization of the paper is as follows. In section 2 we introduce a supersymmetric version of the Zn gauge theory based on higher form abelian gauge super elds. Component expressions for the super elds and actions are readily available in the literature. We then derive the abelian version of the codimension-two supersymmetric defect operator as a classical supersymmetric con guration for a vector supermultiplet on the round ddimensional sphere. We use this con guration to write down an ordinary supersymmetric Wilson type codimension-two operator for the tensor multiplet, using the supersymmetric version of (1.3), and perform localization in the presence of this operator and the BF term. We then derive the resulting e ect on any matrix model to which the original vector multiplet could potentially be coupled. In section 3, we show that the matrix model for the n-copy theory on the round Sd, after a speci c modi cation by such abelian defects, is equal to the matrix model on the squashed Sd, including all classical and one-loop contributions. In the process, we show how the moduli for the squashed sphere can be thought of as n sets of moduli for the round sphere, which are sewn together by the defect. We end with a discussion of possible applications. 2 Supersymmetric defects from supersymmetric Zn gauge theory In this section, we de ne the supersymmetric Zn gauge theory, and supersymmetric codimension-two defects, which we later use to calculate the supersymmetric Renyi entropy. We then calculate the expectation values of these defects using localization. We treat supersymmetric gauge theories of type 3d N = 2, 4d N = 2, and 5d N = 1. 7By SUSY-BF, we mean an untwisted supersymmetric version of BF theory of the type considered in [21]. We construct a supersymmetric Zn gauge theory, and a supersymmetric codimension-two defect, by introducing a pair of dynamical supermultiplets: a vector multiplet V and a (d 2)-form multiplet E. The eld strength for the latter sits in the familiar linear multiplet, denoted by G. We introduce the supersymmetric analogues of the terms used in the discrete gauge theory and The terms on the right hand side are schematic superspace integrals, for which we write component expressions later. The operator (2.2) can also be written as ik Z 2 exp G Vdefect ; where Vdefect is a background con guration for a vector multiplet invariant under a subset of the supersymmetries, whose component expression will be worked out below, and the integral is over the entire superspace. Integrating out G results in a supersymmetric delta function setting We call the coe cient the vortex charge. k n V = Vdefect : in Z 2 G V ; I D ; 2.2 The 3d N = 2 codimension-two defect In three dimensions, both V and E are ordinary vector multiplets. The super eld G is the eld strength super eld which can be used to write the Yang-Mills term in 3d.8 The codimension-two defect is a vortex loop of the type examined, for instance, in [23]. Its supersymmetric version was analyzed in [24, 25]. The calculation of the supersymmetric Renyi entropy in three dimensions using defects was carried out in [6]. We brie y review it below. The localization calculation for an N = 2 theory on the round S3 reduces the path integral to an integration over a single Lie algebra valued scalar: the constant mode of the real elds and D appearing in the N = 2 vector multiplet [26, 27]. The resulting matrix model expression for the partition function will be given in section 3.3. Speci cally, the uctuation determinant for an abelian vector multiplet is 1. 8This is a real linear super eld, which is sometimes denoted by . (2.1) (2.2) (2.3) (2.4) (2.5) (2.6) where [ ] is the volume form on a maximal S1 S3, and [ ]D is the Poincare dual to = nk , this con guration can be imposed on a dynamical vector multiplet by considering an additional vector multiplet with connection B, an o -diagonal Chern-Simons SB(3Fd) = and a supersymmetric abelian Wilson loop of the form A supersymmetric defect which mimics the e ects of computing the supersymmetric Renyi entropy can be introduced by considering an additional background abelian vector multiplet A ; A; DA; fermions : The defect con guration is of the form dA = Localization reduces the above terms to9 Integrating over the new modulus B then sets I W (3d) (k) exp SB(3Fd) W (3d) (k) exp 2 i n A B ; exp 2 k B ; ! ! A = i : k n { 6 { Note that this vev for A is o the original contour of integration and represents an imaginary mass term for chiral multiplets to which the A vector multiplet is coupled. We will argue later that an imaginary Higgs type mass for a dynamical vector multiplet can also be thought of this way. The imaginary mass is the entire e ect of the original defect on the localization computation. By continuity, the same is true for an arbitrary . We will show below that this continues to hold, for appropriate defects, in four and in ve dimensions. 2.3 The 4d N = 2 codimension-two defect We consider a 4d N = 2 theory consisting of vector multiplets and hypermultiplets. We would like to show that the e ect of inserting a codimension-two defect is equivalent to the introduction of an imaginary mass term. 9We universally denote the Lie algebra valued scalar zero mode as . We begin by reviewing some aspects of the localization calculation for an N = 2 theory on the round, and on the branched four-sphere following [10, 28, 29]. We refer the reader to these papers for explicit actions and localizing terms. The authors of [10] considered a smooth resolution of the branched four-sphere, the resolved four-sphere, which is deformation equivalent to it, and on which localization computations can be performed. In this section, we retain an overall scale `, associated with the size of the four-sphere, which will later be set to 1. We use the following coordinates on an n-fold covering of S4 (the branched sphere) and take a basis for the Cli ord algebra where i are the Pauli matrices. to the reality condition [29] a = i = i i ; 0 a 0 a ! ; i i i ; 4 = 4 = 1 ; e1 = ` d ; e2 = n` sin d ; e3 = ` cos d ; e4 = ` cos sin d ; (2.14) An N = 2 supersymmetry is generated by a foursome of Weyl spinors A; _ A, subject A ( A)y = AB B ; A _ _ A y = _ _ AB _B : The subscript A is an SU(2)R index. Indices ; _ indicate a spinor transforming as a doublet under the left and right SU (2) factors of Spin (4). Indices ; _ ; A are raised with ; _ _ ; AB such that 12 = 1 and AB = AB. In order to preserve rigid supersymmetry, one must solve the Killing spinor equation. The relevant Killing spinor equation for the round/branched four-sphere is 1 8 1 8 b a) A iASU(2)R B iASU(2)R B A A B = B = i i A0 ; A0 ; Here, we have introduced an SU(2)R background connection ASU(2)R , and set all other supergravity background elds besides the metric to zero. The solution is given by ASU(2)R = n 1 2 1 0 ! 0 1 d ; { 7 { (2.13) (2.15) (2.16) (2.17) (2.18) (2.19) 0 can be extracted by contracting with / . In order to use the transformation generated by to perform localization, one should impose an additional constraint to ensure that the square of the transformation does not contain scale or U(1)R transformations This constraint has been taken into account above. As shown in [10, 29], one can introduce supergravity backgrounds for the resolved foursphere such that the same Killing spinors are preserved. Moreover, one may use the same localizing term for the vector and hypermultiplets in the presence of defects/squashing as one does in the round sphere case. These localizing terms, which are described in [28, 29], imply that all components of a hypermultiplet must vanish on the moduli space. The contour of integration for vector multiplet scalars compatible with the localizing terms is and The localizing terms yield the following moduli space of zero modes for a vector multiplet y = ; DAB y = DAB : ` 2 = = i = 0), and anti-instantons at the south pole ( = 0; = =2). The complete matrix model expression for the partition function is given in section 3.3. 2.3.2 Introducing the defect We would like to introduce a surface defect into the computation on the round sphere. The data for a surface defect can be embedded in a background N = 2 abelian vector multiplet A ; ; ; DAB; A; A ; where A is the gauge eld, and are complex scalar elds whose relationship depends on the contour of integration, DAB is an SU (2)R triplet of auxiliary scalars, and A; A independent Weyl fermions which are SU (2)R doublets. We want a supersymmetric defect, on the round four-sphere, supported only on the two-sphere at = 0. Such a defect would have a eld strength Fdefect = ( ) d ^ d : { 8 { In order to regularize the above con guration, we introduce a background such that10 g ( ) is a smooth function satisfying g ( ) ; < The supersymmetry variation of the gaugino in an abelian vector multiplet is [29] A = A = 1 2 1 2 F F A + 2D= A + 2D= A + DAB A + DAB B + 2 D= A ; B + 2 D= A : We can complete (2.27) to a supersymmetric con guration by introducing additional bosonic backgrounds i In this background, the gaugino variations vanish with an arbitrary spinor from (2.20). In the limit ! 0 we get g ( ) ! 1 ; Note that both the smooth and singular con gurations are o the original contour of integration for the vector multiplet (2.22) (D12 is imaginary). 2.3.3 The supersymmetric Lagrange multiplier Following the logic of the Zn gauge theory, we will impose the con guration for the background vector multiplet using a supersymmetric Lagrange multiplier. We make the vector multiplet dynamical and introduce an N = 2 abelian tensor multiplet E E ; G; G; LAB; fermions ; whose eld strength sits in the linear multiplet G [30].11 We couple E to the vector multiplet (actually its eld strength) using a supersymmetric BF term [30] SB(4Fd) = 2 1 2 1 4 " G + G DABLAB + E F + fermions : (2.35) 11We use the letters E and G to indicate both the supermultiplets and some of their components. We hope this does not cause too much confusion. (2.26) (2.27) (2.28) (2.29) (2.30) (2.31) (2.32) (2.33) (2.34) We then introduce a Wilson surface operator for the tensor multiplet by adding an additional copy of (2.35), but replacing all elds in the vector multiplet with their values from (2.32) and (2.33), and with exp Integrating out the tensor multiplet would result in a supersymmetric delta function setting all elements of the vector multiplet to their values for the surface defect. Instead of doing this, we rst perform the localization. 2.3.4 Localization of the tensor multiplet The localizing term for the abelian vector multiplet can be taken from [29]. For the tensor multiplet, we must consider the transformation of the fermions in this multiplet, denoted 'A; 'A.12 The variations are (2.36) (2.37) (2.39) (2.40) (2.41) (2.42) (2.43) (2.38) 12We have decomposed each spinor 'i from [30] as a doublet 'A; 'A. Note that the de nition of the Levi-Civita symbol in [30] contains an extra factor of \i" compared to the usual de nition: "1234 = 1. Examination of the gravitino transformation in [30] yields the following dictionary in relation to [29] i ! A! A ; i ! 2 1 = D A! A = 2i 0A! 0A : Superscripts on both sides indicate SU(2)R transformations. Note the factor of 2 di erence with the used in e.g. [31, 32]. 'A = Stensor localizing term = h ( ')yA 'A + ( ')yA 'Ai : Z S4 1 2 " G = Gy ; LAB y = LAB ; L11 y = L22 ; Im L12 = 0 : H G = Gr + iGi : We take the localizing term multiplet elds (2.35) which yields We also de ne We will use reality conditions for G; G; LAB appropriate for the coupling to the vector H = 0 ; L11 = L22 = Gr = 0 ; Gi = = G0 : (2.46) with G0 a real constant. 2.3.5 Modi cation of the matrix model We will couple the vector multiplet representing the surface defect to the physical vector multiplets and hypermultiplets of the N = 2 SCFT as detailed in section 3.1. After localization, the matrix model representing the S4 partition function is modi ed by the presence of the defect. We would like to show that this modi cation amounts to giving the hypermultiplets or the vector multiplets an imaginary mass, given by discuss all contributions of V and E to the matrix model. n i k . Below we Classical contribution. The multiplets V and E have a classical action given by (2.35) and (2.36). After localization, these terms give insertions in the matrix model. Using the vevs for the scalar elds and we get = = i ; 2 D12 = G = G = i G0 ; L12 = ` ; ` 2 G0 ; exp SB(4Fd) Wsurface (k) ! ! exp 2 i`4 n G0 ; exp 2 `3 k G0 : i ` The bosonic part of the resulting localizing term is Z S4 " +2 The vector eld v is such that The localization locus is thus 1 The quadratic approximation to the localizing term (2.44) for the tensor multiplet around the locus (2.46) is independent of G0. The one-loop determinant is thus a G0 independent number, which is furthermore equal to the partition function of a free tensor multiplet on the round four-sphere. This multiplet can be dualized into an uncharged massless hypermultiplet whose partition function can be deduced from the expressions in 3.5. The one-loop determinant for A is trivial. H = ?dE = 0 ; implies that we can set up to tensor gauge transformations. When there are non-trivial two-cycles, the form E is closed and the tensor gauge transformations imply that E is gauge equivalent to 0 whenever it represents an integral class. The remaining moduli of E can be identi ed with the values of the possible Wilson surfaces. The gauge eld A also has moduli in this situation. A smooth instanton con guration for A may exist, where A is a non-trivial connection with integral ux on the two-cycle. The analysis above applies only to smooth con gurations for A; E and their gauge transformation parameters. Experience shows that we should allow singular con gurations for A, as we did for the non-abelian gauge elds. The coupling of E to the eld strength FA means that we should consider allowing singular con gurations for E as well, at least at the poles. In fact, the con guration (2.32) which we are trying to reproduce has a singularity on the entire maximal two-sphere. If we allow E to have the singularities at the same position as A, then the coupling (2.50) (2.51) (2.52) (2.53) (2.54) (2.55) (2.56) Non-perturbative contributions. In the absence of two-cycles, the equation I I E = H = 0 ; exp in Z F Ainst ^ [ ] ; induces a non-vanishing classical contribution upon localization in Z 2 in Z 2 FA ^ E FA ^ E ! in Z 2 V G ; in Z 2 F Ainst ^ Eclassical : the form trivial The form E is again closed and the tensor gauge transformations again imply that E is gauge equivalent to 0 whenever it represents an integral class. The remaining moduli of E, parameterizing singular closed two-forms localized at the poles modulo integral forms, are angular parameters. Upon localization, there is a contribution to the matrix model of n Z FA ^ E ! n Z F Ainst ^ [ ] ; A where F inst is the eld strength of some singular instanton con guration at the poles and [ ] is a representative of any singular two-form at the poles, which is de ned only up to integral classes. The part of the electric defect containing an integral over E is, however, because E is being paired with a trivial cycle represented by @ . Integration of A over all then restricts F inst to vanish. This means that we only need to work in the instanton number 0 sector for A. Integrating out. In addition to the above, there is a remaining integration over G0. This integral, and the one over , can be done explicitly. The result is simply to set or in the units of the rest of the sections This was what we set out to show. The parameter entering the one-loop and instanton contributions in the matrix model, described in section 3, is im, where m is identi ed with the of a background vector multiplet. Hence, our surface operator is equivalent to an imaginary mass HJEP1(207) equivalently, a shift of the parameters by 2.4 The 5d N = 1 codimension-two defect We describe the implementation of codimension-two defect operators in ve-dimensional N = 1 theories. Most of the analysis is similar to the four-dimensional case, so we will be brief. Conventions and notations are the same as in [14]. 2.4.1 Killing spinors on the ve-sphere Supersymmetric eld theories on curved spaces are systematically obtained in the rigid limit of the N = 1 supergravity in ve dimensions which has an SU(2)R symmetry whose indices are denoted by A; B as in the 4d case. There are SU(2)R gauge eld V AB, SU(2)R triplet scalar eld tAB and the other elds which are irrelevant to the following discussion in the Weyl multiplet [33]. The variations of the fermions in the multiplet have to vanish for preserving supersymmetries on given background elds and the solutions are the Killing spinors. We set the radius of the ve-sphere to one from the beginning to simplify the discussion. The round unit ve-sphere allows the Killing spinor in the coordinates ds2S5 = d 2 + sin2 d 2 + cos2 ds2S3 ; when the background elds are set to [14, 34] 1 2 tAB = ( 3)AB ; V AB = ( 3)AB d ; others = 0 : With the hermitian gamma matrices in tensor product forms 1 = 1 (2.57) (2.58) (2.59) (2.60) (2.61) (2.62) (2.63) ; = i 2 i i eld describing the defect is whose eld strength becomes We want a codimension-two surface defect at = 0 on the round sphere. Introducing a smoothing function g ( ) as in the four-dimensional case (2.28), the background gauge a spinor A in ve dimensions is also written as tensor products of spinors A and two and three dimensions A = A A : The Killing spinor on the round sphere (2.61) is given by (2.64) (2.65) (2.66) (2.67) (2.68) (2.69) A defect and (2.70) (2.71) where 1;2 are constant spinors and are the Killing spinors on a unit three-sphere Codimension-two defects N = 1 abelian vector multiplet A surface defect in a avor symmetry is speci ed as a singular con guration in a background a real scalar, Y AB an SU(2)R triplet scalar and A an SU(2)R This con guration is supersymmetric if the real scalar defect, the gaugino the triplet scalar YdAefBect take the following forms (Ydefect)AB = g0 ( )( 3)AB ; defect = A defect = 0 : i 2 1 4 This con guration is invariant under the supersymmetric transformation, especially one sees 0 = A defect = A(Fdefect) + (Ydefect)AB B : One can replace g0 ( ) with the delta function ( ) in the singular limit ! 0. fLAB; E ; N; 'Ag ; consisting of an SU(2)R triplet scalar LAB, an antisymmetric tensor gauge eld E , a real scalar N and an SU(2)R-Majorana fermion 'A, through a supersymmetric BF coupling [33] We will implement defects in a avor symmetry by coupling the abelian vector multiplet to the linear multiplet It is gauge invariant for an integer n as E transforms as gaugeE supersymmetric transformation laws of the linear multiplet with only the backgrounds tAB and V AB turned on are given by 10i A'BtAB ; 4tBC LCB A 6t(AC LB)C B + ik Z pg 2 ik Z S3=0 i 2 (Y defect)BA LAB + (L11 L22) !3 + 4 1 F defectE 1 2 E # ; In addition, we introduce a surface operator for the linear multiplet with the BF coupling to the defect abelian vector multiplet where !3 is the volume form of the unit three-sphere and is the Hodge operator in ve dimensions. Choosing = 2 the defect operator is gauge invariant for an integer k. The linear multiplet plays a role of the Lagrange multiplier and integrating it out in the path integral with the BF term and the Wilson surface results in setting the vector Instead of doing so, we localize the tensor multiplet on the round sphere with the LAB = 2i (A'B) ; E = 2i N = 2 ' ; (D ') 'A = i(D LAB) Wsurface(k) = exp 1 2 where we de ned L^AB multiplet to the defect con guration. localizing term whose bosonic part is written as Vt(elnosco)r = i(D LAB) B 4tBC LCB A 6t(AC LB)C B ) A + N A (D LAB)(D LBA) + (N 8L^AA)2 + 18 L^(AB)L^(AB) ; = exp B " 1 2 i 2 i 2 1 4 tAC LCB. 1 4 SB(5Fd) = In localizing the tensor multiplet, we impose the reality condition for LAB that is equivalent to (LAB)y = LBA ; (L^AA)y = L^AA ; (L^(AB))y = L(AB) : This choice makes the bosonic part of the localizing term be semi-positive de nite and the tensor multiplet localizes to D LAB = 0 ; which on the round sphere yields Localizing the vector multiplet to the xed locus N = 4(L11 the BF term (2.73) and the surface operator (2.75) end up with exp SB(5Fd) Wsurface(k) ! ! = exp exp i : k n i 2 2 2 2 n N0 ; kN0 : (2.78) (2.79) (2.80) (2.81) (2.82) (2.83) (2.84) (3.1) (3.2) the integration over N0 sets we will con rm in section 3. 3 Squashing from defects This value is consistent with the mass shift by the parameter (2.60) as in the 4d case as We are now in a position to demonstrate the relationship between the supersymmetric Renyi entropy and supersymmetric codimension-two defects. In the original setup, the supersymmetric Renyi entropy is de ned as Snsusy 1 1 n log Znsusy (Z1) n ; where Zn is the appropriate supersymmetry preserving partition function on the branched d-sphere, or the squashed d-sphere. The schematic form of the localization calculation for Zn is Zn = X Z moduli hZnclassical (moduli) Znpert (moduli)i ; (3.3) (3.4) where the sum/integral is over the moduli space of supersymmetric zero modes, Znclassical is the exponential of minus the Euclidean action evaluated on the moduli space, and Znpert is the one-loop determinant obtained by evaluating the path integral over non-zero modes in the quadratic approximation around the moduli space. The partition function for the n-copy theory on the round sphere is simply We would like to demonstrate that Zn can be computed using Z1 and the insertion of Where the right hand side is the partition function of the n-copy theory on the round sphere in the presence of a speci c codimension-two defect described in the next subsection. After the linear eld rede nition, the result is Zndefect X Z n moduli " The outer sum/integral is over n copies of the moduli. The symbol moduli is a placeholder for the sewing operation which identi es how the squashed sphere moduli space fractionalizes into n copies. Zclassical is the classical contribution of a single copy of the theory 1 evaluated on the round sphere. Zpert-defect (k; moduli) is the perturbative contribution, in 1 the quadratic approximation around the moduli space, in the presence of the appropriate defect for the kth copy. 3.1 Coupling defects to the n-copy theory The super eld V , which carries the information about the supersymmetric defect, must be coupled to the physical elds of the n-copy theory. For matter multiplets, this is the usual minimal coupling of V to chiral multiplets or hypermultiplets. We need not consider the e ect of the defect on the other terms in the action involving matter elds, since these vanish at the level of the quadratic approximation around the localization locus for any of the setups we consider. The coupling of a at connection carried by V to physical gauge elds can be accomplished by formally performing the eld rede nition in the introduction. Since non-abelian gauge elds are not free elds at nite gauge coupling, the resulting action would inevitably look like a gauge non-invariant mess. However, gauge invariance, with the caveats already mentioned, is guaranteed by the ability to undo the eld rede nition. If the gauge group is SU(N ), there is a physical procedure which implements the right defect and makes clear the form of the coupling to a background vector, at the level of the quadratic approximation to the moduli space. We use the language of 4d N = 2, but the same applies to any of the theories under consideration. First, consider the enlarged gauge group U(nN ), where the gauge group of the n-copy theory, SU(N )n, is embedded as a block diagonal subgroup The required codimension-two defect, before the eld rede nition, can be viewed as a at connection, on Sdn@ , which is represented by a one-form with holonomy data [35, 36]. vector multiplet once in a chosen direction. To preserve supersymmetry, we consider a Gukov-Witten type surface defect with the holonomy (3.7) specifying the To go back to the n-copy theory, while keeping the defect, one should rst Higgs U(nN ) down to SU(N )n by giving an appropriate large vev to the adjoint scalar in the 0 SU(N )0 B B B B . . . 0 SU(N )1 0 . . . 0 . . . 0 0 . . . SU(N )n 1 1 C CC : C A BBB ... ... B 0 0 B 1 0 0 1 0 0 . . . ... CCC ; XHiggs = BBBB 0... t...1 . 0 0 . . . 0 0 . . tn 1 1 C CC : C A (3.6) (3.7) (3.8) Upon taking ti ! 1, all modes not coming from the original n-copy theory are in nitely massive and do not contribute to the computation. XHiggs should really be considered only up to permutations of the ti, which are a part of the Weyl group of the theory. In fact, the holonomy (3.7) acting on XHiggs produces such a permutation. Although we do not show this explicitly, we take this to mean that the Gukov-Witten operator with this data preserves the same supersymmetry as XHiggs. After a change of variables, which is in this case a constant SU(nN ) gauge transformation, the elds in the vector multiplet for SU(N )k acquire a monodromy exp (2 ik=n) around @ .13 The physical e ect of the defect on vector multiplets can now be examined more carefully at the level of the quadratic approximation to the moduli space of the remaining light elds. Since the action in this approximation is quadratic, the monodromy can be traded for a coupling to a background vector multiplet with a speci c pro le. This pro le is singular, and determined by the value of the monodromy and by supersymmetry. It is this multiplet, denoted by V , which arises in our realization of the supersymmetric Zn gauge theory. 13Note that the diagonal elements of SU(N )k also acquire this monodromy. Had we tried to implement the defect using a Gukov-Witten type surface defect in each SU(N )k, this would not have been so. 3.3 We collect the expressions for the matrix models associated to the squashed sphere partition functions in three, four and ve dimensions. Our conventions for integration over the Lie algebra are in appendix A. Special functions in the one-loop determinants and instanton contributions are de ned in appendices B and C, respectively. The matrix model for a 3d N = 2 theory on the squashed sphere is [41, 42] Zsusy 3d = 1 Z rankG Y jW j i=1 d i p!1!2 e !1i!2 Tr( 2) Y 2 + # of chirals Y I=1 S2 (i ( )j!) S2 ( i ( )j!) Y 2RI S2 i ( )+imI + j!j I ! 2 + are the positive roots of g. to the Ith chiral multiplet. jW j is the size of the Weyl group. I is the R-charge of the Ith chiral multiplet. is the Chern-Simons level. ! = (!1; !2) are squashing parameters, and i runs over the Cartan of the Lie algebra g of the group G, which we have assumed is U(N ). Our conventions are such that i are real (see appendix A). denotes a weight in the representation RI associated j!j X !i = j!1 + !2j : The supersymmetric Renyi entropy is computed using ! = 1; n1 . We will set the mass parameters mI to zero. The matrix model for a 4d N = 2 theory on the squashed sphere is [29] Zsusy 4d = jW j Y 2 + 1 Z rank G Y i=1 d i p!1!2 e !1!82g2Y2 M Tr( 2) (i ( )j!) ( i ( )j!) Y 2R Zi(n4sdt) q(4d); i ; mf0 ; !1; !2 i ( ) + im + j!j ! 2 1 2 ; gYM is the Yang-Mills coupling. gYM and YM appear also in the instanton part of the matrix model. q(4d) is de ned in appendix C. R is now the total representation of the hypermultiplets. We have set all hypermultiplet masses to a common value: m. (3.19) 1 ; (3.20) (3.21) The supersymmetric Renyi entropy is computed using ! = parameter m for all hypermultiplets set to a common value 1; n1 , with the mass The result for a 5d N = 1 theory on the squashed sphere is still conjectural [34, 43{49]. (3.22) (3.23) (3.24) (3.25) (3.26) 2 Zinst q; i ; mf0 ; !1; !2; !3 Zinst q; i ; mf0 ; !3; !1; !2 Zinst q; i ; mf0 ; !2; !3; !1 ; 2 is the Chern-Simons coupling. and gYM appear also in the instanton part. q is de ned in appendix C. into [51] The derivative of the triple sine function in the integral measure can be written S30(0j!) = 3(!1; !2; !3)2 1(!1) 1(!2) 1(!3) ; 2(!1; !2) 2(!2; !3) 2(!3; !1) where r(!) is the Stirling modular form Note that 1(!) = p2 =!. metric Renyi entropy is computed using ! = 1; 1; n1 , and setting ! = (!1; !2; !3) are squashing parameters, and j!j = j!1 + !2 + !3j. The supersym1 Z rank G Y i=1 jW j Y 2 + We follow the form in [50] Zsusy 5d = z!0 z r(zj!) j!j 2 : : 3.4 The scalar moduli space and classical contributions All of the theories we consider have a moduli space which is partially given by the vev of a Lie algebra valued scalar. The scalar has eigenvalues ~ . Integration over is what makes the result of the localization procedure into a matrix model. In our setup for the n-copy theory, there is one such k and one integration for each copy. There are also classical contributions to the matrix model which depend on . The mode which parametrizes is a part of the original theory which cannot be treated as free, even after localization. Therefore, the values of k are subject to the boundary conditions implied by the original codimension-one de nition of the replica defect. It is Starting from the n-copy theory on the round sphere, after using the delta functions to set all the k equal, we recover the factor of n. Perturbative contributions The defect operator interpretation of the supersymmetric Renyi entropy was originally observed in [6] by rewriting the perturbative partition function on the n-fold cover as ncopies of the partition functions on a round three-sphere with vortex loops inserted on each copy. (See also [16, 17] for related works in two dimensions). We extend this interpretation to higher dimensions and show the perturbative parts of the partition functions in the 4d and 5d supersymmetric Renyi entropies also have similar structures. The non-perturbative contributions arise in higher dimensions will be discussed separately in section 3.6. First, let us review the story in three dimensions [6]. There are no non-perturbative contributions in the matrix model (3.19) and we only need to deal with the one-loop partition functions appearing as the double sine functions S2 from the vector and matter multiplets. The identity (B.7) for the double sine function yields that the one-loop partition function of a multiplet with R-charge I can be decomposed as a product of those in the presence of a supersymmetric abelian vortex loop [25] S2 i ( ) + j!j 2 I 1; 1 n Y S2 i ( ) + I 2 1 n k n 1 + + 1; 1 1 = = n 1 k=0 n 1 k=0 Y S2 i ( ) + qk3d vortex + I 1; 1 1 ; 1 ; ec(Qi !i 1)Tr( p) ; Y ! 1 = n; Y ! 1 = 1: i i (3.27) (3.28) (3.29) (3.30) (3.31) (3.32) trivial to see that this implies that all k are equal.16 Equivalently, the sewing operation for this set of moduli consists of a set of delta functions, in the matrix model for the n-copy theory in the presence of the defect, which enforce this equality mscoadlaurli = n 2 rank G Y Y k=0 i=1 (( k)i ( k+1)i) : The classical contributions depending only on are products of expressions of the form for some constant c. In the theory considered on the squashed sphere, we have HJEP1(207) while for the round sphere where qk3d vortex is introduced to be q3d vortex = k I 2 1 n 1 + k n : 16This only makes sense if one identi es the gauge transformations at the interface. This also implies the identi cation of the residual gauge transformations acting in the di erent matrix models. Comparing it with the shift of the modulus (2.12), we interpret the decomposition as a manifestation of the introduction of a supersymmetric abelian vortex loop, described in section 2.2, of charge qkvortex supported on the entangling surface ( = 0) on each copy of a round sphere. Note that qkvortex di ers from (2.12) by a term proportional to the R-charge I for matter multiplets. This means that the vortex loops for the supersymmetric Renyi entropy are dressed by the R-symmetry ux. One can work out a similar decomposition for the perturbative part in the 4d N = 2 matrix model (3.21) with a slight modi cation. The matter one-loop partition function is represented by the function, which enjoys decomposition into the product of the n-copies with the help of the identity (B.12): i (a) + im + j!j 1; 2 1 1 Y k=0 i (a) + 1 + qkvortex 1; 1 1 ; where qkvortex is to be interpreted as the charge of a codimension-two surface defect k n ; (3.33) (3.34) under the choice of the shifted mass (3.14) and the relation (A.5). It agrees with the shift of the modulus (2.60) induced by the insertion of a supersymmetric abelian surface operator of charge k in section 2.3. Repeating the analogous procedure to the 5d N one-loop partition function with the identity (B.7): = 1 theory one nds the matter S3 i ( ) + im + j!j 1; 1; 2 1 n 1 = Y S3 i ( ) + 1 + qkvortex 1; 1; 1 1 ; (3.35) where we introduce the charges qkvortex by (3.34) and the mass shift (3.16). Once again it can be interpreted as a supersymmetric codimension-two surface defect of charge k described in section 2.4. It is straightforward to apply the same argument to vector multiplets in any dimensions to read o the surface charges and the results (3.32) and (3.34) still hold with I = 0. 3.6 Non-perturbative contributions | instantons and contact-instantons The partition functions on the four-sphere and the ve-sphere receive non-perturbative contributions from instantons and contact-instantons, respectively. These are supersymmetric con gurations localized at xed loci of the equivariant action generated by the square of the supersymmetry. For the four-sphere, the xed points are at the north and south pole and give rise to instanton and anti-instanton contributions, respectively. The partition function includes contributions from contact instantons [43], which are extended along the ber of (3.36) S 1 ! S 5 ! CP 2 ; Yli k = 1 Ylp k = 0 k = 2 HJEP1(207) and localized at three points on the base.17 The supersymmetric instanton contributions are computed by the Nekrasov partition function [52, 53]. We nd it convenient to express both types of contributions in terms of the 5d, or q-deformed, version of the Nekrasov partition function, which we review in appendix C. The 4d undeformed partition function can be recovered by taking an appropriate limit. Instanton and contact-instanton contributions introduce new moduli, classical contributions, and perturbative contributions into the calculation. We begin by discussing how the moduli of the replicas in the n-copy theory are sewn up to produce those of the original theory. We assert a speci c pattern for the fractionalization of a particular instanton conguration, given by a vector of Young diagrams. We then show that, given this pattern, the classical and perturbative contributions recombine to yield (3.4). Similar decompositions of partitions appear in the context of instantons on ALE spaces (cf. [54]). A relationship between the instanton partition function on a Zn orbifold of C2, acting on just one C factor, and surface operators appears in [55, 56] (see also the review [57]). It is possible that the results we need for the covering space associated to the n-copy theory can be recovered from the latter papers. Speci cally, this seems plausible given the connection between the supersymmetric Renyi entropy at n and at 1=n. However, we need 5d results and a very speci c surface operator, so we derive the necessary fractionalization and relationships between the determinants in this context. 3.6.1 Sewing of instantons After employing equivariant localization, the instanton moduli space localizes to a set of points, each given by a vector of partitions [52, 53]. The partitions describing the instanton moduli space of the theory on the branched sphere fractionalize to yield partitions for each 17To the best of our understanding, the precise form of these contributions is still conjectural. We will use the form considered in e.g. [50] and nd that it works well. of the replicas in the n-copy theory. This happens by splitting one set of Young diagrams, yielding a partition vector Y~, into n Young diagrams, yielding partitions n Y~ (k)on 1 , having the same combined number of boxes. This process is illustrated in gure 1. The diagrams can be split along the vertical or along the horizontal, depending on which deformation parameter i is taken to be n times smaller. Since the instanton partition function is invariant under the simultaneous transposition of the diagrams and 1 $ 2, it su ces k=0 to consider the situation in gure 1. For the contribution of a hypermultiplet in the fundamental representation, each element of the uctuation determinant around an instanton can be associated to a box in the Young diagram, and gure 1 describes the decomposition into replicas. The contribution from a vector multiplet, or from an adjoint hypermultiplet, involves pairs of partitions in the vector Y~. We nd that the decomposition into replicas follows the pattern HJEP1(207) n 1 Y k1;2=0 (Y1; Y2) (Y1 (k1) ; Y2 (k2)) jk1 k2=k mod n : (3.37) ZY5~d;-CS ~a; ; 1; 2 1 2 = exp 42 i 2 n 1 k=0 = Y exp 42 i X X a l X X l (p;t)2Yl(k) s 1 n a l k n The moduli space of contact-instantons can also fractionalize along the additional S1 direction. This process is simpler, amounting to decomposing the Kaluza-Klein (KK) momentum, and is described in section 3.6.5. It does not involve the partitions. Classical and Chern-Simons contributions depending on instantons The classical weight in the instanton partition function of a con guration with instanton number j Y~j in a 5d SCFT is The number of boxes in the Young diagrams on the right pane of gure 1 sum to the number of boxes on the left, and therefore the combined weight from each of the replicas matches that of the original theory qj Y~j : n 1 k=0 ~ qjYj = Y qj Y~(k)j : In the presence of a 5d Chern-Simons term, we also need to split the contribution 2 ZY5~d;-CS (~a; 1; 2; ) = exp 4i X X (al (s 1) 1 We do this rst for the case where 1 = 1=n by setting s = np (3.38) (3.39) (3.40) (t k 3 (t 1) 5 ; 3 where in the second line we have reparametrized (3.42) (3.43) (3.44) (3.45) (3.46) (3.47) The expression on the second line can be thought of as the contributions from the n-replicas, where the additional shift is the e ect of the monodromy brought on by the defect. Since we associate this shift with k ! n 1 k : l ! a qsurface = k n ; n ; we see that the contribution from the Higgs type mass enters the Chern-Simons term in the same way as it enters the uctuation determinant for the fundamental hypermultiplet considered below. The contribution from the kth replica is sensitive only to the partition represented by the Young diagrams Y~ (k). It might seem strange to see the mass shift appear at all in a classical contribution. Note, however, that the combination appearing in the exponential comes from evaluating the classical Chern-Simons term at the positions of the poles for the integral over the scalar associated to the auxiliary U jYj symmetry [58{60]. This ~ position is shifted by the Higgs type mass term. Fundamental hypermultiplets uctuations We now demonstrate the relationship between the contribution of a hypermultiplet on the squashed sphere, with deformation parameters ( 1; 2; ) = n1 ; 1; 2 , and n hypermultiplets in the presence of defects with deformation parameters ( 1; 2; ) = (1; 1; 2 ). At the Qm10 = t : f Expressing the squashed sphere contribution using the round sphere values for t; q etc., conformal point we get = = 4 4 n 1 2 N Y Y Y 1 n 1 2 N Y Y Y 1 k=0 l=1 p=1 l=1 j=1 Ql qYlj+1 t n ; q k=0 l=1 j=1 Ql qYlj+1 t n ; q Ql q t n ; q j 1 j 1 1 ; Ql q t n ; q j 1 1 j 1 1 1 3 5 j=np k ; Qm1f (k) Ql q t (p 1); q Qm1f (k) Ql qYl;np k+1 t (p 1); q 1 1 3 5 ; Qm1f (k) t nk 1+ n1 : We now identify the terms in the square parentheses with the contribution of uctuations of a hypermultiplet in the presence of a codimension-two defect ~ Y Zhyper-defect (~a; 1; 1; 2 ; k) 2 N 4 Y Y 1 l=1 p=1 Qm1f (k) Ql q t (p+1); q Qm1f (k) Ql qYl;np+k+1 t (p+1); q 1 1 3 5 ; horizontal position given by ~ Y The expression for Zhyper-defect (k) di ers from the expression Zhyper in two ways ~ Y 1. The fugacity, or mass parameter, involving the background vector is shifted in the We ascribe this to the e ect of the codimension-two defect on the uctuations. 2. The kth such contribution is sensitive only to the boxes of the Young diagram with We ascribe this to the fractionalization of the instanton moduli corresponding to the partition. These are the same e ects visible for the Chern-Simons contribution. We conclude that n 1 k=0 ~ Y = Y Zhyper-defect (~a; 1; 1; 2 ; k) : 3.6.4 Adjoint hypermultiplet or vector multiplet uctuations An adjoint hypermultiplet contributes to the uctuation determinant around an instanton con guration as (3.48) (3.49) (3.50) (3.51) (3.52) (3.53) (3.54) Reparametrizing the product over k as k ! n k, we get implying n qvortex = k : i = k mod n : Y Zadjoint hyper ~a; mf0 ; 1; 2; ~ Y (l;i)6=(m;j) Qm0 Ql Qm1 qYli Ymj tj i+1; q f Qm10 Ql Qm1 qYli Ymj tj i; q f 1 1 This involves pairs of partitions Yl and Ym. The virtue of the form of the uctuation determinant written above is that t appears raised only to a power corresponding the Y = Y k=0 = Y k=0 4 4 n 1 2i j=n(p q)+k column indices i; j. As such, the determinant can be decomposed in a way similar to the fundamental hypermultiplet Zadjoint hyper ~a; n1 ; 1; 1; 2 ~ Y Ql Qm1 qYli Ymj t n1 (j i); q (l;i);(m;j) Ql Qm1 qYli Ymj t n1 (j i+1); q n 1 2j i=n(p q)+k 1 1 Ql Qm1 t n1 (j i+1); q Ql Qm1 t n1 (j i); q 1 ; 1 (3.55) (3.56) 1 ; 1 (3.57) (3.58) (3.59) 1 : 1 such that Zadjoint hyper ~a; mf0 ; n1 ; 1; 2 ~ Y n 1 k=0 ~ Y = Y Zadjoint-defect (~a; 1; 1; 2 ; k) : We now consider the contribution of a vector multiplet. Its uctuation determinant is inverse to that of an adjoint hypermultiplet with zero mass, as was the case for the perturbative contribution. As we did there, we keep a \mass fugacity" to keep track of the deformation brought on by the defect. Since this mass is now associated to the Higgs vev, it is not shifted from mf = 0 by the Okuda-Pestun prescription. Repeating the calculation for the adjoint hypermultiplet using the same manipulations, we get Zv~ector ~a; mf0 ; n1 ; 1; 2 Y = Y Zv~ector-defect (~a; 1; 1; 2 ; k) ; Y n 1 k=0 Ql Qm1 qYli Ymj tp q+ nk ; q Ql Qm1 qYmj Yli tp q+1 nk ; q Ql Qm1 tp q+1 nk ; q Ql Qm1 tp q+ nk ; q 1 1 3 1 5 ; 1 Qmf (k) Ql Qm1 qYli Ymj tp q; q Qm1f (k) Ql Qm1 qYmj Yli tp q+1; q 1 1 Qm1f (k) Ql Qm1 tp q+1; q Qmf (k) Ql Qm1 tp q; q 3 1 5 : 1 In the second line, we have replaced in two of the factors k ! n 1 k. If we de ne ~ Y Zadjoint -defect (~a; 1; 1; 2 ; k) i j=n(p q)+k Y Qm1f (k) Ql Qm1qYmj Yli tp q+1; q 1 1 Qm1f (k) Ql Qm1 tp q+1; q Qmf (k) Ql Qm1 tp q; q with Zv~ector-defect (~a; 1; 1; 2 ; k) Y i j=n(p q)+k Y (l;i)6=(m;j) Qmf (k) Ql Qm1 qYli Ymj tp q; q Qm1f (k) Ql Qm1 qYmj Yli tp q+1; q 1 1 Qm1f (k) Ql Qm1 tp q+1; q Qmf (k) Ql Qm1 tp q; q Since the original product is over pairs of Young diagrams corresponding to l and m, the decomposition of an adjoint hypermultiplet or a vector multiplet into n parts is not as simple as in gure 1. Instead, each pair of diagrams of the theory on the squashed sphere splits into n pairs for each of the n copies. 1 ei (`Ym (s) 1 (aYl (s)+1) 2+al am)i Y h 1 ei ( (`Yl (t)+1) 1+aYm (t) 2+al am)i ; the defect decomposition, in this case, follows simply from the identity n 1 Y h k=0 1 e2 i( + nk )i = 1 e2 in : A similar expression exists for the hypermultiplet contribution. Its decomposition follows from the same method.18 To relate this decomposition to the KK decomposition, we use the regularized in nite The 5d squashed sphere has one more contribution, not of the type above. The third point contributes a sum over contact-instantons with deformation parameters ( 1; 2; ) = (1; 1; 2 n), i.e. is n times as large as it would be on the round sphere. In order to decompose this contribution, it is useful to write the determinant part of the q-deformed instanton partition function as a product over Kaluza Klein modes coming from the extra circle. Starting from the expressions in [61] (3.60) HJEP1(207) where NlY;m = Y h ~ s2Yl product the form form Nc l;m t2Ym Z3d or 4d !1 !2 : (3.61) (3.62) (3.63) (3.64) (3.65) (3.66) 1 Y m= 1 (m + a) = 1 e2 ia ; Im (a) > 0 : vev of the scalar modulus. 3.7 The defect decomposition can now be thought of as writing the quantum number m in m = np + k : The partitions of the various copies are simply identi ed in this case, in analogy with the The supersymmetric Renyi entropy in three and four dimensions satis es an interesting property stemming from the fact that for a superconformal theory Zn3d or 4d = Z13=dnor 4d : This follows simply from two facts: 1. Conformal invariance implies a dependence on the squashing parameters !1=2 of the 18The mass shift given by mf0 is immaterial in this case. (3.67) (3.68) (3.69) !1 $ !2 : 1 = 1=n ; 2 = 1 ; 1 = n ; 2 = 1 ; Taking !1 = 1=n and !2 = 1 yields (3.65). The same trick does not work in ve dimensions. This relationship can be thought of as an interacting supersymmetric version of the BoseFermi duality in three dimensions [62], which does not hold for Renyi entropies in higher dimensions without introducing supersymmetry. One can calculate the nth supersymmetric Renyi entropy from the partition function on the branched sphere. From the point of view of the instanton partition function, taking corresponds to counting instantons on a space which is branched over a codimension-two surface. Taking 2. There is a trivial change of coordinates which exchanges on the other hand, corresponds to counting instantons on an orbifold. It is interesting that the two counts are related. 4 We have shown that the supersymmetric Renyi entropy (SRE) can be computed using supersymmetric codimension-two defects. After giving a microscopic de nition of the defect operators, we computed the expectation values of these defects using localization.19 We showed that the e ect of such defects on the matrix models calculating the partition function on the round sphere amounted to imaginary mass terms. We made a conjecture regarding the details of the sewing operation needed to complete the picture for the moduli, scalar vevs, instantons and contact-instantons, encountered in localization. We then showed the equality with the squashed sphere partition function. Although we explicitly only showed agreement of the partition functions representing the SRE, the decomposition into defects seems to work at the level of the matrix model ingredients, and for any deformation parameters !i. It is reasonable to conjecture that it works at the level of the 5d holomorphic blocks and gluing [50, 63, 64]. If this is the case, a relationship similar to the one described here should hold for the partition functions on four-manifolds and ve-manifolds of the type described in e.g. [65, 66]. In the context of holographic duality, the Ryu-Takanayagi prescription [67, 68] allows us to compute the entanglement entropy in a CFT, in a particular limit corresponding to classical gravity in the bulk, using a minimal area surface in AdS which is homologous to a given entangling region . Corrections to this computation have recently been conjectured in [69, 70]. A variant for the Renyi entropy was put forth in [71]. Somewhat similar 19The supersymmetric codimension-two defects coincide, in three and in four dimensions, with speci c versions of the operators de ned in [24] and [35] respectively. Five-dimensional versions were considered in e.g. [40]. prescriptions are used to compute the expectation values of supersymmetric non-local operators (see e.g. [72{78]). The authors, and others, have long suspected that there is a relationship between these computations. We do not, however, know of a concrete example of such a relationship. We hope that the de nition of the supersymmetric defect operator version of the SRE calculation can be used to nd one. This may involve going, rst, to a dual picture in the SCFT. For instance, the codimension-two defects realizing the SRE in a 3d N = 2 theory are vortex loops, which, in certain situations, are dual to a Wilson loop under 3d mirror symmetry [24]. Acknowledgments We would like to thank S. Hellerman, C. Herzog, K. Hosomichi, D. Ja eris, R. Myers, T. Okuda, A. Sheshmani, Y. Tachikawa and B. Willett for valuable discussions. The work of T.N. was supported in part by JSPS Grant-in-Aid for Young Scientists (B) No. 15K17628 and JSPS Grant-in-Aid for Scienti c Research (A) No.16H02182. The work of I.Y. was supported by World Premier International Research Center Initiative (WPI), MEXT, Japan. A Conventions We summarize our conventions for gauge theories and the matrix models resulting from the localization procedure in three, four, and ve dimensions. To begin with, we set an overall scale associated with the size of S3;4;5 HJEP1(207) Dimensionful parameters such as 1;2; !1;2;3; a; m; etc. are expressed using this scale. We use physics conventions for the gauge and avor symmetry groups. The generators of the Lie algebra u (N ) are taken to be Hermitian matrices, and factors of i appear in appropriate places in the eld strength. Consequently, integration over the Cartan subalgebra means an N -dimensional real integral over variables denoted ~ , which are the eigenvalues of a matrix . In 5d, ~ is related to the scalar vev as In 4d we have In 3d, where the real adjoint scalar in the vector multiplet is also denoted , we have This convention extends to mass parameters, which are vevs for scalars in background vector multiplets. The physical mass of a chiral multiplet or hypermultiplet is a real number m, which bears the same relation to the background vev as does to the dynamical ` = 1 : h i = = i : 2 h i = : (A.1) (A.2) (A.3) (A.4) We summarize the de nitions and identities for the special functions appear in the text. Multiple gamma function. For ! = (!1; ; !r) zeta function is de ned by 0 and z 2 C, the multiple Hurwitz where n = (n1; ; nr) 0. The integral representation is r(s; z; !) := X(n ! + z) s ; n 0 r(s; z; !) = e zt e !it) ts 1dt : r s; z; !1; ; !r 1; !Nr r s; z + N k!r ; ! : For an integer N , one can prove the identity The Barnes multiple gamma function r(zj!) is de ned by vev. The deformation parameters a and mf , which are used when discussing the instanton contributions to the partition function in 4d and 5d, are set to a = i ; mf = i m : Our conventions for spinors and supersymmetry transformations are di erent in different dimensions. However, supersymmetry transformation parameters are always taken to be commuting spinors. B Special functions (A.5) (B.1) (B.2) (B.3) (B.4) (B.5) (B.6) HJEP1(207) Qir=1(1 N 1 X k=0 Multiple sine function. One can de ne the r-ple sine function Sr(zj!) by with j!j = Pir=1 !i. It satis es the following identities [79]: Sr(zj!) := r(zj!) 1 r(j!j zj!)( 1)r ; Sr(j!j zj!) = Sr(zj!)( 1)r 1 Sr(N zj!) = N = Y 0 ki N 1 Y 0 ki N 1;k6=0 Sr z + Sr k ! N k ! N ! ; ! ; Sr(czjc!) = Sr(zj!) ; for c > 0 : ; !i 1; !i+1; The formula (B.3) yields an additional identity Sr z !1; k=0 Y Sr z + k!r ! N : This is the generalization of the identity for the hyperbolic gamma function found in [6]. The function. The double gamma function is used to de ne the function [29, 80] (zj!1; !2) := 2 2 j!j !1; !2 ( 2(zj!1; !2) 2(j!j 2 zj!1; !2)) 1 : satis es several identities where (z) := (z)= (1 z) and the scaling law Some literatures including [29, 80] use (z + !1j!1; !2) = ! !2z2 1 2 (z + !2j!1; !2) = ! !2z1 1 1 (z=!2) (zj!1; !2) ; (z=!1) (zj!1; !2) ; (j!j 2z)2 (czjc!1; c!2) = c 4!1!2 (zj!1; !2) : b(z) := (zjb; 1=b) : which is sometimes denoted (z) without the subscript. The formula (B.3) yields z !1; !N2 N 1 Y k=0 z + N k!2 !1; !2 : (B.7) (B.8) (B.9) (B.10) (B.11) (B.12) (C.1) C The instanton partition function Nekrasov's instanton partition function, [52, 53], is the equivariant volume of the instanton moduli space with respect to the action of U(1)a U(1) 1 U(1) 2 : The three factors correspond to (constant) gauge transformations and to rotations in two orthogonal two-planes inside R4, respectively. The q-deformed version of the partition function counts instantons extended along an additional S1 factor in the geometry of circumference . The undeformed partition function can be recovered by letting the size of this S1 shrink to 0. Our expressions for the instanton partition function are taken from [61]. We use a 5d parameter which can be used to take the 4d limit, and our conventions di er from those in [61] by the substitutions 1 ! i 1 ; 2 ! i 2 ; a ! i a : (C.2) The q-deformed version of the instanton partition function for G = U(N ) and in the presence of hypermultiplets can be expressed as follows [61, 81] Zinst (q;~a; m~f ; 1; 2; ) = X qj Y~jZY5~d;-CS (~a; 1; 2; ) Z Y~ (~a; m~f ; 1; 2; ) ; (C.3) ~ Y Y Y N 1 s2Yl 1 Y h 1 t2Ym Zfund hyper (~a; mf ; 1; 2; ) = Y Y ~ Y Qm1f Ql q t j; q Qm1f Ql qYlj+1 t j; q 1 1 : Another expression for the vector contribution is Zv~ector (~a; 1; 2; ) = Y Y ~ NlY;m = ei [`Ym (s) 1 (aYl (s)+1) 2+al am]i ei [ (`Yl (t)+1) 1+aYm (t) 2+al am]i : l=1 Zv~ector (~a; 1; 2; ) = Y Zadjoint hyper (~a; mf ; 1; 2; ) = ~ Y Ql Qm1 qYli Ymj tj i; q Ql Qm1 qYli Ymj tj i+1; q Ql Qm1 tj i+1; q Ql Qm1 tj i; q 1 1 1 ; (C.4) Qmf Ql Qm1 qYli Ymj tj i+1; q Qm1f Ql Qm1 qYli Ymj tj i; q 1 1 Qm1f Ql Qm1 tj i; q Qmf Ql Qm1 tj i+1; q The symbols above are de ned as follows: Y~ is an N -vector of partitions Yl. A partition is a non-increasing sequence of nonnegative integers which stabilizes at zero Yl = fYl 1 Yl 2 : : : Yl nl+1 = 0 = Yl nl+2 = Yl nl+3 = : : :g : 2 X Yl2i ; i X Yli : l;i The sum in (C.3) is over all such partitions. A partition Yl can be identi ed with a Young diagram whose ith column is of height Yli. We denote the partition corresponding to the transposed Young diagram as Yt. l For a box s 2 Yl with coordinates s = (i; j), we de ne the leg length and arm length `Yl (s) Yltj i ; aYl (s) Yli j : (C.10) 1 ; 1 (C.5) (C.6) (C.7) (C.8) (C.9) ~a is an N -vector of deformation parameters corresponding to the equivariant action of the gauge group on the instanton moduli space. In the partition functions we compute, they are integrated over the imaginary axis and identi ed with the vev of a scalar eld in the vector multiplet. m~f is an Nf -dimensional vector of mass deformation parameters associated to hypermultiplets. When all of the hypermultiplets are in the fundamental representation of the gauge group, m~f transforms as a fundamental of the avor symmetry group SU(Nf ). Mass deformations should be viewed as coming from a vev for a background vector multiplet. Physical masses are the imaginary part of this vev. q e e i 1 ; Ql ei al ; Qmf e i mf : (C.11) This de nition di ers from [61] by (C.2). The q-Pochhammer symbol is de ned as q is a classical contribution equal to20 { in 5d we have { In a 4d calculation one uses where e2 i is minus the exponential of the one instanton action of the conformal theory with coupling constant If the theory is not conformal, then21 20Later versions of [61] include a di erent convention for the counting parameter, essentially rede ning There is a similar factor included in the Chern-Simons contribution. We will not use these rede nitions. 21We follow [57]. h_ (G) is the dual Coxeter number and k (R) is the quadratic Casimir, normalized such that k (adjoint) = 2h_. For SU(N ), we have h_ = N and k (fund) = 1. The combination 2h_ (G) k (R) is the coe cient of the one-loop beta function for the 4d N = 2 theory with hypermultiplets in the representation R. (x; q)1 1 Y (1 p=0 x qp) : 8 2 gY2M : q(4d) e2 i ; = 2 YM + 4 i 2 gYM 2h_(G) k(R) ; q ! q e 1+ 2 2 : (C.12) (C.14) (C.15) (C.16) (C.17) (C.13) In the presence of a 5d Chern-Simons term with parameter , we have [81] l : An alternative version is [61, 82] 2 ZY5~d;-CS (~a; 1; 2; ) = exp 4i 3 1) 1 (C.21) where is the holomorphic dynamical scale. According to [57], the relationship between the 5d and 4d partition functions is q = q(4d) ( i )2h_(G) k(R) ; and Zi(n4sdt) q(4d); ~a; m~f ; 1; 2 1 = !1 ; 2 = !2 ; 1 = !2 ; 2 = !1 ; !1 = 1 ; !2 = (!1; !2; !3) = 1; 1; { 37 { = li!m0 Zinst q(4d) ( i )2h_(G) k(R) ; ~a; m~f ; 1; 2; N l=1 X X l (s;t)2Yl 1 n n (C.18) : (C.19) (C.20) (C.22) (C.23) (C.24) (C.25) (C.26) The parameters 1;2 are associated with the -deformation in the 4d theory. 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Tatsuma Nishioka, Itamar Yaakov. Supersymmetric Rényi entropy and defect operators, Journal of High Energy Physics, 2017, 71, DOI: 10.1007/JHEP11(2017)071