Entwinement in discretely gauged theories

Journal of High Energy Physics, Dec 2016

We develop the notion of “entwinement” to characterize the amount of quantum entanglement between internal, discretely gauged degrees of freedom in a quantum field theory. This concept originated in the program of reconstructing spacetime from entanglement in holographic duality. We define entwinement formally in terms of a novel replica method which uses twist operators charged in a representation of the discrete gauge group. In terms of these twist operators we define a non-local, gauge-invariant object whose expectation value computes entwinement in a standard replica limit. We apply our method to the computation of entwinement in symmetric orbifold conformal field theories in 1+1 dimensions, which have an S N gauging. Such a theory appears in the weak coupling limit of the D1-D5 string theory which is dual to AdS3 at strong coupling. In this context, we show how certain kinds of entwinement measure the lengths, in units of the AdS scale, of non-minimal geodesics present in certain excited states of the system which are gravitationally described as conical defects and the M = 0 BTZ black hole. The possible types of entwinement that can be computed define a very large new class of quantities characterizing the fine structure of quantum wavefunctions.

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Entwinement in discretely gauged theories

Received: September Entwinement in discretely gauged theories V. Balasubramanian 0 1 3 4 6 7 8 A. Bernamonti 0 1 2 3 5 7 8 B. Craps 0 1 3 4 7 8 T. De Jonckheere 0 1 3 4 7 8 F. Galli 0 1 2 3 5 7 8 0 31 Caroline Street North , ON N2L 2Y5 , Canada 1 Pleinlaan 2 , B-1050 Brussels , Belgium 2 Institute for Theoretical Physics, Katholieke Universiteit Leuven 3 Philadelphia , PA 19104 , U.S.A 4 Theoretische Natuurkunde, Vrije Universiteit Brussel, and International Solvay Institutes 5 Perimeter Institute for Theoretical Physics 6 David Rittenhouse Laboratory, University of Pennsylvania 7 Open Access , c The Authors 8 Celestijnenlaan 200D , B-3001 Leuven , Belgium tum entanglement between internal, discretely gauged degrees of freedom in a quantum eld theory. This concept originated in the program of reconstructing spacetime from entanglement in holographic duality. We de ne entwinement formally in terms of a novel replica method which uses twist operators charged in a representation of the discrete gauge group. In terms of these twist operators we de ne a non-local, gauge-invariant object whose expectation value computes entwinement in a standard replica limit. We apply our method to the computation of entwinement in symmetric orbifold conformal eld theories in 1+1 dimensions, which have an SN gauging. Such a theory appears in the weak coupling limit of the D1-D5 string theory which is dual to AdS3 at strong coupling. In this context, we show how certain kinds of entwinement measure the lengths, in units of the AdS scale, of non-minimal geodesics present in certain excited states of the system which are gravitationally described as conical defects and the M = 0 BTZ black hole. The possible types of entwinement that can be computed de ne a very large new class of quantities characterizing the ne structure of quantum wavefunctions. ArXiv ePrint: 1609.03991 AdS-CFT Correspondence; Gauge Symmetry; Gauge-gravity correspondence 1 Introduction 2 3 4 De ning entwinement Replica trick | generalities Entwinement in symmetric orbifold CFTs Single interval entwinement Entanglement entropy of a spatial region D1-D5 CFT Discussion and outlook Conical defects Zero mass BTZ black hole A Twist correlator Introduction When a quantum system nds itself in a pure state j i, the entanglement between a part A of the system and its complement A is quanti ed by the entanglement entropy. In most applications, A and A describe the degrees of freedom in complementary spatial regions. In systems with localized degrees of freedom such as spin chain models or local quantum eld theory, this corresponds to a natural separation of the total Hilbert space. However, it is also fruitful to consider other ways of separating the Hilbert space. In [1, 2], the degrees of freedom of a local quantum eld theory were separated into high and low spatial momentum modes. It was demonstrated that in a generic interacting eld theory, even in the vacuum state, the long wavelength (low energy) degrees of freedom necessarily nd themselves in a nontrivial reduced density matrix because of entanglement with the short wavelength (high energy) degrees of freedom. This gives rise to tanglement entropy, which was computed explicitly in perturbative scalar The more traditional way of describing the low energy degrees of freedom is Wilsonian renormalization. In this language, the vacuum state of a Wilsonian low energy theory is necessarily a density matrix with nite entropy [1, 2]. This phenomenon of UV-IR entanglement in quantum eld theories could be particularly important in theories of gravity (in which ultraviolet and infrared degrees of freedom are known to couple in nontrivial ways), as well as in the ground states of strongly correlated electronic systems (see, e.g. [3]). This also raises the question whether it is fruitful to consider other non-spatial ways of dividing the degrees of freedom of quantum eld theories. One interesting way to separate energy scales is by imagining a collection of local observers who have a nite duration T over which they can make measurements. This is a natural situation to consider, as it describes the practical constraints of most measurements. Intuitively, given Heisenberg's energy-time uncertainty relation, such observers will be insensitive to energies smaller than the inverse duration of the experiment, so that those low energy degrees of freedom are effectively traced out, turning the accessible part of the state into a nontrivial density matrix. Thinking in this way, [4{7] proposed a new information theoretic quantity, the di erential entropy, as a measure of UV-IR entanglement, at least for two-dimensional theories. These e ects should become stronger for theories with an energy gap that is smaller than the inverse spatial size of the system. For example, consider relativistic theories where the local degrees of freedom are matrices, e.g. SU(N ) gauge theories. In such systems the energy gap can be much smaller than the inverse spatial size of the system so that even a set of observers with enough time to observe the entire spatial domain will not have access to the lowest energy excitations. Another example which is easier to visualize is a \long string" theory, where strings or spin chains are multiply wound around a spatial circle, allowing for excitation wavelengths that exceed the system size. In both these examples, the key to the physics lies in entanglement between \internal" degrees of freedom (matrix components, or strands of string) that are not spatially organized. One way to study the entanglement of gauge degrees of freedom in an SU(N ) theory is to break the gauge group into SU(m) m) while allowing for interactions between the two sectors. This could be realized holographically by separating a stack of N branes into a stack of m and one of N m branes and studying entangling surfaces in the AdSd S10 d geometry which arises in the low energy limit. Such a set-up was rst considered in [8] and later re ned in [9{11]. The authors of [9, 10] also considered global symmetries and in case of an SO(11 d) global symmetry they proposed a quantity in the eld theory which would holographically be represented by the area of caps on the internal S10 d. One important complication that we have glossed over so far is that in systems exhibiting gauge symmetry, even the association of degrees of freedom to spatial regions is subtle. For example, some of the fundamental degrees of freedom, such as Wilson loops, are not local in space, making it more complicated to split up the Hilbert space according to spatial regions. Interesting work on how to de ne entanglement entropy in gauge theories has recently appeared (see e.g. [12{21]), but a complete understanding is still lacking. These questions about entanglement in quantum eld theory are also linked to equally deep questions about the nature of black hole horizons and the holographic emergence of spacetime. It was proposed in [22{24] that the entanglement entropy of a spatial region A in the eld theory is proportional to the area of the minimal surface in AdS space that ends on the boundary of A. Furthermore, [4{7] showed that the area of closed surfaces in the bulk of AdS can be related to a measure of UV-IR entanglement, the di erential entropy discussed above, at least for two-dimensional boundary theories and higher dimensional cases with translational symmetries | some of the limitations were discussed in [25]. Finally, in [26{31] it was suggested that spacetime connectedness is related to entanglement of the underlying quantum degrees of freedom, and that the linearized equations of motion of gravity can be derived from the dynamics of entanglement perturbations. In general, can all of spacetime geometry be reconstructed from spatial entanglement entropy in the AdS/CFT correspondence? At least when we do not consider bulk quantum corrections to the entanglement entropy [32], the answer is no | in some asymptotically AdS spacetimes, the minimal surfaces anchored on the boundary that geometrically reproduce the entanglement entropy will not penetrate a region [33, 34] which has been called the entanglement shadow [35, 36]. It is argued in [35] that in such systems entanglement can be dominated by \internal" degrees of freedom (e.g. the matrix components, or strands of string discussed above) that are not spatially organized, and that these entanglements can measure the areas of non-minimal, extremal surfaces that can penetrate part of the entanglement shadows of the gravitational dual. In the examples arising in the AdS/CFT correspondence, such internal degrees of freedom are usually gauged. Thus, reconstructing the emergent space in gauge/gravity duality will involve entanglement between \internal", gauged degrees of freedom | a notion that was named entwinement in [35]. While we will not address the question of which part of a general spacetime can be probed by extremal surfaces (see, for instance, [33, 34, 36]), it is clear that entwinement will often allow the reconstruction of a larger part of spacetime than spatial entanglement entropy. Entwinement also plays a key role in the description of holographic spacetimes using methods of integral geometry based in kinematic space [37{39]. In summary, both in quantum eld theory and in quantum gravity, we are driven to consider a new notion of \entwinement" | non-spatial quantum entanglement between gauged degrees of freedom. In this paper we will de ne entwinement formally in discretely gauged theories, and discuss how it can be explicitly computed. Section 2 develops the general formalism. For two-dimensional theories, we de ne entwinement in terms of a replica method using twist operators that are charged under the discrete gauge group. We use these operators to construct a new non-local, gauge invariant object whose expectation value is de ned to be the entwinement in a standard replica limit. Section 3 applies this formalism to symmetric orbifold conformal eld theories in two dimensions. By explicitly applying uniformization maps, obtained by generalizing a construction of [40], to the Riemann surfaces arising from the replica method we directly compute entwinement in generic microstates of these theories. We also comment on how to recover the usual spatial entanglement entropy as a special limit of entwinement. In section 4, we apply our formalism to the weak coupling limit of the D1-D5 CFT, a theory with a holographic dual. Ordinary spatial entanglement entropy in generic classes of the D1-D5 CFT was considered before in [40{42]. For generic microstates, entanglement entropy was computed approximately using both a short interval expansion [41] and large c methods [42]. In [40], a speci c state corresponding to a local quench was considered, and the evolution of the entanglement entropy was computed using a uniformization map | we generalize the construction for use with other microstates. Although this uniformization map will work for computing single interval entwinement in general microstates, in section 4 to compare with holography we focus on two examples of microstates, which are demonstrate that the lengths of non-minimal geodesics that penetrate entanglement shadows of the spacetime are computed by certain entwinements. In the same way as spatial entanglement entropy corresponds to minimal extremal area surfaces in the bulk, here entwinement provides a direct eld theory interpretation for non-minimal extremal surfaces. The paper concludes with a discussion of possible directions forward to develop the notion of entwinement in more general situations, and comments on the relation with the appearance of non-minimal geodesics in the semiclassical Virasoro conformal blocks discussed in [43]. A number of technical results are collected in appendices. De ning entwinement In a gauge theory, states are required to be symmetric under identi cations by the gauge group. The main complication in discussing entanglement entropy in gauge theories is that the Hilbert space does not factorize. For example, for a U(1) gauge theory there is a Gauss law constraint which requires that the electric ux entering a region should determine the ux leaving it. We will be interested in situations where the gauge symmetry is discrete. Recently various authors have developed a formalism for dealing with this lack of factorization in gauge theories. One method is to consider an extended Hilbert space in which the gauge constraints are temporarily relaxed [12, 13, 16{18, 20, 21]. A way of achieving this is by introducing \edge modes" on the entangling surface [13]. In this approach, the Hilbert space splits into superselection sectors de ned by the uxes at the entangling surface, making the reduced density matrix block-diagonal. The entanglement entropy then becomes a sum of two contributions, a Shannon entropy associated to the distribution over superselection sections and a distillable piece arising from entanglement within each superselection sector. In the same spirit, the approach to entwinement suggested in [35] was to ungauge the theory, compute, and then symmetrize. An alternative approach is to de ne a subalgebra of gauge invariant observables OA associated to the region A. Then, given a density matrix for the full theory, the reduced density matrix A is de ned as the element of the subalgebra of region A such that splits into blocks according to superselection sectors determined by the center of the subalgebra. An algebraic approach to entwinement was recently proposed in [44], where it was shown that for a spin system the entwinement could be recovered from a state-dependent A third approach, which works for states that can be constructed by a Euclidean path integral, is the replica trick. In this case, the entanglement entropy is computed by analytically continuing the Renyi entropies. In two dimensions this approach is particularly convenient and the Renyi entropies can be de ned in terms of the correlation functions of twist operators that splice together replicated copies of the CFT. At least for the case of 2d Yang-Mills theory in de Sitter space, it was veri ed in [16] that the replica method gives the same result as the extended Hilbert space method described above. Below we will de ne entwinement along these lines. N sets of elds, where each set can be viewed as coordinates on one copy of the manifold M , together with companion fermions in case of a supersymmetric theory. The SN indicates that we identify con gurations that di er by permutations of these N sets of elds. This is similar to the way one treats indistinguishable particles in quantum mechanics: wavefunctions need to be appropriately symmetrized under permutations. The SN identi cation is really a discrete gauge symmetry. We can gauge x the local symmetry and think of the elds as changing continuously from point to point (i.e., each of the N copies of M has a continuous string embedded in it). The theory has so-called \twisted sectors" in which strings are only periodic up to permutations. A twisted sector is labeled by a conjugacy class, which is characterized by the lengths of its permutation cycles: there will be Nm cycles of length m such that P because it can be visualized as a string winding m times. We will refer to each winding of the long string as a \strand". The conventional spatial entanglement entropy of an angular interval of size be thought of as the entanglement entropy of the union of intervals of size in each of the N strands of the system. Following the proposal of [35], we want to de ne entwinement as the entanglement of intervals that extend over some strands and not others. For example, one can talk about the entwinement of an interval on a single strand. If one considers the entwinement of a union of identical intervals in each strand, then it reduces to the conventional spatial entanglement. Because there is a gauge symmetry, we cannot invariantly specify which strand we are talking about. But, as we will argue below, we can meaningfully talk about things like \the entanglement of one and a half connected strands". Likewise, while one cannot invariantly ask for the entanglement of a particular strand, one can ask for the entanglement of a single strand if we do not specify which one it is. This invariance can be made manifest by simply averaging the computation of entanglement of a single strand over all the strands. It is worth emphasizing that this average is not the same thing as the entanglement of the union of such intervals. Below, we will give a mathematical de nition of such quantities. Replica trick | generalities A useful method for computing entanglement entropy in two-dimensional conformal eld theories is the replica trick. In this method, the entanglement entropy of an interval [0; ] is computed from the reduced density matrix for this region by taking a limit of the Renyi entropies: S( ) = lim Consider a two-dimensional CFT in the plane in Euclidean signature. In radial quantization, circles of xed radius become equal time slices. An operator acting at the origin creates a pure state, and we can integral with the operator inserted and xed boundary conditions at the unit circle. The density matrix ( ; 0) corresponding to such a pure state is then computed by inserting operators at the origin and at in nity in the path integral, and imposing boundary condiin full blue lines) with boundary conditions on the lower cut and 0 on the upper cut. The dashed lines de ne the complementary interval along which boundary conditions are matched = 0. The prepares the state. and 0 on the interior and the exterior of the unit circle, respectively. The reduced density matrix for the interval [0; ] is computed by tracing over the complementary part of the unit circle (i.e., setting there). This leaves us with a path integral over the entire plane, except over the arc corresponding to the interval [0; ], as depicted in gure 1. To compute Tr( n) we consider n copies of the plane cyclically glued together over the cut [0; ] producing an n-sheeted Riemann surface as in gure 2 (left). This can alternatively be obtained as a correlator of Renyi twist operators (n) computed on a single sheet of the n-fold cover of the theory, \replicas". Each of the n copies of the CFT is placed in the same state. The Renyi twist cyclically splices together the n CFT copies, such that dragging a eld from one CFT around the cut produces a eld in the next copy of the CFT. Below we will illustrate this procedure in symmetric product CFTs with target space We will then generalize the de nition to apply to more general discretely gauged theories in two dimensions. M . As discussed above, this CFT can be regarded as having N elementary strands spliced together into series of cycles (\long strings") determined by the twisted sector. The twist operator can therefore be regarded as a product of elementary twists (n) = g i(n)g 1 = (n) splices together the n replica copies of the ith strand. Each elementary twist is in the fundamental representation of SN . Thus we can write for any g 2 SN where g(i) is the strand produced by permuting i by the action of g. The twist operator appearing in the computation of entanglement entropy (2.2) is a product of all the elementary twists and hence is invariant under the action of SN . The twists con guration that computes the entanglement entropy is depicted in gure 3 (left). dashed arrows denote how to sew elds across the cuts. (Right) Correlator in the plane. The insertions represent Renyi twist operators, while the insertions de ne the replicated state. The entanglement entropy is computed by inserting Renyi twist operators at the endpoints of the interval on every strand. The entangling region can be visualized in the long string picture as a union of disjoint intervals on all strands. (Right) Con guration of twists corresponding to the bilocal operator of single interval entwinement. The entangling region extends across di erent strands of We can de ne entwinement formally in terms of the elementary twists. Take i to be an elementary twist operator for strand i and consider the bilocal combination where for compactness we have only written the holomorphic coordinate and ~ i is the conjugate twist. In this bilocal operator the i is taken around the complex plane relative to ~ i by an amount + 2 `. In a speci c state of the symmetric orbifold the ith strand is generally spliced with k other strands into a long string. In such a state the twist be inserted on a di erent strand, as represented on the right of gure 3. We then consider the bilocal, gauge-invariant quantity where jSN j is the cardinality of SN , i is any reference strand of the CFT, and g(i) is the strand to which i is transported when all the strands are permuted by g 2 SN . Because we are summing over all permutations in SN , the nal quantity is independent of i. Its expectation value computes the Renyi analog of entwinement of single intervals. When ` > k (k being the number of strands of the speci c cycle, i.e. long string, in which the strand g(i) lives) we mean the operator in the sum in (2.5) to represent the twisted boundary conditions of the replicated set of elds on the full long string. Intuitively, we can imagine starting from a short interval on a single strand and putting twisted boundary conditions on the elds inside the interval. Enlarging the interval until it eventually covers the full string, i.e. elds on the full string. Further increasing the interval to + 2 ` > 2 k does not change this picture and just keeps all elds on the long string twisted, nothing more. Keeping this in mind, we can de ne the entwinement as E`( ) = lim For symmetric product orbifolds this is just a formal way of saying that we are calculating the entanglement entropy of a connected set of partial strands in a long string. This de nition of single interval entwinement can be generalized to multi-interval entwinements by taking a product of operators like (2.5) de ned at di erent locations and strands. A particular example of multi-interval entwinement is entanglement, where we take a product of the same interval with < 2 in each of the N strands. The formalism described above is general. We can consider theories with any discrete gauge symmetry H, and elementary twist operators in any representation R of H that These can be used to de ne bilocal, gauge invariant twist operators of the form (2.5) and products of such operators. Entwinements de ned as expectation values of these quantities are a very general new class of gauge-invariant objects than can be used to characterize quantum wavefunctions in two-dimensional theories. Conceptually we can also talk about the entanglement of subsets of degrees of freedom in a spatial region even in higher dimensional theories on any manifold, but we need a formalism for calculating such quantities e ciently. In the next section we will use the replica method to explicitly compute entwinements in symmetric orbifold conformal eld theories in two dimensions. In the following we analyze symmetric orbifold CFTs on a circle of length L. These are obtained starting with a seed CFT with target space M and central charge c. The orbifold the SN identi cation, states need only be periodic up to the action of a group element. In a sector twisted by h 2 SN , the boundary conditions are i(L) = i = 1; : : : ; N ; where here i collectively indicates the elds in the i-th copy of the CFT. All physical states should be invariant under the action of SN . Since acting with a group element g maps the sector twisted by h to that twisted by ghg 1, twisted sectors should really be labeled by conjugacy classes [h], as mentioned in the previous section. Twisted states can be conveniently obtained through the action of orbifold twist operators on untwisted states. An orbifold twist operator m(0) at the origin of the complex plane causes m copies of the target space M to be linked together by the periodicity j (ze2 i) = j0 (z) ; j0 = (j + 1) each with period 2 , into a single long string with period 2m . We will be interested in twisted states of the form We wish to de ne entwinement in such twisted states of the symmetric product CFT. In radial quantization we can specify a connected entangling region at a in twisted states of the form (3.3) can be computed using the replica trick, by inserting elementary replica twist operators, i. Such twists act each on a single strand out of the N strands in the CFT and can be thought of as connecting the n-fold Renyi replicates of that strand. The elementary replica twists, i, have conformal weights m acts on a di erent subset of the N copies of the target space M . Thus, there will be Nm long strings of period 2m m mNm = N . In radial quantization this prepares a state on the spatial circle, and the corresponding out state is h j = h0j m=1 z;z!1 where ~m has opposite action to m. The twists transform as primaries with conformal j i = hm = hm = Hn = Hn = We insert the elementary twists at the endpoints of the chosen interval of length and average over the symmetric group, which moves the left boundary of the interval over all strands while keeping the length of the interval xed. The state of the replicated theory, j i, is obtained inserting orbifold twist operators for each of the n replica copies of j i = i.e., by taking products of (3.3) for each of the n replicated theories. In terms of j i, we E`( ) = lim h j ~ (gn(i))(1) (gn(i))(e2 i`x; e 2 i`x)j i ; where we have taken the entwinement interval to extend between 1 and x in the complex over SN , the result is independent of the arbitrary choice of the initial strand i, which can run from 1 to N . This correlator is a four-point function in the cyclic orbifold theory At the practical level then, the computation above simpli es drastically. Consider a strands, which we relabel here as 1; 2; : : : ; m for convenience. Let us de ne a notation [ m]n = (n) = (1:::m) (m+1:::2m) : : : (m(n 1)+1:::mn) ; (k;k+m;:::;k+(n 1)m) in the cycles notation (:::) of Smn. where the subscripts k and k + ` are understood modulo m because of the cyclic symmetry of the m-stranded long string. Then we have to compute terms like As discussed in the previous section, if the long string is shorter than the interval in question, i.e., 2 m < + 2 `, (3.10) is understood as computing how the entire long string is entangled with the rest of the system. Single interval entwinement To evaluate the entwinement of a single interval we must therefore compute the correlator h[~m(1)]n ~ (kn)(1) (kn+)`(x; x)[ m(0)]ni : The branching structure of the correlator is e ectively mn-dimensional as illustrated in gure 4, and we can label the twist elds in terms of these mn sheets involved in the Representation of the branching structure of a correlator of the form (3.11) in the The correlator (3.11) can be evaluated through a uniformization map to a covering space. To compute the map we extend a calculation of [40], which dealt with the case the Riemann-Hurwitz formula g = 1 X(ri s + 1 = mn + 1 = 0 : of sheets involved in the correlator. An explicit formula for such twist correlation function in terms of the properties of the uniformization map to a genus zero surface is worked out in appendix D of [45] (see also [46, 47]). In appendix A, we review these results and apply them to the computation of single interval entwinement. Our nal result is (see eq. (A.26) in the appendix) h[~m(1)]n ~ (kn)(1) (kn+)`(x; x)[ m(0)]ni = To obtain the entwinement of an interval of opening angle on the spatial circle w+L, we only need to relate the result (3.15) obtained on the plane to the computation L=(2 ). Using the map z = e L : j icylinder = operators and work out the sum over all elements in SN , we obtain E`( ) = lim UV to regulate the twist The sum in the rst term extends over long strings with ` + 1 and more strands, because shorter strings are completely covered by intervals of length + 2 `. The contribution from each of these shorter strings computes the entanglement of the string with the rest of the theory. Since the short string is by construction disconnected from all other strands in this particular twisted sector, its entanglement entropy will not have the dominant UVdivergent contributions that are present for the longer strings with ` + 1 and more strands. In formulas, if one sets 2 m up to a contribution of order UV, the occurences of result, which is small compared to the cuto -dependent terms arising from long strings longer than the interval. Entanglement entropy of a spatial region The entanglement entropy of an interval is a speci c case of computing entwinement. The gauge invariant twist operators are decomposed into products of twist operators on each strand as in (2.2), and the entanglement entropy of a spatial interval can be expressed as S( ) = lim In the perspective of entwinement, entanglement entropy coincides with the entwinement of the union over all strands of an interval that ts within a single strand. Clearly, as this quantity is already gauge invariant, the sum over SN appearing in the entwinement de nition exactly cancels the normalization jSN j 1. In fact the entanglement entropy for a state of the form (3.3) is not known in general. the OPE of the elementary twists ~ i(n)(1) i(n)(x; x) For instance the branching structure of the correlator leads generically to a covering space of non-trivial genus, and thus one cannot straightforwardly apply the same techniques we used for computing entwinement. However, in the limit of a short interval (x; x ! 1), using 1 =j1 x 2Hn the correlator factorizes. Via the conformal map to the cylinder, the result reproduces the short interval expansion entanglement for a single factor in the symmetric product orbifold theory, while entangle! 0) of the entwinement entropy simultaneously involves elds in all N factors. D1-D5 CFT A well-known example of a symmetric orbifold CFT is the D1-D5 CFT. This is realized in type IIB string theory compacti ed on S1 K3), with N1 D1-branes wrapping the circle and N5 D5-branes wrapping the entire compact product space. The near horizon geometry of the D1-D5 brane system is AdS3 T 4, and one can formulate supersymmetric sigma model with SU(2) SU(2) R-symmetry, corresponding to the isometry group of the S3, another SU(2) SU(2) global symmetry and central charge equal to 6N1N5 (see for instance [45] for a review). The moduli space of the CFT contains an orbifold point where the theory consists of N N1N5 copies of a c = 6 free CFT of 4 real We will work at the orbifold point of the D1-D5 CFT and focus on the Ramond ground states. These can be constructed by multiplying together bosonic and fermionic twist operators to achieve a total twist of N . The theory contains eight bosonic and eight fermionic twists labeled in terms of the global symmetries. Since we are only interested in computing correlators of bosonic quantities that do not carry R-charge, we can simplify the discussion and generically consider the normalized symmetric orbifold microstates (3.3). In this section we will consider two examples of such states, which in the large N limit are dual to conical defects and zero mass BTZ black holes in the bulk (see [48] for discussion of the map between Ramond ground states of the D1-D5 system and AdS3 gravity).1 Points in the moduli space with a geometric supergravity description are actually far from the orbifold point where we perform our computations, and agreement with semiclassical gravity is not to be expected a priori. However protected BPS quantities can be computed exactly at the orbifold point and it has been proposed that agreement should extend also to observables computed in terms of covering space constructions [49]. Conical defects Simple Ramond ground states of the D1-D5 CFT are of the form [46, 47, 50] j i = [ m~ (0)]N=m~ j0i ; ds2 = is a periodic coordinate and the AdS radius is related to the length of the spatial obtained from empty AdS3 in global coordinates via Z m~ identi cations. For a single interval of opening angle on the boundary, there exist multiple geodesics connecting the endpoints of the interval. These have regulated lengths [35] L`( ) = 2RAdS log The index ` = 0; : : : ; m~ 1 counts the number of times the geodesics winds around the conical defect at r = 0. Here r 1 is an IR gravitational cuto . Identifying it with the eld 1The BTZ black hole is actually dual to an ensemble of states; we will comment on this point and on the notion of typical states in section 4.2. 1 = 2L RAdS and substituting c = 6 in (3.17), we nd the relation UV between entwinement in the state (4.1) and geodesic lengths E`( ) = L`( ) S( ) = L0( ) Therefore in this case, the eld theory notion of entwinement computes the length of geodesics in AdS, in agreement with the idea advanced in [35] that non-minimal geodesics in the bulk capture the entanglement of internal degrees of freedom. Ordinary entanglement entropy on the other hand is related to the length of minimal geodesics, in agreement with the Ryu-Takayanagi formula [22] Zero mass BTZ black hole One could wonder whether the length of long geodesics also captures entwinement in a black hole background. We will show this is indeed the case in the zero mass BTZ black hole ds2 = which arises in the m~ ! hole has a horizon of zero size which coincides with the singularity. rather to an ensemble of states of the D1-D5 CFT with xed N . Following [50], instead of working with the microcanonical ensemble, it is more convenient to work in the canonical description where Nm as well as the total number of strands uctuate, but the ensemble average is xed to N . The average number of m-cycles is [50] In the large N limit, typical states in the ensemble have individual twist distributions that lie very close to (4.7) and expectation values of observables in a typical state deviate by only a small amount from those computed in the ensemble. In the following, we therefore compute entwinement in a typical state with representative distribution (4.7), rather than in the ensemble. where the inverse ctitious temperature is determined in terms of the average N of the total number of strands, as hNmi = N = The single interval entwinement in a typical microstate is again given by (3.17). As N , we obtain we prove in appendix B, for xed and `, the sum over m is dominated by the terms with Black holes admit a region outside the horizon which is not penetrated by minimal geodesics, which we can call the entanglement shadow of the black hole. Just as in the case of the conical defect, non-minimal geodesics penetrate the entanglement shadow. The non-minimal geodesics wind around the horizon and the bigger their winding number, the zero size, but nevertheless it has a nite entanglement shadow. The lengths of non-minimal L`( ) = 2RAdS log E`( ) = L`( ) Using again that c = 6 and r are related by 1 = 2L RAdS , we recover that entwinement and geodesic length UV N , the relation between entwinement and geodesic length breaks down. But the corresponding long geodesics, which wind very many times around the black hole, approach the horizon to within a Planck length. It is not clear that they are well de ned in the quantum theory where we expect classical geometry to be ill-de ned at the Planck scale. Hence it is not surprising that the lengths of these geodesics do not match the corresponding entwinements. Discussion and outlook We have veri ed the correspondence between single-interval entwinement and lengths of to speci c states of the D1-D5 system. We studied these con gurations because there are explicit constructions of the corresponding states in the literature [50]. Furthermore, these states are BPS-protected ground states in the Ramond sector of the theory [48, 49], so that we can expect non-renormalization of some quantities as we deform the theory away from its orbifold point. Of course this does not mean that all correlation functions extrapolate from weak to strong coupling, but we were essentially computing partition sums after a conformal map, which might help explain the agreements we found between eld theory and gravity. It has also been seen that certain graviton correlators computed at the orbifold point do match the gravitational results which are related to the strongly coupled theory [50]. Possible general reasons for such matching are discussed in [49]. It should be possible to extend our computations in at least two interesting directions. First, we can BTZ black hole and the conical defects considered here were two speci c examples of such states, but a more general class is discussed in [50], including candidate states describing \black ring" geometries. Another interesting extension is to consider rotating, but extremal, AdS3 black holes. These are represented in the D1-D5 CFT by adding energy to the left moving sector alone. One approach would be to add a small left-moving temperature; another would be to perturb the theory with a holomorphic stress tensor. Both of these are settings where it would be very interesting to compute both spatial entanglement and While in this paper we have mostly focused on 2d symmetric orbifold CFTs, our de nition (2.6) of entwinement can be extended to more general discretely gauged theories. twisted by the generator of N , the target space coordinates satisfy X( ; 2 ) = cos Y ( ; 2 ) = sin X( ; 0) + cos We can extend the range of the coordinate in X( ; ) to 0 < 4 by de ning A twisted sector string is then determined by the \long string" pro le X( ; ) with 0 < 4 , which satis es the boundary condition X( ; 4 ) = X( ; 0) + 2 cos Entwinement de ned in analogy with (2.6) then quanti es how one part of this long string is entangled with its complement. There does seem to be an important di erence, however, with symmetric orbifold theories. For the above rotation orbifold, we could equally well have de ned entwinement by considering target space coordinates X0; Y 0 related to X; Y by rotation in eld space over an arbitrary angle. Entwinement de ned using long string pro les X0( ) would generically be di erent from that de ned using X( ), re ecting the basis dependence of measures of quantum entanglement in general. In contrast, for symmetric orbifolds the requirements that the target space coordinates should be mapped into each other by permutations and should have diagonal kinetic terms does select a preferred notion of entwinement. It is interesting to ask for which gauge theories our de nition of entwinement leads to \natural" quantities, including quantities with a clear geometrical meaning in a holographic dual. Other generalizations worth studying are continuous gauge theories, higher dimensional theories and matrix models. Note that entwinement, as we de ned it, measured the lengths of geodesics in units of the AdS scale, rather than in units of the Planck length (which is related to the three dimensional Newton constant GN ). In the original formulation of holographic entanglement entropy [22, 23], it seemed natural that GN should appear in the formulas, in analogy with black hole entropy which is measured by horizon area in units of GN . Of course eld theories with a holographic dual having a classical description generally have a large number of local degrees of freedom arising from e.g. dynamical variables that are large matrices. The entwinement that we are de ning seeks to piece apart the entanglement of some of these local degrees of freedom (e.g. parts of the local matrices) with other elements of the Hilbert space. As such, we are extracting the elementary parts out of which spatial entanglement arises in such quantum eld theories. These elementary entanglements are correspondingly smaller, and are thus related to geodesic lengths measured relative to a length scale that is much larger than the Planck length. One might ask if single interval entwinement can ever be of the same order of magnitude as the spatial entanglement as we make the interval size larger. At least in the 2d CFT states we considered, this cannot happen because entanglement between di erent long strings is negligible and within a single long string entanglement only depends logarithmically on the interval size. This could be di erent in excited or thermal states where there could be extensive contributions to entanglement, or in theories with less local interactions such as matrix models. It is also interesting that the fundamental object (2.6) from which we construct entwinement is non-local. This recalls the discussion in [51] of the relevance of non-local observables in eld theory for reconstructing local physics in AdS space in a gauge-invariant manner. One of our goals in this paper has been to de ne the eld theoretic dual of extremal, non-minimal geodesics in AdS3. These geodesics also appear in the semiclassical CFT computation of Renyi entropies in terms of the conformal block expansion of heavy-heavylight-light correlators [43]. There it was shown that the single interval entanglement entropy in a state created by heavy operator insertions, and dual to an AdS3 conical defect or BTZ, is well approximated by the semiclassical identity block and reproduces the Ryu-Takayanagi minimal geodesic result. This leading answer for the four-point function however has monodromies as the Renyi twists are moved around the heavy operators in the CFT. These monodromies transform the minimal geodesic result into quantities related to the length of non-minimal geodesics. Therefore in this context non-minimal geodesics are also related in the dual CFT to analytic properties of semiclassical Virasoro blocks. Acknowledgments We thank Alexandre Belin, Netta Engelhardt, Jutho Haegeman, Volkher Scholz, Norbert Schuch, Karel Van Acoleyen, Henri Verschelde, and Frank Verstraete for very helpful discussions. We are also particularly grateful to Bartek Czech, Lampros Lamprou, Sam McCandlish, and Jamie Sully for discussing details of their ongoing work on related questions with us. This work was supported in part by a grant from the Simons Foundation (#385592, Vijay Balasubramanian) through the It From Qubit Simons Collaboration, by the Belgian Federal Science Policy O ce through the Interuniversity Attraction Pole P7/37, by FWO-Vlaanderen through projects G020714N, G044016N and Odysseus grant G.001.12, by the European Research Council grant no. ERC-2013-CoG 616732 HoloQosmos, by COST Action MP1210 The String Theory Universe, and by Vrije Universiteit Brussel through the Strategic Research Program \High-Energy Physics". It was performed h[~m(1)]n ~ (1n)(1) (1n+)`(x; x)[ m(0)]ni ; Consider a general correlator of P + Q normalized twists, h p1 (z1) p2 (z2) : : : pP (zP ) q1 (1) q2 (1) : : : qQ (1)i; where pi; qj denote the lengths of the symmetric group cycles of the corresponding operators, inserted in the nite z-plane and at in nity, respectively. We restrict attention to correlators of genus zero, g = 1 XP(pi 1 X(qj s + 1 = 0 ; where, as in the main text, s denotes the total number of copies. correct monodromies corresponding to the twist insertions in part at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1066293. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research & Innovation. T.D.J. is Aspirant FWO-Vlaanderen. F.G. is a Postdoctoral Researcher of FWO-Vlaanderen and acknowledges support from a JuMo grant of KU Leuven. V.B., A.B., and F.G. thank YITP for support during the program \Quantum Information in String Theory and Many-body Systems" and conference \Quantum Matter, Spacetime and Information". V.B. and T.D.J. also thank the Perimeter Institute for hospitality during the It From Qubit workshop and school. B.C. thanks the organizers of the Nordita program \Black Holes and Emergent Spacetime" for hospitality while this work was in progress. Twist correlator In this appendix we brie y review the result described in [45] for general bare twists correlators with covering space of spherical genus, and apply them explicitly to the computation of the correlator in eq. (3.11), 8i = 1; : : : ; P 8j = 2; : : : ; Q to leading order near each branch point. The correlator (A.2) can then be shown to depend only on the coe cients ai; bj and parameters pi; qj and to be given by [45] Specializing to the correlator (A.1), we have is the number of distinct images of in nity and c denotes the central charge of the seed CFT. 8i = 1; : : : ; n 8j = 2; : : : ; n and a map that satis es these properties is In order to completely x the map that computes the correlator (3.11), we choose the branch requiring that for two elementary replica twists in (A.1) acting on the same strand, increase the interval size and take x ` times around the unit circle in the complex plane, z = x = Am : A = e m x m : In the t-plane, the insertions map to tk = F = s + Q t ! t11 = 1 1)n = (A so that the conformal map can also be written as From this expression we can directly determine the coe cients b1 = bj = 6 z = 1)m i=1 tj1 = 1)n = n Y(t To work out the remaining coe cients ai, we consider the rst derivative t = ti, and the rst n points we have 0; 1; ti the rst derivative vanishes. In fact, the rst m 1 derivatives vanish near z = z = 1 + z = x + m! dtm t=ti n! dtn t=0 n! dtn t=1 ai = an+1 = m( 1) (n+1)(A an+2 = mAm 1(A To evaluate explicitly the products, observe that from (A.11), (A.12), (A.15) we can derive the following identities 1)nj = k=1 1 k=1;k6=i t=tk from which we read Substituting in (A.5), we obtain h[~m(1)]n ~ (1n)(1) (1n+)`(x; x)[ m(0)]ni = m 6c (1 n)n 6c (n+1) j k=1 Y jtkjn 2 = n j j k=1 j=1;j6=k tj j = 1j = jAj Y jtkj = k=1 jA k=1 jA k=1 jA j=2 k=1 k = k=1 1 = n n 1 j=2 k=2;k6=j Using these identities we can write the expressions (A.14) and (A.17) as Y jaij = nmn jAjm(n 1) jA Y jbj j = n m(n+2) jA and substituting in the correlator gives h[~m(1)]n ~ (1n)(1) (1n+)`(x; x)[ m(0)]ni = Dominant contributions to the entwinement of M = 0 BTZ In this appendix, we show that in a typical state of the zero mass BTZ black hole the sum appearing in the correlator (3.17) is dominated by strings with m small. The total number of strings of length m in a typical state is We assume we are working at large enough N , such that deviations from typicality are m = 1, and by rede ning m Nm = N = 1 = N = Y tj1 n 2 = n It follows that < 1=2: 8 N 2 = 8N N 0 1=2e 2 N 0 1=2 In the limit N ! 1 this becomes the integral 1 = N which can be split as 1 = N large N limit. indeed strings with m N ) dominate the sum (B.3). To complete the proof, we observe that for xed is of the same order regardless of m. Therefore for all nite ` not scaling with N . Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. [arXiv:1108.3568] [INSPIRE]. Dynamics, arXiv:1412.3148 [INSPIRE]. (2014) 256404 [arXiv:1404.7545] [INSPIRE]. Entanglement Spectrum of Bosons and Fermions with Interactions, Phys. Rev. Lett. 113 spacetime, JHEP 10 (2013) 220 [arXiv:1305.0856] [INSPIRE]. (2014) 044 [arXiv:1403.3416] [INSPIRE]. (2014) 149 [arXiv:1408.4770] [INSPIRE]. and Generalized Holographic Entanglement Entropy, JHEP 04 (2014) 185 [arXiv:1403.1393] [INSPIRE]. Phys. Rev. D 91 (2015) 086005 [arXiv:1501.00003] [INSPIRE]. 85 (2012) 085004 [arXiv:1109.0036] [INSPIRE]. Phys. Rev. D 89 (2014) 085012 [arXiv:1312.1183] [INSPIRE]. (2014) 214003 [arXiv:1406.7304] [INSPIRE]. JHEP 09 (2015) 069 [arXiv:1501.02593] [INSPIRE]. [22] S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE]. entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE]. (2014) 066004 [arXiv:1405.6185] [INSPIRE]. loss, JHEP 09 (2016) 065 [arXiv:1606.01119] [INSPIRE]. e ective geometries of the D1 [1] V. Balasubramanian , M.B. McDermott and M. Van Raamsdonk , Momentum-space entanglement and renormalization in quantum eld theory , Phys. Rev. D 86 (2012) 045014 [2] C. Agon , V. Balasubramanian , S. Kasko and A. Lawrence , Coarse Grained Quantum [3] R. Lundgren , J. Blair , M. Greiter , A. Lauchli , G.A. Fiete and R. Thomale , Momentum-Space [4] V. Balasubramanian , B. Czech , B.D. Chowdhury and J. de Boer, The entropy of a hole in [5] V. Balasubramanian , B.D. Chowdhury , B. Czech , J. de Boer and M.P. Heller , Bulk curves from boundary data in holography , Phys. Rev. D 89 ( 2014 ) 086004 [arXiv:1310.4204] [6] R.C. Myers , J. Rao and S. Sugishita , Holographic Holes in Higher Dimensions, JHEP 06 [7] M. Headrick , R.C. Myers and J. Wien , Holographic Holes and Di erential Entropy , JHEP 10 [8] A. Mollabashi , N. Shiba and T. Takayanagi , Entanglement between Two Interacting CFTs [9] A. Karch and C.F. Uhlemann , Holographic entanglement entropy and the internal space , [10] M. Taylor , Generalized entanglement entropy , JHEP 07 ( 2016 ) 040 [arXiv:1507.06410] [11] M.R. Mohammadi Moza ar and A. Mollabashi , On the Entanglement Between Interacting Scalar Field Theories , JHEP 03 ( 2016 ) 015 [arXiv:1509.03829] [INSPIRE]. [12] P.V. Buividovich and M.I. Polikarpov , Entanglement entropy in gauge theories and the holographic principle for electric strings , Phys. Lett . B 670 ( 2008 ) 141 [arXiv:0806.3376] [13] W. Donnelly , Decomposition of entanglement entropy in lattice gauge theory , Phys. Rev . D [14] H. Casini , M. Huerta and J.A. Rosabal , Remarks on entanglement entropy for gauge elds , [15] D. Radicevic , Notes on Entanglement in Abelian Gauge Theories, arXiv:1404. 1391 [16] W. Donnelly , Entanglement entropy and nonabelian gauge symmetry , Class. Quant. Grav . 31 [18] S. Ghosh , R.M. Soni and S.P. Trivedi , On The Entanglement Entropy For Gauge Theories, [19] D. Radicevic , Entanglement in Weakly Coupled Lattice Gauge Theories , JHEP 04 ( 2016 ) 163 [20] R.M. Soni and S.P. Trivedi , Aspects of Entanglement Entropy for Gauge Theories , JHEP 01 [21] K. Van Acoleyen , N. Bultinck , J. Haegeman , M. Marien , V.B. Scholz and F. Verstraete , The entanglement of distillation for gauge theories , Phys. Rev. Lett . 117 ( 2016 ) 131602 [23] S. Ryu and T. Takayanagi , Aspects of Holographic Entanglement Entropy, JHEP 08 ( 2006 ) [24] V.E. Hubeny , M. Rangamani and T. Takayanagi , A Covariant holographic entanglement [25] N. Engelhardt and S. Fischetti , Covariant Constraints on Hole-ography , Class. Quant. Grav. [26] M. Van Raamsdonk , Building up spacetime with quantum entanglement , Gen. Rel. Grav . 42 [27] E. Bianchi and R.C. Myers , On the Architecture of Spacetime Geometry , Class. Quant. Grav. [28] N. Lashkari , M.B. McDermott and M. Van Raamsdonk , Gravitational dynamics from entanglement 'thermodynamics' , JHEP 04 ( 2014 ) 195 [arXiv:1308.3716] [INSPIRE]. [29] T. Faulkner , M. Guica , T. Hartman , R.C. Myers and M. Van Raamsdonk , Gravitation from Entanglement in Holographic CFTs , JHEP 03 ( 2014 ) 051 [arXiv:1312.7856] [INSPIRE]. [30] J. Maldacena and L. Susskind , Cool horizons for entangled black holes, Fortsch . Phys. 61 [31] B. Swingle and M. Van Raamsdonk , Universality of Gravity from Entanglement , [32] T. Faulkner , A. Lewkowycz and J. Maldacena , Quantum corrections to holographic entanglement entropy , JHEP 11 ( 2013 ) 074 [arXiv:1307.2892] [INSPIRE]. [33] V.E. Hubeny , Extremal surfaces as bulk probes in AdS/CFT , JHEP 07 ( 2012 ) 093 [34] N. Engelhardt and A.C. Wall , Extremal Surface Barriers , JHEP 03 ( 2014 ) 068 [35] V. Balasubramanian , B.D. Chowdhury , B. Czech and J. de Boer, Entwinement and the emergence of spacetime , JHEP 01 ( 2015 ) 048 [arXiv:1406.5859] [INSPIRE]. [36] B. Freivogel , R.A. Je erson , L. Kabir, B. Mosk and I.-S. Yang , Casting Shadows on Holographic Reconstruction, Phys. Rev. D 91 ( 2015 ) 086013 [arXiv:1412.5175] [INSPIRE]. [37] B. Czech and L. Lamprou , Holographic de nition of points and distances , Phys. Rev. D 90 ( 2014 ) 106005 [arXiv:1409.4473] [INSPIRE]. 10 ( 2015 ) 175 [arXiv:1505.05515] [INSPIRE]. JHEP 07 ( 2016 ) 100 [arXiv:1512.01548] [INSPIRE]. [38] B. Czech , L. Lamprou , S. McCandlish and J. Sully , Integral Geometry and Holography , JHEP [39] B. Czech , L. Lamprou , S. McCandlish and J. Sully , Tensor Networks from Kinematic Space, [40] C.T. Asplund and S.G. Avery , Evolution of Entanglement Entropy in the D1 System , Phys. Rev . D 84 ( 2011 ) 124053 [arXiv:1108.2510] [INSPIRE]. [41] S. Giusto and R. Russo , Entanglement Entropy and D1 D5 geometries , Phys. Rev. D 90 [42] A. Galliani , S. Giusto , E. Moscato and R. Russo , Correlators at large c without information [43] C.T. Asplund , A. Bernamonti , F. Galli and T. Hartman , Holographic Entanglement Entropy from 2d CFT: Heavy States and Local Quenches , JHEP 02 ( 2015 ) 171 [arXiv:1410.1392] [44] J. Lin , A Toy Model of Entwinement, arXiv: 1608 . 02040 [INSPIRE]. [45] S.G. Avery , Using the D1D5 CFT to Understand Black Holes , arXiv:1012.0072 [INSPIRE]. [46] O. Lunin and S.D. Mathur , Correlation functions for M N = SN orbifolds, Commun . Math. Phys. 219 ( 2001 ) 399 [hep-th /0006196] [INSPIRE]. [47] O. Lunin and S.D. Mathur , Three point functions for M N = SN orbifolds with N = 4 supersymmetry, Commun . Math. Phys. 227 ( 2002 ) 385 [hep-th /0103169] [INSPIRE]. [48] V. Balasubramanian , J. de Boer , E. Keski-Vakkuri and S.F. Ross , Supersymmetric conical defects: Towards a string theoretic description of black hole formation , Phys. Rev. D 64 [49] E.J. Martinec and W. McElgin , Exciting AdS orbifolds , JHEP 10 ( 2002 ) 050 [50] V. Balasubramanian , P. Kraus and M. Shigemori , Massless black holes and black rings as D5 system , Class. Quant. Grav. 22 ( 2005 ) 4803

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V. Balasubramanian, A. Bernamonti, B. Craps, T. De Jonckheere, F. Galli. Entwinement in discretely gauged theories, Journal of High Energy Physics, 2016, 94, DOI: 10.1007/JHEP12(2016)094