Supersymmetric Renyi Entropy in and

Amit Giveon and David Kutasov

Racah Institute of Physics, The Hebrew University

Jerusalem 91904, Israel

EFI and Department of Physics, University of Chicago

5640 S. Ellis Av., Chicago, IL 60637, USA

We show that in any two dimensional conformal field theory with supersymmetry one can define a supersymmetric analog of the usual Renyi entropy of a spatial region . It differs from the Renyi entropy by a universal function (which we compute) of the central charge, Renyi parameter and the geometric parameters of . In the limit it coincides with the entanglement entropy. Thus, it contains the same information as the Renyi entropy but its computation only involves correlation functions of chiral and anti-chiral operators. We also show that this quantity appears naturally in string theory on .

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1. Introduction

In recent years there has been a lot of work on spatial entanglement in quantum field theory; see e.g. [1,2] for reviews. In a prototypical construction one divides space into two complementary regions, and , and traces the density matrix of the vacuum,^{†}^{†} One can replace the vacuum state by any other pure or mixed state in this construction, but we will not do that here. , over the degrees of freedom in region , thus forming the reduced density matrix

This density matrix contains information about the entanglement between the degrees of freedom in regions and in the state . A convenient way to access this information is via the Renyi entropy

where is initially a positive integer. One can often continue (1.2) to arbitrary real , and in particular take the limit , which gives the entanglement entropy

This is known as the replica trick.

In general, calculating the entropies (1.2), (1.3) is hard, even for free field theories. However, in dimensional conformal field theory (CFT) the problem simplifies somewhat. One can take the region to be the union of (ordered) disjoint intervals

and its complement in . The problem of calculating the Renyi entropy (1.2) for integer can then be mapped to one involving correlation functions of certain twist fields . These twist fields can be thought of as living in the symmetric product CFT

where is the original CFT with central charge , and is an arbitrary integer. The operator is the lowest dimension operator in a twisted sector. The generator of takes the ’th copy of to the ’th, with the ’th copy mapped back to the first. The dimension of is given by [3-6]

One can show [2] that the Renyi entropy (1.2) is given^{†}^{†} Up to an overall constant that will not play a role in our discussion. by

where all fields are inserted at a particular time , on which nothing depends.

For one interval (, , , ), one has

which leads to

with an arbitrary constant. For intervals one has to evaluate point functions of twist fields, which is in general more involved, but can in principle be done using orbifold CFT techniques.

The construction above is general, but it is natural to ask whether it simplifies in CFT’s with additional symmetries. In this note, we will discuss this question for the case where the additional symmetry is supersymmetry. supersymmetry usually does not buy one much in CFT, but theories with superconformal symmetry do enjoy many special properties. In particular, the superconformal algebra contains a conserved current, and there is a special class of operators, known as chiral operators, that belong to short multiplets, and have the property that their scaling dimensions and R-charges satisfy the relation^{†}^{†} There are also anti-chiral operators that satisfy the opposite relation .

Chiral operators form a ring – the chiral ring – which controls many properties of their correlation functions.

The operator that figures in the discussion of the Renyi entropy (1.7) is not chiral. In fact, as we review below, the orbifold that enters the construction of [2] is in general not supersymmetric, even if the theory is. This is analogous to the fact that when one computes the thermal free energy for a supersymmetric field theory by putting Euclidean time on a circle, supersymmetry is broken by the boundary conditions for fermions on the circle.

In this note we will show that the information contained in the correlation function (1.7) can be obtained by studying a supersymmetric analog of (1.5), in which the role of the operator is played by a certain closely related operator , which is chiral. The analog of (1.7) is a correlation function of chiral operators inserted at the points , and anti-chiral operators inserted at . We will refer to the resulting point function as supersymmetric Renyi entropy (SRE), in analogy to (1.7). We will show that it differs from its non-supersymmetric analog (1.7) by a universal function of the parameters and thus contains the same information. At the same time it might be easier to compute by using the special properties of chiral operators in SCFT.

Another advantage of the supersymmetric Renyi entropy is that in SCFT’s that have an dual, it can be calculated in the bulk. In string theory on with Neveu-Schwarz field, the operators are described by physical vertex operators, and thus the SRE can be calculated in the full string theory rather than in the gravity limit, as in [1].

In the rest of this note we describe the construction of the chiral operators , and the relation of their correlation functions which give the supersymmetric Renyi entropy to the usual Renyi entropy (1.7). In section 2 we study the special case where the CFT consists of one free chiral superfield. In section 3 we extend the discussion to a general SCFT. In section 4 we discuss the AdS/CFT dual of SRE. Section 5 contains some comments.

2. Warm-up exercise: a single free chiral superfield

In this section we consider the case where the SCFT , whose Renyi entropy we would like to compute, consists of one free chiral superfield. The bottom component of this superfield is a complex scalar ; its superpartner under the left (right) moving supersymmetry is a complex left (right) moving fermion . The left-moving superconformal generators take the form , . The central charge of this theory is . The structure of the right-moving sector is similar.

We start by reviewing the results of [2] for this case. To construct the twist field , we start with copies of the SCFT , i.e. chiral superfields^{†}^{†} Here and below we will mostly focus on the left-movers; the right-moving sector will be added later. , and consider the cyclic orbifold . The generator of takes . However, the scalars and fermions obey different periodicity conditions in ,

One can think of the label in (2.1) as parametrizing a discretized circle, and of (2.1) as specifying the periodicity of the fields around the circle. The bosons are always periodic, while the fermions are (anti) periodic for (even) odd . In analogy to the case of the thermal free energy, we expect supersymmetry to be broken by the boundary conditions for even , and to remain unbroken for odd .

As is standard for orbifolds, we can diagonalize the action by a discrete Fourier transform,

where labels the discrete momentum. The bosonic and fermonic fields (2.2) are multiplied by a phase under ,

The transformation properties (2.3) can be thought of as due to an insertion of a twist field, which has the property that as the fields , encircle it, they are multiplied by the appropriate phase. For the scalar field , the corresponding twist field has dimension given by (see e.g. [7])

To construct the fermionic twist fields, it is convenient to bosonize the fermions , and write them as

where are canonically normalized left-moving scalar fields. In terms of , the fermionic twist field for takes the form

Its dimension is given by

The operator discussed in section 1 takes in this case the form^{†}^{†} After adding the right-movers we have
, where are the right-moving analogs of .

Adding up the dimensions (2.4), (2.7), we find

in agreement with (1.6) for .

As mentioned above, we expect this construction to break supersymmetry for even . Each of the copies of the SCFT labeled by contains superconformal generators , , but the total superconformal generators,

do not have zero modes in this case. A quick way to see that is to note that in (2.2) the momentum on the discretized circle is integer for the bosonic field , while for the fermionic field it takes value in . Thus, for even it is half integer, so SUSY is broken by boundary conditions. Equivalently, one notes that applying to (first line of (2.2)) gives . For even , the latter is not one of the modes on the second line of (2.2). Note that the current,

does survive the orbifold, but since for even there are no supercharges for it to act on, it gives rise to a standard global symmetry of the orbifold theory. The twist field is not charged under this symmetry.

So far we discussed the cyclic orbifold . In terms of the symmetric product CFT (1.5), one can view it as a sector of that CFT, corresponding to a particular conjugacy class, . The orbifold (1.5) has cycles with both even and odd ; hence, it is not supersymmetric when defined via (2.1).

To summarize, the calculation of the Renyi entropy for a single chiral superfield involves point functions (where is the number of intervals, as above) of the operators (2.8), which contain bosonic and fermionic twist fields. It is worth pointing out that the difficult part of the calculation is the one associated with the bosonic twist fields. The fermionic ones are written in terms of free fields (2.6), and thus their correlation functions can be computed exactly using standard methods.

We would like to modify the construction of [2] such that the superconformal symmetry generated by (2.10) remains unbroken. The discussion above makes it clear how to proceed. We need to modify the conditions (2.1) such that the fermions and bosons have the same periodicity. Thus, we want the fermions to be periodic, , and their Fourier expansion should be the same as that of the bosons in (2.2).

The second line of (2.3) is now modified to . To implement this we replace the twist field (2.6) with

whose dimension is

As before, the index goes over the range . Note that for the operator (2.12) is not the lowest dimension operator which implements the right twist for the fermion . That operator is obtained by taking in (2.12). However, has the advantage that the modified twist field,

is constructed out of the chiral operators . Indeed, they have dimension (2.4), (2.13) , so (1.10) is satisfied. The resulting dimension and R-charge of are and R-charge (2.11)

strictly larger than (2.9), as expected.

We can now consider the correlation function

which is obtained from (1.7) by replacing the usual twist fields with their chiral analogs. The supersymmetric Renyi entropy is obtained by writing the analog of (1.2),

For one interval, one has (in similar notation to (1.8))

which leads to the SRE

We see that (a) The SRE is independent of in this case. (b) It is equal to the standard entanglement entropy (1.9) (with ).

As we will see below, property (a) is special to the one interval case, but (b) is a special case of the more general statement that

i.e. the supersymmetric Renyi entropy gives the standard entanglement entropy in the limit .

For the general case of intervals, the two Renyi entropies defined via (1.2), (1.7), and (2.16), (2.17), are difficult to compute. However, as noted above, the complicated part is the point function of bosonic twist operators , which is the same in the two cases. Therefore, when computing the difference

it cancels. The ratio of correlation functions in (2.21) only involves the fermionic twist fields (2.6), (2.12), and thus can be computed exactly using free field techniques. One finds (after adding the right-moving contributions to , ; see footnote 5)

Looking back at (2.9), (2.15), we see that

Plugging this into (2.21) we finally have

We see that (a) While each of the two Renyi entropies on the l.h.s. of (2.24) is complicated, the difference between them is simple. (b) As , approaches the entanglement entropy (1.3) for the region (1.4), and according to (2.24) so does .

In the next section we will see that the discussion of this section can be extended to a general SCFT. For the purpose of this generalization it is useful to present the results of this section in a more algebraic way.

As mentioned above, the twist operator (2.8) has vanishing charge under the current of , (2.11). In fact, it satisfies the stronger property that its OPE with is regular, i.e. it belongs to the coset . This means that if we define a canonically normalized scalar field via , and consider the operators

their dimension and R-charge are given by

It is natural to ask whether we can adjust such that the operator satisfies the chirality condition (1.10). A short calculation leads to^{†}^{†} There is another solution that does not have the right limit.

The resulting R-charge and dimension are precisely those of , (2.15). In fact, it is easy to check directly that

Thus, the chiral operator obtained from the construction of (2.25) – (2.27) is precisely the one that gives rise to the SRE (2.16), (2.17). As we will see in the next section, the decomposition (2.28) leads directly to (2.24).

3. The general case

In this section we extend the construction of the previous section to a general SCFT with central charge . The Renyi entropy of this theory is again obtained by studying the cyclic orbifold . The lowest dimension operator in the twisted sector, (1.7), has in this case the scaling dimension (1.6) and it commutes with the current

as in the free field case analyzed in section 2. We can thus generalize the construction described at the end of section 2 to this case. We define a canonically normalized scalar field via

and consider the operator (2.25), whose dimension is given by (2.26), while the R-charge is now

As there, we can tune such that the operator is chiral. This leads to

The corresponding operator,

is the chiral operator whose correlation functions determine the SRE via (2.16), (2.17). Its dimension and R-charge are given by

Note that this is in agreement with the chiral spectrum of symmetric products as analyzed e.g. in section 5.1 of [8].

For one interval, one finds the analog of (2.19),

which is again independent of and equal to the usual entanglement entropy given by (1.9). For an arbitrary number of intervals, one needs to generalize the calculation (2.22). The key point is that the numerator of that expression splits into two decoupled factors,

The first is the contribution to the correlator of , the second is the CFT contribution. The ratio (2.21) receives contributions only from the latter, and is given again by (2.22). The difference of dimensions takes in this case the form

which reduces to (2.23) for . The corresponding difference of entropies is

As in section 2, we see that the difference between the two Renyi entropies goes to zero as , and for general they differ by a simple universal^{†}^{†} In the sense that it is independent of any details of the SCFT . function of the parameters .

4. String theory on

A large class of two dimensional CFT’s is obtained by studying string theory on . In this case, we can make use of the fact that the worldsheet CFT on the background (supported by Neveu-Schwarz field) can be described as a WZW model on the group manifold. To get theories with superconformal symmetry, it is natural to consider string backgrounds of the form

where is a worldsheet SCFT with superconformal symmetry.^{†}^{†} Not to be confused with the superconformal symmetry of the spacetime, or boundary, SCFT, which is our focus here.
The level of the WZW model is related to the worldsheet central charge of ; see e.g. [9] for the details. In this class of models one can impose a chiral GSO projection that leads to superconformal symmetry in spacetime
[10,11].

It is natural to ask how to calculate the Renyi entropy of the spacetime SCFT in these models. In this section we will argue that string theory on (4.1) computes the supersymmetric Renyi entropy discussed in the previous sections. A useful observation for that purpose, [8], is that the spectrum of chiral operators in the spacetime SCFT is the same as that on the symmetric product (1.5). Here is a SCFT with central charge , whose properties are implicitly determined by the worldsheet SCFT in (4.1), and is an integer related to the string coupling, . For example, for string theory on , which can be viewed as a special case of (4.1) (after imposing a chiral GSO projection in the latter), one has ; more precisely, is in the moduli space of this symmetric orbifold.

Based on this and other observations, the authors of [8] proposed that the spacetime SCFT corresponding to (4.1) is^{†}^{†} Actually, [8] proposed that it is in the moduli space of the symmetric product, but in general this symmetric product does not have any marginal operators, so if the proposal is correct, it is the symmetric product. the symmetric product (1.5). If this is correct, the only non-trivial Renyi entropy one needs to compute is that for . Thus, one needs to identify in string theory on the appropriate twist operators.

There is a very natural candidate for the chiral operators . As discussed in [8], chiral operators in the twisted sector of the spacetime SCFT (1.5) are described by vertex operators with winding , in the sense of [12]. In particular, starting with a chiral operator with spacetime dimension in the short string sector, one can construct a sequence of chiral operators in sectors with winding , whose dimensions are given by

For the special case this leads to an operator with the same quantum numbers as (3.6), and it is natural to identify the two. Computing the correlation functions (2.16) of these operators thus computes the SRE of .

Physical vertex operators corresponding to chiral operators in the untwisted sector of the spacetime orbifold are obtained by dressing the chiral operators in the internal worldsheet theory by an appropriate factor from the piece [9,13,8]. The vertex operators corresponding to chiral operators in the twisted sectors of the spacetime theory are obtained by acting on the untwisted ones with a simple ‘chiral twist’ operator in the worldsheet theory; see [8] for the detailed construction.

In particular, the spacetime identity operator, [14,15], is obtained from the identity in . Consequently, the vertex operators for discussed above live purely in the factor, and thus do not contain any information about the particular SCFT in (4.1) that enters the construction, except for the level , which determines the spacetime central charge of . Thus, any SCFT with a description in terms of string theory on with NS field has the property that its Renyi entropy is universal, at least to leading order in .

5. Discussion

In this note we showed that in any two dimensional conformal field theory with SUSY one can obtain the Renyi entropy (1.2) for any spatial region (1.4) by calculating a correlation function of chiral and anti-chiral operators. This was done by defining the chiral operators (3.5), and showing that their correlation function (2.16), which leads to the supersymmetric Renyi entropy (2.17), satisfies the relation (3.10) and thus determines the Renyi entropy .

We further showed that the operators are present in string theory on with superconformal symmetry in spacetime, and thus in any SCFT with an string dual one can compute the correlators (2.16), and hence the Renyi entropy (1.2). The resulting entropy is universal – the only data it depends on is the parameters . In particular, it is independent of the detailed structure of the string background (4.1).

There is a number of natural extensions of our results. One is to use the chirality of the operators to compute the correlation functions (2.16), and thus shed new light on the Renyi entropy. Another is to calculate these correlation functions in string theory on using worldsheet techniques. It would also be interesting to generalize the discussion to the case of RG flows that preserve supersymmetry.

Another interesting direction involves the relation of our results to [16]. In that paper the authors defined a supersymmetric version of Renyi entropy in three dimensional theories with SUSY via partition functions on branched covers of the three-sphere. This construction has been further developed in [17], and generalized to four [18,19] and five [20] dimensions; its gravity dual was discussed in [17,21,19,22]. It would be interesting to generalize this work to two dimensions, and see whether it gives the same quantity as our supersymmetric Renyi entropy. If that’s the case, it would be interesting to understand our result (3.10) from this perspective, and study possible generalizations of this result to higher dimensions.

Acknowledgements: The work of AG is supported in part by the I-CORE Program of the Planning and Budgeting Committee and the Israel Science Foundation (Center No. 1937/12), and by a center of excellence supported by the Israel Science Foundation (grant number 1989/14). DK is supported in part by DOE grant DE FG02-13ER41958. DK thanks Tel Aviv University and the Hebrew University for hospitality during part of this work.

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