Large- transitions of the connectivity index
The connectivity index, defined as the number of decoupled components of a separable quantum system, can change under deformations of the Hamiltonian or during the dynamical change of the system under renormalization group flow. Such changes signal a rearrangement of correlations of different degrees of freedom across spacetime and field theory space. In this paper we quantify such processes by studying the behavior of entanglement entropy in a specific example: the RG flow in the Coulomb branch of large- superconformal field theories. We find evidence that the transition from the non-separable phase of the Higgsed gauge theory in the UV to the separable phase of deformed decoupled CFTs in the IR exhibits sharp features in the middle of the RG flow in the large- limit. The entanglement entropy on a sphere with radius exhibits the formation of a separatrix on the co-dimension-two Ryu-Takayanagi surface in multi-centered brane geometries above a critical value of . We discuss how other measures of entanglement and separability based on the relative quantum entropy and quantum mutual information might detect such transitions between non-separable and separable phases and how they would help describe some of the key properties of the IR physics of such flows.
\affiliationCrete Center for Theoretical Physics and
Crete Center for Quantum Complexity and Nanotechnology,
Department of Physics, University of Crete, 71303, Greece \emailAddaprile@physics.uoc.gr \emailAddniarchos@physics.uoc.gr \preprintCCQCN-2014-47
\keywordsEntanglement entropy, relative quantum entropy, holography, Coulomb branch, conformal field theory
In any quantum system we can arbitrarily partition the total Hilbert space into two subspaces and . For a given configuration we can ask to what extent states in are entangled with states in , or how strongly observables computed in are correlated with observables in . This is an interesting question that can reveal useful information about the state of the system and its dynamical properties.
In a well studied example we take a system defined in spatial dimensions and separate the degrees of freedom inside a spatial region from the degrees of freedom in the complement . A natural measure of the entanglement of the two sets of degrees of freedom (in Hilbert spaces and ) is the entanglement entropy defined as the von-Neumann entropy of the reduced density matrix
is obtained by tracing the total density matrix over the states of the outside Hilbert space
is an interesting quantity that has played a central role in many recent developments. For example, when applied to -dimensional relativistic conformal field theories its dependence on the characteristic size of the region holds information about basic constants of the theory, the central charge in dimensions  (see e.g.  for a review), or the -function in dimensions  etc.
Another possibility is to partition the system in field theory space, namely split the degrees of freedom at each point of spacetime into two subsets. This type of partitioning arises naturally, for example, when we have two distinct quantum systems with Hamiltonians and interacting weakly via an interaction Hamiltonian , but it can also be considered more generally without reference to a specific type of dynamics.
The first question we want to ask in this paper is the following. Given an arbitrary split of the degrees of freedom of a quantum system, e.g. a quantum field theory, in spacetime and/or in field theory space, can we define a meaningful measure of the entanglement or strength of correlation between the subsystems. Several well known measures from quantum information theory that quantify the notion of separability, e.g. measures based on the relative quantum entropy, turn out to be very well suited for this purpose. We will review the relevant concepts, and give specific definitions, in section 2.
The second question we want to raise concerns the behavior of such measures under deformations of the theory, or under the dynamical change of the parameters of the system under the renormalization group (RG) flow.
For example, there are many cases where by tuning the parameters of a theory, or by looking at the system at
different energies, the interaction coupling in between two subsystems becomes weak or even turns off.
In the latter case the subsystem Hamiltonians and decouple completely. Any observable computed
in this product theory (e.g. an arbitrary correlation function) factorizes in a (sum of) products of observables of
theory 1 and theory 2. It is useful to introduce a connectivity index
With any of the above definitions the connectivity index can decrease when turns on, or increase when turns off. The measures of entanglement mentioned above will behave accordingly. It is possible, however, to encounter more subtle situations where many of the effects mediated by are suppressed until a finite value of the interaction coupling. We will argue that RG flows in the Coulomb branch of large- superconformal field theories (SCFTs) provide interesting examples of this type.
For definiteness, let us consider the Coulomb branch of the four-dimensional super-Yang-Mills (SYM) theory. In the ultraviolet (UV) we have an gauge theory with the apparent connectivity index 1. Turning on the vacuum expectation values of the adjoint scalars we move away from the origin of the Coulomb branch, the gauge group is Higgsed, say to , and there is an RG flow to the infrared (IR) where an gauge theory decouples from an gauge theory. In the far IR the connectivity index counts 2 decoupled components with order degrees of freedom and another component associated with the decoupled degrees of freedom of the part of the theory. At low energies the leading order direct interaction between the two IR CFTs is mediated by an irrelevant double-trace dimension 8 operator [6, 7] (see section 4.3 for specific expressions). Being irrelevant this operator turns off at the extreme IR. As we explain in section 3 the part also mediates interactions and plays an interesting role in the low energy dynamics.
The interest in the large- limit stems from the following observation. If we could isolate the dynamics of the IR CFTs from the dynamics of the part, we would be able to argue at leading order in the expansion that the multi-trace operators that mediate interaction between the two IR CFTs do not contribute to the anomalous dimension of any combination of their energy-momentum tensors and despite the deformation both energy-momentum tensors remain independently conserved. That would be evidence that the system remains in a separable state in a vicinity of the IR fixed point at leading order in . In the actual RG flow, however, one cannot isolate the dynamics of the part. Since the latter mediates interactions that allow energy to flow from the to the IR CFTs the system is expected to be in a non-separable state with connectivity index 1 infinitesimally away from the extreme IR. It is interesting to find a quantitative measure that expresses how strongly the IR separability is broken by such (-mediated) interactions and to explore how one connects this type of infrared physics to the UV physics of a strongly non-separable Higgsed gauge theory in the UV.
One observable that we consider in the main text, in order to examine these questions, is the entanglement entropy (1) for a spherical geometry with radius . As changes from 0 to we probe physics from the UV to the IR. In the large- limit we can evaluate the entanglement entropy with the generalized Ryu-Takayanagi prescription [8, 9, 10] by determining a minimal co-dimension-2 hypersurface in the multi-throat geometry of separated stacks of branes. We perform this analysis quite generally for the 4d SYM theory on D3 branes, the 3d ABJM theory on M2 branes and for the 2d CFT on D1-D5 branes. In all cases we find that the Ryu-Takayanagi minimal hypersurface exhibits a separatrix at a radius where it shows signs of critical behavior. This is evidence of an interesting sharp feature that occurs in the middle of the RG flow.
In section 2 we define other measures of entanglement based on the concept of relative quantum entropy. Currently, we do not know how to compute these measures holographically from gravity in the large- limit, but we discuss possible behaviors in sections 8 and 9. Eventually, one would like to determine how these measures capture the quantum field theory dynamics that is summarized in section 3.
The main computational results of the paper are presented in sections 4-7. Section 5 contains a description of the qualitative features of the Ryu-Takayanagi surface in multi-centered geometries. The reader can consult this section for a quick overview of the results that arise by studying the holographic entanglement entropy in this context. Concrete quantitative results based on the analysis of the equations of the Ryu-Takayanagi minimal surface are presented in sections 6, 7. For instance, in section 6 we notice that the UV expansion of the holographic entanglement entropy does not receive contributions from the lowest order harmonics. This is a gravity prediction for a corresponding statement about entanglement entropy in the large- superconformal field theories that we consider.
2 Separability, relative quantum entropy and other useful concepts
Assume that we have a -dimensional quantum system with Hilbert space and we partition both in spacetime and field theory space. In spacetime we separate states supported inside a spatial region from states in the complement . In field theory space, we separate (at each point of spacetime) degrees of freedom of a subsystem 1 from degrees of freedom of a subsystem 2. Then, the reduced density matrix (2) is a matrix that lives in the product Hilbert space . We are interested in a measure that quantifies the entanglement of the states of the two subsystems 1 and 2. We will focus on the properties of the density matrix keeping the additional dependence on the size of the region as a useful way to keep track of the entanglement across different length (or energy) scales.
A standard definition in quantum information theory (see  for a review) postulates that the state represented by is separable if it can be written as a sum of product states in the form
with , . If not, is called entangled. In the special case with a single propability coefficient non-zero, i.e. when
the state is called simply separable. This is the case mentioned in the introduction where no correlations between subsystems 1 and 2 exist.
Testing for separability is in general a very hard problem. However, it is possible to formulate a measure that quantifies how far from separability a quantum system is by using the concept of relative quantum entropy. For any two density matrices the relative quantum entropy of with respect to is defined as
One can prove the Klein inequality (see e.g. ), which states that is a positive-definite quantity that vanishes only when , i.e. when the states and are indistinguishable. On the other extreme, the relative quantum entropy is infinite when the two states are perfectly distinguishable. This fact played a useful role in the recent work .
One can use the relative quantum entropy to define a measure of how far a system is from separability. The usual approach defines the following quantity
which is called relative entropy of entanglement. The minimum is obtained by sampling over the whole space of separable states. is zero if and only if is a separable state.
Since we are interested in simply separable states we can modify this definition in an obvious way by taking the minimum over the simply separable states. In what follows, however, we consider instead a related quantity that we define as follows. Concentrating on the specific context of our density matrix , and a partitioning into two complementary subsystems 1 and 2, we consider the relative quantum entropy
where is defined as the tensor product of the reduced density matrices
This quantity vanishes if and only if our system is completely decoupled into the two subsystems 1 and 2. In fact, one can show that the definition (7) is simply the quantum mutual information
As a concept, separability is very well adapted to describe properties related to the connectivity index and its behavior under changes of the system, e.g. under renormalization group flows that lead to Hilbert space fragmentation. We will soon examine these properties in a specific context of large- quantum field theories.
3 Hilbert space fragmentation in quantum field theory
There are several common mechanisms in quantum field theory that change the connectivity index.
For example, in strongly coupled gauge theories an operator will frequently hit the unitarity bound
and decouple from the remaining degrees of freedom as a free field.
In this paper we will examine closely the case of gauge group Higgsing in the Coulomb branch of large- superconformal field theories. A concrete example of the general setup has the following ingredients. The UV conformal field theory CFT is a gauge theory with gauge group . It flows by Higgsing to an IR conformal field theory which is a product of decoupled theories, e.g. CFT CFTCFTCFT. CFT is a gauge theory with gauge group , and CFT is a gauge theory with gauge group . CFT denotes collectively a gauge theory with a set of free decoupled massless fields. The massless scalar fields in this set express the moduli whose vacuum expectation value Higgses the UV gauge theory and sets the vacuum state.
It is interesting to consider the low-energy effective description of this theory. At small energies above the extreme IR the direct product theory is deformed by irrelevant interactions of three different types
The first type includes the operators and , which are single-trace operators in
CFT and CFT respectively (with irrelevant couplings , of order ).
The second type involves a double-trace operator of the form ,
where and are single-trace in CFT and CFT.
Explicit examples of such operators and the corresponding irrelevant interactions will be provided in the next section 4.3 for SYM theory.
So far our discussion is valid at any .
We notice that the non-abelian IR CFTs, CFT and CFT, communicate directly
only by multi-trace operators, as dictated by gauge invariance
(a point emphasized in ), and indirectly via the interaction with abelian
degrees of freedom of CFT. At finite both types of interactions contribute to the precise manner
in which the system passes from a non-separable UV state to an extreme IR separable state.
However, in the large- limit
The main purpose of this paper is to quantify this transition using the measures of entanglement presented in the previous section 2 and to explore potentially new features associated with the large- limit. We will focus on large- quantum field theories with a weakly curved gravitational dual.
Entanglement entropy. The entanglement entropy of large- conformal field theories with a weakly curved gravitational dual can be computed efficiently using the Ryu-Takayanagi prescription in the AdS/CFT correspondence. This computation, which will be performed in the next four sections, involves the analysis of a minimal co-dimension-2 surface in multi-centered brane geometries in ten- or eleven-dimensional supergravities. The non-standard feature of this computation is that the minimal surface embeds non-trivially along the compact manifolds transverse to AdS. We will see that the above-mentioned transitions of the connectivity index are closely related to the formation of a separatrix in the geometry of the Ryu-Takayanagi minimal surface.
Relative entropy of entanglement and quantum mutual information. In section 2 we presented two measures of separability, the relative entropy of entanglement (6) and the quantum mutual information (9). Currently, we are not aware of an efficient computational method for such quantities in interacting quantum field theories, either directly in quantum field theory or holographically. Nevertheless, the above discussion indicates that we should anticipate the following features.
To specify we define subsystem 1 as the subsystem associated with the degrees of freedom of the IR CFT. The subsystem 2 (that refers to the complementary Hilbert space) includes the remaining degrees of freedom of the full theory. In the effective IR description subsystem 2 includes the degrees of freedom of CFT and CFT. Since we are considering a non-trivial RG flow the relative quantum entropy on a sphere of radius will be a non-trivial function of . Complete decoupling in the extreme IR implies that vanishes at and increases as decreases towards (that probes the extreme UV). The increasing positive magnitude of
is a measure of the increasing strength of correlation of the degrees of freedom of the IR CFT with the rest of the system at high energies. The general discussion in the beginning of this section suggests that this increase is suppressed in the large- limit at low energies because the effects of inter-system interactions mediated by multi-trace operators are suppressed. It is of interest to understand if this expectation is verified by the explicit computation of , and to determine precisely how interpolates between the extreme UV and IR descriptions that exhibit a different connectivity index. We anticipate a qualitatively similar behavior from other measures of separability, e.g. the relative entropy of entanglement . An efficient computational method for the relative quantum entropy would be helpful in addressing these issues, but goes beyond the immediate goals of this paper.
4 Coulomb branch of SCFTs and multi-centered geometries
In this preparatory section we collect useful facts and notation for the geometries involved in the holographic computation of the entanglement entropy in the Coulomb branch of superconformal field theories.
4.1 Notation and main features of multi-centered brane geometries
We focus on supersymmetric conformal field theories with a weakly curved gravitational dual
in string/M-theory. The gravitational description of the Coulomb branch of these theories is directly
related to the geometry of a discrete collection of flat parallel D/M-branes in 10 or 11-dimensional supergravity.
This geometry is uniquely specified by a single harmonic function , where
are the coordinates transverse to the brane volume. The supergravity solution also carries charge under the
corresponding -form gauge fields and generically sources the dilaton
In this paper, we will focus on multi-centered geometries given by:
D3 branes in dimensions, relevant for the SYM theory,
M2 branes in dimensions, relevant for the ABJM Chern-Simons-Matter theories ,
D1-D5 bound states in . The D5 branes wrap the compact manifold (usually taken as or ) and give rise at low energies to an interacting -dimensional superconformal field theory.
The corresponding geometries in asymptotically flat space
The harmonic functions , are
The vectors locate the position of the different stacks of branes in the transverse space. We are considering stacks of D3 (M2) branes, each one made out of D3 branes ( M2 branes). and are the string coupling constant and string Regge slope. is the eleven-dimensional Planck length.
For the D1-D5 system, the two harmonic functions and are:
where is essentially the volume of , i.e. . It will be technically convenient to focus on D1-D5 bound states with parameters that obey the relation
This restriction guarantees that the dilaton , given by the relation , will be constant in the near-horizon limit.
Near-horizon limit. For the D3 and D1-D5 branes, the decoupling limit  is defined by sending , and keeping the ratios and fixed. As a result, the in the harmonic functions drops out, and the geometry remains finite in units of . Under the assumption (19), the product simplifies
For the M2 branes the decoupling limit is obtained by sending and keeping and fixed.
In summary, the D1-D5 system is now described by the function , and the D3 and M2 backgrounds are described by
The resulting geometry interpolates between an space at , that captures the UV fixed point with connectivity index 1, and a decoupled product of spaces as gets scaled towards the centers . The latter describes the extreme IR fixed point with connectivity index .
4.2 UV physics
For the cases we analyze the asymptotic geometry is an space with for the D1-D5, M2 and D3 brane systems respectively. The radius of each space is
These UV geometries are dual to the microscopic -dimensional superconformal field theories mentioned in the beginning of the previous subsection. For concreteness, let us focus for the moment on the most emblematic case, i.e. the duality between string theory on and SYM theory.
The multi-centered D3 brane solutions are dual to a configuration in SYM in which the gauge group has been Higgsed down to (). Conformal invariance, as well as the R-symmetry of the theory, are broken by the non-vanishing vacuum expectation value of the gauge-invariant chiral operators
where are totally symmetric traceless rank tensors of the -charged real adjoint scalars of the theory. These modes arise in the gravity dual from a Kaluza-Klein decomposition of the transverse space. By analyzing the asymptotic, large , behavior of these modes in the multi-centered geometry one can determine the vacuum expectation value of the operators (25)[18, 19, 20].
4.3 IR physics
The decoupled product of gauge theories that arises in the extreme infrared of the Coulomb branch translates, in the dual multi-centered geometry, into a decoupled product of string theories on the spacetimes. Each of these spacetimes arises from the full multi-centered geometry by taking the limit that isolates the gravitational dynamics near the -th center. The radius of the -th spacetime is weighted by the single coefficient , or , respectively. The factors are decoupled sectors of singleton degrees of freedom that reside on the common holographic boundary of the spacetimes.
As explained in section 3, in the IR description of the RG flow the non-abelian IR CFTs interact off criticality via an infinite set of irrelevant multi-trace interactions, and via irrelevant interactions mediated by the abelian singleton degrees of freedom —the Lagrangian in equation (10). For example, in the case of SYM theory the leading single-trace operator for the -th non-abelian IR theory (see equation (10)) is a dimension 8 operator of the form [6, 7, 21]
The coefficient is proportional to the sum
Note that (26) is also the type of interaction that appears in the small field strength expansion of the Dirac-Born-Infeld action that describes the exit from the near-horizon throat. In the current context the single-trace interaction (26) describes how the throat in question connects with the rest of the geometry.
which mediate the direct inter-CFT interactions mentioned in equation (10).
Finally, there are interactions of the non-abelian degrees of freedom with the abelian singleton degrees of freedom. Part of the singleton degrees of freedom are the massless scalar fields associated with the moduli . Expanding (27) around the values of the given vacuum state produces irrelevant single-trace interactions of the form
This makes the single-trace couplings dynamical.
Holographically, in this description we are working in the bulk with an explicit UV cutoff and we are dealing with a set of UV-deformed gravity theories coupled in two ways: by mixed boundary conditions and by explicit boundary degrees of freedom (the singletons) that make the sources of some of the bulk fields dynamical. A similar picture of coupled throat geometries was proposed some time ago in [22, 23].
5 Holographic Entanglement entropy
In a field theory in dimensions, the static entanglement entropy of a space-like region is defined as the von-Neumann entropy of the density matrix which is obtained by tracing out the degrees of freedom in the complement of (see equations (1), (2)).
For conformal theories living on the boundary of , the Ryu-Takayanagi prescription (RT) computes the holographic entanglement entropy (HEE) by considering the area of a -dimensional minimal surface in , whose boundary is . We will refer to this surface as . There is a beautiful derivation of the correctness of this prescription for spherical entangling surfaces. By conformally mapping the density matrix to a thermal density matrix, the authors of  showed that the thermal entropy of the dual hyperbolic black hole coincides with the HEE computed à la Ryu-Takayanagi. The relation between the entropy of and the minimal area condition was further investigated and clarified in .
where is the induced metric of a minimal co-dimension-2 surface in the full string theory or M-theory background. The surface is again specified to have as its boundary. This generalized prescription is the prescription we will apply in the computation that follows. In our setup, the dilaton field is a constant for all the cases we will consider; the D3, M2 and D1-D5 branes. This statement is obvious for D3 and M2 branes, and follows from the assumption (19) in the case of the D1-D5 bound states.
It is clear that for spaces, the Ryu-Takayanagi prescription is in perfect agreement with (30). When there is no dependence on the transverse sphere, the problem of a minimal surface that wraps reduces to the problem of finding in . The Newton constant in is related to through the formula
A typical class of examples in which the prescription (30) is non-trivial are the confining backgrounds of [26, 27], for which the entanglement entropy was studied in . These backgrounds are of the type , where is a cone over a certain compact manifold . The volume of may shrink along the radial coordinate of the cone, and since wraps , it will be sensitive to the dynamics of these extra dimensions along the RG flow.
Similarly, the multi-centered geometries of interest in this paper are not product spaces globally. They become locally spacetimes only in certain asymptotic regions. If the dimension of was different from , other data would be needed to determine it, and the surface would not be unique for a given . An example appears in the holographic computation of the Wilson loop in .
The remainder of this section provides a qualitative description of the surface in the multi-centered backgrounds described previously. We consider spherical entangling surfaces when , and intervals when . It is useful to choose space-like coordinates adapted to these geometries. In dimensions , we choose spherical coordinates: , where is the radius of the sphere and are angles. In one dimension we use a similar notation: is the spatial field theory coordinate that runs along the real line. The entangling region is described by the equation . This means that where for , and for .
The main example we will consider in detail is the case of the two-centered geometry. The two-centered geometries are conveniently described by hyper-cylindrical coordinates in the transverse space. The branes are separated along a direction , and the space orthogonal to is described by hyper-spherical coordinates . In this setting, the functions of the previous section will depend both on and . The origin is taken to be the center of mass. We can also introduce polar coordinate in the plane,
with and . For coincident branes, , the coordinate becomes the radial coordinate of , and becomes the polar angle of the -sphere.
The minimal surface is static with Dirichlet boundary conditions in the time direction, which will not play any further role. The coordinates describing the co-dimension-2 surface are chosen as follows
The embedding in the -dimensional background is specified by the function , where and . This function is an interesting object because it mixes the evolution along a field theory direction, , with the change of the geometry along the transverse space direction . The non-trivial dependence on originates from which are explicit functions of .
The behavior of can be understood qualitatively by regarding as a map from to the plane . We imagine foliating the surface by fixing a certain , drawing the curve in the plane , and moving in the interval . For example, in , the solution is given by the Ryu-Takayanagi surface which is independent, therefore , and the map draws circles of radius . From this simple analysis we are able to infer three out of the four boundary conditions that fix a generic -dependent solution on :
We will discuss the boundary condition at in a moment.
To start thinking about in two-centered solutions, it is useful to first consider the limit . In this limit the surface probes the physics of the deep IR of the field theory where the UV gauge group has been Higgsed and the energy scales of interest are well below the mass of the massive bosons. In the gravity dual this limit zooms into the vicinity of the two centers which can be regarded as decoupled. The surface is then given by the union , where . At this point, it is important to recall that has a turning point at , i.e. for any . The fact that is the turning point follows from the symmetries of the entangling surface and from the assumption that is convex.
When is finite, but large enough for to probe the IR throats, the picture we have just described will be approximately valid only locally close to each of the two centers. In a neighborhood of the map draws approximately small disconnected circles around the position of each stack of branes (points and in the plane in Figure 1). The curve generalizes the notion of turning point in the Ryu-Takayanagi surface and obeys the boundary conditions
The overall picture in the IR is summarized by the brown curves in Figure 1.
The above description refers to the IR patch of the surface associated to a space-like region of large enough radius . In the opposite regime, we can ask what happens at (), when the curve is close to the UV boundary. Because the boundary is , this curve is again approximately -independent and the associated map draws a large circle in the plane (captured by the blue curves in Figure 1).
The inevitable conclusion of the above analysis is that, although the surface is always simply connected, the topology of the curves may change as we vary . When is large enough, the minimal surface will have a UV patch where is topologically , and an IR patch where is topologically . For such a surface there is necessarily a branch point. The curve at which this branch point belongs will be referred to from now on as the separatrix. This is sketched as the red line in Figure 1.
The topology change that we described above does not occur for surfaces with small enough that can only probe the UV part of the full geometry. For such surfaces the curves are topologically for any . It is clear that the discriminating quantity between the existence of the topology change or not, for a given , is the turning point curve . Accordingly, we will distinguish between the following two phases:
Phase A, for , where the topology of is . In this case we can describe with single-valued coordinates.
Phase B, for , where the topology of is . In that case a separatrix exists and when moves below the separatrix, becomes double-valued.
The counterpart of the transition between these phases in field theory is a transition of the behavior of the entanglement entropy as a function of at .
The qualitative behavior of for multi-centered geometries can be deduced by following the same logic as in the two-centered solution. However, in the general case it will not be possible to restrict the discussion to a certain plane , and one has to consider the full transverse space.
6 UV expansion of the entanglement entropy
In this section we will study more explicitly the HEE of phase A. The equation of motion of is a non-linear, quite challenging, PDE. Yet, we are able to obtain a series expansion of the solution by expanding in a small dimensionless parameter that combines the mass scale of symmetry breaking (equivalently the center separation in the geometry) and the sphere radius . Our perturbative solution is analytic in the variables and , and at zeroth order coincides with the AdS solution. The perturbative solution does not allow us to detect analytically the formation of the separatrix as we approach , but it confirms the qualitative description of the previous section.
By direct integration of the generalized HEE functional we obtain a series of finite corrections to the entanglement entropy. Perhaps suprisingly, the translation of the result to field theory language suggests that the lowest chiral primary operators do not contribute to these corrections.
6.1 Minimal surface action and its equations of motion
In phase A the variable is single-valued as a function of , thus we can write the induced metric on by using the coordinates (32). Referring to the components of the background metrics (12), (13) and (14), with the generic notation,
the induced metric on , in the coordinates (32), is given by
The HEE functional is then
where the Lagrangian can be put into a form valid for all cases of interest here (the D3, M2 and D1-D5 branes),
In (38) we defined the functions
In the two-centered geometries we fix the origin of the axis at the center of mass of the system, namely we set
After the implementation of the condition (40), the Euler-Lagrange equation following from (38) depends only on a single dimensionful parameter, for example. Schematically, the single PDE that we need to solve is the equation of motion of
The explicit form of this equation is provided in Appendix A.
6.2 Perturbative UV Solution
Before entering the details of the calculation, we review the solution making manifest the underlying scale invariance. This is our starting point towards a perturbative solution of the non-linear PDE (41) that follows from (38).
It is convenient to work with the variable , in the cases of D3 and D1-D5 branes, and for the M2 branes. The UV boundary is now at . In our conventions, the metric of is written as
It should be noted that with our choice of spherical entangling surfaces, the embedding function is independent of . In the r.h.s. of (44) we wrote in a conformal fashion: we isolated the pre-factor , and defined the function that depends only on the dimensionless combination . The pre-factor captures the weight of under rescaling of . We also notice that the equation (43) has weight zero; in particular, the corresponding equation for has no dependence.
Now the idea is to consider a UV ansatz for of the type,
As expected, by plugging (45) into the equation of motion we obtain an equation for which depends only on the dimensionless parameter for D3 and D1-D5 branes with , and for M2 branes with . The limit is well defined and gives back (43). Around it we can solve the equation for in perturbation theory. Schematically, our problem becomes
In (46) we restored the label to stress that the perturbative solution depends on the number of dimensions. The functions capture the two-center deformation of the UV solution. Solving for still requires finding the solution of a set of PDEs. However, this problem is tractable and analytic solutions can be obtained.
For D3 and D1-D5 branes it is possible to write down simple explicit formulae. Results for the M2 branes are more involved due to the fact that the UV comes in horospherical coordinates. However, the algorithm to find the perturbative solution is valid for generic .
For , the functions solve a PDE of the form,
where are forcing terms whose explicit dependence is inherited from . At fixed , the forcing term is determined by the lower order solutions for . We find the first non-trivial , and solve for . Then we proceed to compute , solve for , and continue by iteration. An important observation is that upon the change of variable , the forcing terms become polynomials in with -dependent coefficients. Therefore, the ansatz
which is compatible with the boundary conditions at , solves the dependence in (47). The set of functions is just a rewriting of the standard Fourier basis in a way that is compatible with our boundary conditions. For any of the form (48), the PDE (