Advances in High Energy Physics

Volume 2013 (2013), Article ID 604637, 34 pages

http://dx.doi.org/10.1155/2013/604637

Research Article

Matrix Theory, AdS/CFT, and Gauge/Gravity Correspondence

Shan Hu¹ and Dimitri Nanopoulos1,2,3

¹George P. and Cynthia W. Mitchell Institute for Fundamental Physics, Texas A&M University, College Station, TX 77843, USA

²Astroparticle Physics Group, Houston Advanced Research Center (HARC), Mitchell Campus, Woodlands, TX 77381, USA

³Division of Nature Sciences, Academy of Athens, 28 panepistimiou Avenue, Athens 10679, Greece

Received 29 April 2013; Accepted 10 August 2013

Academic Editor: Kingman Cheung

Copyright © 2013 Shan Hu and Dimitri Nanopoulos. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

With   being fixed, , the free energy of the Matrix theory on a supergravity background is a functional of , . We try to relate this functional with , the effective action of , where is translation invariant along . The vertex function is then associated with the connected correlation function of the current densities. From , one can construct an effective action for the arbitrary matrix configuration . is and thus independent. If , will give the supergravity interactions among theory objects with no light-cone momentum exchange. We then discuss the Matrix theory dual of the background generated by branes with the definite as well as the gauge theory dual of the background arising from the reduction. Finally, for SYM₄ with the background field , we give a possible way to induce the radial dependent field .

1. Introduction

Up to present, two kinds of nonperturbative formulation of /string theory are developed. The one is Matrix theory. The typical examples are BFSS Matrix model [1] and the plane wave Matrix model (PWMM) [2], describing a sector of M theory with the definite light-cone momentum on flat background and pp-wave background, respectively. M theory on a generic weakly curved background is described by BFSS Matrix model with the corresponding vertex operator perturbations added [3, 4]. PWMM could just be derived in this way [5]. For backgrounds that cannot be taken as the perturbations of the flat spacetime, the corresponding Matrix models are also known provided that certain amount of supersymmetries is preserved. The other is the correspondence [6–9], for which the correspondence is intensively-studied. gives a nonperturbative description of string theory on . It is natural to expect that with the vertex operator perturbations added then describes the string theory on with the corresponding field perturbations turned on. Although Matrix theory and AdS/CFT are obtained in entirely different ways, both of them use a gauge theory to describe M/string theory on a particular background. If the M theory background is , the dual gauge theory will become , which is Matrix theory compactified on [10–12].

As the nonperturbative description of M/string theory on a particular background, the Hilbert space of the gauge theory should be isomorphic to the Hilbert space of the M/string theory on that background. For , the one-to-one correspondence should exist between the state (spectrum) of the second-quantized string theory on and the state (spectrum) of . For Matrix theory, it is easier to establish the correspondence between configurations. The transition amplitude between the matrix configurations should be equal to the transition amplitude between their M theory counterparts. When , Matrix theory is the discrete regularization of the supermembrane theory in light-cone gauge [13]. The matching is explicit. It is natural that the transition amplitude for membranes is defined in the same way as the transition amplitude for strings, since the former, when wrapping with the vanishing radius, reduces to the latter. Matrix theory compactified on gives the Matrix string theory [14, 15]. The off-diagonal degrees of freedom are KK modes of membrane along [16]. In strong coupling limit (the radius of approaches ), matrices commute (KK modes could be dropped), and Matrix string theory reduces to the second quantized type IIA string theory in light-cone gauge. The configurations are multistring configurations with the transition amplitude given by the integration of all intermediate string joining and splitting processes [14, 15].

Since the state, the spectrum, and the transition amplitude are all in one-to-one correspondence, the partition function of the gauge theory equals the partition function of the M/string theory. For , we have [8, 9]

where is the radius of the time direction. The zero temperature partition function is only the functional of , for which [8, 9]

is the on-shell supergravity solution with the boundary value . is the type IIB supergravity action on . If the gravity dual of with the source added is the type IIB string theory on with the background field turned on, in zero temperature limit, the partition function will only contain the contribution of the ground state geometry, so .

Except for (2), also has another expression on . Let be the free energy of the type IIB string theory on and the partition function of , it is expected that

where . When the background field varies, undergoes a change of the coupling constants. Correspondingly, there is also a change of the coupling constants for strings living in , which is in fact a modification of the background. For (3) to be valid, a one-to-one correspondence should exist:

On side, the string free energy cannot be defined without a definite background . Also, for the state (spectrum) correspondence to be valid, the definite background is necessary; otherwise, it is impossible to determine the string spectrum. If (2) and (4) both hold,

The free energy of the string theory on a given background equals the effective action of the background fields.

For Matrix theory, similarly,

is the supergravity background and is the translation invariant along the direction. is the partition function of the M theory sector with the light-cone momentum , which is supposed to be the supermembrane. Equation (6) is trivially satisfied since Matrix theory is just the regularization of the supermembrane with the definite light-cone momentum. When ,

is the -parameterized functional of with the covariance.

In string theory, one can also calculate the free energy of the strings on a given background :

In [17, 18], it was shown that, for satisfying the free field equation, could be taken as the effective action of the renormalized background field ; that is, .

It is tempting to establish a relation between and the effective action of the supergravity. However, Matrix theory, no matter if it was taken as the DLCQ formulation of the M theory or as the discrete regularization of the supermembrane theory in light-cone gauge, only describes the sub-Hilbert space of the M theory with the definite light-cone momentum without capturing all the information of the covariant theory. For different , is different. Nevertheless, let , where , , represent fields with zero, one, and two indices, respectively; one can find that . Let with being the radius of , , , and . On the other hand, for supergravity fields that are translation invariant along , suppose is the effective action of , and there is also . The dependence of is consistent with .

One may want to consider the complete partition function with all light-cone momentum taken into account, which is roughly . Each only differs by a rescaling of , so the summation does not give more information. It is enough to consider with the definite . In fact, is the translation invariant along ; as a result, a sector with the definite has the enough degrees of freedom to produce . The complete M theory degrees of freedom including sectors with all light-cone momentum is necessary only when is the field with the spacetime dependence.

For the arbitrary matrix configuration , we may define via

with solved from

It is easy to see that is independent.

To describe the supergravity interactions among theory objects with no light-cone momentum exchange, and one may define through

with being the zero mode of the supergravity along and the classical action of supergravity. The integrating out of F induces the effective action for the theory object with the supergravity interaction (without transferring the light-cone momentum) taken into account.

where is the connected Green’s function of supergravity in light-cone gauge with the zero light-cone momentum and is the current density of the configuration coupling with the supergravity field . Under a Legendre transformation, could be written as

with solved from

So, if , .

Although M theory/Matrix theory correspondence and AdS/CFT correspondence are very different, it is possible to construct the connection between the two. In [19–21], it was shown that PWMM expanded around the certain BPS states gives , , and , while the backreaction of the corresponding BPS states on pp-wave produces the gravity dual. We will investigate the correspondence in more detail.

In (7), is the generic supergravity field with the isometry. , if is indeed equal to , gives the effective action of the field . On the other hand, in (2), only the field is given, from which the field is obtained from the equation of motion or from the RG flow. is the action of . This is the holography of AdS/CFT. One may want to turn on the arbitrary on and try to find the corresponding gauge dual. The dual gauge theory may not be , since can only encode a subset of fields, which are in one-to-one correspondence with the fields. In fact, since the transverse space of is other than , the gauge theory dual may have the scalar fields other than . Suppose the coordinate of is , for a scalar with the spherical harmonic of ; the operator counterpart is

can be the arbitrary function. In , we only have to represent such fields. Nevertheless, for with the scalar field and the background , a , transformation can be made, under which the partition function remains invariant. If the with the scalar field and the background is the gauge theory description of the string theory on with the background , its partition function will then equal with being the supergravity action. So we arrive at (2). is a Weyl transformation, under which must evolve as to preserve the partition function. We will show that, for such , , so if is the action of the supergravity, will be the on-shell solution. The discussion can also be extended to and . With no source term added, under the transformation, the induced fields give the near horizon geometry of and , respectively. The holography in AdS/CFT is very similar to the holography in noncritical string coupling with being the gravity and with replaced by .

This paper is organized as follows. In Section 2, we consider the free energy of the Matrix theory on supergravity background that is translation invariant along the direction, and its relation with the effective action of . In Section 3, we consider Matrix theory on the configuration representing branes and its gravity dual. The discussion will then be specified to the PWMM, from which, , , and can be obtained [19–21]. In Section 4, we give a possible way to induce the radial dependent fields from the background fields in .

2. Free Energy and the Effective Action of Supergravity

In this section, we will consider , the free energy of the Matrix theory on a generic supergravity background that is translation invariant along . Since the supermembrane action in light-cone gauge only contains one free parameter , as the discrete regularization of the supermembrane action, Matrix theory action also has one free parameter which could be taken as , the radius of . , and is fixed. The concrete dependence of is , where , , represent fields with zero, one, and two indices. As a result, . On the other hand, due to the coordinate invariance, the dependence of the effective action of the supergravity field is also . From , we can define , with solved through . is , or equivalently, , independent. could be taken as the effective action of the matrix configuration . In fact, at the one-loop level, and the standard effective action of the Matrix theory coincide. If , we will have with being the effective action describing the supergravity interactions among the M theory objects with the zero light-cone momentum exchange.

2.1. The Action of the Matrix Theory on a Generic Background

The Matrix theory action in flat spacetime is

where , , and are hermitian matrices with , and . , ,   is the Planck length.

, , and have the dimension of length, so each commutator is multiplied by a factor to make the action dimensionless. With the replacement , , , we get the action

in which is cancelled and , , , and are all dimensionless. In the following, we will still adopt this convention, so will not appear explicitly.

The supergravity field, after the gauge fixing, has the nonzero components (, , , ), (, ), and (, ) [22]. Based on the Hamiltonian in [22], one can write down the action of the bosonic membrane on such supergravity background:

where and . Note that it is that appears. The Matrix theory version is

where and . Without the gauge fixing, , , , , , so terms involving them can also be added:

and similarly for , which is the sum of the 4th order terms , , and . The current densities for , , , , were derived in [3, 23–25] through the calculation of the one-loop effective action. Otherwise, in the light-cone quantization of the membrane, one may add and into the action, which could couple with the indexed fields.

In all these terms, it is , , , , , that are involved. (Similarly, in supermembrane action, it is , , , , , that will appear.)   is the radius of . Under the parameterization transformation , ,

the action is invariant. This is consistent with the coordinate transformation

, the supersymmetric extension of (19) and (20), is not constructed yet. With given, the current density for can be defined as

In particular, when is the flat background, for which, the only nonvanishing fields are and , reduces to the localized vertex operator of the supergravity field . In [3, 4, 23–25], the vertex operators for various supergravity fields are constructed. Although the exact is unknown, with the vertex operators at hand, one can write down the Matrix theory action on weakly curved background in linear gravity approximation [3, 4, 23, 24]:

where, for example,

are background fields with the -number coordinate replaced by the matrix coordinate . The background fields are only the functions of ; they are the zero modes of the supergravity along .

The vertex operator can take three different forms. First, it can be the operator defined in a SYM theory, just as that in AdS/CFT. Then, the background fields, like , should be expanded as the Taylor series:

with the derivative of at [3, 23–25].

Since all background fields enter into the action in the form of , the lives at . When

undergoes a translation. Nothing specifies where the should be, so one may put it at any point in . Similar to AdS/CFT, there is a one-to-one correspondence between operators and fields. However, no holography is present here. Fields living on are Taylor series coefficients, which uniquely determines the background. The background fields are arbitrary and are not necessarily on shell. On the other hand, in AdS/CFT, fields living on CFT are boundary values, from which the full background is solved through the equations of motion or the RG flow. In contrast to the chiral primary operators in AdS/CFT, the moment operators do not need to be traceless. As a result, couples with the field, while only couples with the field.

In the second form, the vertex operator is defined in spacetime:

where are matrices and for uniformity; we have set . This is a matrix generalization of the -function. In special situations, when all of the are diagonal, that is, , (29) becomes

With the generalized -function, it is straightforward to write down the current densities for various fields. For , we have

It is not convenient to deal with the -function. One may want to do a Fourier transformation, which gives the third representation of the vertex operator:

2.2. Partition Function of Matrix Theory on Curved Background

Suppose the exact form of is given, and the partition function of Matrix theory on supergravity background is

where is the background field (, , , , , ), (, , , ), and (, , ) mentioned before, and collectively represents .

In gauge/string correspondence, partition function is an important quantity, the value of which should be equal on both sides. For correspondence, it is expected that

should hold, where is the free energy of the strings on , and are the partition function of and the partition function of the second quantized type IIB string theory on , respectively. For the present situation, the comparison is relatively trivial. On one side, we have a gauge theory with the partition function given by (33); on the other side, the -theory sector with the light-cone momentum is described by the Matrix model with , for which the partition function is again (33).

Suppose is the membrane action on background . should be general covariant, so, for , , and , there is , where is the field coming from the coordinate transformation:

For the bosonic action, we can see this is indeed the case. If , that is, the path integral measure is coordinate independent, we will have

After the matrix regularization, the membrane configurations and   become the matrix configurations and , and there is

. At least restricted to (35), is the diffeomorphism invariant functional of .

Let , , represent fields with zero, one, and two indices. Since

For the supergravity fields , which are translation invariant along the direction, , the supergravity effective action is .   is invariant under the coordinate transformation:

so

The radius of is absorbed in , , as is in (39). In this respect, is consistent with .

Let , where is the flat background with , , and the rest fields being zero. :

where ( collectively represents supergravity fields. For example, .)

For the same , the supergravity effective action is

is the vertex function of the supergravity. If , there will be

In linear gravity approximation, , where is the current density coupling with as is defined in (23):

is the connected correlation function of the current density, in contrast to

which is the correlation function of the current density. In (42), is expanded around the flat background . One can of course expand on a different background, giving rise to the different .

In terms of ,

Note that , , so

The correlation function is translation invariant. if .

Let us first consider the one point function

is the vacuum expectation value of the current density, when the background field is , and is a constant in spacetime. is invariant, so should also be invariant. As a result,

if the traceless condition is imposed. due to the supersymmetry. The nonvanishing current densities are , , and . In particular, is the vacuum expectation value of the light-cone momentum density. For the generic value of ,

is the vacuum expectation value of the current density in presence of the background field. In string theory, the vanishing of the one point function, the tadpole, for vertex operators gives the equations of motion for background fields. Similarly, here, if is the effective action of the supergravity fields, on SUGRA solution background, there will be except for , whose vertex operator is the same as the tachyon in bosonic string. We will return to this problem later.

2.3. Another Effective Action of Matrix Theory

For the given , is uniquely determined. Conversely, different may result in the same . This is quite like the source-gravity coupled system. For the given gravity field, the density of the source can be obtained through . On the other hand, with the given source, the gravity solution is not unique. Nevertheless, with the proper boundary condition imposed, there is always a privileged solution. We will choose the boundary condition so that, for , . could be interpreted as the field generated by the current density . Other boundary conditions correspond to adding the external supergravity background, for example, the plane wave background, in addition to fields generated by source. We will discuss this situation later.

Then, there is a one-to-one correspondence between and , and so a Legendre transformation is possible. Before that, we will first define :

is solved from the equation

or equivalently,

In some sense, is the field generated by . Take a derivative of (54) with respect to ; using (55), we get

where is solved from (55). The variation on the right-hand side of (57) only acts on with being fixed.

Recall that the dependence of only comes from and , so if the solution of (56) is , with replaced by , the solution will become . , , and remain invariant. is -independent. In supermembrane picture, suppose there are two supermembrane theories with the light-cone momentum and , and consider the fields generated by the same configuration . Let , , in this frame, the generated fields are the same, changing back, we get the relation , , . When plugged into , the dependence is canceled, so the obtained is the same.

With these properties collected, we may consider the possible interpretation of and . If is the effective action of supergravity, since is the matrix theory action on background , will be the action of the source-gravity coupled system and is on shell with respect to supergravity and thus could be taken as the effective action of the configuration . is the quantum corrected equation of motion, which differs from the classical equation of motion in that the background fields in former are generated by the configuration itself, while the background fields in latter are given. For time-independent , the stationary point of the effective action gives the vacuum configuration. In this case, is equivalent to the requirement that the branes should not exert force to each other, which is the no force condition for BPS configurations [26]. All of the above statements are based on the assumption that is the effective action of supergravity, which, however, is unproved.

Let us continue to explore the properties of and . For simplicity, let , or, in other words, let the indexed fields absorb . In a weakly curved background, , , where

is the Legendre transformation of :

Let

then

For the given ,

Equation (52) gives the current density generated by the field ; (62) gives the field generated by the current density . If is the connected Green’s function of supergravity, will be the vacuum expectation value of the supergravity field in presence of the source . In classical level, could be calculated by with the classical action of supergravity:

so

is the inverse of . Subsequently,

The relation between and shows that and are the connected Green’s function and the vertex function of a particular quantum field theory. gives the change of the supergravity field with respect to the current density, so it is natural to take it as the propagator of the supergraviton. We will see some evidence for it.

In (54), let ,

could be written as (We only write , but one should be aware that the fermionic kinetic term also exists.)

with taking the form as that in (27). For the given , solve from , and expand around . . Consider

is solved as the functional of . Let since

with

In particular, when , , reduces to

The corresponding is the same as the one-loop contribution of the standard effective action in background gauge:

subject to the condition

has been calculated for the arbitrary satisfying [3, 25]:

with being the free supergraviton propagator. Compared with (61), . If the result can be extended to the -point Green’s functions and to the full-loop calculation, there will be

where is the connected full Green's function of supergravity. and will then become the -point vertex function and the effective action of supergravity, respectively.

In quantum field theory, unlike the -matrix, the effective action is not the observable and thus is not uniquely defined. Different gauges and the parameterization give the different effective actions. and differ from a parameterization transformation. . However, to compare the Matrix theory with supergravity, we do need a privileged effective action. On supergravity side, in light-cone gauge, the expected effective action is

The effective action defined in (61) is also the Taylor series of and thus could be compared with (76) directly. The standard effective action is expanded as the Taylor series of . A careful reorganization is needed to get , but it is unclear whether it is always possible to do so. Moreover, under a Legendre transformation, (76) becomes

with

is the effective action of supergravity. Equation (77) is almost the same as (53). The only problem is whether or not.

2.4. Free Energy of String and the Effective Action of the Background Fields

In string theory, there is a similar story. In [17, 18], it was shown that the effective action of the supergravity could be taken as the renormalized free energy of the strings on background :

where   satisfies the free field equation.

Equation (79) looks consistent with our philosophy: the free energy of the strings/membranes on a given background gives the effective action of the background fields. However, there is a difference: is the renormalized field other than the bare field. In fact, in (79), a particular Weyl gauge is always imposed, and so, also has the dependence on :

When , , although is not the necessary condition.

Suppose is the effective action of the string modes:

Consider with . Since , will evolve along the RG flow; that is, . A special property of is that it will finally reach an IR fixed point , . In fact, since satisfies the first-order equation, the RG flow may bring it to an IR fixed point. Since , .

is calculated from via

. Since , is equivalent to [18]. In [18], it was shown that , which means . There is a one-to-one correspondence between the solution space of and the solution space of . So, for all with , . In -model approach, only for such , the conformal factor decouples; thus, the calculated string free energy is physical. In this subspace, .

with usually depends on . However, the -dependence drops out for . In fact,

is the -matrix functional [27]. For the on-shell ,

is the connected scattering amplitude. It is only in Polyakov approach can we construct the -matrix functional in this way since it intrinsically involves some kind of renormalization, which is equivalent to the subtraction of the massless pole exchange contribution [18, 27].

To conclude, for satisfying , . However, the off-shell extension of the effective action and the free energy are both ambiguous in Sigma-model approach.

Another support for the identification of the string theory free energy on a particular background and the effective action of the background fields comes from AdS/CFT. is a nonperturbative description of the type IIB string theory on . It is expected that the Hilbert spaces of both sides are isomorphic to each other. Similar with the Chern-Simons/topological string correspondence, , where is the free energy of the string theory on . The question is what will be the string dual of if the background metric of is other than for very small . A natural expectation is that such is dual to the type IIB string theory on with a little modification of the background metric that is entirely determined by . Correspondingly, , where is the free energy of the string theory on with the background perturbation being turned on. Note that, for the stringy explanation of the partition function to be possible, the type IIB dual must have the definite background since the string partition function is always defined on a given background. Also, for the state correspondence to be valid, the dual type IIB string theory should have the definite background; otherwise, it is impossible to determine the string spectrum. Now, return to the original topic. Gauge theory calculation gives , where is the supergravity effective action for the modified background fields, so if holds.

There is a naive way to interpret . Suppose the classical supergravity action is , from the field theory's point of view:

where is the fluctuation on or, in other words, supergravitons living on background . The effective action is the sum of the connected 1PI vacuum-vacuum diagrams of the supergravitons on background . The elementary propagator and the vertices can be read from . Now, consider the string theory on background , we may have

where is the sum of the irreducible vacuum-vacuum string diagrams on background . Note that, in string diagram, there is nither a concept of 1PI nor 1PR. Also, there is no classical action like to determine the basic constitution of the diagram. The integration simply covers all possible string configurations. Equation (86) can be taken as the stringy refined version of (85). In (86), we secretly assumed that the unphysical worldsheet conformal factor is decoupled. In Polyakov approach, this is possible only when is the solution of . The cancelation of the Weyl anomaly gives the e.o.m for the effective action of the supergravity, including the corrections, so should be the effective action with the corrections taken into account.

A natural -theory extension is

where is the supermembrane action on supergravity background . The membrane is already the second quantized object, so the left-hand side of (87) is other than . For the generic background , should be covariantm, and, so, the worldvolume metric must be introduced and integrated, making the supermembrane theory nonrenormalizable. For which is translation invariant along , the light-cone gauge can be imposed. The configurations are then truncated to those with the light-cone momentum . The integration out of such configurations on background gives the effective action of with the radius of .

3. Matrix Theory on a Particular Vacuum

If in (42) is the effective action of the supergravity field that is translation invariant along , for the on-shell , there will be . However, for , which is obviously on shell, the current densities , , and are nonvanishing. (Note that the vertex operators for and are quite like the vertex operators for tachyon and dilaton in bosonic string theory, in which, there is also a tadpole in flat spacetime [28].)

With , one may try to solve from , and then take this value other than , as the background to do the expansion. are constants, so the generated are also constants (although the constant does not really solve the equation) and do not represent the substantial change of the background. We will simply neglect these tadpoles. should be distinguished from , which is the light-cone momentum density of a supergraviton localized at and, of course, will produce the nontrivial .

In (54), is the functional of solved through (55). For the arbitrary , we may define

is the action of the source-gravity coupled system and is not necessarily on shell with respect to gravity. If gives the Matrix theory description of theory on background , will describe theory on background , in presence of the brane . If satisfies the equations of motion, that is, ,

starts from the quadratic term. For satisfying , then gives a description of theory on a background generated by the brane . In this case, :

The expectation values of all current densities vanish. There is no tadpole.

The symmetry of Matrix theory is destroyed by . In particular, if

the preserved gauge symmetry is . Nevertheless, the original symmetry still has its manifestation. For all ,

so for , transforms like a gauge field.

3.1. Applied to PWMM

In the following, we will focus on a special example: the plane wave matrix model [2]. PWMM is a Matrix theory description of -theory on pp-wave background:

The background preserves supersymmetries, while the rest supergravity solutions with supersymmetries are flat spacetime, , and [29]. In pp-wave, the dynamics of the -theory sector with the light-cone momentum is described by the matrix model:

, . PWMM also preserves supersymmetries and has the same symmetry group as that of the pp-wave background.

The classical supersymmetric solutions of the action are [2]

form the dimensional representation of . Suppose , , can be decomposed as

where stands for the spin irreducible representation of :

The solution can be taken as a set of coincident fuzzy spheres with the radius [2]. The gauge symmetry is broken into . All fuzzy spheres are concentric. When , the fuzzy spheres become a set of membrane spheres given by with . Each spherical membrane has , , so they are also called the giant gravitons.

The solution can also be interpreted as the spherical branes with a dual assignment of the light-cone momentum [30]. Any partition of may be represented by a Young diagram whose column lengths are the elements in the partition. In the interpretation, such a diagram corresponds to a state with one membrane for each column with the number of boxes in the column being the number of units of momentum. In the dual interpretation, it is the rows of the Young diagram that correspond to the individual , with the row lengths giving the number of units of momentum carried by each . In both cases, the total light-cone momentum is always . If are finite, they are fuzzy branes. When , the fuzzy become the spherical wrapping given by . In patricular, the trivial vacuum represents a single brane.

All of the supersymmetric solutions preserve nonlinearly realized supersymmetries. They are the BPS states on pp-wave background. Although the background and the Lagrangian are both maximally supersymmetric, there is no state in matrix model preserving all of the supersymmetries. The reason is that all states have the same nonzero light-cone momentum , which itself would destroy the linearly realized supersymmetries. The situation is different in in which, we do have a vacuum preserving supersymmetries, representing the ground state of the string theory on . (Brane like) BPS states are giant gravitons on [31–33].

Similar to the PWMM, “tiny graviton matrix model” (TGMT) is proposed as the nonperturbative description of the type IIB string theory on pp-wave background [34–36]. TGMT can also be taken as the DLCQ of the type IIB string theory on , capturing the physics seen from the infinite momentum frame (IMF) since, in IMF, is viewed as the pp-wave [34–36]. Although TGMT preserves 32 supersymmetries, the vacuum configurations, carrying the definite light-cone momentum, are all BPS, matching exactly with the BPS states on type IIB pp-wave background, which are giant gravitons (spherical branes) and type IIB strings. Lifted to , the light-cone dimension is replaced by , while the TGMT, which is a discrete regularization of the branes in type IIB picture, becomes the regularization of the branes [35]. Note that the 4-form field in type IIB pp-wave, when lifted to , becomes the 6-form field coupling electrically with the branes, so it is natural to construct the matrix model via the discrete regularization of branes other than the usual branes. On type IIB pp-wave, branes are branes wrapping , while the type IIB strings are membranes wrapping one of . In contrast to the PWMM, in TGMT, the trivial vacuum represents type IIB string (with origin), while the nontrivial vacuum represents branes (with origin), which is probably because the PWMM and the TGMT are constructed from and , respectively. Recall that, in PWMM, the fuzzy configuration may have the and dual interpretations [30], it is interesting to figure out whether a similar dual interpretation also exists for configurations in TGMT.

With denoting the supersymmetric vacuum in (96), theory on pp-wave background in presence of the brane is described by the Matrix model with the action

Since , starts from the quadratic term.

For simplicity, in the following, we will assume is absorbed into the fields. If the backreaction of the brane on pp-wave background is turned on, the field will be generated:

where

pp-wave with added respects the symmetry, so the generated as well as the corresponding will have the same symmetry.

With the fields given, in principle, one can write down explicitly. The classical supersymmetric solutions of are still (96). To see this, note that the spherical giant gravitons are BPS states on pp-wave, so they should not exert force to each other [26], or, in other words, giant gravitons are still stable even if the backreaction of the other giant gravitons on pp-wave is turned on. Indeed, in [37], the giant graviton on the backreacted geometry is analyzed. The stable configuration is still the same as that in the pp-wave case. In [38], the quantum effective action of PWMM around was calculated at the one-loop level. is also the stationary point of the effective action. We may have . With the pp-wave background replaced by the backreacted geometry, is modified to

still starts from the quadratic term since .

Notice that, although any configuration in (96) may be the classical vacuum of , is special because the vacuum expectation values of current densities vanish only for but not for the generic .

The geometry produced by BPS states of PWMM was constructed in [19, 39]. The geometry has a bubble structure containing noncontractible 7 cycles and 4 cycles supporting and fluxes. The geometry is smooth without singularity and, then, sourceless. This is an explicit realization of the geometric transition [40]. The backreaction makes the the worldvolume of the branes shrink and the transverse sphere blow up. As a result, although we start from the source-gravity coupled action, the obtained supergravity solution is smooth and satisfies the sourceless equations too. The brane action as well as the current density is zero on the generated supergravity background.

Return to (100),

is the momentum eigenstate in direction, so the generated background is translation invariant along . In large limit, the local structure of the giant gravitons is not important while the asymptotic geometry is just the pp-wave with the perturbation roughly given by [39]

which is the field produced by the supergraviton which is static in space, carrying the definite light-cone momentum [41]. . (It is not because is absorbed into and .)  With the coordinate transformation , , (104) becomes

while the radius of is now .

Under the reduction, in string frame,

In particular, for (105),

is the near-horizon geometry of the branes [41].

The type IIA solution coming from the reduction of the field was constructed in [19, 37]. When , the perturbation, which is the reduction of (104) along , is the near-horizon geometry of coincident -branes. The appearance of the near-horizon geometry is because of the null reduction. The reduction of (104) along gives the brane solution [41]. Different from AdS/CFT, here, no near-horizon limit is taken, and the brane solution itself becomes the near-horizon geometry when reduced along .

3.2. The Gauge Theory Dual from the Matrix Model

According to the previous proposal, the Matrix theory description of the theory on background (104) in presence of the supergraviton with is , where is the field in (104),   (if is the field in (105), .) Cosider

is the BFSS action in (17). On the other hand, according to AdS/CFT, the gauge theory description of the type IIA string theory on background (104) is with the action . Since (104) becomes (107) under the reduction, it is desirable to find a limit to make (108) reduce to .

Consider the coordinate transformation , , under which

Correspondingly, in Matrix theory action , with a field redefinition and also a rescaling to make the radius of become , the background fields appearing in action are just . . With a further rescaling , . In , it is

that will appear.

Return to (108). For any , is equivalent to , in which . In limit, ,

arrives. Note that, for finite , , so by taking the limit, the background field that matters lives at the region, far away from the source.

For PWMM, , and , all fields are marginal, so, for the arbitrary [5],

Now consider the background coming from the backreaction of the giant graviton on pp-wave. Suppose is a vacuum in (96) with

The radii of the spherical membranes are , . reduced along gives the smooth type IIA solution that is constructed in [19]. The generic solution of type IIA supergravity with symmetry is characterized by a function and is given as [19]

where the dot and the prime represent the derivatives with respect to and , respectively. can be taken as an electrostatic potential for an axially symmetric system with conducting disks and a background potential. is the distance from the center axis, and is the coordinate in the direction along the center axis. , where is the background potential and is determined by a configuration of conducting disks. Each symmetric theory is specified by ; each vacuum of the theory is specified by a configuration of conducting disks.

From (115), one can also get the uplifted solution. For example,

For PWMM,

Only the region is meaningful. There is an infinitely large conducting disk sitting at . Vacuum (97) corresponds to disks located at . The electric charges on these disks are equal to , respectively. is the radius of the spherical membranes. The correspondence between the spacetime coordinates and the PWMM fields is , , .

With the given disk configuration, the corresponding potential is

which, when plugged into (115) and (116), gives the solution as well as the solution . stands for the space coordinates since the solution is translation invariant along and .

Under the coordinate transformation , , the disks are now located at , while the fields will transform as , . As a result, . and can be obtained by plugging

into (115) and (116). Except for , we also have with given by (110). Plug

into (115) and (116); one can get and the corresponding . In particular, for PWMM, , , so as we have seen before. Both and satisfy cylindrically symmetric Laplace equation, so , and are also the type IIA supergravity solutions. In fact, , and are related to each other by the coordinate transformation. .

The limit of interest is , , and fixed. In this limit, the concentric spherical membranes all have the infinite radii with the difference . For the finite , , , , we need to consider the behavior of around . Since , near , can be approximated as the potential generated by dipoles located at :

. Indeed, with plugged into (116), , ,   being fixed, the solution finally approaches the pp-wave. As a result, . Even if the background field is , in the limit taken, we still get PWMM.

On the other hand, plug into the reduced solution (115); ; the part now matters. Let :

The solution has the dependence on the disk configuration. Equation (122) is . With , ,  , we will get . By taking the limit, the relevant region is , , which is again far from the source. The field approaches the pp-wave, but the reduced field still depends on the disk configuration. Similarly, for (104), suppose ; then, when , the background is fat, while the associated background (107) is still the near-horizon geometry but becomes singular now. Nevertheless, let , , fixed; (107) can be written as

This is similar to the limit taken in AdS/CFT [7].

Now, consider the exact form of . For the generic background , the bosonic part of is

The relation between and the reduced background is

In the limit with , , fixed, if , , the rest is finite, there will be

With these relations,

For , ,

So, in the limit taken, could be identified with , the action of on the background . The discussion can also be extended to the full with the fermionic part included.

Specified to the background generated by on pp-wave, (126) is satisfied for the solution in (122), so . Indeed, one can verify the validity of (127) by directly plugging (122) in it. In the limit taken, PWMM can be taken as the living on background (122). Equation (104) also satisfies the above criteria. Actually,

where , , and so forth are the near-horizon geometry of the branes in (107) with .

In the limit with , , fixed, (122) gives the solution in the region of finite and . To study the region near the spherical membrane shells with , finite, a change of variables can be made. As is shown in [42], in this limit,

where is the background potential for . Alternatively, we may directly plug

into (116) and (115), taking the limit at the end. Let ; the background is

which has the topology of . The background is

, and now have the same prefactor. Equations (122) and (133) can be taken as the background for and gauge theories, respectively.

On gauge theory side, to study the fluctuation around the spherical membranes, we should expand around . In [20], it was shown that in the limit of , fixed, (i) PWMM around a certain vacuum is equivalent to around each vacuum and (ii) around a certain vacuum with a periodicity imposed is equivalent to around each vacuum. In particular, can be realized as the PWMM around a certain vacuum with a periodicity condition imposed.

Concretely, , where is the action of the    with :

is the radius of which is parameterized by . .   represents the vacuum:

Region I and II correspond to and , respectively. is identified with the in (113). The gauge group is spontaneously broken to . Equation (135) corresponds to disks located at with the electric charges on each equal to . The previous in PWMM now becomes due to the redefinition.

The correspondence between the spacetime coordinates and the fields is , , . Equation (134) is the on flat background. Plug (133) into (134), similar to (127), for bosonic part, we have

where , , . The action remains the same, so (134) is also the action of on background (133). The conclusion holds for fermionic part except for a rescaling of .

Indeed, as we will show in Appendix A, such phenomenon is very common. Generically, the action of on flat background is equal to the action of the on the near-horizon geometry of the branes. (For , the worldvolume theories of branes do not decouple from the bulk, as is discussed in [7].) For and on and , such requirement can even offer some clues for the structure of the field theory.

is the function of the radial directions. For the given field configuration , let

then

Compared with (134), with , , the background fields on will get the radial dependence:

This is some kind of realization of the holography, on which, we will discuss more in the next section. One special feature here is that the background fields depend on two radial directions and . Let ; in (133) can be written as

with living at . However, with the energy indentified with , the RG flow cannot give , which is not just the function of . Instead, we will make the , transformation to recover dependence of the background fields in .

For PWMM, we have

is the reduction of the supergravity effective action of the field generated by the brane on pp-wave. In the limit with , , fixed, under the change of variables ,

where is the type IIA action for the above-mentioned supergravity solution dual to a vacuum of . Then,

As is demonstrated in [19, 20], from and the corresponding type IIA solution, it is also possible to get the and the associated type IIB solution. In (135), suppose

where runs from to . Expanding the action (134) around this vacuum and imposing the condition on all of the field fluctuations, one will get [20]

where

is the action of the SYM on .

This is a special example of Taylor’s prescription for the compactification (the -duality) in matrix models [43]. The new ingredient is the nontrivial gauge field, which makes a nontrivial fibration of over rather than a direct product; as a result, it is other than that is obtained [20].

On gravity side, start from the trivial vacuum of , for which there is only a single disk with the electric charge ; compactify the direction; the disk configuration in covering space will contain the infinite copies of disks with the period , corresponding to the vacuum (144). The type IIA geometry generated by (144), after a -duality transformation, becomes the type IIB geometry [19]. On field theory side, since is compactified, the field fluctuations should respect the periodicity condition . Taylor’s prescription for the compactification also involves the -duality transformation, making in type IIA become in type IIB. Equation (143) turns into

For the trivial vacuum of , represents background.

Finally, the geometry arising from the backreaction of the giant gravitons with the definite light-cone momentum on type IIB pp-wave background was also constructed in [39]. It is tempting to find the corresponding gauge theory dual. one attempt is to expand the TGMT [34–36] around the corresponding BPS configuration, and then take the certain limit. TGMT is the discrete regularization of the branes, so the resulted gauge theory should be a gauge theory as is required since the geometry is generated by spherical branes on pp-wave.

4. Holography in AdS/CFT

is the gauge dual of the string theory on . A natural question is what will be the gauge dual of the string theory if the field perturbation is added to . For BFSS matrix model, we have matrices , representing nine transverse coordinates, so for a scalar field

we may get the operator realization

with replaced by matrix . Adding to the Lagrangian gives a matrix model on a background with the scalar field turned on. The situation for is a little different. In BFSS model, the original background is flat; all of the fields could be expanded as the Taylor series in Cartesian coordinates. For , the original background is ; fields are expanded in terms of the spherical harmonics on . For scalar ,

where is a spherical harmonic of rank . is a scalar field on transforming in the irrep of . It is necessary to find the operator correspondence of . In , we have , which, however, is the operator realization of .

Naively, let , ; and may be the operators corresponding to and , respectively. For the field , for all , formally,

Functions are coupling constants of the gauge theory living in . Similar to (28), when the gauge theory moves along , , , could be decomposed into the part and the traceless part :

. The configuration is equivalent to . , .

One may take as the operator corresponding to and consider the gauge theory with the vertex operator perturbation realized as (in fact, we will choose instead of with . , where )

can be the arbitrary function. Gauge theory like this is difficult to approach directly. We still prefer with added, which, after a suitable transition, will become a gauge theory with the operator . The field is not arbitrary anymore but is determined by .

This is quite similar to the noncritical string coupling with gravity. For critical string with coordinates, any background fields can be represented by the vertex operator perturbations on the string worldsheet action. For noncritical string with, for example, coordinates, only the vertex operator perturbations corresponding to fields can be constructed. However, if the noncritical string is coupled to the conformal mode of the gravity so that the total number of degrees of freedom is , after a property transformation with the partition function kept invariant, the theory will become the critical string coupling with the background fields. The fields are induced from the fields with the conformal mode acting as the dimension. For , the role of the conformal mode is played by .

4.1. Noncritical String Coupling with Gravity

Let us have a simple review of the noncritical string [44–49]. Consider the bosonic string living in dimensional spacetime. The corresponding nonlinear sigma model action is

. The partition function is defined as

The theory is conformal invariant since all are integrated. The integration over the small gives the divergence, to cure which a cutoff should be introduced, destroying the conformal invariance. Under the conformal gauge fixing , , is a small metric giving the cut-off scale

where is the Liouville action. With , , the partition function becomes

where

The original dimensional background field , after the gravitational dressing, becomes the dimensional field . . One can also make a change of the variables to move the boundary from to :

In , .

Since , for ,

With the cutoff being introduced, the conformal transformation should be accompanied by a change of the boundary:

With the cutoff being removed, .

is the action of the bosonic string coupling with the background fields . The conformal invariance indicates that should be the on-shell solution of gravity. To arrive at this result, it is important that, in (157), the change of the metric is always compensated by the adjustment of the background fields, keeping the form of the sigma model invariant.

4.2. Inducing the Radial Dependent Fields

Now, consider with the action

Select so that can be expressed as with . is the infinite matrix representing the maximum . The partition function is

Due to the Weyl invariance,

Suppose

with no counterterm added,   is conformal invariant, so at least for finite and , . . However, this may not be the case when or when . will still be kept for the infinite conformal transformation.

Nevertheless, with , we always have . With parameterized as ,

From (166) to (167), the integration over in is converted to the integration over in . The partition function of on flat background is equal to the partition function of a gauge theory on a curved background. From (167), one may read the background metric:

. , , and . because we take as the standard matrix. For finite , . With being the radius of , (167) could be taken as the partition function of a gauge theory on .

Equations (166) and (167) have the direct extension to and for and . For , has the weight , and the Weyl transformation is . The induced metric is

with . For , has the weight , and the Weyl transformation is . The induced metric is

with . Equations (169) and (170) are the near-horizon geometries of and , respectively.

More generically, as we will show in Appendix A, for , , and on , , and , the action on flat background and the action on the near-horizon geometry of , , and are the same. As a result, for and with the near-horizon geometry given by (169) and (170), and are always accompanied by and . For with the near-horizon geometry

the transformation will then make , . The coupling constant as well as the metric now gets the radial dependence. , and the weight of is .

In the above situations, the transformations are all made for . This is not always the case. Consider the SCFT and the geometry [50, 51]. The scalars , are related to the transverse , for which the metric is . There is no radial dependent prefactor for to absorb, so has the weight . In fact, the chiral primaries are constructed from the fermions that have the weight and are representations of the -symmetry group associated with .

Return to , with the operator perturbation added:

where . is the arbitrary metric. The partition function is (here, the path integral measure depends on both and . However, except for , no other field will enter into the path integral measure directly. Even though, the transformation of the path integral measure still has the dependence on other external fields. An explicit example is the chiral anomaly in gauge theory. We will discuss the path integral measure in more detail in Appendix B)

For finite , , ,

We still want to do a transformation:

When , may not be zero. Instead of and , we can find the suitably adjusted and so that

In (176), the integration over , , and has already been carried out; both sides only depend on the function . Obviously, , . We also require

The functions and are induced from the fields and . For finite , , .

More generically, with the background field turned on, for the given function , since , is a UV to IR transformation, the effect of which can be cancelled by the adjustment of the coupling constants, keeping the partition function invariant. , . With all possible operators included, the change of the scale can always be compensated by the change of the background fields, leaving the form of the Lagrangian invariant:

In (178), with specified to , the induced action contains the term with given by . . since is the infinite matrix. The corresponding field is and , where is the spherical harmonic on . Equation (178) is the partition function of the gauge theory on with the background fields and turned on.

On gravity side, the asymptotic expansion of the gravity solution is [52]

where . Let , with , , and .

For finite , . Similarly, for the solution of the scalar field [53],

with . .

with . When , for finite . For - coupled system, the subleading terms are determined by both and . and are fields on with finite. It is expected that the transformation will give and in (180) and (182).

From (177),

As a result,

With , the induced fields become and . Also, since

we have

where and are fields induced from and , respectively. There will be , . Indeed, in (182) and (180), with and replaced by and , one may get and . , , is a diffeomorphism transformation of the gravity solution and [54].

Equation (184) is valid for induced from . For the generic functions , we only have

for some functions . For constant , under the scale transformation ,

In special cases, if , , the background is “conformal”. For , this requires , and thus , . is the marginal field.

For fields which are “conformal”,

and so

As a result,

and then

is on shell.

Restricted to , usually, , is not “conformal”. Instead of (191), if we require

still,

is also on shell.

is the functional of . In general,

since nothing could guarantee that the one-point function of the would vanish in presence of the arbitrary source . The physical field is , so there is also

The difference between and is that is replaced by , while the field becomes . has the natural interpretation as a gauge theory on with the external field

turned on, where and can be the arbitrary functions. and are physical fields on . According to the previous philosophy,

is the effective action of the type IIB supergravity on .

For a special subset of fields that can be derived from the fields ,

and then

that is, is the on-shell solution of the supergravity. For the given , there are two ways to get the induced : one is through (176) and the other is to find the solution of , imposing as the boundary condition.

Finally, we arrive at

Starting from , with the source term added, after a transformation, we get , the action of the gauge theory on , whose free energy may be equal to . During the transition, are carefully adjusted to make the partition function of and remain the same, then we arrive at (201), where is induced from and is on shell with respect to .

4.3. Imposing the Cutoff

where   is the Lagrangian of the supergravity. is divergent. One may impose a cutoff , ,

On gauge theory side, the one-loop effective action is also divergent. With a cutoff being introduced [55],

where , , is the Laplace operator with the metric . Obviously, . For small but finite [56],

so

The renormalized is finite. This is the realization of the holographic renormalization [52] on gauge theory side. With , , [52]

is the conformal anomaly [57]. On the other hand,

so

and could be compared with the local part and the nonlocal part of the gravity action in [58]. The nonlocal part contains the logarithmical divergence [59], which is just .

On gravity side, the corresponding gravity solution has the near boundary expansion

At , the metric is . The on-shell gravity action with the cutoff is

which is entirely determined by the boundary value .

The renormalized is finite [52]. Since , , [52]

In both gauge theory and gravity, subtracting of the infinity introduces the conformal anomaly.

In (204), is the UV cutoff in gauge theory. It is desirable to find a direct and exact way to impose it. In the following, we will consider a cutoff imposed on , which, although has some relevance with , is still not it.

Take other than as the upper bound of ; the partition function is

For finite and , . There is also

since the field induced from is . From and , the induced fields have the same boundary value .

For the infinitesimal function ,

where

With , we may have

where

Also, we may define , with

if

For finite and , . For infinite , for example, and with ,

, so .

Although (219) and (204) look similar, the two kinds of cutoffs are different. In particular,

because the part also has the contribution to .

Let

then, . From (206) or (212), one can see

For finite and , (226) cannot be directly identified with and . However, when , ,

is the conformal anomaly, which, with the infinite part subtracted, becomes .

In the previous subsection, the radial dependent function is induced via (176). Nevertheless, it can also be induced from , with being the upper bound of the integration. Take a particular finite matrix as the standard so that the arbitrary configuration of can be represented by with :

Starting from , with increasing, will also increase, so should change accordingly to make the partition function invariant. Namely, we have

with

For finite , . When , , the simple scaling relation may not be valid. For finite and , . If

there will be

So, in (229), with fixed and replaced by , the induced remains the same. Also, with fixed and replaced by , the induced field will be . is the unique function trajectory that does not depend on the chosen.

We can compare it with the previously mentioned noncritical string coupling with gravity:

where is the dimensional background field. is the cutoff metric. With replaced by ,

.   is a conformal transformation, which should be accompanied by to make the partition function invariant. . is a function trajectory that does not depend on the chosen. A further transformation gives

in which is induced.

Just as (224), cannot be identified with the dimensional gravity action with a cutoff. The gravity action interpretation is possible only when ; that is, the cutoff is removed.

Here, is directly related to the worldsheet metric, . gives the RG flow of the dimensional fields. In gauge theory case, represents the radial direction, along which all fields, including the metric , will evolve. gives the radial evolution of the fields, which cannot be directly identified with the RG flow. To discuss the RG flow, we should consider the renormalized , for which (see, e.g., [60])

where represents the local anomalies. For the renormalization scale ,

is the anomalous dimension. is the function.

5. Conclusion

With being fixed and varying, the free energy of the Matrix theory on a -translation invariant supergravity background is the functional of and ; that is, . Under the coordinate transformation

with , if , , . preserves part of the diffeomorphism invariance. can be compared with , the effective action of the supergravity for the same field . On field theory side, naively, is calculated from

with being the classical action of the supergravity. can be taken as the sum of the 1PI graphs of the vacuum-vacuum amplitude for supergravitons living on background .(Of course, (239) is not well-defined.) In theory, the basic objects are membranes other than particles. is the sum of the membrane configurations on background . In some sense, gives an theory refined version of the in (239).

In Matrix theory, the problem is that we only consider the theory sector with the definite light-cone momentum and the background that is translation invariant along . For the same but different , also has the or, equivalently, dependence. Nevertheless, , with being the radius of the . In fact, for on the same background, there is also . In , is only a parameter encoding the scale of , so it is enough to consider the sector with the definite . in Matrix theory is kept fixed in order to maintain the consistency with membrane theory, which is only characterized by one parameter .

On field theory side, to describe the supergravity interactions among theory objects with no light-cone momentum exchange, one may consider given by

where is the zero mode of the supergravity along . The integration out of gives the effective action for the theory object with the supergravity interactions (without transferring the light-cone momentum) all taken into account. Under the Legendre transformation, we may get , where is given by (239). . With being replaced by , one can define another effective action , which is totally constructed from the Matrix theory with no input like added. is independent, which is expected, since the scattering amplitudes with the zero light-cone momentum transfer should not depend on due to the Lorentz invariance [61]. When , at the one-loop order, and the standard Matrix theory effective action are the same.

A special property of Matrix theory is that various brane configurations have the natural matrix realization, so it is possible to construct some AdS/CFT type gauge/gravity correspondences from it. In [20], , , and are all obtained by expanding the PWMM around the particular BPS states, while the backreaction of the BPS states on pp-wave after the reduction produces the dual geometry [19]. A special type IIA limit is taken in both gauge theory and gravity side. In this limit, the backreacted geometry approaches the pp-wave, so the Matrix theory dual is still PWMM. On the other hand, the reduced geometry (122) is a -type solution. The action of the PWMM equals the action of the on background (122). To study the fluctuations around the BPS states, the and the redefinition should be made on gravity and the Matrix theory side. The geometry then becomes (133), a -type solution. Correspondingly, becomes the action of , which, with the background (133) plugged in, remains invariant. With the and the gravity dual at hand, a further -duality-like transformation gives and [19, 21].

is the nonpertubative definition of the type IIB string theory on . It is unlikely such gauge/string correspondence will suddenly vanish just because the metric in deviates a little. If the correspondence still exists, the gravity dual will be the type IIB string theory on with a little background perturbation turned on. Then a one-to-one correspondence should exist between the field in and the field on . The natural candidate of is the gravity solution on with the boundary condition. For the correspondence to be valid, it is necessary to derive merely from . In the simplest situation, when the background metric in gauge theory is the standard , the near-horizon geometry of , , and could be induced from , , and SCFT₆ by a transformation. It is expected that the same method, when applied to with the arbitrary turned on, will give the corresponding . With , , becomes a gauge theory living in background times the transverse . The free energy can be expressed as the functional of ; that is, . , so if is the gravity action as is in Matrix theory case, will be the on-shell solution.

Appendices

A. The Action on the Flat Background and the Near-Horizon Geometry

The action of the -dimensional SYM theory on background could be written as

where , , . The near-horizon geometry of the branes is

With replaced by the matrix , the gauge theory on the near-horizon geometry background has the action

where . With , except for a rescaling of , . The SYM action in and the SYM action in the near-horizon geometry are the same. Before and after the backreaction, the action remains invariant.

Similarly, the near-horizon geometries of and branes are

respectively. For both and ,

But

Instead, for , (the 3-algebra was proposed in [62–64] to construct the model for two coincident branes)

We see another necessity of the sextic potential. For SCFT, the dimension of the scalar field is , so the potential term should contain six scalars. For , (3-algebra like this appeared in [65])

The scalar dimension is , so, effectively, the potential should contain two-dimension scalars and two-dimension scalars.

B. Path Integral Measure and the External Fields

For simplicity, consider the scalar field with the action

where . . Following [66], we will define the scalar density field to eliminate the dependence of the path integral measure on :

In terms of ,

The partition function is then

with no entering into the path integral measure.

where . If ; then,

with being the conformal anomaly. Note that to arrive at (B.6), we potentially assumed

On gravity side, this is equivalent to assume that , so

in (B.8) could be compared with (227). From (B.5),

indicating that will depend on both and . Indeed, the direct calculation of the conformal anomaly gives composed by the gravity part () and the matter part (). Correspondingly, the transformation of the path integral measure will also depend on and , although neither of them enters into explicitly.

Similarly, when the theory is coupled to the external gauge field , although does not enter into the path integral measure, the Jacobian of the path integral measure under the -symmetry transformation gives the -symmetry anomaly which is the function of .

Finally, on gravity side, one can read the -symmetry anomaly from the type IIB supergravity action directly but should make the regularization of the action first to get the conformal anomaly. Correspondingly, on gauge theory side, the -symmetry anomaly exists originally, while the conformal anomaly is introduced by regularization.

Acknowledgments

The work was supported in part by the Mitchell-Heep chair in High Energy Physics (CMC) and by the DOE Grant DEFG03-95-Er-40917.

References

T. Banks, W. Fischler, S. H. Shenker, and L. Susskind, “M theory as a matrix modela conjecture,” Physical Review D, vol. 55, no. 8, pp. 5112–5128, 1997.

D. Berenstein, J. Maldacena, and H. Nastase, “Strings in flat space and pp waves from super Yang Mills,” Journal of High Energy Physics, vol. 2002, no. 4, article 13, 2002.

W. Taylor and M. Van Raamsdonk, “Supergravity currents and linearized interactions for matrix theory configurations with fermionic backgrounds,” Journal of High Energy Physics, vol. 1999, no. 4, article 013, 1999.

A. Dasgupta, H. Nicolai, and J. Plefka, “Vertex operators for the supermembrane,” The Journal of High Energy Physics, no. 5, article 7, 28, 2000.

K. Dasgupta, M. M. Sheikh-Jabbari, and M. Van Raamsdonk, “Matrix perturbation theory for M-theory on a pp-wave,” Journal of High Energy Physics, vol. 2002, no. 5, article 56, 2002.

J. Maldacena, “The large limit of superconformal field theories and supergravity,” Advances in Theoretical and Mathematical Physics, vol. 2, no. 2, pp. 231–252, 1998.

N. Itzhaki, J. M. Maldacena, J. Sonnenschein, and S. Yankielowicz, “Supergravity and the large limit of theories with sixteen supercharges,” Physical Review D, vol. 58, no. 4, Article ID 046004, 1998.

S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, “Gauge theory correlators from non-critical string theory,” Physics Letters B, vol. 428, no. 1-2, pp. 105–114, 1998.

E. Witten, “Anti de Sitter space and holography,” Advances in Theoretical and Mathematical Physics, vol. 2, no. 2, pp. 253–291, 1998.

S. Sethi and L. Susskind, “Rotational invariance in the M(atrix) formulation of type IIB theory,” Physics Letters B, vol. 400, no. 3-4, pp. 265–268, 1997.

L. Susskind, “T duality in M(atrix) theory and S duality in field theory,” http://xxx.lanl.gov/abs/hep-th/9611164

O. J. Ganor, S. Ramgoolam, and W. Taylor, “Branes, fluxes and duality in M(atrix)-theory,” Nuclear Physics B, vol. 492, no. 1-2, pp. 191–204, 1997.

B. de Wit, J. Hoppe, and H. Nicolai, “On the quantum mechanics of supermembranes,” Nuclear Physics B, vol. 305, no. 4, pp. 545–581, 1988.

L. Motl, “Proposals on non-perturbative superstring interactions,” http://lanl.arxiv.org/abs/hep-th/9701025

R. Dijkgraaf, E. Verlinde, and H. Verlinde, “Matrix string theory,” Nuclear Physics B, vol. 500, no. 1–3, pp. 43–61, 1997.

Y. Sekino and T. Yoneya, “From supermembrane to matrix string,” Nuclear Physics B, vol. 619, no. 1–3, pp. 22–50, 2001.

E. S. Fradkin and A. A. Tseytlin, “Effective field theory from quantized strings,” Physics Letters B, vol. 158, no. 4, pp. 316–322, 1985.

A. A. Tseytlin, “Sigma model approach to string theory effective actions with tachyons,” Journal of Mathematical Physics, vol. 42, no. 7, pp. 2854–2871, 2001.

H. Lin and J. Maldacena, “Fivebranes from gauge theory,” Physical Review D, vol. 74, no. 8, Article ID 084014, 40 pages, 2006.

G. Ishiki, S. Shimasaki, Y. Takayama, and A. Tsuchiya, “Embedding of theories with symmetry into the plane wave matrix model,” Journal of High Energy Physics, vol. 20069, no. 11, article 089, 2006.

T. Ishii, G. Ishiki, S. Shimasaki, and A. Tsuchiya, “ super Yang-Mills theory from the plane wave matrix model,” Physical Review D, vol. 78, no. 10, Article ID 106001, 21 pages, 2008.

B. deWit, K. Peeters, and J. Plefka, “Superspace geometry for supermembrane backgrounds,” Nuclear Physics B, vol. 532, Article ID 980320, pp. 99–123, 1998.

W. Taylor and M. Van Raamsdonk, “Multiple -branes in weakly curved backgrounds,” Nuclear Physics B, vol. 558, no. 1-2, pp. 63–95, 1999.

W. Taylor and M. Van Raamsdonk, “Multiple -branes in weak background fields,” Nuclear Physics B, vol. 573, no. 3, pp. 703–734, 2000.

D. Kabat and W. Taylor, “Linearized supergravity from Matrix theory,” Physics Letters B, vol. 426, no. 3-4, pp. 297–305, 1998.

A. A. Tseytlin, “‘No-force’ condition and BPS combinations of -branes in and dimensions,” Nuclear Physics B, vol. 487, no. 1-2, pp. 141–154, 1997.

A. Jevicki and C. Lee, “S-matrix generating functional and effective action,” Physical Review D, vol. 37, no. 6, pp. 1485–1491, 1988.

E. S. Fradkin and A. A. Tseytlin, “Quantum string theory effective action,” Nuclear Physics B, vol. 261, no. 1, pp. 1–27, 1985.

J. Figueroa-O'Farrill and G. Papadopoulos, “Homogeneous fluxes, branes and a maximally supersymmetric solution of M-theory,” Journal of High Energy Physics, vol. 2001, no. 08, article 036, 2001.

J. Maldacena, M. M. Sheikh-Jabbari, and M. Van Raamsdonk, “Tranverse fivebranes in Matrix theory,” Journal of High Energy Physics, vol. 2003, no. 1, article 038, 2003.

V. Balasubramanian, M. Berkooz, A. Naqvi, and M. J. Strassler, “Giant gravitons in conformal field theory,” Journal of High Energy Physics, vol. 2002, no. 4, article 34, 2002.

A. Hashimoto, S. Hirano, and N. Itzhaki, “Large branes in AdS and their field theory dual,” The Journal of High Energy Physics, vol. 2000, no. 8, article 51, 2000.

J. McGreevy, L. Susskind, and N. Toumbas, “Invasion of the giant gravitons from anti-de Sitter space,” The Journal of High Energy Physics, vol. 2000, no. 6, article 8, 2000.

M. M. Sheikh-Jabbari, “Tiny graviton matrix theory: DLCQ of IIB plane-wave string theory, a conjecture,” Journal of High Energy Physics, vol. 2004, no. 9, article 17, 2004.

M. M. Sheikh-Jabbari and M. Torabian, “Classification of all 1/2 BPS solutions of the tiny graviton matrix theory,” Journal of High Energy Physics, vol. 2005, no. 4, article 001, 2005.

M. Ali-Akbari, M. M. Sheikh-Jabbari, and M. Torabian, “Tiny graviton matrix theory/SYM correspondence: analysis of BPS states,” Physical Review D, vol. 74, no. 6, Article ID 066005, 2006.

H. Lin, “The supergravity dual of the BMN matrix model,” Journal of High Energy Physics, vol. 2004, no. 12, article 001, 2004.

K. Sugiyama and K. Yoshida, “Giant graviton and quantum stability in the Matrix model on a pp-wave background,” Physical Review D, vol. 66, no. 8, Article ID 085022, 9 pages, 2002.

H. Lin, O. Lunin, and J. Maldacena, “Bubbling AdS space and 1/2 BPS geometries,” Journal of High Energy Physics, vol. 2004, no. 10, article 025, 2004.

F. Cachazo, K. A. Intriligator, and C. Vafa, “A large N duality via a geometric transition,” Nuclear Physics B, vol. 603, Article ID 010306, 3 pages, 2001.

K. Becker, M. Becker, J. Polchinski, and A. Tseytlin, “Higher order graviton scattering in M(atrix) theory,” Physical Review D, vol. 56, no. 6, pp. R3174–R3178, 1997.

H. Ling, A. R. Mohazab, H.-H. Shieh, G. van Anders, and M. Van Raamsdonk, “Little string theory from a double-scaled matrix model,” Journal of High Energy Physics, vol. 2006, no. 10, article 018, 2006.

W. Taylor, “D-brane field theory on compact spaces,” Physics Letters B, vol. 394, no. 3-4, pp. 283–287, 1997.

I. Antoniadis, C. Bachas, J. Ellis, and D. V. Nanopoulos, “Cosmological string theories and discrete inflation,” Physics Letters B, vol. 211, no. 4, pp. 393–399, 1988.

I. Antoniadis, C. Bachas, J. Ellis, and D. V. Nanopoulos, “An expanding universe in string theory,” Nuclear Physics B, vol. 328, no. 1, pp. 117–139, 1989.

I. Antoniadis, C. Bachas, J. Ellis, and D. V. Nanopoulos, “Comments on cosmological string solutions,” Physics Letters B, vol. 257, no. 3-4, pp. 278–284, 1991.

J. Distler and H. Kawai, “Conformal field theory and 2D quantum gravity,” Nuclear Physics B, vol. 321, no. 2, pp. 509–527, 1989.

J. Polchinski, “A two-dimensional model for quantum gravity,” Nuclear Physics B, vol. 324, no. 1, pp. 123–140, 1989.

A. Dhar and S. R. Wadia, “Noncritical strings, RG flows and holography,” Nuclear Physics B, vol. 590, no. 1-2, pp. 261–272, 2000.

A. Strominger and C. Vafa, “Microscopic origin of the Bekenstein-Hawking entropy,” Physics Letters B, vol. 379, no. 1–4, pp. 99–104, 1996.

J. R. David, G. Mandal, and S. R. Wadia, “Microscopic formulation of black holes in string theory,” Physics Reports, vol. 369, no. 6, pp. 549–686, 2002.

M. Henningson and K. Skenderis, “The holographic Weyl anomaly,” The Journal of High Energy Physics, no. 7, article 23, 1998.

S. de Haro, K. Skenderis, and S. N. Solodukhin, “Holographic reconstruction of spacetime and renormalization in the AdS/CFT correspondence,” Communications in Mathematical Physics, vol. 217, no. 3, pp. 595–622, 2001.

C. Imbimbo, A. Schwimmer, S. Theisen, and S. Yankielowicz, “Diffeomorphisms and holographic anomalies,” Classical and Quantum Gravity, vol. 17, no. 5, pp. 1129–1138, 2000.

N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space, vol. 7 of Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, UK, 1982.

B. S. DeWitt, Dynamical Theory of Groups and Fields, Gordon and Breach Science Publishers, New York, NY, USA, 1965.

M. J. Duff, “Twenty years of the Weyl anomaly,” Classical and Quantum Gravity, vol. 11, no. 6, pp. 1387–1403, 1994.

J. De Boer, E. Verlinde, and H. Verlinde, “On the holographic renormalization group,” Journal of High Energy Physics, vol. 4, no. 8, article 003, pp. 1–15, 2000.

M. Fukuma, S. Matsuura, and T. Sakai, “A note on the Weyl anomaly in the holographic renormalization group,” Progress of Theoretical Physics, vol. 104, no. 5, pp. 1089–1108, 2000.

J. Erdmenger, “Field-theoretical interpretation of the holographic renormalization group,” Physical Review D, vol. 64, no. 8, Article ID 085012, 12 pages, 2001.

T. Banks, “Tasi lectures on matrix theory,” http://arxiv.org/abs/hep-th/9911068

J. Bagger and N. Lambert, “Modeling multiple M2-branes,” Physical Review D, vol. 75, no. 4, Article ID 045020, 7 pages, 2007.

J. Bagger and N. Lambert, “Gauge symmetry and supersymmetry of multiple M2-branes,” Physical Review D, vol. 77, no. 6, Article ID 065008, 6 pages, 2008.

A. Gustavsson, “Algebraic structures on parallel M2-branes,” Nuclear Physics B, vol. 811, no. 1-2, pp. 66–76, 2009.

N. Lambert and C. Papageorgakis, “Nonabelian (2,0) tensor multiplets and 3-algebras,” Journal of High Energy Physics, vol. 2010, no. 8, article 083, 2010.

K. Fujikawa, “Comment on chiral and conformal anomalies,” Physical Review Letters, vol. 44, no. 26, pp. 1733–1736, 1980.