#### Abstract

With â€‰â€‰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 SYM_{4} 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 [6â€“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 [10â€“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 Greenâ€™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 [19â€“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 [19â€“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 . , ,â€‰â€‰ 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, 23â€“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.)â€‰â€‰ 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, 23â€“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, 23â€“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 â€‰â€‰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 . â€‰ 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 Greenâ€™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: