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

With N → ∞ being fixed, R → ∞, the free energy of the Matrix theory on a supergravity background F is a functional of F, W = W(R, F). We try to relate this functional with Seff (R, F), the effective action of F, where F is translation invariant along x .The vertex function is then associated with the connected correlation function of the current densities. FromW(R, F), one can construct an effective action Γ(Y) for the arbitrary matrix configuration Y. Γ(Y) is R, and thus p+ independent. If W(R, F) = Seff (R, F), Γ(Y) will give the supergravity interactions among M theory objects with no light-cone momentum exchange. We then discuss the Matrix theory dual of the 11d background generated by branes with the definite p+ as well as the gauge theory dual of the 10d background arising from the x− reduction. Finally, for SYM


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 AdS/CFT correspondence [6][7][8][9], for which the AdS 5 /SYM 4 correspondence is intensively-studied.SYM 4 gives a nonperturbative description of string theory on AdS 5 × 5 .It is natural to expect that SYM 4 with the 4 vertex operator perturbations added then describes the string theory on AdS 5 ×  5 with the corresponding 5 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  11− ×   , the dual gauge theory will become SYM +1 , which is Matrix theory compactified on   [10][11][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 AdS 5 /SYM 4 , the one-to-one correspondence should exist between the state (spectrum) of the second-quantized string theory on AdS 5 × 5 and the state (spectrum) of SYM 4 .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  1 with the vanishing radius, reduces to the latter.Matrix theory compactified on  1 gives the Matrix string theory [14,15].The off-diagonal degrees of freedom are KK modes of membrane along  1 [16].In strong coupling limit (the radius of  1 approaches 0), matrices commute (KK modes could be dropped), and Matrix string theory reduces to the second quantized type IIA string theory in lightcone 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 SYM 4 , we have [8,9] 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  eff (, ).The complete M theory degrees of freedom including sectors with all lightcone momentum is necessary only when  is the field with the 11 spacetime dependence.
Advances in High Energy Physics To describe the supergravity interactions among  theory objects with no light-cone momentum exchange, and one may define Γ  () through  Γ  () = ∫ []   , ()− cla (,) , (11) with  being the zero mode of the 11 supergravity along  − and  cla the classical action of 11 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.

The Action of the Matrix Theory on a Generic Background.
The Matrix theory action in flat spacetime is where   , , and  0 are × hermitian matrices with  → ∞, and  = 1 ⋅ ⋅ ⋅ 9.  → ∞,  + = /,   is the Planck length.
Since all background fields enter into the action in the form of ( + , 0, . . ., 0), the () SYM 1 lives at ( + , 0, . . ., 0).When   →   −   ,  ( + , 0, . . ., 0) →  ( + ,  1 , . . .,  9 ) , Advances in High Energy Physics SYM 1 undergoes a translation.Nothing specifies where the SYM 1 should be, so one may put it at any point in  9 .Similar to AdS/CFT, there is a one-to-one correspondence between operators and fields.However, no holography is present here.Fields living on SYM 1 are Taylor series coefficients, which uniquely determines the 11 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 Φ  1 ⋅⋅⋅  in AdS/CFT, the moment operators   1 ⋅⋅⋅  do not need to be traceless.As a result, {1,   1 , . . .,   1 ⋅⋅⋅  , . ..} couples with the 10 field, while {1, Φ  1 , . . ., Φ  1 ⋅⋅⋅  , . ..}only couples with the 9 field.
In the second form, the vertex operator is defined in 10 spacetime: where   are  ×  matrices and for uniformity; we have set  0 =  + 1 × .This is a matrix generalization of the function.In special situations, when all of the   are diagonal, that is, With the generalized -function, it is straightforward to write down the current densities for various fields.For ℎ + , 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:
In gauge/string correspondence, partition function is an important quantity, the value of which should be equal on both sides.For AdS 5 /CFT 4 correspondence, it is expected that should hold, where  = ∑  ℎ =0   ℎ is the free energy of the strings on AdS 5 ×  5 ,  SYM 4 and exp{−} are the partition function of SYM 4 and the partition function of the second quantized type IIB string theory on AdS 5 ×  5 , 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 ( + ,  1 ,  2 ) and   ( + ,  1 ,  2 ) become the matrix configurations ( + ) and   ( + ), and there is (, ) = (  ,   ).At least restricted to (35), (, ) is the diffeomorphism invariant functional of .
V  is the vacuum expectation value of the current density, when the background field is , and is a constant in spacetime.  is (9) invariant, so V  should also be  (9) invariant.As a result, V   = 0 if the traceless condition is imposed.V  + = V  − = 0 due to the supersymmetry.The nonvanishing current densities are V ℎ ++ , V ℎ −− , and V ℎ +− .In particular, V ℎ ++ =  + / 9 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 ⟨  ()⟩  = 0 except for ℎ ++ , whose vertex operator is the same as the tachyon in bosonic string.We will return to this problem later.

Another Effective Action of Matrix Theory.
For the given (), V  () is uniquely determined.Conversely, different () may result in the same V  ().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 V  () = 0, F() = 0. F() could be interpreted as the field generated by the current density V  ().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 V  and , and so a Legendre transformation is possible.Before that, we will first define Γ(): is solved from the equation 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.
With these properties collected, we may consider the possible interpretation of Γ() and (, ).If (, ) is the effective action of supergravity, since  , () is the matrix theory action on background , Γ() 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 .Γ()/ =  , ()/ = 0 is the quantum corrected equation of motion, which differs from the classical equation of motion  0 ()/ = 0 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, Γ()/ =  , ()/ = 0 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 Γ() and (, ).For simplicity, let  = 1, or, in other words, let the  − indexed fields absorb .In a weakly curved background,   () =  0 () + ∫  10  F() () (), Γ() =  0 () + Γ(V  ), where is the Legendre transformation of (): Let then For the given V  (), Equation (52) gives the current density V  () generated by the field (); (62) gives the field () generated by the current density V  ().If Λ  is the connected Green's function of supergravity, () will be the vacuum expectation value of the supergravity field  in presence of the source V  .In classical level, () could be calculated by   /() = −V  () with   the classical action of supergravity: so Advances in High Energy Physics The relation between Λ  and Γ   shows that Λ  and Γ   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 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.Γ() and Γ eff () differ from a parameterization transformation.Γ() ̸ = Γ eff ().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 Γ eff () 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 eff () is the effective action of supergravity.Equation ( 77) is almost the same as (53).The only problem is whether () =  eff () or not.

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: F is the renormalized field other than the bare field.In fact, in (79), a particular Weyl gauge   =  2 ĝ is always imposed, and so,   (, ) also has the dependence on : When () = 0,   (, ) =   (), although () = 0 is not the necessary condition.Suppose  eff () is the effective action of the string modes: Consider  0 with Δ 0 = 0. Since ( 0 ) ̸ = 0,  will evolve along the RG flow; that is,   ( 0 , ) =   [(  ),   ].A special property of  0 is that it will finally reach an IR fixed point F,   ( 0 , ) =   ( F, ∞).In fact, since  0 satisfies the first-order  equation, the RG flow may bring it to an IR fixed point.
(, ) with () ̸ = 0 usually depends on .However, the -dependence drops out for  0 .In fact, is the -matrix functional [27].For the on-shell  1 , . . .,   , 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].
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.SYM 4 is a nonperturbative description of the type IIB string theory on AdS 5 ×  5 .It is expected that the Hilbert spaces of both sides are isomorphic to each other.Similar with the Chern-Simons/topological string correspondence,  SYM 4 =    , where   is the free energy of the string theory on AdS 5 ×  5 .The question is what will be the string dual of SYM 4 if the background metric of SYM 4 is  ] +  ] () other than  ] for very small  ] ().A natural expectation is that such SYM 4 is dual to the type IIB string theory on AdS 5 ×  5 with a little modification of the background metric  ] (, ) that is entirely determined by  ] ().
, where   [ ] (, )] is the free energy of the string theory on AdS 5 × 5 with the background perturbation  ] (, ) being turned on.Note that, for the stringy explanation of the SYM 4 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  eff [ ] (, )] is the supergravity effective action for the modified background fields, so There is a naive way to interpret   () =  eff ().Suppose the classical supergravity action is   (), from the field theory's point of view: where F 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   ( + F).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   ( + F) 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  eff ()/ = 0.The cancelation of the Weyl anomaly gives the e.o.m for the effective action of the supergravity, including the   corrections, so  eff () should be the effective action with the   corrections taken into account.
A natural -theory extension is where    () is the supermembrane action on 11 supergravity background .The membrane is already the second quantized object, so the left-hand side of (87) is   eff () other than  eff ().For the generic 11 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  − ∼ 1/ + .

Matrix Theory on a Particular Vacuum
If (, ) in ( 42) is the effective action of the 11 supergravity field () that is translation invariant along  − , for the on-shell (), there will be (, )/() = −V  () = 0.However, for  = , which is obviously on shell, the current densities V  ++ , V  −− , and V  +− 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 (, )/  () = 0, and then take this value other than  +− () = −1,  −− () =  ++ () = 0 as the background to do the expansion.V   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.V  ++ should be distinguished from  , ()/ ++ ()| =0 = (/) 10 (), which is the lightcone momentum density of a supergraviton localized at (, 0, . . ., 0) and, of course, will produce the nontrivial  −− ().
In (54),  = (, ) is the functional of  solved through (55).For the arbitrary , we may define The expectation values of all current densities vanish.There is no tadpole.

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 11 pp-wave background: The background preserves 32 supersymmetries, while the rest 11 supergravity solutions with 32 supersymmetries are flat spacetime, AdS 4 ×  7 , and AdS 7 ×  4 [29].In pp-wave, the dynamics of the -theory sector with the light-cone momentum  + = / is described by the () matrix model: → ∞,  → ∞.PWMM also preserves 32 supersymmetries and has the same symmetry group as that of the pp-wave background.
The solution can also be interpreted as the spherical 5 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 2 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 5 interpretation, it is the rows of the Young diagram that correspond to the individual 5, with the row lengths giving the number of units of momentum carried by each 5.In both cases, the total lightcone momentum is always /.If   are finite, they are fuzzy 5 branes.When   → ∞, the fuzzy 5 become the spherical 5 wrapping  5 given by ∑ 9 =4 (   ) 2 =  2  .In patricular, the trivial vacuum X = 0 represents a single 5 brane.
All of the supersymmetric solutions preserve 16 nonlinearly realized supersymmetries.They are the 1/2 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 32 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 SYM 4 in which, we do have a vacuum preserving Advances in High Energy Physics 32 supersymmetries, representing the ground state of the string theory on AdS 5 ×  5 .(Brane like) 1/2 BPS states are giant gravitons on AdS 5 ×  5 [31][32][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][35][36].TGMT can also be taken as the DLCQ of the type IIB string theory on AdS 5 ×  5 , capturing the physics seen from the infinite momentum frame (IMF) since, in IMF, AdS 5 × 5 is viewed as the pp-wave [34][35][36].Although TGMT preserves 32 supersymmetries, the vacuum configurations, carrying the definite light-cone momentum, are all 1/2 BPS, matching exactly with the 1/2 BPS states on type IIB ppwave background, which are giant gravitons (spherical 3 branes) and type IIB strings.Lifted to 11, the light-cone dimension is replaced by  2 , while the TGMT, which is a discrete regularization of the 3 branes in type IIB picture, becomes the regularization of the 5 branes [35].Note that the 4-form field in type IIB pp-wave, when lifted to 11, becomes the 6-form field coupling electrically with the 5 branes, so it is natural to construct the matrix model via the discrete regularization of 5 branes other than the usual 2 branes.On type IIB pp-wave, 3 branes are 5 branes wrapping  2 , while the type IIB strings are membranes wrapping one of  1 .In contrast to the PWMM, in TGMT, the trivial vacuum  = 0 represents type IIB string (with 2 origin), while the nontrivial vacuum represents 3 branes (with 5 origin), which is probably because the PWMM and the TGMT are constructed from 2 and 5, respectively.Recall that, in PWMM, the fuzzy configuration may have the 2 and 5 dual interpretations [30], it is interesting to figure out whether a similar 1-3 dual interpretation also exists for configurations in TGMT.
With Ŷ denoting the supersymmetric vacuum in (96),  theory on pp-wave background in presence of the brane Ŷ is described by the Matrix model with the action Since ( PW  ()/)| = Ŷ = 0,  PW  ( Ŷ; ) starts from the quadratic term.
For simplicity, in the following, we will assume  is absorbed into the fields.If the backreaction of the brane Ŷ on pp-wave background is turned on, the field ( Ŷ) will be generated: where pp-wave with Ŷ added respects the (2 | 4) symmetry, so the generated ( Ŷ) as well as the corresponding  PW ( Ŷ) () will have the same symmetry.
With the fields ( Ŷ) given, in principle, one can write down  PW ( Ŷ) () explicitly.The classical supersymmetric solutions of  PW ( Ŷ) () are still (96).To see this, note that the spherical giant gravitons are 1/2 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 Ŷ was calculated at the one-loop level.Ŷ is also the stationary point of the effective action.We may have Γ PW ()/| = Ŷ =  PW ( Ŷ) ()/| = Ŷ = 0.With the pp-wave background replaced by the backreacted geometry,  PW  ( Ŷ; ) is modified to PW ( Ŷ) ( Ŷ; ) still starts from the quadratic term since Notice that, although any configuration Ŷ in (96) may be the classical vacuum of  PW ( Ŷ) (), Ŷ is special because the vacuum expectation values of current densities vanish only for  PW ( Ŷ) ( Ŷ; ) but not for the generic  PW ( Ŷ) ( Ŷ ; ).The 11 geometry produced by 1/2 BPS states of PWMM was constructed in [19,39].The geometry has a bubble structure containing noncontractible 7 cycles and 4 cycles supporting  4 and  7 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 sourcegravity 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), Ŷ is the momentum eigenstate in  − direction, so the generated 11 background ( Ŷ) is translation invariant along  − .In large  limit, the local structure of the giant gravitons is not important while the asymptotic geometry is just the ppwave with the perturbation roughly given by [39] Advances in High Energy Physics 15 which is the field produced by the supergraviton which is static in 9 space, carrying the definite light-cone momentum [41]. = 30 while the radius of  − is now  −2/3  .Under the  − reduction, in string frame, In particular, for (105), is the near-horizon geometry of the 0 branes [41].
The type IIA solution coming from the  − reduction of the 11 field ( Ŷ) was constructed in [19,37].When  → ∞, the perturbation, which is the reduction of (104) along  − , is the near-horizon geometry of  coincident 0-branes.The appearance of the near-horizon geometry is because of the null reduction.The reduction of (104) along  + −  − gives the 0 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  − .(

The Gauge Theory
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 SYM 1 with the action  =1 ().Since (104) becomes (107) under the  − reduction, it is desirable to find a limit to make (108) reduce to  =1 ().
SYM 1 arrives.Note that, for finite , lim  → 0 / = ∞, so by taking the  → 0 limit, the background field that matters lives at the  → ∞ region, far away from the source.For PWMM,  2  ++ (/) =  ++ (), and  + (/) =  + (), all fields are marginal, so, for the arbitrary  [5], Advances in High Energy Physics Now consider the background ( Ŷ) coming from the backreaction of the giant graviton Ŷ on pp-wave.Suppose Ŷ is a vacuum in (96) with The radii of the spherical membranes are   ∼ ( +   )/,  = 1, . . ., . ( Ŷ) reduced along  − gives the smooth type IIA solution that is constructed in [19].The generic solution of type IIA supergravity with (2 | 4) symmetry is characterized by a function (, ) and is given as [19] where the dot and the prime represent the derivatives with respect to log  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.(, ) =   (, ) + V  (, ), where   is the background potential and V  is determined by a configuration of conducting disks.Each (2 | 4) symmetric theory is specified by   ; each vacuum of the theory is specified by a configuration of conducting disks.
In the limit with  → ∞,  → ∞,  3 / =  2 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 SYM × 2 .Alternatively, we may directly plug into ( 116) and (115), taking the limit at the end.Let V / (/, ( + )/) = ( 2 , ); the 11 background  * (, ) is which has the topology of  ×  7 ×  2 × .The 10 background  * 10 (, ) is +  + , and Ω 2 2 now have the same prefactor.Equations ( 122) and ( 133) can be taken as the background for 1 and 3 gauge theories, respectively.
On gauge theory side, to study the fluctuation around the spherical membranes, we should expand  PW =1 / 3 around Ŷ. In [20], it was shown that in the limit of  → ∞,  3 / fixed, (i) PWMM around a certain vacuum is equivalent to SYM × 2 around each vacuum and (ii) SYM × 2 around a certain vacuum with a periodicity imposed is equivalent to SYM × 3 /  around each vacuum.In particular, SYM × 3 can be realized as the PWMM around a certain vacuum with a periodicity condition imposed.
Indeed, as we will show in Appendix A, such phenomenon is very common.Generically, the action of SYM +1 on flat background is equal to the action of the SYM +1 on the near-horizon geometry of the  branes.(For  ≥ 6, the worldvolume theories of  branes do not decouple from the bulk, as is discussed in [7].)For SCFT 3 and SCFT 6 on 2 and 5, such requirement can even offer some clues for the structure of the field theory.

𝑢 = 𝑢(𝜌
then Compared with (134), with   → X , Φ → Φ, the 3 background fields on  ×  2 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  = ( 2 +  2 ) 1/2 ;  2 10 in (133) can be written as with SYM × 2 living at  = ∞.However, with the energy  indentified with , the RG flow cannot give ( 2 , ), which is not just the function of .Instead, we will make the   → X , Φ → Φ transformation to recover  dependence of the 3 background fields in  × 2 .
For PWMM, we have PW () is the  − reduction of the 11 supergravity effective action of the field generated by the brane Ŷ on pp-wave.In the limit with  → ∞,  → ∞,  3 / =  2 fixed, under the change of variables  →  + , where  × 2 () is the type IIA action for the abovementioned supergravity solution dual to a vacuum of SYM × 2 .Then, As is demonstrated in [19,20], from SYM × 2 and the corresponding type IIA solution, it is also possible to get the SYM × where  runs from −∞ to ∞. Expanding the action (134) around this vacuum and imposing the condition  (+1,+1) =  (,) on all of the field fluctuations, one will get [20] where is the action of the () SYM on  ×  3 .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  1 over  2 rather than a direct product; as a result, it is  3 other than  2 ×  1 that is obtained [20].
On gravity side, start from the trivial vacuum of SYM × 2 , for which there is only a single disk with the electric charge ( 2 /8); compactify the  direction; the disk configuration in covering space will contain the infinite copies of disks with the period /2, corresponding to the vacuum (144).The type IIA geometry generated by (144), after a -duality transformation, becomes the type IIB geometry AdS 5 ×  5 [19].On field theory side, since  is compactified, the field fluctuations should respect the periodicity condition  (+1,+1) =  (,) .Taylor's prescription for the compactification also involves the -duality transformation, making SYM × 2 in type IIA become  × 3  in type IIB.Equation (143) turns into For the trivial vacuum of  × 3 ,  represents AdS 5 ×  5 background.
Finally, the geometry arising from the backreaction of the 3 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][35][36] around the corresponding 1/2 BPS configuration, and then take the certain limit.TGMT is the discrete regularization of the 3 branes, so the resulted gauge theory should be a 4 gauge theory as is required since the geometry is generated by spherical 3 branes on pp-wave.
One may take   1 ⋅⋅⋅  Tr(  1  ⋅ ⋅ ⋅     ) as the operator corresponding to   (/||) and consider the gauge theory with the vertex operator perturbation realized as (in fact, we will choose X instead of   with det ℎ  (, ) can be the arbitrary 5 function.Gauge theory like this is difficult to approach directly.We still prefer SYM 4 with ℎ  ()  1 ⋅⋅⋅  Tr(  1 ⋅ ⋅ ⋅    ) added, which, after a suitable transition, will become a gauge theory with the operator ℎ  (, )  1 ⋅⋅⋅  Tr( X 1 ⋅ ⋅ ⋅ X  ).The 5 field ℎ  (, ) is not arbitrary anymore but is determined by ℎ  ().This is quite similar to the noncritical string coupling with 2 gravity.For critical string with 26 coordinates, any 26 background fields can be represented by the vertex operator perturbations on the string worldsheet action.For noncritical string with, for example, 25 coordinates, only the vertex operator perturbations corresponding to 25 fields can be constructed.However, if the noncritical string is coupled to the conformal mode of the 2 gravity so that the total number of degrees of freedom is 26, after a property transformation with the partition function kept invariant, the theory will become the critical string coupling with the 26 background fields.The 26 fields are induced from the 25 fields with the conformal mode acting as the 26th dimension.For SYM 4 , the role of the conformal mode is played by .

Noncritical String Coupling with 2𝑑
Gravity.Let us have a simple review of the noncritical string [44][45][46][47][48][49].Consider the bosonic string living in  dimensional spacetime.The corresponding nonlinear sigma model action is , ] = 1, . . ., .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  =   ĝ,  ≥ 0, ĝ is a small metric giving the cut-off scale where   (; ĝ) is the Liouville action.With    →  ĝ,    →  ĝ, the partition function becomes where The original  dimensional background field (), after the gravitational dressing, becomes the  + 1 dimensional field F(, ).F(, 0) = ().One can also make a change of the variables to move the boundary from  = 0 to  = −∞: In Ŝ , F () = ( + ∞).
Since  =  +  − ĝ, for  ≥ 0, With the cutoff being introduced, the conformal transformation should be accompanied by a change of the boundary: With the cutoff being removed, ( ĝ) = ( − ĝ).
For g-Φ coupled system, the subleading terms are determined by both  0 and  0 .Φ(, ) and g ] (, ) are fields on AdS 5 with  3 finite.It is expected that the  → X transformation will give  ] (, ) and (, ) in ( 180) and (182).
(, ) is also on shell.
4 is the conformal anomaly [57].On the other hand, loc (g ] (0)) = A 0 /4 + A 2 /2 and  nonloc (g ] (0)) 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  ∞  4 .On gravity side, the corresponding gravity solution has the near boundary expansion At  = , the metric is ( 0 , ).The on-shell gravity action with the cutoff  =  is which is entirely determined by the boundary value ( 0 , ).
Take  =   X other than X as the upper bound of ; the partition function is For finite  ]0 and , ( ]0 ;   ) = ( 2  ]0 ; ).There is also where With   ( 0 ; ) = ∫  4   ( 0 ;) (), we may have ( For finite  and  ]0 ,   ( 0 ;) () = 0.For infinite , for example, ( ]0 ;   X) and ( 2  ]0 ; X) with  = −∞,  ( ]0 ;   X) = ln Ẑ ( ] ( + ) ; X; 0) ,  ( 2  ]0 ; X) = ln Ẑ (  ] () ; X; 0) .In the previous subsection, the radial dependent function () is induced via (176).Nevertheless, it can also be induced from ( ]0 ; ), 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  ∈ (−∞, +∞):  ( ]0 ; ) =  ( ]0 ;    * ) = ln  ( ]0 ;    * ) .(228) Starting from  = 0, with  increasing, ( ]0 ;    * ) will also increase, so  ]0 should change accordingly to make the partition function invariant.Namely, we have (234) PW ( + Ŷ) becomes the action of SYM × 2 , which, with the background (133) plugged in, remains invariant.With the SYM × 2 and the gravity dual at hand, a further -duality-like transformation gives SYM × 3 and AdS 5 ×  5 [19,21].SYM 4 is the nonpertubative definition of the type IIB string theory on AdS 5 ×  5 .It is unlikely such gauge/string correspondence will suddenly vanish just because the metric in SYM 4 deviates  ] a little.If the correspondence still exists, the gravity dual will be the type IIB string theory on AdS 5 ×  5 with a little background perturbation turned on.Then a one-to-one correspondence should exist between the 4 field  0 () in SYM 4 and the 5 field F(, ) on AdS 5 .The natural candidate of F(, ) is the gravity solution on AdS 5 with  0 () the boundary condition.For the correspondence to be valid, it is necessary to derive F(, ) merely from SYM 4 .In the simplest situation, when the background metric in gauge theory is the standard  ] , the near-horizon geometry of 3, 2, and 5 could be induced from SYM 4 , SCFT 3 , and SCFT 6 by a  → X transformation.It is expected that the same method, when applied to SYM 4 with the arbitrary  0 () turned on, will give the corresponding F(, ).With  → X,  0 () → (, ), SYM 4 becomes a gauge theory living in 5 background F(, ) times the transverse  5 .The free energy can be expressed as the functional of F(, ); that is,  0 ( 0 ) = (F).(F)/F(, ) = 0, so if (F) is the 10 gravity action as is in Matrix theory case, F(, ) will be the on-shell solution.On gravity side, this is equivalent to assume that lim  → 0 ( ]0 ,  0 , ) = lim  → 0 ( ]0 ,  0 ,  2 ), so indicating that   will depend on both  ]0 and  0 .Indeed, the direct calculation of the conformal anomaly gives   =   ( ]0 ,  0 ) composed by the gravity part ( ]0 ) and the matter part ( 0 ).Correspondingly, the transformation of the path integral measure will also depend on  ]0 and  0 , although neither of them enters into  explicitly.Similarly, when the theory is coupled to the external (4)  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.
2, ) is the function of the radial directions.For the given field configuration [Φ,  4 , . . .,  9 ], let Energy Physics 23 Ŝ(, ; ĝ) is the action of the bosonic string coupling with the background fields F(, ).The conformal invariance indicates that F(, ) should be the on-shell solution of gravity.To arrive at this result, it is important that, in (157), the change of the 2 metric is always compensated by the adjustment of the background fields, keeping the form of the sigma model invariant.