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

Abstract

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 𝑆_eff(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. From 𝑊(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)=𝑆_eff(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 SYM₄ with the 4𝑑 background field 𝐹₀, we give a possible way to induce the radial dependent 5𝑑 field 𝐹(𝜎).

1. Introduction

Up to present, two kinds of nonperturbative formulation of 𝑀/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 [6–9], for which the AdS₅/SYM₄ correspondence is intensively-studied. SYM₄ gives a nonperturbative description of string theory on AdS₅×𝑆⁵. It is natural to expect that SYM₄ with the 4𝑑 vertex operator perturbations added then describes the string theory on AdS₅×𝑆⁵ with the corresponding 5𝑑 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 𝑅_11−𝑛×𝑇^𝑛, the dual gauge theory will become SYM_𝑛+1, which is Matrix theory compactified on 𝑇^𝑛 [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 AdS₅/SYM₄, the one-to-one correspondence should exist between the state (spectrum) of the second-quantized string theory on AdS₅×𝑆⁵ and the state (spectrum) of SYM₄. 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 𝑁→∞, 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 𝑆¹ with the vanishing radius, reduces to the latter. Matrix theory compactified on 𝑆¹ gives the Matrix string theory [14, 15]. The off-diagonal degrees of freedom are KK modes of membrane along 𝑆¹ [16]. In strong coupling limit (the radius of 𝑆¹ approaches 0), 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 SYM₄, we have [8, 9] 𝑍_SYM4𝛽,𝜙₀=𝑍_AdS5×𝑆⁵𝛽,𝜙₀, (1) where 𝛽=1/𝑇 is the radius of the time direction. The zero temperature partition function is only the functional of 𝜙₀, for which [8, 9] 𝑍_SYM4𝜙₀=𝑒^{𝑊[𝜙(𝜙0)]}. (2)𝜙(𝜙₀) is the on-shell supergravity solution with the boundary value 𝜙₀. 𝑊 is the type IIB supergravity action on AdS₅×𝑆⁵. If the gravity dual of SYM₄ with the source 𝜙₀ added is the type IIB string theory on AdS₅×𝑆⁵ with the background field 𝜙(𝜙₀) turned on, in zero temperature limit, the partition function will only contain the contribution of the ground state geometry, so 𝑍_SYM4(𝜙₀)=𝑒^{𝑊[𝜙(𝜙0)]}.

Except for (2), 𝑍_SYM4(𝜙₀) also has another expression on AdS₅×𝑆⁵. Let 𝑊_𝑠 be the free energy of the type IIB string theory on AdS₅×𝑆⁵ and 𝑍_SYM4 the partition function of SYM₄, it is expected that 𝑍_SYM4=𝑒^𝑊𝑠, (3) where 𝑊=∑_𝑛ℎ=0𝑊_𝑛ℎ. When the background field 𝜙₀ varies, SYM₄ undergoes a change of the coupling constants. Correspondingly, there is also a change of the coupling constants for strings living in AdS₅×𝑆⁵, which is in fact a modification of the 5𝑑 background. For (3) to be valid, a one-to-one correspondence 𝜙₀↔𝜙(𝜙₀) should exist: 𝑍_SYM4𝜙₀=𝑒^{𝑊𝑠[𝜙(𝜙0)]}. (4) On AdS₅ side, the string free energy 𝑊_𝑠[𝜙(𝜙₀)] cannot be defined without a definite 5𝑑 background 𝜙(𝜙₀). Also, for the state (spectrum) correspondence to be valid, the definite 5𝑑 background is necessary; otherwise, it is impossible to determine the string spectrum. If (2) and (4) both hold, 𝑊𝜙𝜙₀=𝑊_𝑠𝜙𝜙₀. (5) The free energy of the string theory on a given background equals the effective action of the background fields.

For Matrix theory, similarly, 𝑍_Matrix𝛽,𝑝⁺,𝐹=𝑍_𝑀𝛽,𝑝⁺,𝐹. (6)𝐹 is the 11𝑑 supergravity background and is the translation invariant along the 𝑥⁻ direction. 𝑍_𝑀(𝛽,𝑝⁺,𝐹) is the partition function of the M theory sector with the light-cone momentum 𝑝⁺, 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 𝛽=∞, 𝑍_Matrix𝑝⁺,𝐹=𝑒^{𝑊(𝑝+,𝐹)}. (7)𝑊(𝑝⁺,𝐹) is the 𝑝⁺-parameterized functional of 𝐹 with the 10𝑑 covariance.

In string theory, one can also calculate the free energy of the strings on a given background 𝐹: 𝑊_𝑠(𝐹)=^∞_𝑚=0𝑒^{𝜙(2−2𝑚)}_{𝑀2−2𝑚}[𝑑𝑋]𝑒^−𝑆𝐹. (8) In [17, 18], it was shown that, for 𝐹 satisfying the free field equation, 𝑊_𝑠(𝐹) could be taken as the effective action of the renormalized background field 𝐹(𝐹); that is, 𝑊_𝑠(𝐹)=𝑆_eff[𝐹(𝐹)].

It is tempting to establish a relation between 𝑊(𝑝⁺,𝐹) and the effective action of the 11𝑑 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 𝑝⁺ without capturing all the information of the covariant theory. For different 𝑝⁺, 𝑊(𝑝⁺,𝐹) is different. Nevertheless, let 𝐹={𝐹₀,𝐹₋,𝐹₋₋}, where 𝐹₀, 𝐹₋, 𝐹₋₋ represent fields with zero, one, and two 𝑥⁻ indices, respectively; one can find that 𝑊(𝑝⁺,𝐹)=𝑊(𝐹₀,𝐹₋/𝑝⁺,𝐹₋₋/𝑝⁺²). Let 𝑝⁺=𝑁/𝑅 with 𝑅 being the radius of 𝑥⁻, 𝑁→∞, 𝑅→∞, and 𝑊(𝑅,𝐹)=𝑊(𝐹₀,𝑅𝐹₋,𝑅²𝐹₋₋). On the other hand, for 11𝑑 supergravity fields 𝐹 that are translation invariant along 𝑥⁻, suppose 𝑆_eff(𝑅,𝐹) is the effective action of 𝐹, and there is also 𝑆_eff(𝑅,𝐹)=𝑆_eff(𝐹₀,𝑅𝐹₋,𝑅²𝐹₋₋). The 𝑅 dependence of 𝑊(𝑅,𝐹) is consistent with 𝑆_eff(𝑅,𝐹).

One may want to consider the complete partition function with all light-cone momentum taken into account, which is roughly 𝑒^𝑊(𝐹)=∫𝑑𝑝⁺𝑒^{𝑊(𝑝+,𝐹)}. Each 𝑊(𝑝⁺,𝐹) only differs by a rescaling of (𝐹₋,𝐹₋₋), so the summation does not give more information. It is enough to consider 𝑊(𝑝⁺,𝐹) with the definite 𝑝⁺. In fact, 𝐹 is the translation invariant along 𝑥⁻; as a result, a sector with the definite 𝑝⁺ has the enough degrees of freedom to produce 𝑆_eff(𝑅,𝐹). The complete M theory degrees of freedom including sectors with all light-cone momentum is necessary only when 𝐹 is the field with the 11𝑑 spacetime dependence.

For the arbitrary matrix configuration 𝑌, we may define Γ(𝑌) via 𝑒^Γ(𝑌)=𝑑𝑌𝑒^{−𝑆𝑅,𝐹(}𝑌_{)+𝑆𝑅,𝐹(𝑌)}=𝑒^{𝑊(𝑅,𝐹)+𝑆𝑅,𝐹(𝑌)}, (9) with 𝐹(𝑥) solved from −𝛿𝑊(𝑅,𝐹)𝛿𝐹(𝑥)_𝐹=𝛿𝑆_𝑅,𝐹(𝑌)𝛿𝐹(𝑥)_𝐹. (10) It is easy to see that Γ(𝑌) is 𝑅 independent.

To describe the supergravity interactions among 𝑀 theory objects with no light-cone momentum exchange, and one may define Γ_𝑔(𝑌) through 𝑒^Γ𝑔(𝑌)=[𝑑𝐹]𝑒^{𝑆𝑅,𝐹(𝑌)−𝑆cla(𝑅,𝐹)}, (11) with 𝐹 being the zero mode of the 11𝑑 supergravity along 𝑥⁻ and 𝑆_cla the classical action of 11𝑑 supergravity. The integrating out of F induces the effective action for the 𝑀 theory object 𝑌 with the supergravity interaction (without transferring the light-cone momentum) taken into account. Γ_𝑔(𝑌)=_𝑛1𝑛!𝑑¹⁰𝑥₁⋯𝑑¹⁰𝑥_𝑛𝐺_Fc𝑥₁,…,𝑥_𝑛×𝑉_𝐹(𝑌)𝑥₁⋯𝑉_𝐹(𝑌)𝑥_𝑛, (12) where 𝐺_Fc(𝑥₁,…,𝑥_𝑛) is the connected Green’s function of supergravity in light-cone gauge with the zero light-cone momentum and 𝑉_𝐹(𝑌)(𝑥) is the current density of the configuration 𝑌 coupling with the supergravity field 𝐹(𝑥). Under a Legendre transformation, Γ_𝑔(𝑌) could be written as Γ_𝑔(𝑌)=𝑆_eff()+𝑅,𝐹𝑑¹⁰𝑥𝐹(𝑥)𝑉_{𝑅,𝐹(𝑌)}(𝑥)∼𝑆_eff(𝑅,𝐹)+𝑆_𝑅,𝐹(𝑌), (13) with 𝐹(𝑥) solved from −𝛿𝑆_eff(𝑅,𝐹)𝛿𝐹(𝑥)_𝐹=𝑉_{𝑅,𝐹(𝑌)}(𝑥)=𝛿𝑆_𝑅,𝐹(𝑌)𝛿𝐹(𝑥)_𝐹. (14) So, if 𝑊(𝑅,𝐹)=𝑆_eff(𝑅,𝐹), Γ(𝑌)=Γ_𝑔(𝑌).

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 1/2 BPS states gives SYM_𝑅×𝑆², SYM_𝑅×𝑆³_/𝑍𝑘, and SYM_𝑅×𝑆³, while the backreaction of the corresponding 1/2 BPS states on pp-wave produces the gravity dual. We will investigate the correspondence in more detail.

In (7), 𝐹 is the generic 11𝑑 supergravity field with the 𝑥⁻ isometry. 𝑊(𝑅,𝐹), if is indeed equal to 𝑆_eff(𝑅,𝐹), gives the effective action of the field 𝐹. On the other hand, in (2), only the 4𝑑 field 𝜙₀(𝑥) is given, from which the 5𝑑 field 𝜙(𝑥,𝑟) is obtained from the equation of motion or from the RG flow. 𝑊[𝜙(𝜙₀)] is the action of 𝜙(𝑥,𝑟). This is the holography of AdS/CFT. One may want to turn on the arbitrary 𝜙(𝑥,𝑟) on AdS₅ and try to find the corresponding 4𝑑 gauge dual. The dual gauge theory may not be SYM₄, since 𝜙₀ can only encode a subset of 5𝑑 fields, which are in one-to-one correspondence with the 4𝑑 fields. In fact, since the transverse space of AdS₅×𝑆⁵ is 𝑆⁵ other than 𝑅⁶, the gauge theory dual may have the scalar fields 𝑋^𝐼 other than 𝑋^𝐼. Suppose the coordinate of AdS₅×𝑆⁵ is (𝑥,𝑟,Ω), for a 10𝑑 scalar 𝐻_𝑛(𝑥,𝑟,Ω)=ℎ_𝑛(𝑥,𝑟)𝑌^𝑛(Ω) with 𝑌^𝑛(Ω)=𝐶_{𝑎1⋯𝑎𝑛}(𝑦^𝑎1⋯𝑦^𝑎𝑛)/|𝑦|^𝑛 the spherical harmonic of 𝑆⁵; the operator counterpart is 𝐻_𝑛𝑥,𝑟,𝑋=ℎ_𝑛(𝑥,𝑟)𝐶_{𝑎1⋯𝑎𝑛}𝑋^𝑎1⋯𝑋^𝑎𝑛. (15)ℎ_𝑛(𝑥,𝑟) can be the arbitrary 5𝑑 function. In SYM₄, we only have ℎ_𝑛(𝑥)𝐶_{𝑎1⋯𝑎𝑛}(𝑋^𝑎1⋯𝑋^𝑎𝑛) to represent such fields. Nevertheless, for SYM₄ with the scalar field 𝑋 and the 4𝑑 background 𝐹₀(𝑥), a 𝑋→𝑋, 𝐹₀(𝑥)→𝐹(𝑥,𝑟) transformation can be made, under which the partition function remains invariant. If the SYM₄ with the scalar field 𝑋 and the 5𝑑 background 𝐹(𝑥,𝑟) is the gauge theory description of the string theory on AdS₅×𝑆⁵ with the background 𝐹(𝑥,𝑟), its partition function will then equal 𝑒^{𝑊[𝐹(𝑥,𝑟)]} with 𝑊 being the 10𝑑 supergravity action. So we arrive at (2). 𝑋→𝑋 is a Weyl transformation, under which 𝐹₀(𝑥) must evolve as 𝐹(𝑥,𝑟) to preserve the partition function. We will show that, for such 𝐹(𝑥,𝑟), 𝛿𝑊[𝐹(𝑥,𝑟)]/𝛿𝐹(𝑥,𝑟)=0, so if 𝑊[𝐹(𝑥,𝑟)] is the action of the supergravity, 𝐹(𝑥,𝑟) will be the on-shell solution. The discussion can also be extended to SCFT₃ and SCFT₆. With no source term added, under the 𝑋→𝑋 transformation, the induced fields give the near horizon geometry of 𝑀2 and 𝑀5, respectively. The holography in AdS/CFT is very similar to the holography in noncritical string coupling with 2𝑑 being the gravity and with 𝑔→𝑔 replaced by 𝑋→𝑋.

This paper is organized as follows. In Section 2, we consider the free energy of the Matrix theory on supergravity background 𝐹 that is translation invariant along the 𝑥⁻ direction, and its relation with the 11𝑑 effective action of 𝐹. 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, SYM_𝑅×𝑆², SYM_𝑅×𝑆³_/𝑍𝑘, and SYM_𝑅×𝑆³ can be obtained [19–21]. In Section 4, we give a possible way to induce the radial dependent 5𝑑 fields from the 4𝑑 background fields in SYM₄.

2. Free Energy and the Effective Action of Supergravity

In this section, we will consider 𝑊(𝑅,𝐹), the free energy of the Matrix theory on a generic 11𝑑 supergravity background 𝐹 that is translation invariant along 𝑥⁻. Since the supermembrane action in light-cone gauge only contains one free parameter 𝑝⁺, as the discrete regularization of the supermembrane action, Matrix theory action 𝑆_𝑅,𝐹(𝑌) also has one free parameter which could be taken as 𝑅, the radius of 𝑥⁻. 𝑝⁺=𝑁/𝑅, and 𝑁→∞ is fixed. The concrete 𝑅 dependence of 𝑆_𝑅,𝐹(𝑌) is 𝑆_𝑅,𝐹(𝑌)=𝑆_{𝐹0,𝑅𝐹−,𝑅}²_𝐹−−(𝑌), where 𝐹₀, 𝐹₋, 𝐹₋₋ represent fields with zero, one, and two 𝑥⁻ indices. As a result, 𝑊(𝑅,𝐹)=𝑊(𝐹₀,𝑅𝐹₋,𝑅²𝐹₋₋). On the other hand, due to the coordinate invariance, the 𝑅 dependence of the effective action of the supergravity field 𝐹 is also 𝑆_eff(𝑅,𝐹)=𝑆_eff(𝐹₀,𝑅𝐹₋,𝑅²𝐹₋₋). From 𝑊(𝑅,𝐹), we can define Γ(𝑌)=𝑆_𝑅,𝐹(𝑌)+𝑊(𝑅,𝐹), with 𝐹 solved through 𝛿[𝑆_𝑅,𝐹(𝑌)+𝑊(𝑅,𝐹)]/𝛿𝐹=0. Γ(𝑌) is 𝑅, or equivalently, 𝑝⁺, independent. Γ(𝑌) could be taken as the effective action of the matrix configuration 𝑌. In fact, at the one-loop level, Γ(𝑌) and the standard effective action of the Matrix theory coincide. If 𝑊(𝑅,𝐹)=𝑆_eff(𝑅,𝐹), we will have Γ(𝑌)=Γ_𝑔(𝑌) with Γ_𝑔(𝑌) 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 𝑆_𝑅=𝑅𝑑𝑥⁺1×Tr2𝑅²𝐷₊𝑋^𝐼𝐷₊𝑋^𝐼+14𝑙⁶_𝑝𝑋^𝐼,𝑋^𝐽2−𝑖𝑅𝜃𝐷₊1𝜃+𝑙³_𝑝𝜃𝛾_𝐼𝑋^𝐼,𝜃, (16) where 𝑋^𝐼, 𝜃, and 𝐴₀ are 𝑁×𝑁 hermitian matrices with 𝑁→∞, and 𝐼=1⋯9. 𝑅→∞, 𝑝⁺=𝑁/𝑅,  𝑙_𝑝 is the Planck length.

𝑋^𝐼, 𝑥⁺, and 𝑅 have the dimension of length, so each commutator is multiplied by a factor 1/𝑙³_𝑝 to make the action dimensionless. With the replacement 𝑋^𝐼→𝑙_𝑝𝑋^𝐼, 𝑥⁺→𝑙_𝑝𝑥⁺, 𝑅→𝑙_𝑝𝑅, we get the action 𝑆_𝑅=𝑅𝑑𝑥⁺1×Tr2𝑅²𝐷₊𝑋^𝐼𝐷₊𝑋^𝐼+14𝑋^𝐼,𝑋^𝐽2−𝑖𝑅𝜃𝐷₊𝜃+𝜃𝛾_𝐼𝑋^𝐼,𝜃, (17) in which 𝑙_𝑝 is cancelled and 𝑋^𝐼, 𝜃, 𝑥⁺, and 𝑅 are all dimensionless. In the following, we will still adopt this convention, so 𝑙_𝑝 will not appear explicitly.

The 11𝑑 supergravity field, after the gauge fixing, has the nonzero components (𝑔₊₋, 𝑔₊₊, 𝑔_𝐼+, 𝑔_𝐼𝐽), (𝐶_𝐼𝐽+, 𝐶_𝐼𝐽𝐾), and (𝜓₊, 𝜓_𝐼) [22]. Based on the Hamiltonian in [22], one can write down the action of the bosonic membrane on such supergravity background: 𝑆^𝑏_𝐹0=𝑑𝑥⁺𝑑²𝜎𝑔₊₋𝑃⁺×𝑃⁺²𝐷₊𝑋^𝐼𝐷₊𝑋^𝐽2𝑔²₊₋𝑔_𝐼𝐽+𝑃⁺𝐷₊𝑋^𝐼𝑔₊₋𝐶^𝐽+𝑃⁺𝑔^𝐽₊𝑔₊₋𝑔_𝐼𝐽+𝑃⁺²𝑔₊₊2𝑔²₊₋−𝑋^𝐼,𝑋^𝐽𝑋^𝐾,𝑋^𝐿𝑔_𝐼𝐾𝑔_𝐽𝐿+𝑃⁺𝐶₊𝑔₊₋, (18) where 𝐶^𝐼={𝑋^𝐽,𝑋^𝐾}𝐶^𝐼_𝐽𝐾 and 𝐶₊={𝑋^𝐽,𝑋^𝐾}𝐶_+𝐽𝐾. Note that it is 𝑔₊₋/𝑃⁺ that appears. The Matrix theory version is 𝑆^𝑏_𝐹0=𝑑𝑥⁺×Tr𝑅𝑔₊₋𝐷₊𝑋^𝐼𝐷₊𝑋^𝐽2𝑅𝑔₊₋²𝑔_𝐼𝐽+𝐷₊𝑋^𝐼𝑅𝑔₊₋𝐶^𝐽+𝑔^𝐽₊𝑅𝑔₊₋𝑔_𝐼𝐽+14𝑋^𝐼,𝑋^𝐽𝑋^𝐾,𝑋^𝐿𝑔_𝐼𝐾𝑔_𝐽𝐿+𝑔₊₊2𝑅𝑔₊₋²+𝐶₊𝑅𝑔₊₋, (19) where 𝐶^𝐼=−(𝑖/2)[𝑋^𝐽,𝑋^𝐾]𝐶^𝐼_𝐽𝐾 and 𝐶₊=−(𝑖/2)[𝑋^𝐽,𝑋^𝐾]𝐶_+𝐽𝐾. Without the gauge fixing, 𝑔_𝐼−, 𝑔₋₋, 𝐶_−𝐼𝐽, 𝐶_+−𝐼, 𝜓₋≠0, so terms involving them can also be added: 𝑆^𝑏_𝑔𝐼−=𝑑𝑥⁺×Tr𝑅𝑔₊₋𝐷₊𝑋^𝐼𝐷₊𝑋^𝐽𝐷₊𝑋^𝐾2𝑅𝑔₊₋³𝑔_𝐽𝐾−𝐷₊𝑋^𝐼4𝑅𝑔₊₋𝑋^𝐾,𝑋^𝐿𝑋^𝑀,𝑋^𝑁×𝑔_𝐾𝑀𝑔_𝐿𝑁+𝐷₊𝑋^𝐿𝑅𝑔₊₋𝑋^𝐽,𝑋^𝑀𝑋^𝐼,𝑋^𝑁×𝑔_𝐿𝐽𝑔_𝑀𝑁×𝑅𝑔_𝐼− (20) and similarly for 𝑆^𝑏_𝑔−−, which is the sum of the 4th order terms (𝐷₊𝑋)⁴, (𝐷₊𝑋)²[𝑋,𝑋]², and [𝑋,𝑋]⁴. The current densities for 𝑔₋₋, 𝑔_𝐼−, 𝐶_−𝐼𝐽, 𝐶_+−𝐼, 𝜓₋ 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 𝐷₊𝑋⁻=(1/2)𝐷₊𝑋^𝐼𝐷₊𝑋_𝐼+(1/𝜈²){𝑋^𝐼,𝑋^𝐽}{𝑋_𝐼,𝑋_𝐽} and 𝐷_𝑎𝑋⁻=𝐷₊𝑋^𝐼𝐷_𝑎𝑋_𝐼 into the action, which could couple with the 𝑥⁻ indexed fields.

In all these terms, it is 𝑅𝑔₊₋, 𝑅𝑔_𝐼−, 𝑅²𝑔₋₋, 𝑅𝐶_−𝐼𝐽, 𝑅𝐶_+−𝐼, 𝑅𝜓₋ that are involved. (Similarly, in supermembrane action, it is 𝑔₊₋/𝑃⁺, 𝑔_𝐼−/𝑃⁺, 𝑔₋₋/𝑃⁺², 𝐶_−𝐼𝐽/𝑃⁺, 𝐶_+−𝐼/𝑃⁺, 𝜓₋/𝑃⁺ that will appear.)  𝑅 is the radius of 𝑥⁻. Under the parameterization transformation 𝑥⁺=𝑓(𝑥^′+), 𝑥⁻=𝛼𝑥^′−⇔𝑅=𝛼𝑅^′, 𝑔^′₊₊=𝑓^′2𝑥^′+𝑔₊₊,𝑔^′₊₋=𝛼𝑓^′𝑥^′+𝑔₊₋,𝑔^′_+𝐼=𝑓^′𝑥^′+𝑔_+𝐼,𝑔^′_−𝐼=𝛼𝑔_−𝐼,𝑔^′₋₋=𝛼²𝑔₋₋,𝐶^′_+−𝐽=𝛼𝑓^′𝑥^′+𝐶_+−𝐽,𝐶^′_+𝐼𝐽=𝑓^′𝑥^′+𝐶_+𝐼𝐽,𝐶^′_−𝐼𝐽=𝛼𝐶_−𝐼𝐽, (21) the action is invariant. This is consistent with the coordinate transformation 𝑑𝑠²=𝑔₊₊𝑑𝑓𝑥^′+𝑑𝑓𝑥^′++2𝑔_𝐼+𝑑𝑥^𝐼𝑑𝑓𝑥^′++𝑔_𝐼𝐽𝑑𝑥^𝐼𝑑𝑥^𝐽+2𝛼𝑔₊₋𝑑𝑓𝑥^′+𝑑𝑥^′−+2𝛼𝑔_𝐼−𝑑𝑥^𝐼𝑑𝑥^′−+𝛼²𝑔₋₋𝑑𝑥^′−𝑑𝑥^′−=𝑔₊₊𝑓^′2𝑥^′+𝑑𝑥^′+𝑑𝑥^′++2𝑓^′𝑥^′+𝑔_𝐼+𝑑𝑥^𝐼𝑥^′++𝑔_𝐼𝐽𝑑𝑥^𝐼𝑑𝑥^𝐽+2𝛼𝑔₊₋𝑓^′𝑥^′+𝑑𝑥^′+𝑑𝑥^′−+2𝛼𝑔_𝐼−𝑑𝑥^𝐼𝑑𝑥^′−+𝛼²𝑔₋₋𝑑𝑥^′−𝑑𝑥^′−,𝐶=𝐶_𝐼𝐽𝐾𝑑𝑥^𝐼∧𝑑𝑥^𝐽∧𝑑𝑥^𝐾+𝐶_𝐼𝐽+𝑑𝑥^𝐼∧𝑑𝑥^𝐽∧𝑑𝑓𝑥^′++𝛼𝐶_𝐼𝐽−𝑑𝑥^𝐼∧𝑑𝑥^𝐽∧𝑑𝑥^′−+𝛼𝐶_𝐼+−𝑑𝑥^𝐼∧𝑑𝑓𝑥^′+∧𝑑𝑥^′−=𝐶_𝐼𝐽𝐾𝑑𝑥^𝐼∧𝑑𝑥^𝐽∧𝑑𝑥^𝐾+𝐶_𝐼𝐽+𝑓^′𝑥^′+𝑑𝑥^𝐼∧𝑑𝑥^𝐽∧𝑑𝑥^′++𝛼𝐶_𝐼𝐽−𝑑𝑥^𝐼∧𝑑𝑥^𝐽∧𝑑𝑥⁻+𝛼𝐶_𝐼+−𝑓^′𝑥^′+𝑑𝑥^𝐼∧𝑑𝑥^′+∧𝑑𝑥⁻. (22)

𝑆_𝑅,𝐹, the supersymmetric extension of (19) and (20), is not constructed yet. With 𝑆_𝑅,𝐹 given, the current density for 𝐹 can be defined as 𝑉_𝐹(𝑌)(𝑥)=𝛿𝑆_𝑅,𝐹(𝑌)𝛿𝐹(𝑥)_𝐹. (23) In particular, when 𝐹 is the flat background, for which, the only nonvanishing fields are 𝑔₊₋=−1 and 𝑔_𝐼𝐽=𝛿_𝐼𝐽, 𝑉_𝐹(𝑌)(𝑥) reduces to the localized vertex operator of the supergravity field 𝐹. In [3, 4, 23–25], the vertex operators for various supergravity fields are constructed. Although the exact 𝑆_𝑅,𝐹 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]: 𝑆_𝑅,𝐹=𝑆_𝑅−𝑅𝑑𝑥⁺×STr𝑉_{ℎ𝐼𝐽(𝑋)}ℎ_𝐼𝐽(𝑋)+𝑉_{ℎ𝐼+(𝑋)}ℎ_𝐼+(𝑋)+𝑉_ℎ++(𝑋)ℎ₊₊(𝑋)+𝑉_{ℎ𝐼−(𝑋)}ℎ_𝐼−(𝑋)+𝑉_{ℎ+−(𝑋)}ℎ₊₋(𝑋)+𝑉_{ℎ−−(𝑋)}ℎ₋₋(𝑋)+𝑉_{𝐶𝐼𝐽𝐾(𝑋)}𝐶_𝐼𝐽𝐾(𝑋)+𝑉_{𝐶+𝐽𝐾(𝑋)}𝐶_+𝐽𝐾(𝑋)+𝑉_{𝐶𝐼𝐽−(𝑋)}𝐶_𝐼𝐽−(𝑋)+𝑉_{𝐶+−𝐾(𝑋)}𝐶_+−𝐾(𝑋)+𝑉_𝜓(𝑋)𝜓(𝑋)+𝑉_𝜓+(𝑋)𝜓₊(𝑋)+𝑉_{𝜓−(𝑋)}𝜓₋(𝑋), (24) where, for example, 𝑉_{ℎ𝐼+(𝑋)}ℎ_𝐼+(𝑋)=𝐷₊𝑋^𝐼𝑅−14𝜃𝛾^𝐼𝐽𝜃𝜕𝜕𝑋^𝐽ℎ_𝐼+(𝑋)𝑅. (25)ℎ_𝐼𝐽(𝑋),…,𝜓₋(𝑋) are 11𝑑 background fields with the 𝑐-number coordinate 𝑥 replaced by the matrix coordinate 𝑋. The background fields are only the functions of (𝑥⁺,𝑥¹,…,𝑥⁹); they are the zero modes of the 11𝑑 supergravity along 𝑥⁻.

The vertex operator can take three different forms. First, it can be the operator defined in a 1𝑑 SYM theory, just as that in AdS/CFT. Then, the background fields, like ℎ_𝐼+(𝑋), should be expanded as the Taylor series: ℎ_𝐼+𝑥⁺,𝑋¹,…,𝑋⁹=^∞_𝑛=01𝜕𝑛!^𝑘1⋯𝜕_𝑘𝑛ℎ_𝐼+𝑥⁺,0,…,0𝑋^𝑘1⋯𝑋^𝑘𝑛, (26) with (𝜕_𝑘1⋯𝜕_𝑘𝑛ℎ_𝐼+)(𝑥⁺,0,…,0) the derivative of ℎ_𝐼+(𝑥⁺,𝑥) at (𝑥⁺,0) [3, 23–25]. STr𝑉_{ℎ𝐼+(𝑋)}ℎ_𝐼+(𝑋=_∞𝑛=0)1𝜕𝑛!^𝑘1⋯𝜕_𝑘𝑛ℎ_𝐼+𝑥⁺1,0,…,0×STr𝑅𝐷₊𝑋^𝐼𝑅−14𝜃𝛾^𝐼𝐽𝜃𝜕𝜕𝑋^𝐽𝑋^𝑘1⋯𝑋^𝑘𝑛=^∞_𝑛=01𝜕𝑛!^𝑘1⋯𝜕_𝑘𝑛ℎ_𝐼+𝑥⁺,0,…,0𝐼^{𝑘1⋯𝑘𝑛}_ℎ𝐼+𝑥⁺. (27) Since all background fields enter into the action in the form of 𝑓(𝑥⁺,0,…,0), the 𝑈(𝑁)SYM₁ lives at (𝑥⁺,0,…,0). When 𝑋^𝑖⟶𝑋^𝑖−𝑎^𝑖,𝑓𝑥⁺,0,…,0⟶𝑓𝑥⁺,𝑎¹,…,𝑎⁹, (28)SYM₁ undergoes a translation. Nothing specifies where the SYM₁ should be, so one may put it at any point in 𝑅⁹. Similar to AdS/CFT, there is a one-to-one correspondence between operators and fields. However, no holography is present here. Fields living on SYM₁ are Taylor series coefficients, which uniquely determines the 11𝑑 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^{𝑘1⋯𝑘𝑛} in AdS/CFT, the moment operators 𝐼^{𝑘1⋯𝑘𝑛} do not need to be traceless. As a result, {1,𝐼^𝑘1,…,𝐼^{𝑘1⋯𝑘𝑛},…} couples with the 10𝑑 field, while {1,Φ^𝑘1,…,Φ^{𝑘1⋯𝑘𝑛},…} only couples with the 9𝑑 field.

In the second form, the vertex operator is defined in 10𝑑 spacetime: 𝛿(𝑥)𝐹𝑥𝛿𝐹⁺,𝑋¹,…,𝑋⁹=𝛿𝑥⁰−𝑥⁺𝛿⁹𝑥^𝐼−𝑋^𝐼=𝛿¹⁰𝑥^𝜇−𝑋^𝜇, (29) where 𝑋^𝐼 are 𝑁×𝑁 matrices and for uniformity; we have set 𝑋⁰=𝑥⁺𝟙_𝑁×𝑁. This is a matrix generalization of the 𝛿-function. In special situations, when all of the 𝑋^𝐼 are diagonal, that is, 𝑋^𝐼=diag{𝑥^𝐼₁,𝑥^𝐼₂,…,𝑥^𝐼_𝑁}, (29) becomes 𝛿(𝑥)𝐹𝑥𝛿𝐹⁺,𝑋¹,…,𝑋⁹=diag𝛿¹⁰𝑥^𝜇−𝑥^𝜇₁,𝛿¹⁰𝑥^𝜇−𝑥^𝜇₂,…,𝛿¹⁰𝑥^𝜇−𝑥^𝜇_𝑁. (30) With the generalized 𝛿-function, it is straightforward to write down the current densities for various fields. For ℎ_𝐼+, we have 𝑉_ℎ𝐼+(𝑥)=𝛿𝛿ℎ_𝐼+(𝑥)−𝑅𝑑𝑥⁺STr𝑉_{ℎ𝐼+(𝑋)}ℎ_𝐼+(𝑋)=−𝑅𝑑𝑥⁺STr𝑉_{ℎ𝐼+(𝑋)}𝛿¹⁰(𝑥−𝑋). (31)

It is not convenient to deal with the 𝛿-function. One may want to do a Fourier transformation, which gives the third representation of the vertex operator: 𝑉_ℎ𝐼+𝑝=𝑑¹⁰𝑥𝑉_ℎ𝐼+(𝑥)𝑒^𝑖𝑝𝑥=−𝑅𝑑𝑥⁺STr𝑉_{ℎ𝐼+(𝑋)}𝑒^𝑖𝑝𝑋=−𝑅𝑑𝑥⁺−𝐷STr⁺𝑋^𝐼𝑅+𝑖4𝜃𝛾^𝐼𝐽𝜃𝑝_𝐽𝑒^𝑖𝑝𝑋. (32)

2.2. Partition Function of Matrix Theory on Curved Background

Suppose the exact form of 𝑆_𝐹 is given, and the partition function of Matrix theory on supergravity background is 𝑍(𝑅,𝐹)=𝑒^{𝑊(𝑅,𝐹)}=𝑑𝑋𝑑𝜃𝑑𝐴₀𝑒^{−𝑆𝑅,𝐹(𝑋,𝜃,𝐴0)}=[𝑑𝑌]𝑒^{−𝑆𝑅,𝐹(𝑌)}, (33) where 𝐹 is the 11𝑑 background field (𝑔₊₋, 𝑔₊₊, 𝑔_𝐼+, 𝑔_𝐼𝐽, 𝑔₋₋, 𝑔_𝐼−), (𝐶_𝐼𝐽+, 𝐶_𝐼𝐽𝐾, 𝐶_𝐼𝐽−, 𝐶_𝐼+−), and (𝜓₊, 𝜓_𝐼, 𝜓₋) mentioned before, and 𝑌 collectively represents (𝑋^𝐼,𝜃,𝐴₀).

In gauge/string correspondence, partition function is an important quantity, the value of which should be equal on both sides. For AdS₅/CFT₄ correspondence, it is expected that 𝑍_SYM4=exp{−𝐹} (34) should hold, where 𝐹=∑_𝑛ℎ=0𝐹_𝑛ℎ is the free energy of the strings on AdS₅×𝑆⁵, 𝑍_SYM4 and exp{−𝐹} are the partition function of SYM₄ and the partition function of the second quantized type IIB string theory on AdS₅×𝑆⁵, 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 𝑀-theory sector with the light-cone momentum 𝑝⁺ is described by the Matrix model with 𝑝⁺=𝑁/𝑅, for which the partition function is again (33).

Suppose 𝑆^𝑀_𝑃⁺_,𝐹(𝑌) is the membrane action on background 𝐹. 𝑆^𝑀_𝑃⁺_,𝐹(𝑌) should be general covariant, so, for 𝑃⁺=𝑃^′+/𝛼, 𝑥⁺=𝑓(𝑥^′+), and 𝑋=𝑔(𝑋^′), there is 𝑆^𝑀_𝑃⁺_,𝐹(𝑌)=𝑆^𝑀_{𝑃′+,𝐹′}(𝑌^′), where 𝐹^′ is the field coming from the coordinate transformation: 𝑥⁻=𝛼𝑥^′−,𝑥⁺=𝑓𝑥^′+,𝑥=𝑔𝑥^′. (35) For the bosonic action, we can see this is indeed the case. If [𝑑𝑌]=[𝑑𝑌^′], that is, the path integral measure is coordinate independent, we will have [𝑑𝑌]𝑒^{−𝑆𝑀𝑃+,𝐹(𝑌)}=𝑑𝑌^′𝑒^{−𝑆𝑀𝑃′′+,𝐹(𝑌′)}. (36) After the matrix regularization, the membrane configurations 𝑌(𝑥⁺,𝜎¹,𝜎²) and 𝑌^′(𝑥⁺,𝜎¹,𝜎²)  become the matrix configurations𝑌(𝑥⁺) and 𝑌^′(𝑥⁺), and there is 𝑍(𝑅,𝐹)=𝑒^{𝑊(𝑅,𝐹)}=[𝑑𝑌]𝑒^{−𝑆𝑅,𝐹(𝑌)}=𝑑𝑌^′𝑒^{−𝑆𝑅′′,𝐹(𝑌′)}=𝑒^{𝑊(𝑅′,𝐹′)}=𝑍𝑅^′,𝐹^′. (37)𝑊(𝑅,𝐹)=𝑊(𝑅^′,𝐹^′). At least restricted to (35), 𝑊(𝑅,𝐹) is the diffeomorphism invariant functional of 𝐹.

Let 𝐹₀, 𝐹₋, 𝐹₋₋ represent fields with zero, one, and two 𝑥⁻ indices. Since 𝑆_𝑅,𝐹(𝑌)=𝑆_{𝐹0,𝑅𝐹−,𝑅}²_𝐹−−(𝑌),𝑊(𝑅,𝐹)=𝑊𝐹₀,𝑅𝐹₋,𝑅²𝐹₋₋. (38)(39)For the 11𝑑 supergravity fields (𝐹₀,𝐹₋,𝐹₋₋), which are translation invariant along the 𝑥⁻ direction, 𝑥⁻∈[0,2𝜋𝑅), the 11𝑑 supergravity effective action is 𝑆_eff(𝑅;𝐹₀,𝐹₋,𝐹₋₋).  𝑆_eff is invariant under the coordinate transformation: 𝑥⁻⟶𝑥⁻𝑅,𝐹₀⟶𝐹₀,𝐹₋⟶𝑅𝐹₋,𝐹₋₋⟶𝑅²𝐹₋₋, (40) so 𝑆_eff𝑅;𝐹₀,𝐹₋,𝐹₋₋=𝑆_eff1;𝐹₀,𝑅𝐹₋,𝑅²𝐹₋₋. (41) The radius of 𝑥⁻ is absorbed in 𝐹₋, 𝐹₋₋, as is in (39). In this respect, 𝑊 is consistent with 𝑆_eff.

Let 𝐹=𝐹+𝐹, where 𝐹 is the flat background with 𝑔₊₋=−1, 𝑔_𝐼𝐽=𝛿_𝐼𝐽, and the rest fields being zero. 𝑊(𝑅,𝐹)=0: 𝑊(𝑅,𝐹)=𝑑¹⁰𝑥Γ^𝑐_𝑅,𝐹(𝑥)𝐹(𝑥)+1𝑑2!¹⁰𝑥₁×𝑑¹⁰𝑥₂Γ^𝑐_𝑅,𝐹𝑥₁,𝑥₂𝐹𝑥₁𝐹𝑥₂+⋯=_𝑛1𝑑𝑛!¹⁰𝑥₁⋯𝑑¹⁰𝑥_𝑛Γ^𝑐_𝑅,𝐹𝑥₁,…,𝑥_𝑛×𝐹𝑥₁⋯𝐹𝑥_𝑛, (42) where (𝐹 collectively represents supergravity fields. For example, Γ^𝑐_{𝑅,𝑔++,𝑔−−}(𝑥₁,𝑥₂)=𝛿²𝑊(𝑅,𝐹)/𝛿𝑔₊₊(𝑥₁)𝛿𝑔₋₋(𝑥₂)|_𝐹=𝐹.)Γ^𝑐_𝑅,𝐹𝑥₁,…,𝑥_𝑛=𝛿^𝑛𝑊(𝑅,𝐹)𝛿𝐹(𝑥₁)⋯𝛿𝐹𝑥_{𝑛𝐹=𝐹}. (43) For the same 𝐹, the 11𝑑 supergravity effective action 𝑆_eff(𝑅,𝐹) is 𝑆_eff(𝑅,𝐹)=_𝑛1𝑛!₀^2𝜋𝑅𝑑𝑥⁻₁×𝑑𝑥⁺₁𝑑⁹𝐱₁⋯₀^2𝜋𝑅𝑑𝑥⁻_𝑛𝑑𝑥⁺_𝑛𝑑⁹𝐱_𝑛×Γ^𝑐_𝐹𝑥⁻₁,𝑥⁺₁,𝐱₁;…;𝑥⁻_𝑛,𝑥⁺_𝑛,𝐱_𝑛×𝐹𝑥⁺₁,𝐱₁⋯𝐹𝑥⁺_𝑛,𝐱_𝑛. (44)Γ^𝑐_𝐹(𝑥⁻₁,𝑥⁺₁,𝐱₁;…;𝑥⁻_𝑛,𝑥⁺_𝑛,𝐱_𝑛) is the vertex function of the 11𝑑 supergravity. If 𝑆_eff(𝑅,𝐹)=𝑊(𝑅,𝐹), there will be Γ^𝑐_𝑅,𝐹𝑥₁,…,𝑥_𝑛=Γ^𝑐_𝑅,𝐹𝑥⁺₁,𝐱₁;…;𝑥⁺_𝑛,𝐱_𝑛=₀^2𝜋𝑅𝑑𝑥⁻₁⋯₀^2𝜋𝑅𝑑𝑥⁻_𝑛Γ^𝑐_𝐹𝑥⁻₁,𝑥⁺₁,𝐱₁;…;𝑥⁻_𝑛,𝑥⁺_𝑛,𝐱_𝑛. (45)

In linear gravity approximation, 𝛿𝑉_𝐹(𝑥)/𝛿𝐹(𝑦)=0, where 𝑉_𝐹 is the current density coupling with 𝐹(𝑥) as is defined in (23): Γ^𝑐_𝑅,𝐹𝑥₁,…,𝑥_𝑛=(−1)^𝑛𝑉_𝐹𝑥₁⋯𝑉_𝐹𝑥_𝑛𝑐. (46)⟨𝑉_𝐹(𝑥₁)⋯𝑉_𝐹(𝑥_𝑛)⟩_𝑐 is the connected correlation function of the current density, in contrast to 𝑉_𝐹𝑥₁⋯𝑉_𝐹𝑥_𝑛=𝑑𝑋𝑑𝜃𝑑𝐴₀𝑒^−𝑆0𝑉_𝐹𝑥₁⋯𝑉_𝐹𝑥_𝑛=(−1)^𝑛𝛿^𝑛𝑍(𝑅,𝐹)𝛿𝐹𝑥₁⋯𝛿𝐹𝑥_{𝑛𝐹=𝐹}, (47) which is the correlation function of the current density. In (42), 𝑊(𝑅,𝐹) is expanded around the flat background 𝐹. One can of course expand 𝑊(𝑅,𝐹) on a different background, giving rise to the different Γ^𝑐_𝑅,𝐹(𝑥₁,…,𝑥_𝑛).

In terms of 𝑉_𝐹(𝑝), 𝑉_𝐹𝑥₁⋯𝑉_𝐹𝑥_𝑛𝑐=𝑑¹⁰𝑝₁⋯𝑑¹⁰𝑝_𝑛𝑒^{−𝑖(𝑝1𝑥1+⋯+𝑝𝑛𝑥𝑛)}×𝑉_𝐹𝑝₁⋯𝑉_𝐹𝑝_𝑛𝑐 (48) Note that 𝑉_𝐹(𝑋)(𝑥⁺,𝑥^𝐼−𝑎^𝐼)=𝑉_{𝐹(𝑋+𝑎)}(𝑥⁺,𝑥^𝐼), 𝑆₀(𝑋+𝑎)=𝑆₀(𝑋), so 𝑉_𝐹𝑥₁⋯𝑉_𝐹𝑥_𝑛𝑐=𝑉_𝐹𝑥₁+𝑎₁⋯𝑉_𝐹𝑥_𝑛+𝑎_𝑛𝑐. (49) The correlation function is translation invariant. ⟨𝑉_𝐹(𝑝₁)⋯𝑉_𝐹(𝑝_𝑛)⟩_𝑐=0 if 𝑝₁+⋯+𝑝_𝑛≠0.

Let us first consider the one point function 𝑉_𝐹(𝑥())=−𝛿𝑊𝑅,𝐹𝛿𝐹(𝑥)_𝐹=𝐹=𝑣_𝐹. (50)𝑣_𝐹 is the vacuum expectation value of the current density, when the background field is 𝐹, and is a constant in spacetime. 𝑆_𝑅 is 𝑆𝑂(9) invariant, so 𝑣_𝐹 should also be 𝑆𝑂(9) invariant. As a result, 𝑣_𝑔𝐼+=𝑣_𝑔𝐼−=𝑣_{𝐶𝐼𝐽𝐾}=𝑣_{𝐶𝐼𝐽+}=𝑣_{𝐶𝐼𝐽−}=𝑣_𝐶𝐼+−=𝑣_𝜓𝐼=0. (51)𝑣_𝑔𝐼𝐽=0 if the traceless condition is imposed. 𝑣_𝜓+=𝑣_𝜓−=0 due to the supersymmetry. The nonvanishing current densities are 𝑣_ℎ++, 𝑣_ℎ−−, and 𝑣_ℎ+−. In particular, 𝑣_ℎ++=𝑝⁺/𝐿⁹ is the vacuum expectation value of the light-cone momentum density. For the generic value of 𝐹, 𝑉_𝐹(𝑥)_𝐹()=−𝛿𝑊𝑅,𝐹𝛿𝐹(𝑥)_𝐹=𝐹+𝐹=𝑣_𝐹(𝑥)=_𝑛−1()!×𝑑𝑛−1₁₀𝑥₁⋯𝑑¹⁰𝑥_𝑛−1Γ^𝑐_𝑅,𝐹𝑥,𝑥₁,…,𝑥_𝑛−1×𝐹𝑥₁⋯𝐹𝑥_𝑛−1 (52) 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 𝑊(𝑅,𝐹) is the effective action of the supergravity fields, on SUGRA solution background, there will be ⟨𝑉_𝐹(𝑥)⟩_𝐹=0 except for ℎ₊₊, 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 𝐹(𝑥), 𝑣_𝐹(𝑥) is uniquely determined. Conversely, different 𝐹(𝑥) may result in the same 𝑣_𝐹(𝑥). This is quite like the source-gravity coupled system. For the given gravity field, the density of the source can be obtained through 𝛿𝑆_𝑔/𝛿𝑔^𝜇𝜈=−𝑇_𝜇𝜈. 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_𝐹(𝑥)=0, 𝐹(𝑥)=0. 𝐹(𝑥) could be interpreted as the field generated by the current density 𝑣_𝐹(𝑥). 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_𝐹 and 𝐹, and so a Legendre transformation is possible. Before that, we will first define Γ(𝑌): Γ(𝑌)=𝑆_𝑅,𝐹(𝑌)(),𝑒+𝑊𝑅,𝐹_Γ(𝑌)=𝑑𝑌𝑒^{−𝑆𝑅,𝐹(}𝑌_{)+𝑆𝑅,𝐹(𝑌)}. (53)(54)𝐹 is solved from the equation 𝑑𝑌𝛿(𝑥)𝑒𝛿𝐹^{−𝑆𝑅,𝐹(}𝑌_{)+𝑆𝑅,𝐹(𝑌)}=𝑑𝑌𝑒^{−𝑆𝑅,𝐹(}𝑌^)+𝑆_𝑅,𝐹^(𝑌)−𝛿𝑆_𝑅,𝐹𝑌(𝑥)+𝛿𝐹𝛿𝑆_𝑅,𝐹(𝑌)𝛿𝐹(𝑥)=0 (55) or equivalently, 𝑑𝑌𝑒^{−𝑆𝑅,𝐹(}𝑌⁾𝛿𝑆_𝑅,𝐹𝑌(𝑥)=𝛿𝐹𝛿𝑆_𝑅,𝐹(𝑌)(𝑥)()𝛿𝐹=−𝛿𝑊𝑅,𝐹(𝑥).𝛿𝐹 (56) In some sense, 𝐹(𝑥) is the field generated by 𝑌. Take a derivative of (54) with respect to 𝑌; using (55), we get 𝛿Γ(𝑌)=𝛿𝑌𝛿𝑆_𝑅,𝐹(𝑌),𝛿𝑌 (57) where 𝐹 is solved from (55). The variation on the right-hand side of (57) only acts on 𝑌 with 𝐹 being fixed.

Recall that the 𝑅 dependence of 𝑆_𝑅,𝐹(𝑌) only comes from 𝑅𝐹₋ and 𝑅²𝐹₋₋, so if the solution of (56) is (𝐹₀,𝐹₋,𝐹₋₋), with 𝑅 replaced by 𝛼𝑅, the solution will become (𝐹₀,𝐹₋/𝛼,𝐹₋₋/𝛼²). 𝑆_𝑅,𝐹(𝑌), 𝑊(𝑅,𝐹), and Γ(𝑌) remain invariant. Γ(𝑌) is 𝑅-independent. In supermembrane picture, suppose there are two supermembrane theories with the light-cone momentum 𝑃⁺ and 𝑃^′+, and consider the fields generated by the same configuration 𝑌(𝑥⁺,𝜎¹,𝜎²). Let 𝑥^′−→𝑃^′+𝑥^′−/𝑃⁺, 𝑃^′+→𝑃⁺, in this frame, the generated fields are the same, changing back, we get the relation 𝐹₀=𝐹₀, 𝐹₋/𝑃⁺=𝐹^′₋/𝑃^′+, 𝐹₋₋/𝑃⁺²=𝐹^′₋₋/𝑃^′+2. When plugged into 𝑆^𝑀_𝑃⁺_,𝐹(𝑌), the 𝑅 dependence is canceled, so the obtained Γ(𝑌) is the same.

With these properties collected, we may consider the possible interpretation of Γ(𝑌) and 𝑊(𝑅,𝐹). If 𝑊(𝑅,𝐹) is the effective action of supergravity, since 𝑆_𝑅,𝐹(𝑌) is the matrix theory action on background 𝐹, Γ(𝑌) 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 𝑌. 𝛿Γ(𝑌)/𝛿𝑌=𝛿𝑆_𝑅,𝐹(𝑌)/𝛿𝑌=0 is the quantum corrected equation of motion, which differs from the classical equation of motion 𝛿𝑆₀(𝑌)/𝛿𝑌=0 in that the background fields in former are generated by the configuration 𝑌 itself, while the background fields in latter are given. For time-independent 𝑌, the stationary point of the effective action gives the vacuum configuration. In this case, 𝛿Γ(𝑌)/𝛿𝑌=𝛿𝑆_𝑅,𝐹(𝑌)/𝛿𝑌=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 𝑊(𝑅,𝐹) is the effective action of supergravity, which, however, is unproved.

Let us continue to explore the properties of Γ(𝑌) and 𝑊(𝑅,𝐹). For simplicity, let 𝑅=1, or, in other words, let the 𝑥⁻ indexed fields absorb 𝑅. In a weakly curved background, 𝑆_𝐹(𝑌)=𝑆₀(𝑌)+∫𝑑¹⁰𝑥𝐹(𝑥)𝑉_𝐹(𝑌)(𝑥), Γ(𝑌)=𝑆₀(𝑌)+Γ(𝑣_𝐹), where Γ𝑣_𝐹=𝑑¹⁰𝑥𝐹(𝑥)𝑣_𝐹(𝑥)+𝑊𝐹+𝐹 (58) is the Legendre transformation of 𝑊(𝐹): −𝛿𝑊(𝐹)𝛿𝐹(𝑥)𝐹=𝑣_𝐹=𝑉_𝐹(𝑌)(𝑥),𝛿Γ𝑣_𝐹𝛿𝑣_𝐹(𝑥)_𝑣𝐹=𝐹(𝑥). (59) Let Λ_𝐹𝑥₁,…,𝑥_𝑛=𝛿^𝑛Γ𝑣_𝐹𝛿𝑣_𝐹𝑥₁⋯𝛿𝑣_𝐹𝑥_{𝑛𝑣𝐹=0}, (60) then Γ(𝑌)=𝑆₀(𝑌)+_𝑛1×𝑑𝑛!₁₀𝑥₁⋯𝑑¹⁰𝑥_𝑛Λ_𝐹𝑥₁,…,𝑥_𝑛×𝑣_𝐹𝑥₁⋯𝑣_𝐹𝑥_𝑛=𝑆₀(𝑌)+_𝑛1×𝑑𝑛!₁₀𝑥₁⋯𝑑¹⁰𝑥_𝑛Λ_𝐹𝑥₁,…,𝑥_𝑛×𝑉_𝐹(𝑌)𝑥₁⋯𝑉_𝐹(𝑌)𝑥_𝑛. (61) For the given 𝑣_𝐹(𝑥), 𝐹(𝑥)=_𝑛1()!×𝑑𝑛−1₁₀𝑥₁⋯𝑑¹⁰𝑥_𝑛−1Λ_𝐹𝑥,𝑥₁,…,𝑥_𝑛−1×𝑣_𝐹𝑥₁⋯𝑣_𝐹𝑥_𝑛−1. (62) Equation (52) gives the current density 𝑣_𝐹(𝑥) generated by the field 𝐹(𝑥); (62) gives the field 𝐹(𝑥) generated by the current density 𝑣_𝐹(𝑥). If Λ_𝐹 is the connected Green’s function of supergravity, 𝐹(𝑥) will be the vacuum expectation value of the supergravity field 𝐹 in presence of the source 𝑣_𝐹. In classical level, 𝐹(𝑥) could be calculated by 𝛿𝑆_𝑔/𝛿𝐹(𝑥)=−𝑣_𝐹(𝑥) with 𝑆_𝑔 the classical action of supergravity: Γ^𝑐_𝐹(𝑥−𝑧)=−𝛿𝑣_𝐹(𝑥)𝛿𝐹(𝑧)𝐹₌₀,Λ_𝐹=𝛿𝑧−𝑦²Γ𝑣_𝐹𝛿𝑣_𝐹(𝑧)𝛿𝑣_𝐹𝑦_𝑣𝐹=0=𝛿𝐹(𝑧)𝛿𝑣_𝐹𝑦_𝑣𝐹=0, (63) so −𝑑¹⁰𝑧Γ^𝑐_𝐹(𝑥−𝑧)Λ_𝐹𝑧−𝑦=𝛿¹⁰𝑥−𝑦. (64)−Λ_𝐹(𝑥₁−𝑥₂) is the inverse of Γ^𝑐_𝐹(𝑥₁−𝑥₂). Subsequently, Λ_𝐹𝑥₁,𝑥₂,𝑥₃=𝑑¹⁰𝑦₁𝑑¹⁰𝑦₂×𝑑¹⁰𝑦₃Λ_𝐹𝑦₁−𝑥₁Λ_𝐹𝑦₂−𝑥₂×Λ_𝐹𝑦₃−𝑥₃Γ^𝑐_𝐹𝑦₁,𝑦₂,𝑦₃,Γ^𝑐_𝐹𝑥₁,𝑥₂,𝑥₃=−𝑑¹⁰𝑦₁𝑑¹⁰𝑦₂×𝑑¹⁰𝑦₃Γ^𝑐_𝐹𝑦₁−𝑥₁Γ^𝑐_𝐹𝑦₂−𝑥₂×Γ^𝑐_𝐹𝑦₃−𝑥₃Λ_𝐹𝑦₁,𝑦₂,𝑦₃.Λ_𝐹𝑥₁,𝑥₂,𝑥₃,𝑥₄=𝑑¹⁰𝑦₁𝑑¹⁰𝑦₂𝑑¹⁰𝑦₃×𝑑¹⁰𝑦₄Λ_𝐹𝑦₁−𝑥₁Λ_𝐹𝑦₂−𝑥₂×Λ_𝐹𝑦₃−𝑥₃Λ_𝐹𝑦₄−𝑥₄×Γ^𝑐_𝐹𝑦₁,𝑦₂,𝑦₃,𝑦₄+𝑑¹⁰𝑧₁×𝑑¹⁰𝑧₂Λ_𝐹𝑧₁−𝑧₂×Γ^𝑐_𝐹𝑧₁,𝑦₁,𝑦₄×Γ^𝑐_𝐹𝑧₂,𝑦₂,𝑦₃+Γ^𝑐_𝐹𝑧₁,𝑦₂,𝑦₄×Γ^𝑐_𝐹𝑧₂,𝑦₁,𝑦₃+Γ^𝑐_𝐹𝑧₁,𝑦₃,𝑦₄×Γ^𝑐_𝐹𝑧₂,𝑦₁,𝑦₂,Γ^𝑐_𝐹𝑥₁,𝑥₂,𝑥₃,𝑥₄=𝑑¹⁰𝑦₁𝑑¹⁰𝑦₂𝑑¹⁰𝑦₃×𝑑¹⁰𝑦₄Γ^𝑐_𝐹𝑦₁−𝑥₁Γ^𝑐_𝐹𝑦₂−𝑥₂×Γ^𝑐_𝐹𝑦₃−𝑥₃Γ^𝑐_𝐹𝑦₄−𝑥₄×Λ_𝐹𝑦₁,𝑦₂,𝑦₃,𝑦₄+𝑑¹⁰𝑧₁×𝑑¹⁰𝑧₂Γ^𝑐_𝐹𝑧₁−𝑧₂×Λ_𝐹𝑧₁,𝑦₁,𝑦₄×Λ_𝐹𝑧₂,𝑦₂,𝑦₃+Λ_𝐹𝑧₁,𝑦₂,𝑦₄×Λ_𝐹𝑧₂,𝑦₁,𝑦₃+Λ_𝐹𝑧₁,𝑦₃,𝑦₄×Λ_𝐹𝑧₂,𝑦₁,𝑦₂. (65) The relation between Λ_𝐹 and Γ^𝑐_𝐹 shows that Λ_𝐹 and Γ^𝑐_𝐹 are the connected Green’s function and the vertex function of a particular quantum field theory. Λ_𝐹(𝑧−𝑦)=(𝛿𝐹(𝑧)/𝛿𝑣_𝐹(𝑦))|_𝑣𝐹=0 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 𝑌=𝑌+𝜂, 𝑒^Γ(𝑌)=𝑑𝜂𝑒^{−𝑆𝐹(𝑌+𝜂)+𝑆𝐹(𝑌)}.−𝑆_𝐹𝑌+𝜂+𝑆_𝐹(𝑌)=−𝑑𝜏𝛿𝑆_𝐹(𝑌)(𝜏)𝜂(𝜏)−1𝛿𝑌2𝑑𝜏𝑑𝜏^′𝛿²𝑆_𝐹(𝑌)𝛿𝑌(𝜏)𝛿𝑌𝜏^′×𝜂(𝜏)𝜂𝜏^′+⋯=−𝑆_𝐹⁽¹⁾𝑌,𝜂−𝑆_𝐹⁽²⁾𝑌,𝜂+⋯. (66)𝑆_𝐹 could be written as (We only write 𝜕²_𝜏, but one should be aware that the fermionic kinetic term 𝜕_𝜏 also exists.)𝑆_𝐹(𝑌)=1𝑑𝜏2𝑌𝜕²_𝜏𝑌+𝑉₀(𝑌)+𝑉(𝐹,𝑌)=𝑆₀(𝑌)+𝑑𝜏𝑉(𝐹,𝑌) (67) with 𝑉(𝐹,𝑌) taking the form as that in (27). For the given 𝑌, solve 𝐹₀ from 𝛿𝑆_𝐹0(𝑌)/𝛿𝑌=0, and expand 𝑆_𝐹^(𝑛)(𝑌,𝜂) around 𝐹₀. 𝐹=𝐹₀+𝑓. Consider 𝑆_𝐹⁽¹⁾𝑌,𝜂=𝛿𝑆_𝐹⁽¹⁾(𝑌,𝜂)𝛿𝐹_𝐹0𝑆𝑓+⋯_𝐹(2)𝑌,𝜂=𝑆_𝐹⁽²⁾₀𝑌,𝜂+𝛿𝑆_𝐹⁽²⁾(𝑌,𝜂)𝛿𝐹_𝐹0𝑓+⋯. (68)𝑓=𝑓(𝑌) is solved as the functional of 𝑌. Let Γ(𝑌)=Γ₁(𝑌)+Γ₂(𝑌)+⋯ since −𝑆_𝐹𝑌+𝜂+𝑆_𝐹(𝑌)=−𝑆_𝐹⁽²⁾₀Γ𝑌,𝜂+⋯,₁(𝑌)=ln𝑑𝜂𝑒^{−𝑆𝐹0(2)(𝑌,𝜂)}=lndet𝐴_𝐹0^1/2 (69) with 𝐴_𝐹0𝜏,𝜏^′=𝜕²_𝜏+𝜕²𝑉₀(𝑌)𝜕𝑌²+𝜕²𝑉𝐹₀,𝑌𝜕𝑌²𝛿𝜏−𝜏^′. (70) In particular, when 𝛿𝑆₀(𝑌)/𝛿𝑌=0, 𝐹₀=0, 𝐴_𝐹0 reduces to 𝐴𝜏,𝜏^′=𝜕²_𝜏+𝜕²𝑉₀(𝑌)𝜕𝑌²𝛿𝜏−𝜏^′. (71) The corresponding Γ₁(𝑌) is the same as the one-loop contribution Γ¹_eff(𝑌) of the standard effective action Γ_eff(𝑌) in background gauge: 𝑒^Γeff(𝑌)=𝑑𝜂𝑒^{−𝑆0(𝑌+𝜂)+∫𝑑𝜏Tr[𝐽(𝜏)𝜂(𝜏)]}, (72) subject to the condition 𝛿𝑑𝜂(𝜏)𝑒𝛿𝐽^{−𝑆0(𝑌+𝜂)+∫𝑑𝜏Tr[𝐽(𝜏)𝜂(𝜏)]}=0. (73)

Γ₁(𝑌)=Γ¹_eff(𝑌) has been calculated for the arbitrary 𝑌 satisfying 𝛿𝑆₀(𝑌)/𝛿𝑌=0 [3, 25]: Γ₁(𝑌)=12𝑑¹⁰𝑥₁×𝑑¹⁰𝑥₂𝐺⁰_Fc𝑥₁,𝑥₂𝑉_𝐹(𝑌)𝑥₁𝑉_𝐹(𝑌)𝑥₂, (74) with 𝐺⁰_Fc being the free supergraviton propagator. Compared with (61), Λ_𝐹,1(𝑥₁,𝑥₂)=𝐺⁰_Fc(𝑥₁,𝑥₂). If the result can be extended to the 𝑛-point Green’s functions and to the full-loop calculation, there will be Λ_𝐹𝑥₁,…,𝑥_𝑛=𝐺_Fc𝑥₁,…,𝑥_𝑛, (75) where 𝐺_Fc is the connected full Green's function of supergravity. Γ^𝑐_𝐹(𝑥₁,…,𝑥_𝑛) and 𝑊(𝐹) will then become the 𝑛-point vertex function and the effective action of supergravity, respectively.

In quantum field theory, unlike the 𝑆-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(𝑌) and Γ_eff(𝑌) differ from a parameterization transformation. Γ(𝑌)≠Γ_eff(𝑌). 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 Γ_𝑔[𝑌(𝜏)]=_𝑛1𝑛!𝑑¹⁰𝑥₁⋯𝑑¹⁰𝑥_𝑛𝐺_Fc𝑥₁,…,𝑥_𝑛×𝑉_𝐹(𝑌)𝑥₁⋯𝑉_𝐹(𝑌)𝑥_𝑛. (76) The effective action defined in (61) is also the Taylor series of 𝑉_𝐹(𝑌) and thus could be compared with (76) directly. The standard effective action Γ_eff(𝑌) is expanded as the Taylor series of 𝑌. A careful reorganization is needed to get 𝑉_𝐹(𝑌), but it is unclear whether it is always possible to do so. Moreover, under a Legendre transformation, (76) becomes Γ_𝑔[𝑌(𝜏)]=𝑆_eff(𝐹)+𝑑¹⁰𝑥𝐹(𝑥)𝑉_𝐹(𝑌)(𝑥)∼𝑆_eff(𝐹)+𝑆_𝐹[𝑌(𝜏)] (77) with −𝛿𝑆_eff(𝐹)𝛿𝐹(𝑥)_𝐹=𝑉_𝐹(𝑌)(𝑥)=𝛿𝑆_𝐹(𝑌)𝛿𝐹(𝑥)_𝐹. (78)𝑆_eff(𝐹) is the effective action of supergravity. Equation (77) is almost the same as (53). The only problem is whether 𝑊(𝐹)=𝑆_eff(𝐹) 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 𝐹: 𝑆_eff𝐹(𝐹)=^∞_𝑚=0𝑒^{𝜙(2−2𝑚)}_{𝑀2−2𝑚}[𝑑𝑋]𝑒^−𝑆𝐹=𝑊_𝑠(𝐹), (79) where  𝐹 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 𝑔_𝑎𝑏=𝑒^2𝜎𝑔_𝑎𝑏 is always imposed, and so, 𝑊_𝑠(𝐹,𝜎) also has the dependence on 𝜎: 𝛿𝑊_𝑠(𝐹,𝜎)𝛿𝜎=−𝛿𝑊_𝑠(𝐹,𝜎)𝛽(𝐹).𝛿𝐹 (80) When 𝛽(𝐹)=0, 𝑊_𝑠(𝐹,𝜎)=𝑊_𝑠(𝐹), although 𝛽(𝐹)=0 is not the necessary condition.

Suppose 𝑆_eff(𝐹) is the effective action of the string modes: 𝑆_eff(𝐹)1=−2𝑑^𝐷𝑥𝐹Δ𝐹+𝑆_int(𝐹). (81) Consider 𝐹₀ with Δ𝐹₀=0. Since 𝛽(𝐹₀)≠0, 𝐹 will evolve along the RG flow; that is, 𝑊_𝑠(𝐹₀,𝜎)=𝑊_𝑠[𝐹(𝜎^′),𝜎^′]. A special property of 𝐹₀ is that it will finally reach an IR fixed point 𝐹, 𝑊_𝑠(𝐹₀,𝜎)=𝑊_𝑠(𝐹,∞). In fact, since 𝐹₀ satisfies the first-order 𝛽 equation, the RG flow may bring it to an IR fixed point. Since 𝛽(𝐹)=0, 𝑊_𝑠(𝐹,∞)=𝑊_𝑠(𝐹,𝜎)=𝑊_𝑠(𝐹)=𝑊_𝑠(𝐹₀,𝜎)=𝑊_𝑠(𝐹₀).

𝐹 is calculated from 𝐹₀ via 𝐹=𝐹₀+Δ⁻¹𝛿𝑆_int𝐹𝛿𝐹=𝐹₀(𝑥)+𝑑^𝐷𝑦𝐷𝑥−𝑦,𝐹𝛿𝑆_int𝐹𝛿𝐹𝑦. (82)𝛿𝑆_eff(𝐹)/𝛿𝐹(𝑥)|_𝐹=𝐹=0. Since 𝛿𝑆_eff(𝐹)/𝛿𝐹^𝑖=𝜅_𝑖𝑗𝛽(𝐹^𝑗), 𝛽(𝐹)=0 is equivalent to 𝛿𝑆_eff(𝐹)/𝛿𝐹(𝑥)|_𝐹=𝐹=0 [18]. In [18], it was shown that 𝑊_𝑠(𝐹₀,𝜎)=𝑆_eff(𝐹), which means 𝑊_𝑠(𝐹,𝜎)=𝑊_𝑠(𝐹)=𝑆_eff(𝐹). There is a one-to-one correspondence between the solution space of 𝛿𝑆_eff(𝐹)/𝛿𝐹(𝑥)=0 and the solution space of Δ𝐹=0. So, for all 𝐹 with 𝛿𝑆_eff(𝐹)/𝛿𝐹(𝑥)=0=𝛽(𝐹), 𝑊_𝑠(𝐹,𝜎)=𝑆_eff(𝐹). In 𝜎-model approach, only for such 𝐹, the conformal factor decouples; thus, the calculated string free energy is physical. In this subspace, 𝑊_𝑠(𝐹)=𝑆_eff(𝐹).

𝑊_𝑠(𝐹,𝜎) with 𝛽(𝐹)≠0 usually depends on 𝜎. However, the 𝜎-dependence drops out for 𝐹₀. In fact,𝑊_𝑠𝐹₀,𝜎=𝑊_𝑠𝐹₀=𝑆_eff𝐹=𝑆_eff𝐹₀+Δ⁻¹𝛿𝑆_int𝐹₀𝛿𝐹₀+⋯ (83) is the 𝑆-matrix functional [27]. For the on-shell 𝑝₁,…,𝑝_𝑛, 𝑊_𝑠𝐹₀,𝜎=_𝑛1𝑛!𝑑⁹𝑝₁⋯𝑑⁹𝑝_𝑛𝑆^𝑐_𝐹𝑝₁,…,𝑝_𝑛×𝐹₀𝑝₁⋯𝐹₀𝑝_𝑛,𝐹₀(𝑥)=𝑑⁹𝑝𝐹₀𝑝,𝑆^𝑐_𝐹𝑝₁,…,𝑝_𝑛=1𝑁₀^∞_𝑚=0𝑒^{𝜙(2−2𝑚)}×_{𝑀2−2𝑚}[𝑑𝑋]𝑒^−𝑆0𝑉_𝐹𝑝₁⋯𝑉_𝐹𝑝_𝑛 (84) is the connected scattering amplitude. It is only in Polyakov approach can we construct the 𝑆-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 𝐹 satisfying 𝛿𝑆_eff(𝐹)/𝛿𝐹(𝑥)=0, 𝑊_𝑠(𝐹,𝜎)=𝑊_𝑠(𝐹)=𝑆_eff(𝐹). 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. SYM₄ is a nonperturbative description of the type IIB string theory on AdS₅×𝑆⁵. It is expected that the Hilbert spaces of both sides are isomorphic to each other. Similar with the Chern-Simons/topological string correspondence, 𝑍_SYM4=𝑒^𝑊𝑠, where 𝑊_𝑠 is the free energy of the string theory on AdS₅×𝑆⁵. The question is what will be the string dual of SYM₄ if the background metric of SYM₄ is 𝜂_𝜇𝜈+𝛿𝑔_𝜇𝜈(𝑥) other than 𝜂_𝜇𝜈 for very small 𝛿𝑔_𝜇𝜈(𝑥). A natural expectation is that such SYM₄ is dual to the type IIB string theory on AdS₅×𝑆⁵ with a little modification of the background metric 𝛿𝐺_𝜇𝜈(𝑥,𝜎) that is entirely determined by 𝛿𝑔_𝜇𝜈(𝑥). Correspondingly, 𝑍_SYM4[𝛿𝑔_𝜇𝜈(𝑥)]=𝑒^{𝑊𝑠[𝛿𝐺𝜇𝜈(𝑥,𝜎)]}, where 𝑊_𝑠[𝛿𝐺_𝜇𝜈(𝑥,𝜎)] is the free energy of the string theory on AdS₅×𝑆⁵ with the background perturbation 𝛿𝐺_𝜇𝜈(𝑥,𝜎) being turned on. Note that, for the stringy explanation of the SYM₄ 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 𝑍_SYM4[𝛿𝑔_𝜇𝜈(𝑥)]=𝑒^{𝑆eff[𝛿𝐺𝜇𝜈(𝑥,𝜎)]}, where 𝑆_eff[𝛿𝐺_𝜇𝜈(𝑥,𝜎)] is the supergravity effective action for the modified background fields, so 𝑆_eff[𝛿𝐺_𝜇𝜈(𝑥,𝜎)]=𝑊_𝑠[𝛿𝐺_𝜇𝜈(𝑥,𝜎)] if 𝑍_SYM4=𝑒^𝑊𝑠 holds.

There is a naive way to interpret 𝑊_𝑠(𝐹)=𝑆_eff(𝐹). Suppose the classical supergravity action is 𝑆_𝑐(𝐹), from the field theory's point of view: 𝑒^{𝑆eff(𝐹)}=_1PI𝑑𝐹𝑒^{−𝑆𝑐(𝐹+}𝐹⁾, (85) where 𝐹 is the fluctuation on 𝐹 or, in other words, supergravitons living on background 𝐹. The effective action is the sum of the connected 1PI vacuum-vacuum diagrams of the supergravitons on background 𝐹. The elementary propagator and the vertices can be read from 𝑆_𝑐(𝐹+𝐹). Now, consider the string theory on background 𝐹, we may have 𝑒^{𝑆eff(𝐹)}=𝑍_𝑠(𝐹)=𝑒^{𝑊𝑠(𝐹)}, (86) where 𝑊_𝑠(𝐹) is the sum of the irreducible vacuum-vacuum string diagrams on background 𝐹. Note that, in string diagram, there is nither a concept of 1PI nor 1PR. Also, there is no classical action like 𝑆_𝑐(𝐹+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 𝐹 is the solution of 𝛿𝑆_eff(𝐹)/𝛿𝐹=0. The cancelation of the Weyl anomaly gives the e.o.m for the effective action of the supergravity, including the 𝛼^′ corrections, so 𝑆_eff(𝐹) should be the effective action with the 𝛼^′ corrections taken into account.

A natural 𝑀-theory extension is 𝑒^{𝑆eff(𝐹)}=𝑍_𝑀(𝐹)=[𝑑𝑌]𝑒^{−𝑆𝑀𝐹(𝑌)}, (87) where 𝑆^𝑀_𝐹(𝑌) is the supermembrane action on 11𝑑 supergravity background 𝐹. The membrane is already the second quantized object, so the left-hand side of (87) is 𝑒^{𝑆eff(𝐹)} other than 𝑆_eff(𝐹). For the generic 11𝑑 background 𝐹, 𝑆^𝑀_𝐹(𝑌) should be covariantm, and, so, the worldvolume metric must be introduced and integrated, making the supermembrane theory nonrenormalizable. For 𝐹 which is translation invariant along 𝑥⁻, the light-cone gauge can be imposed. The configurations are then truncated to those with the light-cone momentum 𝑝⁺. The integration out of such configurations on background 𝐹 gives the effective action of 𝐹 with the radius of 𝑥⁻∼1/𝑝⁺.

3. Matrix Theory on a Particular Vacuum

If 𝑊(𝑅,𝐹) in (42) is the effective action of the 11𝑑 supergravity field 𝐹(𝑥) that is translation invariant along 𝑥⁻, for the on-shell 𝐹(𝑥), there will be 𝛿𝑊(𝑅,𝐹)/𝛿𝐹(𝑥)=−𝑣_𝐹(𝑥)=0. However, for 𝐹=𝐹, which is obviously on shell, the current densities 𝑣_𝑔++, 𝑣_𝑔−−, and 𝑣_𝑔+− are nonvanishing. (Note that the vertex operators for 𝑔₊₊ and 𝑔₊₋ 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].) 𝑊()𝑑𝑅,𝐹=−¹⁰𝑥𝑣_𝑔++𝑔₊₊(𝑥)+𝑣_𝑔−−𝑔₋₋(𝑥)+𝑣_𝑔+−𝑔₊₋(𝑥+1)2𝑑¹⁰𝑥₁×𝑑¹⁰𝑥₂Γ^𝑐_{𝑅,𝑔++,𝑔−−}𝑥₁,𝑥₂𝑔₊₊𝑥₁𝑔₋₋𝑥₂+12𝑑¹⁰𝑥₁×𝑑¹⁰𝑥₂Γ^𝑐_{𝑅,𝑔+−,𝑔+−}𝑥₁,𝑥₂𝑔₊₋𝑥₁𝑔₊₋𝑥₂+⋯. (88) With 𝛼,𝛽=+,−, one may try to solve 𝑔_𝛼𝛽(𝑥) from 𝛿𝑊(𝑅,𝐹)/𝛿𝑔_𝛼𝛽(𝑥)=0, and then take this value other than 𝑔₊₋(𝑥)=−1, 𝑔₋₋(𝑥)=𝑔₊₊(𝑥)=0 as the background to do the expansion. 𝑣_𝑔𝛼𝛽 are constants, so the generated 𝑔_𝛼𝛽(𝑥) are also constants (although the constant 𝑔_𝛼𝛽(𝑥) does not really solve the equation) and do not represent the substantial change of the background. We will simply neglect these tadpoles. 𝑣_𝑔++ should be distinguished from 𝛿𝑆_𝑅,𝐹(𝑌)/𝛿𝑔₊₊(𝑥)|_𝑌=0=(𝑁/𝑅)𝛿¹⁰(𝑥), which is the light-cone momentum density of a supergraviton localized at (𝜏,0,…,0) and, of course, will produce the nontrivial 𝑔₋₋(𝑥).

In (54), 𝐹=𝐹(𝑅,𝑌) is the functional of 𝑌 solved through (55). For the arbitrary 𝐹, we may define 𝑒^{Ω(𝑅,𝑌,𝐹)}=𝑑𝑌𝑒^{−𝑆𝑅,𝐹(}𝑌_{)+𝑆𝑅,𝐹(𝑌)}=𝑑𝜂𝑒^{−𝑆𝑅,𝐹(𝑌+𝜂)+𝑆𝑅,𝐹(𝑌)}=𝑑𝜂𝑒^{−𝑆𝑅,𝐹(𝑌;𝜂)}. (89)Ω(𝑅,𝑌,𝐹)=𝑊(𝑅,𝐹)+𝑆_𝑅,𝐹(𝑌) is the action of the source-gravity coupled system and is not necessarily on shell with respect to gravity. If 𝑆_𝑅,𝐹(𝜂) gives the Matrix theory description of 𝑀 theory on background 𝐹, 𝑆_𝑅,𝐹(𝑌;𝜂) will describe 𝑀 theory on background 𝐹, in presence of the brane 𝑌. If 𝑌 satisfies the equations of motion, that is, 𝛿𝑆_𝑅,𝐹(𝑌)/𝛿𝑌=0, 𝑆_𝑅,𝐹𝑌;𝜂=𝑑𝜏𝛿𝑆_𝑅,𝐹(𝑌)(𝜏)𝜂(𝜏)+1𝛿𝑌2𝑑𝜏𝑑𝜏^′𝛿²𝑆_𝑅,𝐹(𝑌)𝛿𝑌(𝜏)𝛿𝑌𝜏^′𝜂(𝜏)𝜂𝜏^′=1+⋯2𝑑𝜏𝑑𝜏^′𝛿²𝑆_𝑅,𝐹(𝑌)𝛿𝑌(𝜏)𝛿𝑌𝜏^′𝜂(𝜏)𝜂𝜏^′+⋯ (90)𝑆_𝑅,𝐹(𝑌;𝜂) starts from the quadratic term. For 𝐹 satisfying 𝛿Ω(𝑅,𝑌,𝐹)/𝛿𝐹(𝑥)=0, 𝑆_𝑅,𝐹(𝑌;𝜂) then gives a description of 𝑀 theory on a background 𝐹 generated by the brane 𝑌. In this case, Ω[𝑅,𝑌,𝐹(𝑌)]=Γ(𝑌): 0=𝑑𝜂𝑒^{−𝑆𝑅,𝐹(𝑌;𝜂)}𝛿𝑆_𝑅,𝐹𝑌;𝜂(𝑥)=𝛿𝐹𝛿𝑆_𝑅,𝐹𝑌;𝜂(𝑥).𝛿𝐹 (91) The expectation values of all current densities vanish. There is no tadpole.

The 𝑈(∞) symmetry of Matrix theory is destroyed by 𝑌. In particular, if 𝑌=diag𝑀₁,…,𝑀_1𝑁1,…,𝑀_𝑚,…,𝑀_𝑚𝑁𝑚, (92) the preserved gauge symmetry is 𝑈(𝑁₁)×⋯×𝑈(𝑁_𝑚). Nevertheless, the original 𝑈(∞) symmetry still has its manifestation. For all 𝑢∈𝑈(∞), 𝜂⟶𝑢𝜂𝑢⁺+𝑢𝑌𝑢⁺−𝑌,𝜂⟶𝜂+𝑖𝛼,𝜂+𝑖[𝛼,𝑌], (93) so for 𝑢∉𝑈(𝑁₁)×⋯×𝑈(𝑁_𝑚), 𝜂 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 𝑀-theory on 11𝑑 pp-wave background: 𝑑𝑠²=−2𝑑𝑥⁺𝑑𝑥⁻+⁹_𝑖=1𝑑𝑥^𝑖𝑑𝑥^𝑖−³_𝑎=1𝜇²9𝑥^𝑎2+⁹_𝑎′=4𝜇²𝑥36^𝑎′2𝑑𝑥⁺𝑑𝑥⁺𝐹₁₂₃₊=𝜇. (94) The background preserves 32 supersymmetries, while the rest 11𝑑 supergravity solutions with 32 supersymmetries are flat spacetime, AdS₄×𝑆⁷, and AdS₇×𝑆⁴ [29]. In pp-wave, the dynamics of the 𝑀-theory sector with the light-cone momentum 𝑝⁺=𝑁/𝑅 is described by the 𝑈(𝑁) matrix model: 𝑆^PW_𝑅=𝑅𝑑𝑥⁺1×Tr4𝑅²𝐷₊𝑋^𝑖2−𝑖𝑅𝜃𝐷₊1𝜃+2𝑋^𝑖,𝑋^𝑗2+2𝜃𝛾_𝑖𝑋^𝑖−𝜇,𝜃6𝑅²𝑋^𝑎2−𝜇12𝑅²𝑋^𝑎′2−𝜇6𝑅𝑖𝜖_𝑎𝑏𝑐𝑋^𝑎𝑋^𝑏𝑋^𝑐−𝜇8𝑅𝜃𝛾₁₂₃𝜃, (95)𝑁→∞, 𝑅→∞. PWMM also preserves 32 supersymmetries and has the same symmetry group as that of the pp-wave background.

The classical supersymmetric solutions of the action are [2] 𝑋^𝑖,𝑋^𝑗𝜇=𝑖𝜖3𝑅^𝑖𝑗𝑘𝑋^𝑘𝑋,𝑖,𝑗,𝑘=1,2,3,_𝑖𝑋̇=0,𝑖=4,…,9,_𝑖=0,𝑖=1,…,9. (96){𝑋¹,𝑋²,𝑋³} form the 𝑁 dimensional representation of 𝑆𝑈(2). Suppose (3𝑅/𝜇)𝑋^𝑖=𝐿^𝑖, 𝑖=1,2,3, 𝐿^𝑖 can be decomposed as 𝐿^𝑖=diag𝐿^𝑖_𝑗1⋯𝐿^𝑖_𝑗1𝑁1𝐿^𝑖_𝑗2⋯𝐿^𝑖_𝑗2𝑁2⋯𝐿^𝑖_𝑗𝑡⋯𝐿^𝑖_{𝑗𝑡𝑁𝑡}, (97) where 𝐿^𝑖_𝑗𝑠(𝑠=1,…,𝑡) stands for the spin 𝑗_𝑠 irreducible representation of 𝑆𝑈(2): 2𝑗₁+1𝑁₁+2𝑗₂+1𝑁₂+⋯+2𝑗_𝑡+1𝑁_𝑡=𝑁. (98)

The solution can be taken as a set of 𝑁_𝑠 coincident fuzzy spheres with the radius (𝜇/3𝑅)𝑗_𝑠(𝑗_𝑠+1) [2]. The 𝑈(𝑁) gauge symmetry is broken into 𝑈(𝑁₁)×𝑈(𝑁₂)×⋯×𝑈(𝑁_𝑡). All fuzzy spheres are concentric. When 𝑗_𝑠→∞, the fuzzy spheres become a set of membrane spheres given by ∑³_𝑖=1(𝑥^𝑖_𝑠)²=𝑟²_𝑠 with 𝑟_𝑠∝(2𝑗_𝑠+1)𝜇/6𝑅. Each spherical membrane has 𝑝⁻=0, 𝑝⁺=(2𝑗+1)/𝑅, so they are also called the giant gravitons.

The solution can also be interpreted as the spherical 𝑀5 branes with a dual assignment of the light-cone momentum [30]. Any partition of 𝑁 may be represented by a Young diagram whose column lengths are the elements in the partition. In the 𝑀2 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 𝑀5 interpretation, it is the rows of the Young diagram that correspond to the individual 𝑀5, with the row lengths giving the number of units of momentum carried by each 𝑀5. In both cases, the total light-cone momentum is always 𝑁/𝑅. If 𝑁_𝑠 are finite, they are fuzzy 𝑀5 branes. When 𝑁_𝑠→∞, the fuzzy 𝑀5 become the spherical 𝑀5 wrapping 𝑆⁵ given by ∑⁹_𝑖=4(𝑥^𝑖_𝑟)²=𝑟²_𝑟. In patricular, the trivial vacuum 𝑋^𝑖=0 represents a single 𝑀5 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 𝑁/𝑅, which itself would destroy the linearly realized supersymmetries. The situation is different in SYM₄ in which, we do have a vacuum preserving 32 supersymmetries, representing the ground state of the string theory on AdS₅×𝑆⁵. (Brane like) 1/2 BPS states are giant gravitons on AdS₅×𝑆⁵ [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 AdS₅×𝑆⁵, capturing the physics seen from the infinite momentum frame (IMF) since, in IMF, AdS₅×𝑆⁵ is viewed as the pp-wave [34–36]. 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 pp-wave background, which are giant gravitons (spherical 𝐷3 branes) and type IIB strings. Lifted to 11𝑑, the light-cone dimension is replaced by 𝑇², while the TGMT, which is a discrete regularization of the 𝐷3 branes in type IIB picture, becomes the regularization of the 𝑀5 branes [35]. Note that the 4-form field in type IIB pp-wave, when lifted to 11𝑑, becomes the 6-form field coupling electrically with the 𝑀5 branes, so it is natural to construct the matrix model via the discrete regularization of 𝑀5 branes other than the usual 𝑀2 branes. On type IIB pp-wave, 𝐷3 branes are 𝑀5 branes wrapping 𝑇², while the type IIB strings are membranes wrapping one of 𝑆¹. In contrast to the PWMM, in TGMT, the trivial vacuum 𝑋=0 represents type IIB string (with 𝑀2 origin), while the nontrivial vacuum represents 𝐷3 branes (with 𝑀5 origin), which is probably because the PWMM and the TGMT are constructed from 𝑀2 and 𝑀5, respectively. Recall that, in PWMM, the fuzzy configuration may have the 𝑀2 and 𝑀5 dual interpretations [30], it is interesting to figure out whether a similar 𝐹1-𝐷3 dual interpretation also exists for configurations in TGMT.

With 𝑌 denoting the supersymmetric vacuum in (96), 𝑀 theory on pp-wave background in presence of the brane 𝑌 is described by the Matrix model with the action 𝑆^PW_𝑅𝑌;𝜂=𝑆^PW_𝑅𝑌+𝜂−𝑆^PW_𝑅𝑌. (99) Since (𝛿𝑆^PW_𝑅(𝑌)/𝛿𝑌)|_𝑌=𝑌=0, 𝑆^PW_𝑅(𝑌;𝜂) starts from the quadratic term.

For simplicity, in the following, we will assume 𝑅 is absorbed into the fields. If the backreaction of the brane 𝑌 on pp-wave background is turned on, the field 𝐹(𝑌) will be generated: 𝑒^ΓPW(𝑌₎=𝑑𝜂𝑒^{−𝑆PW𝐹(}𝑌₎^{(𝜂)+𝑆PW}_𝐹(𝑌₎⁽𝑌₎=𝑒^𝑊PW[𝐹(𝑌_)]+𝑆^PW_𝐹(𝑌₎⁽𝑌₎, (100) where 𝛿𝑆^PW_𝐹(𝑌)𝛿𝐹(𝑥)_𝐹=𝐹(𝑌₎𝑊=−^PW(𝐹)𝛿𝐹(𝑥)_𝐹=𝐹(𝑌₎. (101) pp-wave with 𝑌 added respects the 𝑆𝑈(2∣4) symmetry, so the generated 𝐹(𝑌) as well as the corresponding 𝑆^PW_𝐹(𝑌₎(𝜂) will have the same symmetry.

With the fields 𝐹(𝑌) given, in principle, one can write down 𝑆^PW_𝐹(𝑌₎(𝜂) explicitly. The classical supersymmetric solutions of 𝑆^PW_𝐹(𝑌₎(𝜂) 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 𝑌 was calculated at the one-loop level. 𝑌 is also the stationary point of the effective action. We may have 𝛿Γ^PW(𝑌)/𝛿𝑌|_𝑌=𝑌=𝛿𝑆^PW_𝐹(𝑌₎(𝑌)/𝛿𝑌|_𝑌=𝑌=0. With the pp-wave background replaced by the backreacted geometry, 𝑆^PW_𝑅(𝑌;𝜂) is modified to 𝑆^PW_𝐹𝑌𝑌;𝜂=𝑆^PW_𝐹𝑌𝑌+𝜂−𝑆^PW_𝐹𝑌𝑌. (102)𝑆^PW_𝐹(𝑌₎(𝑌;𝜂) still starts from the quadratic term since 𝛿𝑆^PW_𝐹(𝑌₎(𝑌)/𝛿𝑌|_𝑌=𝑌=0.

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

The 11𝑑 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 𝐹₄ and 𝐹₇ 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 𝑀 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), 𝑆^PW_𝐹𝑌𝑌=0,𝛿𝑆^PW_𝐹(𝑌)𝛿𝐹(𝑥)_𝐹=𝐹(𝑌₎Γ=0,_PW𝑌=𝑊^PW𝐹𝑌,𝛿𝑊^PW(𝐹)𝛿𝐹(𝑥)_𝐹=𝐹(𝑌₎𝑒=0._𝑊PW[𝐹(𝑌_)]=𝑑𝜂𝑒^{−𝑆PW𝐹(}𝑌₎⁽𝑌^;𝜂). (103)

𝑌 is the momentum eigenstate in 𝑥⁻ direction, so the generated 11𝑑 background 𝐹(𝑌) is translation invariant along 𝑥⁻. In large 𝑟 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] 𝑑𝑠²₁₁=2𝑑𝑥⁺𝑑𝑥⁻+𝑔₋₋𝑑𝑥⁻𝑑𝑥⁻+𝑑𝑥^𝑖𝑑𝑥^𝑖𝑔₋₋=𝑄𝑟⁷, (104) which is the field produced by the supergraviton which is static in 9𝑑 space, carrying the definite light-cone momentum [41]. 𝑄=30𝜋³𝑁/𝑅². (It is not 𝑄=30𝜋³𝑙⁹_𝑝𝑁/𝑅² because 𝑙_𝑝 is absorbed into 𝑅 and 𝑟.)  With the coordinate transformation 𝑥→𝑔_𝑠^−1/3𝑥, 𝑥⁻→𝑔_𝑠^2/3𝑥⁻, (104) becomes 𝑑𝑠²₁₁=2𝑔_𝑠^1/3𝑑𝑥⁺𝑑𝑥⁻+𝑔_𝑠^4/3𝑄𝑟⁷𝑑𝑥⁻𝑑𝑥⁻+𝑔_𝑠^−2/3𝑑𝑥^𝑖𝑑𝑥^𝑖, (105) while the radius of 𝑥⁻ is now 𝑔_𝑠^−2/3𝑅.

Under the 𝑥⁻ reduction, in string frame, 𝑑𝑠²=𝑒^−2𝜙/3𝐺_𝑚𝑛𝑑𝑥^𝑚𝑑𝑥^𝑛+𝑒^4𝜙/3𝑑𝑥⁻+𝐴_𝑚𝑑𝑥^𝑚2.𝑔_𝑚𝑛=𝑒^−2𝜙/3𝐺_𝑚𝑛+𝑒^4𝜙/3𝐴_𝑚𝐴_𝑛,𝑔₋₋=𝑒^4𝜙/3,𝑔_𝑚−=𝑒^4𝜙/3𝐴_𝑚. (106) In particular, for (105), 𝑑𝑠²₁₀𝑟=−^7/2𝑄^1/2𝑑𝑥⁺𝑑𝑥⁺+𝑄^1/2𝑟^7/2𝑑𝑥^𝐼𝑑𝑥^𝐼,𝑒^𝜙=𝑔_𝑠𝑄𝑟^73/4,𝐴₊=𝑟⁷𝑔_𝑠𝑄 (107) is the near-horizon geometry of the 𝐷0 branes [41].

The type IIA solution coming from the 𝑥⁻ reduction of the 11𝑑 field 𝐹(𝑌) was constructed in [19, 37]. When 𝑟→∞, the perturbation, which is the reduction of (104) along 𝑥⁻, is the near-horizon geometry of 𝑁 coincident 𝐷0-branes. The appearance of the near-horizon geometry is because of the null reduction. The reduction of (104) along 𝑥⁺−𝑥⁻ gives the 𝐷0 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 𝑥⁻.

3.2. The Gauge Theory Dual from the Matrix Model

According to the previous proposal, the Matrix theory description of the 𝑀 theory on background (104) in presence of the supergraviton with 𝑝⁺=𝑁/𝑅 is 𝑆_𝑅,𝐹(𝑌;𝜂), where 𝐹 is the field in (104), 𝑌=0  (if 𝐹 is the field in (105), 𝑆_𝑅,𝐹→(1/𝑔_𝑠)𝑆_𝑅,𝐹.) Cosider 𝑆_𝑅,𝐹0;𝜂=𝑆_𝑅𝜂+𝑑𝑥⁺Tr𝑔₋₋𝑇⁻⁻𝜂. (108)𝑆_𝑅 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 SYM₁ with the action 𝑆_𝑅=1(𝜂). Since (104) becomes (107) under the 𝑥⁻ reduction, it is desirable to find a limit to make (108) reduce to 𝑆_𝑅=1(𝜂).

Consider the coordinate transformation 𝑥^𝐼→𝑥^𝐼/𝑅, 𝑥⁻→𝑅𝑥⁻, under which 𝐹(𝑥)=𝑔₊₋,𝑔_𝐼𝐽,𝐶_𝐼+−,𝐶_𝐼𝐽𝐾,𝜓_𝐼,𝑔₊₊,𝑔_𝐼+,𝑔₋₋,𝑔_𝐼−,𝐶_𝐼𝐽+,𝐶_𝐼𝐽−,𝜓₊,𝜓₋⟶𝐹_∗(𝑅,𝑥)=𝑅^𝑞𝐹𝑥𝑅=𝑅𝑔₊₋,𝑔_𝐼𝐽𝑅²,𝐶_𝐼+−,𝐶_𝐼𝐽𝐾𝑅³,𝜓_𝐼𝑅,𝑔₊₊,𝑔_𝐼+𝑅,𝑅²𝑔₋₋,𝑔_𝐼−,𝐶_𝐼𝐽+𝑅²,𝐶_𝐼𝐽−𝑅,𝜓₊,𝑅𝜓₋. (109) Correspondingly, in Matrix theory action 𝑆_𝑅,𝐹, with a field redefinition 𝑋^𝐼→𝑋^𝐼/𝑅 and also a 𝑥⁻→𝑅𝑥⁻ rescaling to make the radius of 𝑥⁻ become 1, the background fields appearing in action are just 𝐹_∗(𝑅,𝑋). 𝑆_𝑅,𝐹→𝑆_{1,𝐹∗(𝑅)}. With a further rescaling 𝜃→𝜃/𝑅^3/2, 𝑆_𝑅,𝐹→𝑆_1,𝐹^∗_(𝑅)/𝑅³. In 𝑆_1,𝐹^∗_(𝑅), it is 𝐹^∗(𝑅,𝑋)=𝑅^𝑝𝐹𝑋𝑅=𝑔₊₋,𝑔_𝐼𝐽,𝐶_𝐼+−,𝐶_𝐼𝐽𝐾,𝜓_𝐼,𝑅²𝑔₊₊,𝑅𝑔_𝐼+,𝑔₋₋𝑅²,𝑔_𝐼−𝑅,𝑅𝐶_𝐼𝐽+,𝐶_𝐼𝐽−𝑅,𝑅𝜓₊,𝜓₋𝑅 (110) that will appear.

Return to (108). For any 𝑅, 𝑆_𝑅,𝐹 is equivalent to 𝑆_1,𝐹^∗_(𝑅)/𝑅³, in which 𝑔^∗₋₋(𝑅,𝑥)=𝑔₋₋(𝑟/𝑅)/𝑅²∼𝑅³/𝑟⁷. In 𝑅→0 limit, 𝑔^∗₋₋(𝑅,𝑥)→0, lim_𝑅→0𝑆_𝑅,𝐹⟺lim_𝑅→01𝑅³𝑆_𝑅=1=lim_𝑅→01𝑅³𝑑𝑥⁺1Tr2𝐷₊𝑋^𝐼𝐷₊𝑋^𝐼+14𝑋^𝐼,𝑋^𝐽2−𝑖𝜃𝐷₊𝜃+𝜃𝛾_𝐼𝑋^𝐼,𝜃. (111)SYM₁ arrives. Note that, for finite 𝑥, lim_𝑅→0𝑥/𝑅=∞, so by taking the 𝑅→0 limit, the background field that matters lives at the 𝑟→∞ region, far away from the source.

For PWMM, 𝑅²𝑔₊₊(𝑥/𝑅)=𝑔₊₊(𝑥), and 𝑅𝐶_𝐼𝐽+(𝑥/𝑅)=𝐶_𝐼𝐽+(𝑥), all fields are marginal, so, for the arbitrary 𝑅 [5], 𝑆^PW_𝑅⟺1𝑅³𝑆^PW_𝑅=1=1𝑅³𝑑𝑥⁺1×Tr4𝐷₊𝑋^𝑖2−𝑖𝜃𝐷₊𝜃+12𝑋^𝑖,𝑋^𝑗2+2𝜃𝛾_𝑖𝑋^𝑖−𝜇,𝜃6²𝑋^𝑎2−𝜇12²𝑋^𝑎′2−𝜇6𝑖𝜖_𝑎𝑏𝑐𝑋^𝑎𝑋^𝑏𝑋^𝑐−𝜇8𝜃𝛾₁₂₃𝜃. (112)

Now consider the background 𝐹(𝑌) coming from the backreaction of the giant graviton 𝑌 on pp-wave. Suppose 𝑌 is a vacuum in (96) with 2𝑗_𝑠+1=𝑛+𝛼_𝑠,𝑛⟶∞,𝛼_𝑠finite. (113)(114) The radii of the spherical membranes are 𝑟_𝑠∼(𝑛+𝛼_𝑠)/𝑅, 𝑠=1,…,𝑡. 𝐹(𝑌) reduced along 𝑥⁻ gives the smooth type IIA solution that is constructed in [19]. The generic solution of type IIA supergravity with 𝑆𝑈(2∣4) symmetry is characterized by a function 𝑉(𝜌,𝜂) and is given as [19] 𝑑𝑠²₁₀=𝑉̈−2𝑉̇−𝑉^′′1/2×𝑉̈−4𝑉̈−2𝑉̇𝑑𝑥⁺𝑑𝑥⁺+−2𝑉^′′𝑉̇𝑑𝜌²+𝑑𝜂²+4𝑑Ω²₅𝑉+2^′′𝑉̇Δ𝑑Ω²₂,𝑒^4𝜙=4𝑉̈−2𝑉̇³−𝑉^′′𝑉̇²Δ²,𝐶₁2𝑉̇=−^′𝑉̇𝑉̈−2𝑉̇𝑑𝑥⁺,𝐹₄=𝑑𝐶₃,𝐶₃𝑉̇=−4²𝑉^′′Δ𝑑𝑥⁺∧𝑑²𝐻Ω,₃=𝑑𝐵₂,𝐵₂=𝑉̇𝑉̇^′Δ+𝜂𝑑²̈𝑉̇𝑉Ω,Δ=𝑉−2^′′−𝑉̇^′2, (115) where the dot and the prime represent the derivatives with respect to log𝜌 and 𝜂, respectively. 𝑉 can be taken as an electrostatic potential for an axially symmetric system with conducting disks and a background potential. 𝜌 is the distance from the center axis, and 𝜂 is the coordinate in the direction along the center axis. 𝑉(𝜌,𝜂)=𝑉_𝑏(𝜌,𝜂)+𝑣_𝜂(𝜌,𝜂), where 𝑉_𝑏 is the background potential and 𝑣_𝜂 is determined by a configuration of conducting disks. Each 𝑆𝑈(2∣4) symmetric theory is specified by 𝑉_𝑏; each vacuum of the theory is specified by a configuration of conducting disks.

From (115), one can also get the uplifted 11𝑑 solution. For example, 𝑑𝑠²₁₁=2^2/3𝑉̈−2𝑉̇−𝑉^′′𝑉̇²Δ^21/3𝑑𝑥⁻𝑑𝑥⁻−2^5/3𝑉̇^′𝑉̇−𝑉^′′𝑉̇²Δ^21/3𝑑𝑥⁺𝑑𝑥⁻+2^5/3𝑉̇^1/3𝑉̇^′2−𝑉̈𝑉^′′−𝑉^′′Δ^21/3𝑑𝑥⁺𝑑𝑥⁺+2^2/3𝑉̇Δ^1/3−𝑉^′′1/3−𝑉^′′𝑉̇𝑑𝜌²+𝑑𝜂²+2𝑑Ω²₅+𝑉^′′𝑉̇Δ𝑑Ω²₂. (116)

For PWMM, 𝑉^PW_𝑏𝜌,𝜂=𝜌²2𝜂−3𝜂³. (117) Only the region 𝜂≥0 is meaningful. There is an infinitely large conducting disk sitting at 𝜂=0. Vacuum (97) corresponds to 𝑡 disks located at 𝜂₁=(𝜋/2𝑅)(2𝑗₁+1),𝜂₂=(𝜋/2𝑅)(2𝑗₂+1),…,𝜂_𝑡=(𝜋/2𝑅)(2𝑗_𝑡+1). The electric charges on these disks are equal to (𝜋²/8)𝑁₁,(𝜋²/8)𝑁₂,…,(𝜋²/8)𝑁_𝑡, respectively. 𝜂_𝑠 is the radius of the spherical membranes. The correspondence between the spacetime coordinates and the PWMM fields is (𝑋¹,𝑋²,𝑋³)↔(𝜂,Ω₂), (𝑋⁴,…,𝑋⁹)↔(𝜌,Ω₅), 𝑥⁺↔𝑥⁺.

With the given disk configuration, the corresponding potential is 𝑉𝜌,𝜂=𝜌²2𝜂−3𝜂³+𝑣_𝜂𝜌,𝜂, (118) which, when plugged into (115) and (116), gives the 10𝑑 solution 𝐹₁₀(𝑥) as well as the 11𝑑 solution 𝐹(𝑥). 𝑥 stands for the 9 space coordinates since the solution is translation invariant along 𝑥⁺ and 𝑥⁻.

Under the coordinate transformation 𝑥→𝑥/𝑅, 𝑥⁻→𝑅𝑥⁻, the disks are now located at 𝜂_𝑠=(𝜋/2)(2𝑗_𝑠+1), while the fields will transform as 𝐹(𝑥)→𝐹_∗(𝑅,𝑥)=𝑅^𝑞𝐹(𝑥/𝑅), 𝐹₁₀(𝑥)→𝐹_∗10(𝑅,𝑥). As a result, 𝑆_𝑅,𝐹→𝑆_{1,𝐹∗(𝑅)}. 𝐹_∗10(𝑅,𝑥) and 𝐹_∗(𝑅,𝑥) can be obtained by plugging 𝑉𝜌𝑅,𝜂𝑅=1𝑅³𝜌²2𝜂−3𝜂³+𝑣_𝜂/𝑅𝜌𝑅,𝜂𝑅 (119) into (115) and (116). Except for 𝑆_{1,𝐹∗(𝑅)}, we also have 𝑆_𝑅,𝐹∼𝑆_1,𝐹^∗_(𝑅)/𝑅³ with 𝐹^∗(𝑅,𝑥)=𝑅^𝑝𝐹(𝑥/𝑅) given by (110). Plug 𝑅³𝑉𝜌𝑅,𝜂𝑅=𝜌²2𝜂−3𝜂³+𝑅³𝑣_𝜂/𝑅𝜌𝑅,𝜂𝑅 (120) into (115) and (116); one can get 𝐹^∗₁₀(𝑅,𝑥) and the corresponding 𝐹^∗(𝑅,𝑥). In particular, for PWMM, 𝑉=𝑉^PW_𝑏, 𝑅³𝑉(𝜌/𝑅,𝜂/𝑅)=𝑉(𝜌,𝜂), so 𝐹^∗(𝑅,𝑥)=𝐹(𝑥) as we have seen before. Both 𝑉(𝜌/𝑅,𝜂/𝑅) and 𝑅³𝑉(𝜌/𝑅,𝜂/𝑅) satisfy cylindrically symmetric Laplace equation, so 𝐹_∗(𝑅,𝑥), and 𝐹^∗(𝑅,𝑥) are also the type IIA supergravity solutions. In fact, 𝐹(𝑥), 𝐹_∗(𝑅,𝑥) and 𝐹^∗(𝑅,𝑥) are related to each other by the coordinate transformation. 𝑆_𝑅,𝐹⇔𝑆_{1,𝐹∗(𝑅)}⇔𝑆_1,𝐹^∗_(𝑅)/𝑅³.

The limit of interest is 𝑛→∞, 𝑅→∞, 𝑅³/𝑛=𝛼² and 𝜇 fixed. In this limit, the concentric spherical membranes all have the infinite radii with the difference (𝛼_𝑖−𝛼_𝑗)/𝑅→0. For the finite 𝜌, 𝜂, 𝜌/𝑅→0, 𝜂/𝑅→0, we need to consider the behavior of 𝑣_𝜂(𝜌,𝜂) around 0. Since 𝜂_𝑠/𝑅→∞, near 0, 𝑣_𝜂/𝑅 can be approximated as the potential generated by dipoles located at ±𝜂_𝑠/𝑅: 𝑅³𝑣_𝜂/𝑅𝜌𝑅,𝜂𝑅∼^𝑡_𝑠=1𝑁_𝑠𝑅³𝜂_𝑠−𝜂𝑅²+𝜌𝑅^2−1/2−𝜂_𝑠+𝜂𝑅²+𝜌𝑅^2−1/2∼^𝑡_𝑠=12𝑅⁴𝑁_𝑠𝜂𝜂²_𝑠⟶0. (121)lim𝑅³𝑉(𝜌/𝑅,𝜂/𝑅)=𝜌²𝜂−(2/3)𝜂³. Indeed, with 𝑅³𝑉(𝜌/𝑅,𝜂/𝑅) plugged into (116), 𝑛→∞, 𝑅→∞, 𝑅³/𝑛  being fixed, the 11𝑑 solution finally approaches the pp-wave. As a result, lim𝑆_𝑅,𝐹⇔lim𝑆^PW_𝑅=1/𝑅³. Even if the background field is 𝐹(𝑌), in the limit taken, we still get PWMM.

On the other hand, plug 𝑅³𝑉(𝜌/𝑅,𝜂/𝑅) into the reduced 10𝑑 solution (115); 𝑉̈−2𝑉̇=𝑅³(𝑣̈_𝜂/𝑅−2𝑣̇_𝜂/𝑅); the 𝑣_𝜂/𝑅 part now matters. Let 𝑤(𝜌²,𝜂)=𝑅³𝑣_𝜂/𝑅(𝜌/𝑅,𝜂/𝑅): 𝑑𝑠²₁₀−𝜕=4^2𝜌2𝑤𝜂^−1/2𝑑𝑥⁺𝑑𝑥⁺+𝜕²_𝜌2𝑤𝜂^1/2𝑑𝜌²+𝜌²𝑑Ω²₅+𝑑𝜂²+𝜂²𝑑Ω²₂,𝑒^4𝜙=𝜕²_𝜌2𝑤𝜂³,𝐶₁𝜕=−2^2𝜌2𝑤𝜂⁻¹𝑑𝑥⁺,𝐹₄=𝑑𝐶₃,𝐶₃=−16𝜂³𝑑𝑥⁺∧𝑑²𝐻Ω,₃=𝑑𝐵₂,𝐵₂=4𝜂²𝜕²_𝜌2𝑤−𝜕_𝜌2𝑤+𝜂𝜕_𝜌2𝜕_𝜂𝑤𝑑²Ω. (122) The 10𝑑 solution has the dependence on the disk configuration. Equation (122) is lim𝐹^∗₁₀(𝑅,𝑥). With 𝑒^𝜙→𝑅³𝑒^𝜙, 𝐶₁→𝐶₁/𝑅³,  𝐶₃→𝐶₃/𝑅³, we will get lim𝐹_∗10(𝑅,𝑥). By taking the 𝑅→∞ limit, the relevant region is 𝜌→0, 𝜂→0, which is again far from the source. The 11𝑑 field approaches the pp-wave, but the reduced 10𝑑 field still depends on the disk configuration. Similarly, for (104), suppose 𝑄=𝑁𝑘⁷; then, when 𝑘→0, the 11𝑑 background is fat, while the associated 10𝑑 background (107) is still the near-horizon geometry but becomes singular now. Nevertheless, let 𝑢=𝑟/𝑘, 𝑟→0, 𝑢 fixed; (107) can be written as 𝑑𝑠²₁₀=𝑘²−𝑢^7/2𝑁^1/2𝑑𝑢⁺𝑑𝑢⁺+𝑁^1/2𝑢^7/2𝑑𝑢^𝐼𝑑𝑢^𝐼,𝑒^4𝜙/3=𝑁𝑢⁷,𝐴₊=𝑢⁷𝑁. (123) This is similar to the limit taken in AdS/CFT [7].

Now, consider the exact form of 𝑆_{1,𝐹∗(𝑅)}. For the generic 11𝑑 background 𝐹(𝑥), the bosonic part of 𝑆_{1,𝐹∗(𝑅)} is 𝑆^𝑏_{1,𝐹∗(𝑅)}=𝑑𝑥⁺×Tr𝑔_∗+−𝐷₊𝑋^𝐼𝐷₊𝑋^𝐽2𝑔²_∗+−𝑔_∗𝐼𝐽+𝐷₊𝑋^𝐼𝑔_∗+−𝑐^𝐽_∗+𝑔^𝐽_∗+𝑔_∗+−𝑔_∗𝐼𝐽+14𝑋^𝐼,𝑋^𝐽𝑋^𝐾,𝑋^𝐿𝑔_∗𝐼𝐾𝑔_∗𝐽𝐿+𝑔_∗++2𝑔²_∗+−+𝑐_∗+𝑔_∗+−+𝑆_{𝑔∗𝐼−}+𝑆_{𝑔∗−−}. (124) The relation between 𝐹_∗(𝑅,𝑥) and the 𝑥⁻ reduced 10𝑑 background 𝐹_∗10(𝑅,𝑥) is 𝑔_∗𝑚𝑛=𝑒^{−2𝜙∗/3}𝐺_∗𝑚𝑛+𝑒^4𝜙∗/3𝐴_∗𝑚𝐴_∗𝑛,𝑔_∗−−=𝑒^4𝜙∗/3,𝑔_∗𝑚−=𝑒^4𝜙∗/3𝐴_∗𝑚. (125) In the limit with 𝑛→∞, 𝑅→∞, 𝑅³/𝑛 fixed, if 𝑔^∗₋₋→0, 𝑔^∗_𝐼−=0, the rest 𝐹^∗₁₀(𝑅,𝑥) is finite, there will be 𝑅³𝑆_𝑔𝐼−=0,𝑅³𝑆_𝑔−−𝐺⟶0,_∗++⟶−𝑒^2𝜙∗𝐴_∗+𝐴_∗+,𝐺_∗𝐼𝐽=𝑒^2𝜙∗/3𝑔_∗𝐼𝐽,−𝐺_∗++⟶𝑒^𝜙∗𝐴_∗+. (126) With these relations, lim𝑆^𝑏_{1,𝐹∗(𝑅)}1=lim𝑒^𝜙∗𝑑𝑥⁺Tr−𝐺_∗++×−12𝐺_∗𝐼𝐽𝐺_∗⁺⁺𝐷₊𝑋^𝐼𝐷₊𝑋^𝐽−𝐺_∗𝐼𝐽𝐺_∗⁺⁺𝐷₊𝑋^𝐼𝑔^𝐽_∗++−𝐺_∗^++1/2𝐺_∗𝐼𝐽𝐶^𝐽_∗𝐾𝐿×𝐷₊𝑋^𝐼𝑋^𝐾,𝑋^𝐿+14𝐺_∗𝐼𝐽𝐺_∗𝐾𝐿𝑋^𝐼,𝑋^𝐾𝑋^𝐽,𝑋^𝐿−12𝑒^2𝜙∗/3𝐺_∗⁺⁺𝑔_∗+++−𝐺_∗^++1/2𝐶_∗+𝐽𝐾𝑋^𝐽,𝑋^𝐾=lim𝑆^𝑏_{1,𝐹∗10(𝑅)}. (127) For 𝑔_∗++=𝜇₊₊𝑋^𝐼𝑋^𝐽𝑔_∗𝐼𝐽, 𝑔^𝐽_∗+=𝜈₊𝑋^𝐽, 𝑒^2𝜙∗/3𝐺_∗⁺⁺𝑔_∗++=𝜇₊₊𝐺_∗⁺⁺𝐺_∗𝐼𝐽𝑋^𝐼𝑋^𝐽,𝐺_∗𝐼𝐽𝐺_∗⁺⁺𝐷₊𝑋^𝐼𝑔^𝐽_∗+=𝜈₊𝐺_∗𝐼𝐽𝐺_∗⁺⁺𝐷₊𝑋^𝐼𝑋^𝐽. (128) So, in the limit taken, 𝑆^𝑏_{1,𝐹∗(𝑅)} could be identified with 𝑆^𝑏_{1,𝐹∗10(𝑅)}, the action of SYM₁ on the 10𝑑 background 𝐹_∗10(𝑅,𝑥). The discussion can also be extended to the full 𝑆_{1,𝐹∗(𝑅)} with the fermionic part included.

Specified to the background generated by 𝑌 on pp-wave, (126) is satisfied for the solution in (122), so lim𝑆_{1,𝐹∗(𝑅)}=lim𝑆^PW_𝑅=1/𝑅³=lim𝑆_{1,𝐹∗10(𝑅)}. Indeed, one can verify the validity of (127) by directly plugging (122) in it. In the limit taken, PWMM can be taken as the SYM₁ living on 10𝑑 background (122). Equation (104) also satisfies the above criteria. Actually, 𝑑𝑥⁺1Tr2𝐷₊𝑋^𝐼𝐷₊𝑋^𝐼+14𝑋^𝐼,𝑋^𝐽2−𝑖𝜃𝐷₊𝜃+𝜃𝛾_𝐼𝑋^𝐼=,𝜃𝑑𝑥⁺×Tr−𝐺₊₊𝑒^𝜙−12𝐺_𝐼𝐽𝐺⁺⁺𝐷₊𝑋^𝐼𝐷₊𝑋^𝐽+14𝐺_𝐼𝐽𝐺_𝐾𝐿𝑋^𝐼,𝑋^𝐾𝑋^𝐽,𝑋^𝐿−𝑖𝑒^𝜙−𝐺^++1/2𝜃𝐷₊𝜃+𝑒^𝜙𝐺^1/2_𝐼𝐼𝜃𝛾^𝐼𝑋^𝐼,𝜃, (129) where 𝜙, 𝐺₊₊, and so forth are the near-horizon geometry of the 𝐷0 branes in (107) with 𝑔_𝑠=1.

In the limit with 𝑛→∞, 𝑅→∞, 𝑅³/𝑛=𝛼² fixed, (122) gives the solution in the region of finite 𝜌 and 𝜂. To study the region near the spherical membrane shells with 𝜌, 𝜂−𝑛 finite, a change of variables 𝜂→𝜂+𝑛 can be made. As is shown in [42], in this limit, 1𝑅³𝑉^PW_𝑏=1𝑅³𝜌²−2𝜂+𝑛3𝜂+𝑛³⟶𝑛𝑅³𝜌²−2𝜂²=1𝛼²𝑉^{SYM2𝑅×𝑆}_𝑏, (130) where 𝑉^{SYM2𝑅×𝑆}_𝑏 is the background potential for SYM_𝑅×𝑆². Alternatively, we may directly plug 𝑉𝜌𝑅,𝜂+𝑛𝑅=1𝑅³𝜌²−2𝜂+𝑛3𝜂+𝑛³+𝑣_𝜂/𝑅𝜌𝑅,𝜂+𝑛𝑅 (131) into (116) and (115), taking the limit at the end. Let 𝑣_𝜂/𝑅(𝜌/𝑅,(𝜂+𝑛)/𝑅)=𝑢(𝜌²,𝜂); the 11𝑑 background 𝐹_∗(𝑅,𝑥) is 𝑑𝑠²₁₁=2𝛼²𝑛²𝜕²_𝜌2𝑢^2/3×𝛼⁴𝑛𝜕²_𝜌2𝑢4𝑑𝑥⁻𝑑𝑥⁻−2𝛼²𝑛1+𝜕_𝜌2𝜕_𝜂𝑢𝑑𝑥⁺𝑑𝑥⁻+𝑛²4𝑑𝑥⁺𝑑𝑥⁺−𝑑Ω²₂−4𝑑𝜂²+𝑑𝜌²+𝜌²𝑑Ω²₅, (132) which has the topology of 𝑅×𝑅⁷×𝑆²×𝑅. The 10𝑑 background 𝐹_∗10(𝑅,𝑥) is 𝑑𝑠²₁₀=𝜕²_𝜌2𝑢𝑛^−1/2−4𝑑𝑥⁺𝑑𝑥⁺+𝑑Ω²₂𝜕+4^2𝜌2𝑢𝑛^1/2𝑑𝜂²+𝑑𝜌²+𝜌²𝑑Ω²₅,𝑒^𝜙=𝛼²2𝜕²_𝜌2𝑢𝑛^1/4,𝐶₁2=−1+𝜕_𝜌2𝜕_𝜂𝑢𝛼²𝜕²_𝜌2𝑢𝑑𝑥⁺,𝐹₄=𝑑𝐶₃,𝐶₃=−4𝑛𝛼²𝜕²_𝜌2𝑢𝑑𝑥⁺∧𝑑²𝐻Ω,₃=𝑑𝐵₂,𝐵₂=−1+𝜕_𝜌2𝜕_𝜂𝑢4𝜕²_𝜌2𝑢+𝜂𝑑²Ω. (133)𝑑𝑥⁺𝑑𝑥⁺, and 𝑑Ω²₂ now have the same prefactor. Equations (122) and (133) can be taken as the background for 1𝑑 and 3𝑑 gauge theories, respectively.

On gauge theory side, to study the fluctuation around the spherical membranes, we should expand 𝑆^PW_𝑅=1/𝑅³ around 𝑌. In [20], it was shown that in the limit of 𝑛→∞, 𝑅³/𝑛 fixed, (i) PWMM around a certain vacuum is equivalent to SYM_𝑅×𝑆² around each vacuum and (ii) SYM_𝑅×𝑆² around a certain vacuum with a periodicity imposed is equivalent to SYM_𝑅×𝑆³_/𝑍𝑘 around each vacuum. In particular, SYM_𝑅×𝑆³ can be realized as the PWMM around a certain vacuum with a periodicity condition imposed.

Concretely, (𝑛/𝑅³)𝑆^PW_𝑅=1(𝑌;𝜂)=(𝑛/𝑅³)𝑆^PW_𝑅=1(𝑌+𝜂)=𝑆^𝑅×𝑆2(𝑦+𝑦), where 𝑆^𝑅×𝑆2 is the action of the 𝑈(𝑁)  SYM_𝑅×𝑆² with 𝑁=𝑁₁+⋯+𝑁_𝑡: 𝑆^𝑅×𝑆2=1𝑔²_𝑅×𝑆2𝑑𝑥⁺𝑑Ω₂𝜇²−1×Tr4𝐹_𝑎𝑏𝐹^𝑎𝑏−12𝐷_𝑎Φ𝐷^𝑎Φ−𝜇²2Φ²+𝜇𝐹₁₂1Φ−2𝐷_𝑎𝑋_𝑚𝐷^𝑎𝑋_𝑚−𝜇²8𝑋²_𝑚+14𝑋_𝑚,𝑋_𝑛²+12Φ,𝑋_𝑚²−𝑖2𝜆Γ^𝑎𝐷_𝑎𝜆+𝑖𝜇8𝜆Γ¹²³𝜆−12𝜆Γ³[]−1Φ,𝜆2𝜆Γ^𝑚𝑋_𝑚,𝜆, (134)1/𝜇 is the radius of 𝑆² which is parameterized by (𝜃,𝜙). 𝑔²_𝑅×𝑆2=𝑅³/𝜇³𝑛. 𝑦  represents the vacuum: 𝜇Φ=2diag𝛼₁,…,𝛼_1𝑁1,𝛼₂,…,𝛼_2𝑁2,…,𝛼_𝑡,…,𝛼_𝑡𝑁𝑡,𝐴₁𝐴=0,₂=𝜃tan2𝜃ΦinregionI−cot2ΦinregionII. (135) Region I and II correspond to 0≤𝜃<(𝜋/2)+𝜀 and (𝜋/2)−𝜀<𝜃≤𝜋, respectively. 𝛼_𝑠 is identified with the 𝛼_𝑠 in (113). The 𝑈(𝑁) gauge group is spontaneously broken to 𝑈(𝑁₁)×𝑈(𝑁₂)×⋯×𝑈(𝑁_𝑡). Equation (135) corresponds to 𝑡 disks located at 𝜂₁=(𝜋/2)𝛼₁,𝜂₂=(𝜋/2)𝛼₂,…,𝜂_𝑡=(𝜋/2)𝛼_𝑡 with the electric charges on each equal to (𝜋²/8)𝑁₁,(𝜋²/8)𝑁₂,…,(𝜋²/8)𝑁_𝑡. The previous 𝜂_𝑠=(𝜋/2)(𝑛+𝛼_𝑠) in PWMM now becomes 𝜂_𝑠=(𝜋/2)𝛼_𝑠 due to the 𝜂→𝜂+𝑛 redefinition.

The correspondence between the spacetime coordinates and the SYM_𝑅×𝑆² fields is Φ↔𝜂, (𝑋⁴,…,𝑋⁹)↔(𝜌,Ω₅), (𝑥⁺,Ω₂)↔(𝑥⁺,Ω₂). Equation (134) is the SYM_𝑅×𝑆² on flat background. Plug (133) into (134), similar to (127), for bosonic part, we have 𝑆^𝑅×𝑆2_𝑏=𝜇³𝑑𝑥⁺𝑑Ω₂𝜇²√×Tr−𝐺𝑒^𝜙×−14𝐺^𝑎𝑐𝐺^𝑏𝑑𝐹_𝑎𝑏𝐹_𝑐𝑑−12𝐺^𝑎𝑏𝐺_𝜂𝜂𝐷_𝑎Φ𝐷_𝑏𝜇Φ−²2𝐺^Ω2𝐺_𝜂𝜂Φ²+𝜇(−𝐺)^−1/2𝐺^1/2_𝜂𝜂𝐹₁₂Φ−12𝐺^𝑎𝑏𝐺_𝑚𝑛𝐷_𝑎𝑋^𝑚𝐷_𝑏𝑋^𝑛−𝜇²8𝐺^Ω2𝐺_𝑚𝑛𝑋^𝑚𝑋^𝑛+14𝐺_𝑚𝑝𝐺_𝑛𝑞𝑋^𝑚,𝑋^𝑛𝑋^𝑝,𝑋^𝑞+12𝐺_𝜂𝜂𝐺_𝑚𝑛Φ,𝑋^𝑚Φ,𝑋^𝑛, (136) where 𝐺^𝑎𝑏∼𝐺_𝑚𝑛∼𝐺_𝜂𝜂∼𝐺^Ω2∼(𝜕²_𝜌2𝑢/𝑛)^1/2, 𝐺=det𝐺_𝑎𝑏∼−(𝜕²_𝜌2𝑢/𝑛)^−3/2, 𝑒^𝜙∼𝛼²(𝜕²_𝜌2𝑢/𝑛)^1/4. The action remains the same, so (134) is also the action of SYM_𝑅×𝑆² on background (133). The conclusion holds for fermionic part except for a rescaling of 𝜆.

Indeed, as we will show in Appendix A, such phenomenon is very common. Generically, the action of SYM_𝑝+1 on flat background is equal to the action of the SYM_𝑝+1 on the near-horizon geometry of the 𝐷𝑝 branes. (For 𝑝≥6, the worldvolume theories of 𝐷𝑝 branes do not decouple from the bulk, as is discussed in [7].) For SCFT₃ and SCFT₆ on 𝑀2 and 𝑀5, such requirement can even offer some clues for the structure of the field theory.

𝑢=𝑢(𝜌²,𝜂) is the function of the radial directions. For the given field configuration [Φ,𝑋⁴,…,𝑋⁹], let Φ,𝑋⁴,…,𝑋⁹=𝜕²_𝜌2𝑢𝑛^1/4𝜕Φ,^2𝜌2𝑢𝑛^1/4𝑋⁴𝜕,…,^2𝜌2𝑢𝑛^1/4𝑋⁹; (137) then 𝑆^𝑅×𝑆2_𝑏=𝜇³𝑑𝑥⁺𝑑Ω₂𝜇²√×Tr−𝐺𝑒^𝜙×−14𝐺^𝑎𝑐𝐺^𝑏𝑑𝐹_𝑎𝑏𝐹_𝑐𝑑−12𝐺^𝑎𝑏𝐷_𝑎Φ𝐷_𝑏𝜇Φ−²2𝐺^Ω2Φ²+𝜇(−𝐺)^−1/2𝐹₁₂1Φ−2𝐺^𝑎𝑏𝐷_𝑎𝑋^𝑚𝐷_𝑏𝑋^𝑛−𝜇²8𝐺^Ω2𝑋^𝑚𝑋^𝑛+14𝑋^𝑚,𝑋^𝑛𝑋^𝑝,𝑋^𝑞+12Φ,𝑋^𝑚Φ,𝑋^𝑛. (138) Compared with (134), with 𝑋^𝑚→𝑋^𝑚, Φ→Φ, the 3𝑑 background fields on 𝑅×𝑆² will get the radial dependence: 𝑒^𝜙=1⟶𝑒^𝜙=𝛼²2𝜕²_𝜌2𝑢𝑛^1/4𝜌²,𝜂,𝑑𝑠²₃=−4𝑑𝑥⁺𝑑𝑥⁺+𝑑Ω²₂⟶𝑑𝑠²₃=𝜕²_𝜌2𝑢𝑛^−1/2𝜌²,𝜂−4𝑑𝑥⁺𝑑𝑥⁺+𝑑Ω²₂. (139) 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 𝜌 and 𝜂. Let 𝑟=(𝜌²+𝜂²)^1/2; 𝑑𝑠²₁₀ in (133) can be written as 𝑑𝑠²₁₀=𝜕²_𝜌2𝑢𝑛^−1/2−4𝑑𝑥⁺𝑑𝑥⁺+𝑑Ω²₂𝜕+4^2𝜌2𝑢𝑛^1/2𝑑𝑟²+4𝑟²𝜕²_𝜌2𝑢𝑛^1/2𝑑Ω²₆, (140) with SYM_𝑅×𝑆² living at 𝑟=∞. However, with the energy 𝐸 indentified with 𝑟, the RG flow cannot give 𝑢(𝜌²,𝜂), which is not just the function of 𝑟. Instead, we will make the 𝑋^𝑚→𝑋^𝑚, Φ→Φ transformation to recover 𝜌𝜂 dependence of the 3𝑑 background fields in 𝑆𝑌𝑀_𝑅×𝑆².

For PWMM, we have 𝑒^𝑊PW(𝐹)=𝑑𝜂𝑒^−𝑆PW(𝑌^;𝜂). (141)𝑊^PW(𝐹) is the 𝑥⁻ reduction of the 11𝑑 supergravity effective action of the field generated by the brane 𝑌 on pp-wave. In the limit with 𝑛→∞, 𝑅→∞, 𝑅³/𝑛=𝛼² fixed, under the change of variables 𝜂→𝜂+𝑛, 𝑆^PW𝑌;𝜂⟶𝑆^𝑅×𝑆2𝑊𝑦;𝑦_PW(𝐹)⟶𝑊^𝑅×𝑆2(𝐹), (142) where 𝑊^𝑅×𝑆2(𝐹) is the type IIA action for the above-mentioned supergravity solution dual to a vacuum of SYM_𝑅×𝑆². Then, 𝑒^{𝑊2𝑅×𝑆(𝐹)}=𝑑𝑦𝑒^{−𝑆2𝑅×𝑆(}𝑦_;𝑦). (143)

As is demonstrated in [19, 20], from SYM_𝑅×𝑆² and the corresponding type IIA solution, it is also possible to get the SYM_𝑅×𝑆³ and the associated type IIB solution. In (135), suppose 𝜇Φ=2×diag…,𝑠−1,…,𝑠−1_𝑀,𝑠,…,𝑠_𝑀,𝑠+1,…,𝑠+1_𝑀,…, (144) where 𝑠 runs from −∞ to ∞. Expanding the action (134) around this vacuum and imposing the condition 𝑦^{(𝑠+1,𝑡+1)}=𝑦^(𝑠,𝑡) on all of the field fluctuations, one will get [20] 𝑆^𝑅×𝑆2𝑦+𝑦_{[𝑦(𝑠+1,𝑡+1)=𝑦(𝑠,𝑡)]}=𝑆^𝑅×𝑆3(𝑧)=1𝑔²_𝑅×𝑆3𝑑𝑡𝑑Ω₃(𝜇/2)³−1×Tr4𝐹_𝑎𝑏𝐹^𝑎𝑏−12𝐷_𝑎𝑋_𝑚𝐷^𝑎𝑋_𝑚−112𝑅𝑋²_𝑚−𝑖2𝜆Γ^𝑎𝐷_𝑎1𝜆−2𝜆Γ^𝑚𝑋_𝑚+1,𝜆4𝑋_𝑚,𝑋_𝑛², (145) where 𝑔²_𝑅×𝑆3=4𝜋𝜇𝑔²_𝑅×𝑆2. (146)𝑆^𝑅×𝑆3 is the action of the 𝑈(𝑀) SYM on 𝑅×𝑆³.

This is a special example of Taylor’s prescription for the compactification (the 𝑇-duality) in matrix models [43]. The new ingredient is the nontrivial gauge field, which makes a nontrivial fibration of 𝑆¹ over 𝑆² rather than a direct product; as a result, it is 𝑆³ other than 𝑆²×𝑆¹ that is obtained [20].

On gravity side, start from the trivial vacuum of SYM_𝑅×𝑆², for which there is only a single disk with the electric charge (𝜋²/8)𝑀; compactify the 𝜂 direction; the disk configuration in covering space will contain the infinite copies of disks with the period 𝜋/2, corresponding to the vacuum (144). The type IIA geometry generated by (144), after a 𝑇-duality transformation, becomes the type IIB geometry AdS₅×𝑆⁵ [19]. On field theory side, since 𝜂 is compactified, the field fluctuations should respect the periodicity condition 𝑦^{(𝑠+1,𝑡+1)}=𝑦^(𝑠,𝑡). Taylor’s prescription for the compactification also involves the 𝑇-duality transformation, making SYM_𝑅×𝑆² in type IIA become 𝑆^𝑅×𝑆3 in type IIB. Equation (143) turns into 𝑒^{𝑊3𝑅×𝑆(𝐹)}=[𝑑𝑧]𝑒^{−𝑆2𝑅×𝑆(𝑧)}. (147) For the trivial vacuum of 𝑆^𝑅×𝑆3, 𝐹 represents AdS₅×𝑆⁵ background.

Finally, the geometry arising from the backreaction of the 𝐷3 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 1/2 BPS configuration, and then take the certain limit. TGMT is the discrete regularization of the 𝐷3 branes, so the resulted gauge theory should be a 4𝑑 gauge theory as is required since the geometry is generated by spherical 𝐷3 branes on pp-wave.

4. Holography in AdS/CFT

SYM₄ is the gauge dual of the string theory on AdS₅×𝑆⁵. A natural question is what will be the gauge dual of the string theory if the field perturbation is added to AdS₅×𝑆⁵. For BFSS matrix model, we have matrices 𝑋^𝑖, 𝑖=1,…,9 representing nine transverse coordinates, so for a scalar field 𝜙𝑡,𝑥¹,…,𝑥⁹=^∞_𝑛1=0⋯^∞_𝑛9=01𝑛₁!⋯𝑛₉!×𝜕^𝑛1₁⋯𝜕^𝑛9₉𝜙()×𝑥𝑡,0,…,0_1𝑛1⋯𝑥^9𝑛9, (148) we may get the operator realization Φ(𝑡)=^∞_𝑛1=0⋯^∞_𝑛9=01𝑛₁!⋯𝑛₉!×𝜕^𝑛1₁⋯𝜕^𝑛9₉𝜙()𝑋𝑡,0,…,0×STr^1𝑛1⋯𝑋^9𝑛9, (149) with 𝑥^𝑖 replaced by matrix 𝑋^𝑖. Adding Φ(𝑡) to the Lagrangian gives a matrix model on a background with the scalar field 𝜙(𝑡,𝑥¹,…,𝑥⁹) turned on. The situation for SYM₄ 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 SYM₄, the original background is AdS₅×𝑆⁵; fields are expanded in terms of the spherical harmonics on 𝑆⁵. For scalar ℎ, ℎ𝑥,𝑦=^∞_𝑛=2ℎ_𝑛𝑥,𝑦𝑌^𝑛𝑦𝑦,𝑌^𝑛𝑦𝑦=𝐶_{𝑎1⋯𝑎𝑛}𝑦^𝑎1⋯𝑦^𝑎𝑛𝑦^𝑛, (150)where𝑌^𝑛(𝑦/|𝑦|) is a spherical harmonic of rank 𝑛. ℎ_𝑛(𝑥,|𝑦|) is a scalar field on AdS₅ transforming in the (0,𝑛,0) irrep of 𝑆𝑂(6). It is necessary to find the operator correspondence of 𝑌^𝑛(𝑦/|𝑦|). In SYM₄, we have 𝐶_{𝑎1⋯𝑎𝑛}Tr(𝑋^𝑎1⋯𝑋^𝑎𝑛), which, however, is the operator realization of 𝐶_{𝑎1⋯𝑎𝑛}(𝑦^𝑎1⋯𝑦^𝑎𝑛).

Naively, let 𝑅²=𝑒^2Σ=𝑋^𝑚𝑋^𝑚, Σ=(1/2)ln(𝑋^𝑚𝑋^𝑚); 𝑅 and Σ may be the operators corresponding to 𝑟=|𝑦|=𝑒^𝜎 and 𝜎, respectively. For the 5𝑑 field 𝐹(𝑥,𝑟), for all 𝜎₀=ln𝑟₀, formally, 𝐹(𝑥,𝑅)=𝑓(𝑥,Σ)=𝑓⁽⁰⁾𝑥,𝜎₀+𝑓⁽¹⁾𝑥,𝜎₀Σ−𝜎₀𝐼+12𝑓⁽²⁾𝑥,𝜎₀Σ−𝜎₀𝐼²+⋯=𝑓⁽⁰⁾𝑥,𝜎₀+𝑓⁽¹⁾𝑥,𝜎₀12𝑋ln^𝑚𝑋^𝑚𝑟²₀+12𝑓⁽²⁾𝑥,𝜎₀12𝑋ln^𝑚𝑋^𝑚𝑟²₀²+⋯. (151) Functions 𝑓^(𝑛)(𝑥,𝜎₀) are coupling constants of the 4𝑑 gauge theory living in (𝑥,𝜎₀). Similar to (28), when the gauge theory moves along 𝜎, Σ→Σ−𝑎𝐼, 𝑓^(𝑛)(𝑥,𝜎₀)→𝑓^(𝑛)(𝑥,𝜎₀+𝑎), Σ could be decomposed into the 𝑈(1) part 𝜎 and the traceless part Σ_𝑠=Σ−𝜎𝐼: 1𝜎=𝑁1TrΣ=𝑋2𝑁Trln^𝑚𝑋^𝑚=1𝑋2𝑁lndet^𝑚𝑋^𝑚,Σ_𝑠=12ln𝑋^𝑚_𝑠𝑋^𝑚_𝑠,det𝑋^𝑚_𝑠𝑋^𝑚_𝑠=1, (152)𝑋^𝑚=𝑒^𝜎𝑋^𝑚_𝑠. The configuration (𝑋⁵,…,𝑋¹⁰) is equivalent to (𝑋⁵_𝑠,…,𝑋_𝑠¹⁰,𝜎). 𝜎→𝜎+𝑎, 𝑋^𝑚→𝑒^𝑎𝑋^𝑚.

One may take 𝐶_{𝑎1⋯𝑎𝑛}Tr(𝑋^𝑎1_𝑠⋯𝑋^𝑎𝑛_𝑠) as the operator corresponding to 𝑌^𝑛(𝑦/|𝑦|) and consider the gauge theory with the vertex operator perturbation realized as (in fact, we will choose 𝑋 instead of 𝑋_𝑠 with det(𝑋^𝑚𝑋^𝑚)=∞. 𝐶_{𝑎1⋯𝑎𝑛}Tr(𝑋^𝑎1⋯𝑋^𝑎𝑛)∼𝑌^𝑛(𝑦), where |𝑦|=∞)𝐻_𝑛(𝑥,𝜎)=ℎ_𝑛(𝑥,𝜎)𝐶_{𝑎1⋯𝑎𝑛}Tr𝑋^𝑎1_𝑠⋯𝑋^𝑎𝑛_𝑠; (153)ℎ_𝑛(𝑥,𝜎) can be the arbitrary 5𝑑 function. Gauge theory like this is difficult to approach directly. We still prefer SYM₄ with ℎ_𝑛(𝑥)𝐶_{𝑎1⋯𝑎𝑛}Tr(𝑋^𝑎1⋯𝑋^𝑎𝑛) added, which, after a suitable transition, will become a gauge theory with the operator ℎ_𝑛(𝑥,𝜎)𝐶_{𝑎1⋯𝑎𝑛}Tr(𝑋^𝑎1⋯𝑋^𝑎𝑛). The 5𝑑 field ℎ_𝑛(𝑥,𝜎) is not arbitrary anymore but is determined by ℎ_𝑛(𝑥).

This is quite similar to the noncritical string coupling with 2𝑑 gravity. For critical string with 26 coordinates, any 26𝑑 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 25𝑑 fields can be constructed. However, if the noncritical string is coupled to the conformal mode of the 2𝑑 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 26𝑑 background fields. The 26𝑑 fields are induced from the 25𝑑 fields with the conformal mode acting as the 26th dimension. For SYM₄, the role of the conformal mode is played by 𝜎.

4.1. Noncritical String Coupling with 2𝑑 Gravity

Let us have a simple review of the noncritical string [44–49]. Consider the bosonic string living in 𝑑 dimensional spacetime. The corresponding nonlinear sigma model action is 𝑆=1𝑔,𝑋4𝜋𝛼^′𝑑²𝜉√𝑔𝑔^𝑎𝑏𝜕_𝑎𝑋^𝜇𝜕_𝑏𝑋^𝜈𝐺_𝜇𝜈(𝑋)+𝜖^𝑎𝑏𝜕_𝑎𝑋^𝜇𝜕_𝑏𝑋^𝜈𝐵_𝜇𝜈(𝑋)+𝛼^′𝑅Φ(𝑋)+𝑇(𝑋)+⋯, (154)𝜇,𝜈=1,…,𝑑. The partition function is defined as 𝐷𝑍=^𝑔𝑔𝐷_𝑔𝑋voldiff₂𝑒^{−𝑆(𝑔,𝑋)}.𝐷_𝑔𝑔𝐷_𝑔𝑋voldiff₂𝑒^{−𝑆(𝑔,𝑋)}=𝐷_𝑒𝜔𝑔𝑒^𝜔𝑔𝐷_𝑒𝜔𝑔𝑋voldiff₂𝑒^{−𝑆(𝑒𝜔𝑔,𝑋)}. (155) The theory is conformal invariant since all 𝑔 are integrated. The integration over the small 𝑔 gives the divergence, to cure which a cutoff should be introduced, destroying the conformal invariance. Under the conformal gauge fixing 𝑔=𝑒^𝜑𝑔, 𝜑≥0, 𝑔 is a small metric giving the cut-off scale 𝐷_𝑔𝑔voldiff₂=𝐷_𝑔𝜑volconf₂𝑒^{−𝑆𝐿(𝜑;}𝑔⁾, (156) where 𝑆_𝐿(𝜑;𝑔) is the Liouville action. With 𝐷_𝑔𝑋→𝐷𝑔𝑋, 𝐷_𝑔𝜑→𝐷𝑔𝜑, the partition function becomes 𝑍𝑔,0=^∞_𝜑=0𝐷_𝑔𝜑𝐷_𝑔𝑋volconf₂𝑒^{−𝑆(𝑔,𝑋)−𝑆𝐿(𝜑;}𝑔₎=^∞_𝜑=0𝐷𝑔𝜑𝐷𝑔𝑋volconf₂𝑒⁻𝑆^(𝜑,𝑋;𝑔⁾, (157) where 𝑆𝑔=1𝜑,𝑋;4𝜋𝛼^′𝑑²𝜉𝑔𝑔^𝑎𝑏𝜕_𝑎𝜑𝜕_𝑏𝜑+𝑔^𝑎𝑏𝜕_𝑎𝑋^𝜇𝜕_𝑏𝑋^𝜈𝐺_𝜇𝜈𝑋,𝜑+𝜖^𝑎𝑏𝜕_𝑎𝑋^𝜇𝜕_𝑏𝑋^𝜈𝐵_𝜇𝜈𝑋,𝜑+𝛼^′𝑅Φ𝑋,𝜑+𝑇𝑋,𝜑+⋯. (158) The original 𝑑 dimensional background field 𝐹(𝑋), after the gravitational dressing, becomes the 𝑑+1 dimensional field 𝐹(𝑋,𝜎). 𝐹(𝑋,0)=𝐹(𝑋). One can also make a change of the variables to move the boundary from 𝜑=0 to 𝜑=−∞: 𝑍𝑔,0=^∞_𝜑=0𝐷𝑔𝜑𝐷𝑔𝑋volconf₂𝑒⁻𝑆^(𝜑,𝑋;𝑔⁾=^∞_𝜑=0𝐷𝑔𝜑−∞𝐷𝑔𝑋volconf₂𝑒⁻𝑆_{(𝜑−∞+∞,𝑋;}𝑔₎=^∞_𝜑=−∞𝐷𝑔𝜑𝐷𝑔𝑋volconf₂𝑒⁻𝑆_{(𝜑+∞,𝑋;}𝑔₎=^∞_𝜑=−∞𝐷𝑔𝜑𝐷𝑔𝑋volconf₂𝑒⁻𝑆^′_(𝜑,𝑋;𝑔₎. (159) In 𝑆^′, 𝐹^′(𝜑)=𝐹(𝜑+∞).

Since 𝑔=𝑒^𝜑+𝜔𝑒^−𝜔𝑔, for 𝜔≥0, 𝑍𝑔,0=^∞_𝜑=0𝐷_𝑒−𝜔𝑔𝜑+𝜔𝐷_𝑒−𝜔𝑔𝑋volconf₂𝑒⁻𝑆_{(𝜑+𝜔,𝑋;𝑒}^−𝜔𝑔₎=^∞_𝜑=𝜔𝐷_𝑒−𝜔𝑔𝜑𝐷_𝑒−𝜔𝑔𝑋volconf₂𝑒⁻𝑆_{(𝜑,𝑋;𝑒}^−𝜔𝑔₎=𝑍𝑒^−𝜔𝑔,𝜔. (160) With the cutoff being introduced, the conformal transformation should be accompanied by a change of the boundary: 𝑍𝑒^−𝜔𝑔,0−𝑍𝑔,0=^𝜔_𝜑=0𝐷_𝑒−𝜔𝑔𝜑𝐷_𝑒−𝜔𝑔𝑋volconf₂𝑒⁻𝑆_{(𝜑,𝑋;𝑒}^−𝜔𝑔₎=𝑍𝑒^−𝜔𝑔,0−𝑍𝑒^−𝜔𝑔,𝜔. (161) With the cutoff being removed, 𝑍(𝑔)=𝑍(𝑒^−𝜔𝑔).

𝑆(𝜑,𝑋;𝑔) is the action of the bosonic string coupling with the background fields 𝐹(𝑋,𝜎). The conformal invariance indicates that 𝐹(𝑋,𝜎) should be the on-shell solution of gravity. To arrive at this result, it is important that, in (157), the change of the 2𝑑 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 SYM₄ with the action 𝐼𝑔_𝜇𝜈1,𝑋,𝐴,𝜆=−𝑔²_𝑌𝑀𝑑⁴𝑥−𝑔1×Tr2𝑔^𝜇𝜈𝑔^𝜆𝜎𝐹_𝜇𝜆𝐹_𝜈𝜎+𝑔^𝜇𝜈𝐷_𝜇𝑋^𝑚𝐷_𝜈𝑋^𝑚+16𝑅𝑋²_𝑚−𝑖𝜆Γ^𝜇𝐷_𝜇𝜆−𝜆Γ^𝑚𝑋_𝑚−1,𝜆2𝑋^𝑚,𝑋^𝑛2. (162) Select 𝑋 so that 𝑋 can be expressed as 𝑋=𝑒^𝜎𝑋 with 𝜎∈(−∞,0]. 𝑋 is the infinite matrix representing the maximum 𝑋. The partition function is 𝑍𝑔_𝜇𝜈,𝑋=𝑋𝐷𝑔𝑋𝐷𝑔𝐴𝐷𝑔𝜆exp−𝐼𝑔_𝜇𝜈,𝑋,𝐴,𝜆=𝑒^𝑊(𝑔_𝜇𝜈^,𝑋⁾. (163)

Due to the Weyl invariance, 𝐼𝑔_𝜇𝜈,𝑋,𝐴,𝜆=𝐼𝑒^2𝜔𝑔_𝜇𝜈,𝑒^−𝜔𝑋,𝐴,𝑒^−(3/2)𝜔𝜆. (164) Suppose 𝐷𝑔𝑋𝐷𝑔𝐴𝐷𝑔𝜆=𝐷_𝑒2𝜔𝑔𝑒^−𝜔𝑋𝐷_𝑒2𝜔𝑔×𝐴𝐷_𝑒2𝜔𝑔𝑒^−(3/2)𝜔𝜆𝑒^{𝑆𝑐[𝜔;}𝑔_], (165) with no counterterm added,  SYM₄ is conformal invariant, so at least for finite 𝜔 and 𝑔_𝜇𝜈, 𝑆^𝑐[𝜔;𝑔]=0. 𝑍(𝑔_𝜇𝜈;𝑋)=𝑍(𝑒^−2𝜔𝑔_𝜇𝜈;𝑒^𝜔𝑋). However, this may not be the case when 𝜔=∞ or when 𝑔_𝜇𝜈→0. 𝑆^𝑐[𝜔;𝑔] will still be kept for the infinite conformal transformation.

Nevertheless, with 𝑔_𝜇𝜈=𝜂_𝜇𝜈, we always have 𝑆^𝑐[𝜎;𝑔]=0. With 𝑋 parameterized as 𝑋=𝑒^𝜎𝑋, 𝑍𝑔_𝜇𝜈;𝑋=⁰_𝜎=−∞𝐷𝑔𝑒^𝜎𝑋𝐷𝑔𝑔𝜆𝑔𝐴𝐷×exp−𝐼_𝜇𝜈,𝑒^𝜎=𝑋,𝐴,𝜆_0𝜎=−∞𝐷_𝑒2𝜎𝑔𝑋𝐷_𝑒2𝜎𝑔𝐴𝐷_𝑒2𝜎𝑔𝜆×exp−𝐼𝑒^2𝜎𝑔_𝜇𝜈,𝑋,𝐴,𝜆. (166)(167) From (166) to (167), the integration over 𝜎 in 𝑋 is converted to the integration over 𝜎 in 𝑔. The partition function of SYM₄ 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: 𝑑𝑠²=𝑒^2𝜎𝑑𝑥²₄+𝑟²₃𝑑𝜎²+𝑑Ω²₅=𝑟𝑟₃²𝑑𝑥²₄+𝑟₃𝑟²𝑑𝑟²+𝑟²𝑑Ω²₅. (168)𝑟=𝑒^𝜎𝑟₃. 𝑟∈[0,+∞), 𝑟₃=+∞, and 𝜎∈(−∞,0]. 𝑟₃=+∞ because we take 𝑋 as the standard matrix. For finite 𝑟₃, 𝜎∈(−∞,+∞]. With 𝑟₃ being the radius of AdS₅×𝑆⁵, (167) could be taken as the partition function of a gauge theory on AdS₅×𝑆⁵.

Equations (166) and (167) have the direct extension to SCFT₃ and SCFT₆ for 𝑀2 and 𝑀5. For SCFT₃, 𝑋 has the weight 1/2, and the Weyl transformation is (𝑒^𝜎𝑋,𝑔_𝜇𝜈)→(𝑋,𝑒^4𝜎𝑔_𝜇𝜈). The induced metric is 𝑑𝑠²=𝑒^4𝜎𝑑𝑥²₃+𝑟²₂𝑑𝜎²+𝑑Ω²₇=𝑟𝑟₂⁴𝑑𝑥²₃+𝑟₂𝑟²𝑑𝑟²+𝑟²𝑑Ω²₇ (169) with 𝑟=𝑒^𝜎𝑟₂. For SCFT₆, 𝑋 has the weight 2, and the Weyl transformation is (𝑒^𝜎𝑋,𝑔_𝜇𝜈)→(𝑋,𝑒^𝜎𝑔_𝜇𝜈). The induced metric is 𝑑𝑠²=𝑒^𝜎𝑑𝑥²₆+𝑟²₅𝑑𝜎²+𝑑Ω²₄=𝑟𝑟₅𝑑𝑥²₆+𝑟₅𝑟²𝑑𝑟²+𝑟²𝑑Ω²₄, (170) with 𝑟=𝑒^𝜎𝑟₅. Equations (169) and (170) are the near-horizon geometries of 𝑀2 and 𝑀5, respectively.

More generically, as we will show in Appendix A, for SYM_𝑝+1, SCFT₃, and SCFT₆ on 𝐷𝑝, 𝑀2, and 𝑀5, the action on flat background and the action on the near-horizon geometry of 𝐷𝑝, 𝑀2, and 𝑀5 are the same. As a result, for 𝑀2 and 𝑀5 with the near-horizon geometry given by (169) and (170), 𝑋→(𝑟₂/𝑟)𝑋 and 𝑋→(𝑟₅/𝑟)𝑋 are always accompanied by 𝑑𝑥²₃→(𝑟/𝑟₂)⁴𝑑𝑥²₃ and 𝑑𝑥²₆→(𝑟/𝑟₅)𝑑𝑥²₆. For 𝐷𝑝 with the near-horizon geometry 𝑑𝑠²=𝑟𝑟_𝑝^(7−𝑝)/2𝑑𝑥²_𝑝+1+𝑟𝑟_𝑝^{−(7−𝑝)/2}𝑑𝑟²+𝑟²𝑑Ω²_8−𝑝,𝑒^Φ=𝑔_𝑠𝑟𝑟_𝑝^{(7−𝑝)(𝑝−3)/4}, (171) the 𝑋→(𝑟_𝑝/𝑟)^(7−𝑝)/4𝑋 transformation will then make 𝑑𝑥²_𝑝+1→(𝑟/𝑟_𝑝)^(7−𝑝)/2𝑑𝑥²_𝑝+1, 𝑒^Φ=𝑔_𝑠→𝑒^Φ=𝑔_𝑠(𝑟/𝑟_𝑝)^{(7−𝑝)(𝑝−3)/4}. The coupling constant as well as the metric now gets the radial dependence. ((7−𝑝)/2)/((7−𝑝)/2)=1, and the weight of 𝑋 is 1.

In the above situations, the transformations are all made for 𝑋. This is not always the case. Consider the 2𝑑𝑁=(4,4) SCFT and the 𝐴𝑑𝑆₃×𝑆³×𝑇⁴ geometry [50, 51]. The scalars 𝑋^𝑖, 𝑖=1,2,3,4 are related to the transverse 𝑇⁴, for which the metric is 𝑑𝑥²₄. There is no radial dependent prefactor for 𝑋 to absorb, so 𝑋 has the weight 0. In fact, the chiral primaries are constructed from the fermions Ψ that have the weight 1/2 and are representations of the 𝑅-symmetry group 𝑆𝑂(4) associated with 𝑆³.

Return to SYM₄, with the operator perturbation added: 𝐼𝑔_𝜇𝜈0,𝜙₀𝑔,𝑋,𝐴,𝜆=𝐼_𝜇𝜈0,𝑋,𝐴,𝜆+𝑑⁴𝑥−𝑔_𝜇𝜈0𝜙₀(𝑥)𝑂(𝑥), (172) where 𝑂=𝐶_{𝑎1⋯𝑎𝑛}Tr(𝑋^𝑎1⋯𝑋^𝑎𝑛). 𝑔_𝜇𝜈0 is the arbitrary 4𝑑 metric. The partition function is (here, the path integral measure depends on both 𝑔_𝜇𝜈0 and 𝜙₀. However, except for 𝑔_𝜇𝜈0, 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)𝑍𝑔_𝜇𝜈0,𝜙₀;𝑋=𝑋𝐷_{(𝑔0,𝜙0)}𝑋𝐷_{(𝑔0,𝜙0)}𝐴𝐷_{(𝑔0,𝜙0)}𝜆×exp−𝐼𝑔_𝜇𝜈0,𝜙₀.𝐼𝑔,𝑋,𝐴,𝜆_𝜇𝜈0,𝜙₀𝑒,𝑋,𝐴,𝜆=𝐼^2𝜔𝑔_𝜇𝜈0,𝑒^{(𝑛−4)𝜔}𝜙₀,𝑒^−𝜔𝑋,𝐴,𝑒^−(3/2)𝜔𝜆. (173) For finite 𝜔, 𝑔_𝜇𝜈0, 𝜙₀, 𝑍𝑔_𝜇𝜈0,𝜙₀;𝑋=𝑍𝑒^2𝜔𝑔_𝜇𝜈0,𝑒^{(𝑛−4)𝜔}𝜙₀;𝑒^−𝜔𝑋. (174)

We still want to do a 𝑋→𝑋 transformation: 𝐷_{(𝑔0,𝜙0)}𝑒^𝜎𝑋𝐷_{(𝑔0,𝜙0)}𝐴𝐷_{(𝑔0,𝜙0)}𝜆×exp−𝐼𝑔_𝜇𝜈0,𝜙₀,𝑒^𝜎𝑋,𝐴,𝜆=𝐷_{(𝑒2𝜎𝑔0,𝑒(𝑛−4)𝜎𝜙0)}𝑋𝐷_{(𝑒2𝜎𝑔0,𝑒(𝑛−4)𝜎𝜙0)}×𝐴𝐷_{(𝑒2𝜎𝑔0,𝑒(𝑛−4)𝜎𝜙0)}𝑒^−(3/2)𝜎𝜆×exp−𝐼𝑒^2𝜎𝑔_𝜇𝜈0,𝑒^{(𝑛−4)𝜎}𝜙₀,𝑋,𝐴,𝑒^−(3/2)𝜎𝜆+𝑆^𝑐𝜎;𝑔_𝜇𝜈0,𝜙₀. (175) When 𝜎=−∞, 𝑆^𝑐(𝜎;𝑔_𝜇𝜈0,𝜙₀) may not be zero. Instead of 𝑒^2𝜎𝑔_𝜇𝜈0 and 𝑒^{(𝑛−4)𝜎}𝜙₀, we can find the suitably adjusted 𝑔_𝜇𝜈(𝜎) and 𝜙(𝜎) so that 𝐷_{(𝑔0,𝜙0)}𝑒^𝜎𝑋𝐷_{(𝑔0,𝜙0)}𝐴𝐷_{(𝑔0,𝜙0)}𝜆×exp−𝐼𝑔_𝜇𝜈0,𝜙₀,𝑒^𝜎𝑋,𝐴,𝜆=𝐷_{[𝑔(𝜎),𝜙(𝜎)]}𝑋𝐷_{[𝑔(𝜎),𝜙(𝜎)]}𝐴𝐷_{[𝑔(𝜎),𝜙(𝜎)]}𝜆×exp−𝐼𝑔_𝜇𝜈(𝜎),𝜙(𝜎),𝑋,𝐴,𝜆. (176) In (176), the integration over 𝑋, 𝐴, and 𝜆 has already been carried out; both sides only depend on the 4𝑑 function 𝜎(𝑥). Obviously, 𝑔_𝜇𝜈0=𝑔_𝜇𝜈(0), 𝜙₀=𝜙(0). We also require 𝐷_{[𝑔(𝜎1),𝜙(𝜎1)]}𝑒^𝜎2𝑋𝐷_{[𝑔(𝜎1),𝜙(𝜎1)]}𝐴𝐷_{[𝑔(𝜎1),𝜙(𝜎1)]}𝜆×exp−𝐼𝑔_𝜇𝜈𝜎₁,𝜙𝜎₁,𝑒^𝜎2𝑋,𝐴,𝜆=𝐷_{[𝑔(𝜎1+𝜎2),𝜙(𝜎1+𝜎2)]}𝑋𝐷_{[𝑔(𝜎1+𝜎2),𝜙(𝜎1+𝜎2)]}×𝐴𝐷_{[𝑔(𝜎1+𝜎2),𝜙(𝜎1+𝜎2)]}𝜆×exp−𝐼𝑔_𝜇𝜈𝜎₁+𝜎₂,𝜙𝜎₁+𝜎₂,𝑋,𝐴,𝜆. (177) The 5𝑑 functions 𝑔_𝜇𝜈(𝑥,𝜎) and 𝜙(𝑥,𝜎) are induced from the 4𝑑 fields 𝑔_𝜇𝜈0(𝑥) and 𝜙(𝑥). For finite 𝜎, 𝑔_𝜇𝜈(𝜎)=𝑒^2𝜎𝑔_𝜇𝜈0, 𝜙(𝜎)=𝑒^{(𝑛−4)𝜎}𝜙₀.

More generically, with the 4𝑑 background field 𝐹₀ turned on, for the given 4𝑑 function 𝜎(𝑥), since 𝜎≤0, 𝑒^𝜎𝑋→𝑋 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. 𝑒^𝜎𝑋→𝑋, 𝐹₀→𝐹(𝜎). 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: 𝑍𝑔_𝜇𝜈0,𝐹₀;𝑋=⁰_𝜎=−∞𝐷_{[𝑔(𝜎),𝐹(𝜎)]}𝑋𝐷_{[𝑔(𝜎),𝐹(𝜎)]}𝐴𝐷_{[𝑔(𝜎),𝐹(𝜎)]}𝜆×exp−𝐼𝑔_𝜇𝜈(𝜎)(𝜎),=𝑍𝑔,𝐹𝑋,𝐴,𝜆_𝜇𝜈(𝜎),𝐹(𝜎);𝑋;0. (178)

In (178), with 𝐹 specified to 𝜙, the induced action 𝐼[𝑔_𝜇𝜈(𝜎),𝐹(𝜎),𝑋,𝐴,𝜆] contains the term ∫𝑑⁴𝑥−𝑔(𝜎)𝜙(𝑥,𝜎)𝑂(𝑥) with 𝑂(𝑥) given by 𝑂=𝐶_{𝑎1⋯𝑎𝑛}Tr(𝑋^𝑎1⋯𝑋^𝑎𝑛). 𝑂(𝑥)∼𝐶_{𝑎1⋯𝑎𝑛}(𝑦^𝑎1⋯𝑦^𝑎𝑛). |𝑦|=∞ since 𝑋 is the infinite matrix. The corresponding 10𝑑 field is Φ(𝑥,𝜎,Ω)=|𝑦|^𝑛−4𝜙(𝑥,𝜎)𝑌^𝑛(Ω) and 𝐠_𝜇𝜈(𝑥,𝜎)=|𝑦|⁴𝑔_𝜇𝜈(𝑥,𝜎), where 𝑌^𝑛(Ω) is the spherical harmonic on 𝑆⁵. Equation (178) is the partition function of the gauge theory on AdS₅×𝑆⁵ with the background fields Φ(𝑥,𝜎,Ω) and 𝐠_𝜇𝜈(𝑥,𝜎) turned on.

On gravity side, the asymptotic expansion of the gravity solution 𝐠(𝜎) is [52] 𝐠(𝜎)=𝑒^2𝜎𝑔₀+𝑒^−2𝜎𝑔₂+𝑒^−4𝜎𝑔₄−2𝜎𝑒^−4𝜎ℎ₄+𝑂𝑒^−3𝜎, (179) where 𝜎∈(−∞,+∞). Let 𝜎→𝜎+𝜎_∞, with 𝜎∈(−∞,0], 𝜎_∞=+∞, and 𝑔(𝜎)=𝑒^−2𝜎∞𝐠(𝜎). 𝑔(𝜎)=𝑒^2𝜎𝑔₀+𝑒^{−2(𝜎+𝜎∞)}𝑔₂+𝑒^{−4(𝜎+𝜎∞)}𝑔₄−2𝜎+𝜎_∞𝑒^{−4(𝜎+𝜎∞)}ℎ₄+𝑂𝑒^−5𝜎∞𝑒^−3𝜎. (180) For finite 𝜎, 𝑔(𝜎)=𝑒^2𝜎𝑔₀. Similarly, for the solution of the scalar field Φ(𝜎) [53], Φ(𝜎)=𝑒^{(𝑛−4)𝜎}𝜙₀+𝑒^−2𝜎𝜙₂+⋯+𝑒^{(4−2𝑛)𝜎}𝜙_2𝑛−4−2𝜎𝑒^{(4−2𝑛)𝜎}𝜑_2𝑛−4+𝑂𝑒^{−(𝑛+1)𝜎} (181) with 𝜎∈(−∞,+∞). 𝜙(𝜎)=𝑒^{(4−𝑛)𝜎∞}Φ(𝜎+𝜎_∞). 𝜙(𝜎)=𝑒^{(𝑛−4)𝜎}𝜙₀+𝑒^{−2(𝜎+𝜎∞)}𝜙₂+⋯+𝑒^{(4−2𝑛)(𝜎+𝜎∞)}𝜙_2𝑛−4−2𝜎+𝜎_∞𝑒^{(4−2𝑛)(𝜎+𝜎∞)}𝜑_2𝑛−4+𝑂𝑒^{(3−2𝑛)𝜎∞}𝑒^{−(𝑛+1)𝜎}, (182) with 𝜎∈(−∞,0]. When 𝑛>2, 𝜙(𝜎)=𝑒^{(𝑛−4)𝜎}𝜙₀ for finite 𝜎. For 𝐠-Φ coupled system, the subleading terms are determined by both 𝑔₀ and 𝜙₀. Φ(𝑥,𝜎) and 𝐠_𝜇𝜈(𝑥,𝜎) are fields on AdS₅ with 𝑟₃ finite. It is expected that the 𝑋→𝑋 transformation will give 𝑔_𝜇𝜈(𝑥,𝜎) and 𝜙(𝑥,𝜎) in (180) and (182).

From (177), 𝐷_{[𝑔(𝜎),𝜙(𝜎)]}𝑋𝐷_{[𝑔(𝜎),𝜙(𝜎)]}𝐴𝐷_{[𝑔(𝜎),𝜙(𝜎)]}𝜆×exp−𝐼𝑔_𝜇𝜈(𝜎),𝜙(𝜎),𝑋,𝐴,𝜆=𝐷_{[𝑔(𝜎+𝜔),𝜙(𝜎+𝜔)]}𝑒^−𝜔𝑋𝐷_{[𝑔(𝜎+𝜔),𝜙(𝜎+𝜔)]}×𝐴𝐷_{[𝑔(𝜎+𝜔),𝜙(𝜎+𝜔)]}𝜆×exp−𝐼𝑔_𝜇𝜈(𝜎+𝜔),𝜙(𝜎+𝜔),𝑒^−𝜔𝑋,𝐴,𝜆. (183) As a result, 𝑍𝑔_𝜇𝜈0,𝜙₀;𝑋=𝑍𝑔_𝜇𝜈(𝜎)(𝜎);=,𝜙𝑋;0_0𝜎=−∞𝐷_{[𝑔(𝜎+𝜔),𝜙(𝜎+𝜔)]}𝑒^−𝜔𝑋𝐷_{[𝑔(𝜎+𝜔),𝜙(𝜎+𝜔)]}×𝐴𝐷_{[𝑔(𝜎+𝜔),𝜙(𝜎+𝜔)]}𝜆×exp−𝐼𝑔_𝜇𝜈(𝜎+𝜔),𝜙(𝜎+𝜔),𝑒^−𝜔=𝑍𝑔𝑋,𝐴,𝜆_𝜇𝜈(𝜎+𝜔),𝜙(𝜎+𝜔);𝑒^−𝜔=𝑋;0_{𝜔𝜎=−∞}𝐷_{[𝑔(𝜎),𝜙(𝜎)]}𝑒^−𝜔𝑋𝐷_{[𝑔(𝜎),𝜙(𝜎)]}𝐴𝐷_{[𝑔(𝜎),𝜙(𝜎)]}𝜆×exp−𝐼𝑔_𝜇𝜈(𝜎),𝜙(𝜎),𝑒^−𝜔=𝑍𝑔𝑋,𝐴,𝜆_𝜇𝜈(𝜎),𝜙(𝜎);𝑒^−𝜔𝑋;𝜔. (184) With 𝑋→𝑒^−𝜔𝑋, the induced fields become 𝑔_𝜇𝜈(𝜎+𝜔) and 𝜙(𝜎+𝜔). Also, since 𝑍𝑔_𝜇𝜈0,𝜙₀;𝑋=𝑍𝑒^2𝜔𝑔_𝜇𝜈0,𝑒^{(𝑛−4)𝜔}𝜙₀;𝑒^−𝜔𝑋, (185) we have 𝑍𝑔_𝜇𝜈(𝜎),𝜙(𝜎);𝑋;0=𝑍𝑔^𝜔_𝜇𝜈(𝜎),𝜙^𝜔(𝜎);𝑒^−𝜔𝑋;0, (186) where 𝑔^𝜔_𝜇𝜈(𝜎) and 𝜙^𝜔(𝜎) are fields induced from 𝑒^2𝜔𝑔_𝜇𝜈0 and 𝑒^{(𝑛−4)𝜔}𝜙₀, respectively. There will be 𝑔^𝜔_𝜇𝜈(𝜎)=𝑔_𝜇𝜈(𝜎+𝜔), 𝜙^𝜔(𝜎)=𝜙(𝜎+𝜔). Indeed, in (182) and (180), with 𝜙₀ and 𝑔₀ replaced by 𝑒^{(𝑛−4)𝜔}𝜙₀ and 𝑒^2𝜔𝑔₀, one may get 𝜙(𝜎+𝜔) and 𝑔(𝜎+𝜔). 𝑔₀→𝑒^2𝜔𝑔₀, 𝜙₀→𝑒^{(𝑛−4)𝜔}𝜙₀, 𝜎→𝜎+𝜔 is a diffeomorphism transformation of the gravity solution 𝐠_𝜇𝜈(𝜎) and Φ(𝜎) [54].

Equation (184) is valid for [𝑔_𝜇𝜈(𝑥,𝜎),𝜙(𝑥,𝜎)] induced from [𝑔_𝜇𝜈0(𝑥),𝜙₀(𝑥)]. For the generic 5𝑑 functions [𝑔_𝜇𝜈(𝑥,𝜎),𝜙(𝑥,𝜎)], we only have 𝑍𝑔_𝜇𝜈(),𝜙();=𝑍𝑔𝑥,𝜎𝑥,𝜎𝑋;0_𝜔𝜇𝜈(𝑥,𝜎),𝜙^𝜔(𝑥,𝜎);𝑒^−𝜔𝑋;0 (187) for some 5𝑑 functions [𝑔^𝜔_𝜇𝜈(𝑥,𝜎),𝜙^𝜔(𝑥,𝜎)]. For constant 𝜔, under the scale transformation 𝑥→𝑒^𝜔𝑥, 𝑍𝑔_𝜇𝜈(),𝜙();=𝑍𝑒𝑥,𝜎𝑥,𝜎𝑋;0_−2𝜔𝑔^𝜔_𝜇𝜈𝑒^𝜔𝑥,𝜎,𝜙^𝜔𝑒^𝜔𝑥,𝜎;𝑒^−𝜔𝑋;0. (188) In special cases, if 𝑒^−2𝜔𝑔^𝜔_𝜇𝜈(𝜎)=𝑔_𝜇𝜈(𝜎), 𝜙^𝜔(𝜎)=𝜙(𝜎), the background is “conformal”. For [𝑔_𝜇𝜈(𝑥,𝜎),𝜙(𝑥,𝜎)], this requires 𝑔_𝜇𝜈(𝜎+𝜔)=𝑒^2𝜔𝑔_𝜇𝜈(𝜎), 𝜙(𝜎+𝜔)=𝜙(𝜎) and thus 𝑔_𝜇𝜈(𝜎)=𝑒^2𝜎𝑔_𝜇𝜈0, 𝜙(𝜎)=𝜙₀. 𝜙 is the marginal field.

For fields which are “conformal”, 𝑍𝑔_𝜇𝜈(),𝜙();=𝑍𝑔𝑥,𝜎𝑥,𝜎𝑋;0_𝜇𝜈𝑒^𝜔𝑥,𝜎,𝜙𝑒^𝜔𝑥,𝜎;𝑒^−𝜔𝑋;0, (189) and so 𝛿𝑍𝑔_𝜇𝜈(𝑥,𝜎),𝜙(𝑥,𝜎);𝑋;0𝛿𝜙𝑥₀,𝜎₀=𝛿𝑍𝑔_𝜇𝜈𝑒^𝜔𝑥,𝜎,𝜙𝑒^𝜔𝑥,𝜎;𝑒^−𝜔𝑋;0𝛿𝜙𝑒^𝜔𝑥₀,𝜎₀. (190) As a result, −𝑔𝑥₀,𝜎₀𝑂𝑥₀𝛿𝜎−𝜎₀=⁰_𝜎=−∞𝐷_[𝑔_(𝜎),𝜙_(𝜎)]𝑋𝐷_[𝑔_(𝜎),𝜙_(𝜎)]𝐴𝐷_[𝑔_(𝜎),𝜙_(𝜎)]𝜆×exp−𝐼𝑔_𝜇𝜈(),𝜙(),×𝑥,𝜎𝑥,𝜎𝑋,𝐴,𝜆−𝑔𝑥₀,𝜎₀𝑂𝑥₀𝛿𝜎−𝜎₀=𝑒^4𝜔−𝑔(𝑒^𝜔𝑥₀,𝜎₀)𝑂^𝜔(𝑒^𝜔𝑥₀)𝛿(𝜎−𝜎₀)_𝜔=⁰_𝜎=−∞𝐷_[𝑔_(𝜎),𝜙_(𝜎)]𝑒^−𝜔𝑋𝐷_[𝑔_(𝜎),𝜙_(𝜎)]𝐴𝐷_[𝑔_(𝜎),𝜙_(𝜎)]𝜆×exp−𝐼𝑔_𝜇𝜈𝑒^𝜔𝑥,𝜎,𝜙𝑒^𝜔𝑥,𝜎,𝑒^−𝜔𝑋,𝐴,𝜆×𝑒^{(4−𝑛)𝜔}−𝑔𝑒^𝜔𝑥₀,𝜎₀𝑂𝑒^𝜔𝑥₀𝛿𝜎−𝜎₀=⁰_𝜎=−∞𝐷_[𝑔_(𝜎),𝜙_(𝜎)]𝑋𝐷_[𝑔_(𝜎),𝜙_(𝜎)]𝐴𝐷_[𝑔_(𝜎),𝜙_(𝜎)]𝜆×exp−𝐼𝑔_𝜇𝜈(𝑥,𝜎),𝜙(𝑥,𝜎),𝑋,𝐴,𝜆×𝑒^{(4−𝑛)𝜔}−𝑔𝑒^𝜔𝑥₀,𝜎₀𝑂𝑒^𝜔𝑥₀𝛿𝜎−𝜎₀=𝑒^{(4−𝑛)𝜔}−𝑔𝑒^𝜔𝑥₀,𝜎₀𝑂𝑒^𝜔𝑥₀𝛿𝜎−𝜎₀, (191) and then 0=−𝑔𝑥₀,𝜎₀𝑂𝑥₀𝛿𝜎−𝜎₀=𝛿𝑍𝑔_𝜇𝜈(𝑥,𝜎),𝜙(𝑥,𝜎);𝑋;0𝛿𝜙𝑥₀,𝜎₀. (192)𝜙(𝑥,𝜎) is on shell.

Restricted to [𝑔_𝜇𝜈(𝑥,𝜎),𝜙(𝑥,𝜎)], usually, [𝑒^−2𝜔𝑔_𝜇𝜈(𝜎+𝜔),𝜙(𝜎+𝜔)]≠[𝑔_𝜇𝜈(𝜎),𝜙(𝜎)], [𝑔_𝜇𝜈(𝜎),𝜙(𝜎)] is not “conformal”. Instead of (191), if we require −𝑔𝑥₀,𝜎₀𝑂𝑥₀𝛿𝜎−𝜎₀=⁰_𝜎=−∞𝐷_{[𝑔(𝜎),𝜙(𝜎)]}𝑋𝐷_{[𝑔(𝜎),𝜙(𝜎)]}𝐴𝐷_{[𝑔(𝜎),𝜙(𝜎)]}𝜆×exp−𝐼𝑔_𝜇𝜈()(),×𝑥,𝜎,𝜙𝑥,𝜎𝑋,𝐴,𝜆−𝑔𝑥₀,𝜎₀𝑂𝑥₀𝛿𝜎−𝜎₀=𝑒^{(4−𝑛)𝜔}−𝑔(𝑥₀,𝜎₀)𝑂(𝑥₀)𝑒^{𝑖𝑝(𝜎+𝜔)}_𝜔=⁰_𝜎=−∞𝐷_{[𝑔(𝜎+𝜔),𝜙(𝜎+𝜔)]}𝑒^−𝜔𝑋𝐷_{[𝑔(𝜎+𝜔),𝜙(𝜎+𝜔)]}×𝐴𝐷_{[𝑔(𝜎+𝜔),𝜙(𝜎+𝜔)]}𝜆×exp−𝐼𝑔_𝜇𝜈(𝜎+𝜔),𝜙(𝜎+𝜔),𝑒^−𝜔𝑋,𝐴,𝜆×𝑒^{(4−𝑛)𝜔}−𝑔𝑥₀,𝜎₀𝑂𝑥₀𝑒^{𝑖𝑝(𝜎+𝜔)}=⁰_𝜎=−∞𝐷_{[𝑔(𝜎),𝜙(𝜎)]}𝑋𝐷_{[𝑔(𝜎),𝜙(𝜎)]}𝐴𝐷_{[𝑔(𝜎),𝜙(𝜎)]}𝜆×exp−𝐼𝑔_𝜇𝜈(𝜎),𝜙(𝜎),𝑋,𝐴,𝜆×𝑒^{(4−𝑛)𝜔}−𝑔𝑥₀,𝜎₀𝑂𝑥₀𝑒^{𝑖𝑝(𝜎+𝜔)}=𝑒^{(𝑖𝑝−𝑛+4)𝜔}−𝑔𝑥₀,𝜎₀𝑂𝑥₀𝑒^𝑖𝑝𝜎; (193) still, 0=−𝑔𝑥₀,𝜎₀𝑂𝑥₀𝑒^𝑖𝑝𝜎=−𝑔𝑥₀,𝜎₀𝑂𝑥₀𝛿𝜎−𝜎₀=𝛿𝑍𝑔_𝜇𝜈(𝑥,𝜎),𝜙(𝑥,𝜎);𝑋;0𝛿𝜙𝑥₀,𝜎₀. (194)𝜙(𝑥,𝜎) is also on shell.

𝜙(𝑥,𝜎) is the functional of 𝜙₀(𝑥). In general, 𝛿𝑍𝑔_𝜇𝜈(𝑥,𝜎),𝜙(𝑥,𝜎);𝑋;0𝛿𝜙₀𝑥₀=𝛿𝑍𝑔_𝜇𝜈0(𝑥),𝜙₀(𝑥);𝑋𝛿𝜙₀𝑥₀≠0 (195) since nothing could guarantee that the one-point function of the SYM₄ would vanish in presence of the arbitrary source 𝜙₀(𝑥). The physical field is Φ=𝑒^{(𝑛−4)𝜎∞}𝜙, so there is also 𝛿𝑍𝑔_𝜇𝜈(𝑥,𝜎),𝜙(𝑥,𝜎);𝑋;0𝛿Φ𝑥₀,𝜎₀=0. (196)

The difference between 𝐼[𝑔_𝜇𝜈0(𝑥),𝜙₀(𝑥),𝑋,𝐴,𝜆] and 𝐼[𝑔_𝜇𝜈(𝑥,𝜎),𝜙(𝑥,𝜎),𝑋,𝐴,𝜆] is that 𝑋 is replaced by 𝑋, while the field 𝐹₀(𝑥) becomes 𝐹(𝑥,𝜎). 𝐼[𝑔_𝜇𝜈(𝑥,𝜎),𝜙(𝑥,𝜎),𝑋,𝐴,𝜆] has the natural interpretation as a gauge theory on AdS₅×𝑆⁵ with the external field −𝑔(𝑥,𝜎)𝐹(𝑥,𝜎,Ω)=−𝑔(𝑥,𝜎)𝐹(𝑥,𝜎)𝑌^𝑛=𝑒^𝑛𝜎∞−𝑔(𝑥,𝜎)𝐹(𝑥,𝜎)𝑌^𝑛=−𝐠(𝑥,𝜎)𝐅(𝑥,𝜎)𝑌^𝑛 (197) turned on, where 𝐹(𝑥,𝜎) and 𝑔_𝜇𝜈(𝑥,𝜎) can be the arbitrary 5𝑑 functions. 𝐠_𝜇𝜈=𝑒^4𝜎∞𝑔_𝜇𝜈 and 𝐅=𝑒^{(𝑛−4)𝜎∞}𝐹 are physical fields on AdS₅. According to the previous philosophy, 𝑍𝑔_𝜇𝜈()();𝑆𝑥,𝜎,𝜙𝑥,𝜎𝑋;0=exp_𝑔𝑔_𝜇𝜈(𝑥,𝜎),𝜙(𝑥,𝜎);𝑋;0, (198)𝑆_𝑔[𝑔_𝜇𝜈(𝑥,𝜎),𝜙(𝑥,𝜎);𝑋;0] is the effective action of the type IIB supergravity on AdS₅×𝑆⁵.

For a special subset of fields {𝜙(𝑥,𝜎)} that can be derived from the 4𝑑 fields 𝜙₀(𝑥), 𝛿𝑍𝑔_𝜇𝜈(𝑥,𝜎),𝜙(𝑥,𝜎);𝑋;0𝛿𝜙𝑥₀,𝜎₀=0, (199) and then 𝛿𝑆_𝑔𝑔_𝜇𝜈(𝑥,𝜎),𝜙(𝑥,𝜎);𝑋;0𝛿Φ𝑥₀,𝜎₀=0; (200) that is, Φ(𝑥,𝜎) is the on-shell solution of the supergravity. For the given 𝜙₀(𝑥), there are two ways to get the induced 𝜙(𝑥,𝜎): one is through (176) and the other is to find the solution of 𝛿𝑆_𝑔/𝛿Φ(𝑥₀,𝜎₀)=0, imposing lim_𝜎→0𝑒^{(4−𝑛)𝜎∞}Φ(𝑥,𝜎)=𝜙₀(𝑥) as the boundary condition.

Finally, we arrive at 𝑊𝑔_𝜇𝜈0,𝜙₀;𝑋=ln𝑍𝑔_𝜇𝜈0,𝜙₀;𝑋=𝑆_𝑔𝑔_𝜇𝜈(𝑥,𝜎),𝜙(𝑥,𝜎);𝑋;0. (201) Starting from 𝐼[𝑔_𝜇𝜈0(𝑥),𝜙₀(𝑥),𝑋,𝐴,𝜆], SYM₄ with the source term added, after a 𝑋→𝑋 transformation, we get 𝐼[𝑔_𝜇𝜈(𝑥,𝜎),𝜙(𝑥,𝜎),𝑋,𝐴,𝜆], the action of the gauge theory on AdS₅×𝑆⁵, whose free energy may be equal to 𝑆_𝑔[𝑔_𝜇𝜈(𝑥,𝜎),𝜙(𝑥,𝜎);𝑋;0]. During the transition, 𝐹(𝑥,𝜎) are carefully adjusted to make the partition function of 𝐼[𝑔_𝜇𝜈0(𝑥),𝜙₀(𝑥),𝑋,𝐴,𝜆] and 𝐼[𝑔_𝜇𝜈(𝑥,𝜎),𝜙(𝑥,𝜎),𝑋,𝐴,𝜆] remain the same, then we arrive at (201), where 𝐹(𝑥,𝜎) is induced from 𝐹₀(𝑥) and is on shell with respect to 𝑆_𝑔.

4.3. Imposing the Cutoff

𝑆_𝑔𝑔_𝜇𝜈()();=𝑥,𝜎,𝜙𝑥,𝜎𝑋;0_0−∞𝑑𝜎𝑑⁴𝑥𝐿𝑔_𝜇𝜈(𝑥,𝜎),𝜙(𝑥,𝜎), (202) where  𝐿 is the Lagrangian of the supergravity. 𝑆_𝑔 is divergent. One may impose a cutoff 𝜎, 𝜎=−∞, 𝑆_𝑔𝑔_𝜇𝜈(𝑥,𝜎),𝜙(𝑥,𝜎);𝑋;𝜎=^𝜎_−∞𝑑𝜎𝑑⁴𝑥𝐿𝑔_𝜇𝜈(𝑥,𝜎),𝜙(𝑥,𝜎). (203) On gauge theory side, the one-loop effective action 𝑊(𝑔_𝜇𝜈0)=−lndet(𝐷)/2 is also divergent. With a cutoff 𝜖being introduced [55], 𝑊𝑔_𝜇𝜈0=1;𝜖2^∞_𝜖𝑑𝜌𝜌Tr𝑒^−𝜌𝐷=^𝜎_−∞𝑑𝜎Trexp−𝑒^{−2(𝜎+𝜎∞)}𝐷=^𝜎_−∞𝑑𝜎Trexp−𝑒^−2𝜎𝐃=𝑊𝐠_𝜇𝜈(0);𝜎, (204) where 𝑒^{−2(𝜎+𝜎∞)}=𝜌, 𝐠_𝜇𝜈(0)=𝑒^2𝜎∞𝑔_𝜇𝜈0, 𝐃 is the Laplace operator with the metric 𝐠_𝜇𝜈(0). Obviously, 𝑊(𝑒^2𝜔𝑔_𝜇𝜈0,𝑒^2𝜔𝜖)=𝑊(𝑔_𝜇𝜈0,𝜖)=𝑊(𝑒^2𝜔𝐠_𝜇𝜈(0);𝜎−𝜔)=𝑊(𝐠_𝜇𝜈(0);𝜎). For small but finite 𝜌 [56], Tr𝑒^−𝜌𝐷=^∞_𝑛=0𝐴_2𝑛𝜌^𝑛−2,𝐴_2𝑛=𝑑⁴𝑥√−𝑔𝑎_2𝑛𝑔, (205) so 𝑊𝑔_𝜇𝜈0=lim_𝜖→0𝑊𝑔_𝜇𝜈0,𝜖=lim_𝜖→0𝐴₀4𝜖²+𝐴₂−𝐴2𝜖⁴ln𝜖2+𝑊_ren𝑔_𝜇𝜈0. (206) The renormalized 𝑊_ren(𝑔_𝜇𝜈0) is finite. This is the realization of the holographic renormalization [52] on gauge theory side. With 𝑔_𝜇𝜈0→𝑒^2𝜔𝑔_𝜇𝜈0, 𝜖→𝑒^2𝜔𝜖, [52] 𝑊_ren𝑒^2𝜔𝑔_𝜇𝜈0−𝑊_ren𝑔_𝜇𝜈0=𝜔𝐴₄. (207)𝐴₄ is the conformal anomaly [57]. On the other hand, 𝑊𝐠_𝜇𝜈(0);𝜎=𝐀₀4𝑒^−4𝜎+𝐀₂2𝑒^−2𝜎+𝐀₄𝜎+𝑊_nonloc𝐠_𝜇𝜈(0), (208) so 𝑊𝐠_𝜇𝜈(0)=𝐀;0⁰4+𝐀₂2+𝑊_nonloc𝐠_𝜇𝜈(0)=𝑊_loc𝐠_𝜇𝜈(0)+𝑊_nonloc𝐠_𝜇𝜈(0),𝑊_nonloc𝐠_𝜇𝜈(0)−𝑊_ren𝑔_𝜇𝜈0=𝜎_∞𝐴₄. (209)𝑊_loc(𝐠_𝜇𝜈(0))=𝐀₀/4+𝐀₂/2 and 𝑊_nonloc(𝐠_𝜇𝜈(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 𝜎_∞𝐴₄.

On gravity side, the corresponding gravity solution has the near boundary expansion 𝐺𝑔₀=𝑔,𝜌⁰𝜌+𝑔₂+𝜌𝑔₄+ln𝜌ℎ₄+𝑂𝜌². (210) At 𝜌=𝜖, the metric is 𝐺(𝑔₀,𝜖). The on-shell gravity action with the cutoff 𝜌=𝜖 is 𝑆𝑔_𝜇𝜈0,𝜖=𝑆𝐺𝑔₀,𝜖=^∞_𝜖𝑑𝜌𝐿𝜌,2𝜌 (211) which is entirely determined by the boundary value 𝐺(𝑔₀,𝜖). 𝑆𝑔_𝜇𝜈0=lim_𝜖→0𝑆𝑔_𝜇𝜈0,𝜖=lim_𝜖→0𝑙16𝜋𝐺_𝑁𝑑⁴𝑥√×𝑎−𝑔₀𝑔𝜖²+𝑎₂𝑔𝜖−𝑎₄𝑔ln𝜖+𝑆_ren𝑔_𝜇𝜈0. (212) The renormalized 𝑆_ren(𝑔_𝜇𝜈0) is finite [52]. Since 𝐺(𝑒^2𝜔𝑔₀,𝑒^2𝜔𝜌)=𝐺(𝑔₀,𝜌), 𝐺(𝑒^2𝜔𝑔₀,𝑒^2𝜔𝜖)=𝐺(𝑔₀,𝜖), 𝑆(𝑒^2𝜔𝑔_𝜇𝜈0,𝑒^2𝜔𝜖)=𝑆(𝑔_𝜇𝜈0,𝜖) [52] 𝑆_ren𝑒^2𝜔𝑔_𝜇𝜈0−𝑆_ren𝑔_𝜇𝜈0=𝑙8𝜋𝐺_𝑁𝑑⁴𝑥√−𝑔𝜔𝑎₄𝑔=𝜔𝐴₄. (213) In both gauge theory and gravity, subtracting of the infinity introduces the conformal anomaly.

In (204), 𝜖 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 𝑋, which, although has some relevance with 𝜖, is still not it.

Take 𝑋=𝑒^𝜎𝑋 other than 𝑋 as the upper bound of 𝑋; the partition function is 𝑍𝑔_𝜇𝜈0;𝑋=^𝑋𝐷_𝑔0𝑋𝐷_𝑔0𝐴𝐷_𝑔0𝜆exp−𝐼𝑔_𝜇𝜈0=𝑍𝑔,𝑋,𝐴,𝜆_𝜇𝜈(𝜎);𝑋;𝜎. (214) For finite 𝑔_𝜇𝜈0 and 𝜔, 𝑍(𝑔_𝜇𝜈0;𝑒^𝜔𝑋)=𝑍(𝑒^2𝜔𝑔_𝜇𝜈0;𝑋). There is also 𝑍𝑔_𝜇𝜈0;𝑒^𝜔𝑋=𝑍𝑔_𝜇𝜈(𝜎);𝑋;=𝑍𝑔𝜎+𝜔_𝜇𝜈(𝜎+𝜔);𝑋;𝜎=𝑍𝑒^2𝜔𝑔_𝜇𝜈0;𝑋, (215) since the field induced from 𝑒^2𝜔𝑔_𝜇𝜈0 is 𝑔_𝜇𝜈(𝜎+𝜔). From 𝑍(𝑒^2𝜔𝑔_𝜇𝜈0;𝑋) and 𝑍(𝑔_𝜇𝜈0;𝑒^𝜔𝑋), the induced fields have the same boundary value 𝑔_𝜇𝜈(𝜎+𝜔).

For the infinitesimal 4𝑑 function 𝛿𝜎(𝑥), 𝑍𝑔_𝜇𝜈0;𝑒^𝛿𝜎𝑋=^{𝑒𝛿𝜎𝑋}𝐷_𝑔0𝑋𝐷_𝑔0𝐴𝐷_𝑔0𝜆exp−𝐼𝑔_𝜇𝜈0𝑑,𝑋,𝐴,𝜆=exp⁴𝑥𝐿^𝑎_𝑔0;𝑋(𝑥)𝛿𝜎(𝑥)𝑍𝑔_𝜇𝜈0;𝑋, (216) where 𝐿^𝑎_𝑔0;𝑋(𝑥)=lim_𝛿𝜎→01(𝑥)×𝛿𝜎_{𝑒𝛿𝜎(𝑥)𝑋}𝐷_𝑔0𝑋𝐷_𝑔0𝐴𝐷_𝑔0𝜆×exp−𝐼𝑔_𝜇𝜈0𝑔,𝑋,𝐴,𝜆−𝑍_𝜇𝜈0;𝑋×𝑍⁻¹𝑔_𝜇𝜈0;𝑋. (217) With 𝑆^𝑎(𝑔₀;𝑋)=∫𝑑⁴𝑥𝐿^𝑎_(𝑔0;𝑋)(𝑥), we may have 𝑊𝑔_𝜇𝜈0=^∞₀𝑑𝜎𝑆^𝑎𝑔₀;𝑒^𝜎𝑋+𝑊𝑔_𝜇𝜈0;𝑋=^∞_−∞𝑑𝜎𝑆^𝑎𝑔₀;𝑒^𝜎𝑋=^∞₀𝑑𝜌𝜌𝑆^𝑎𝑔₀;𝑋𝜌, (218) where 𝜌=𝑒^−𝜎𝑊𝑔_𝜇𝜈0;𝑋=⁰_−∞𝑑𝜎𝑆^𝑎𝑔₀;𝑒^𝜎𝑋=^∞₁𝑑𝜌𝜌𝑆^𝑎𝑔₀;𝑋𝜌𝑊𝑔_𝜇𝜈0;𝑒^𝛿𝜎𝑋−𝑊𝑔_𝜇𝜈0;𝑋=𝑑⁴𝑥𝐿^𝑎_𝑔0;𝑋(𝑥)𝛿𝜎(𝑥). (219)(220) Also, we may define 𝐿^𝑏_(𝑔0;𝑋), with 𝑊𝑒^2𝛿𝜎𝑔_𝜇𝜈0;𝑋−𝑊𝑔_𝜇𝜈0;𝑋=𝑑⁴𝑥𝐿^𝑏_𝑔0;𝑋(𝑥)𝛿𝜎(𝑥) (221)if𝑊𝑒^2𝛿𝜎𝑔_𝜇𝜈0;𝑋−𝑊𝑔_𝜇𝜈0;𝑒^𝛿𝜎𝑋=𝑑⁴𝑥𝐿^𝑐_𝑔0;𝑋(𝑥)(𝑥),𝐿𝛿𝜎_{𝑐(𝑔0;𝑋)}(𝑥)=𝐿^𝑏_(𝑔0;𝑋)(𝑥)−𝐿^𝑎_(𝑔0;𝑋)(𝑥). (222) For finite 𝛿𝜎 and 𝑔_𝜇𝜈0, 𝐿^𝑐_(𝑔0;𝑋)(𝑥)=0. For infinite 𝛿𝜎, for example, 𝑊(𝑔_𝜇𝜈0;𝑒^𝜔𝑋) and 𝑊(𝑒^2𝜔𝑔_𝜇𝜈0;𝑋) with 𝜔=−∞, 𝑊𝑔_𝜇𝜈0;𝑒^𝜔𝑋=ln𝑍𝑔_𝜇𝜈();,𝑊𝑒𝜎+𝜔𝑋;0_2𝜔𝑔_𝜇𝜈0;𝑋=ln𝑍𝑔^𝜔_𝜇𝜈(𝜎);𝑋;0. (223)𝑔_𝜇𝜈(𝜔)≠𝑒^2𝜔𝑔_𝜇𝜈0, so 𝑔^𝜔_𝜇𝜈(𝜎)≠𝑔_𝜇𝜈(𝜎+𝜔).

Although (219) and (204) look similar, the two kinds of cutoffs are different. In particular, ln𝑍𝑔_𝜇𝜈0;𝑋=ln𝑍𝑔_𝜇𝜈(𝜎);𝑋;𝜎≠𝑆_𝑔𝑔_𝜇𝜈(𝑥,𝜎);𝑋;𝜎=^𝜎_−∞𝑑𝜎𝑑⁴𝑥𝐿𝑔_𝜇𝜈(𝑥,𝜎) (224) because the 𝑍(𝑔_𝜇𝜈0;𝑋)−𝑍(𝑔_𝜇𝜈0;𝑋) part also has the contribution to 𝑆_𝑔[𝑔_𝜇𝜈(𝑥,𝜎);𝑋;𝜎].

Let 𝑆^𝑏𝑔₀;𝑋=𝑑⁴𝑥𝐿^𝑏_𝑔0;𝑋(𝑥),𝑆^𝑐𝑔₀;𝑋=𝑑⁴𝑥𝐿^𝑐_𝑔0;𝑋(𝑥); (225) then, 𝑆^𝑐(𝑔₀;𝑋)=𝑆^𝑏(𝑔₀;𝑋)−𝑆^𝑎(𝑔₀;𝑋). From (206) or (212), one can see 𝜖𝜕𝑊𝑔𝜕𝜖^𝜇𝜈0𝐴,𝜖=−⁰2𝜖²−𝐴₂−𝐴2𝜖2⁴+𝑂(𝜖)=−𝑔_𝜇𝜈0𝜕𝜕𝑔_𝜇𝜈0𝑊𝑔_𝜇𝜈0,𝜖. (226) For finite 𝑋 and 𝜖, (226) cannot be directly identified with 𝑆^𝑎 and 𝑆^𝑏. However, when 𝑋→𝑋, 𝜖→0, 𝑆^𝑏𝑔₀;𝑋=lim_𝜖→0𝐴₀𝜖²+𝐴₂𝜖+𝐴₄=𝑆^𝑎𝑔₀;𝑋. (227)𝐴₀/𝜖²+𝐴₂/𝜖+𝐴₄ is the conformal anomaly, which, with the infinite part subtracted, becomes 𝐴₄.

In the previous subsection, the radial dependent function 𝐹(𝜎) is induced via (176). Nevertheless, it can also be induced from 𝑊(𝑔_𝜇𝜈0;𝑋), with 𝑋 being the upper bound of the integration. Take a particular finite matrix 𝑋^∗ as the standard so that the arbitrary configuration of 𝑋 can be represented by 𝑋=𝑒^𝜎𝑋^∗ with 𝜎∈(−∞,+∞): 𝑊𝑔_𝜇𝜈0;𝑋=𝑊𝑔_𝜇𝜈0;𝑒^𝜎𝑋^∗=ln𝑍𝑔_𝜇𝜈0;𝑒^𝜎𝑋^∗. (228) Starting from 𝜎=0, with 𝜎 increasing, 𝑊(𝑔_𝜇𝜈0;𝑒^𝜎𝑋^∗) will also increase, so 𝑔_𝜇𝜈0 should change accordingly to make the partition function invariant. Namely, we have 𝑊𝑔_𝜇𝜈0;𝑋^∗=𝑊𝑔_𝜇𝜈(𝜎);𝑒^𝜎𝑋^∗ (229) with 𝑔_𝜇𝜈(0)=𝑔_𝜇𝜈0𝛿𝑊𝛿𝑔_𝜇𝜈(𝜎)𝛿𝑔_𝜇𝜈(𝜎)+𝛿𝜎𝛿𝑊𝛿𝜎=0. (230) For finite 𝜎, 𝑔_𝜇𝜈(𝜎)=𝑒^−2𝜎𝑔_𝜇𝜈0. When 𝜎→∞, 𝑔_𝜇𝜈(𝜎)→0, the simple scaling relation may not be valid. For finite 𝑔_𝜇𝜈0 and 𝜔, 𝑊(𝑔_𝜇𝜈0;𝑒^−𝜔𝑋^∗)=𝑊(𝑒^−2𝜔𝑔_𝜇𝜈0;𝑋^∗). If 𝑊𝑒^−2𝜔𝑔_𝜇𝜈0;𝑋^∗=𝑊𝑔_𝜇𝜈(𝜔+𝜎);𝑒^𝜎𝑋^∗=𝑊𝑔_𝜇𝜈(𝜔);𝑋^∗, (231) there will be 𝑊𝑔_𝜇𝜈0;𝑒^−𝜔𝑋^∗=𝑊𝑔_𝜇𝜈(𝜎);𝑒^𝜎𝑒^−𝜔𝑋^∗. (232) So, in (229), with 𝑔_𝜇𝜈(0)=𝑔_𝜇𝜈0 fixed and𝑋^∗ replaced by 𝑒^−𝜔𝑋^∗, the induced 𝑔_𝜇𝜈(𝜎) remains the same. Also, with 𝑋^∗ fixed and𝑔_𝜇𝜈(0) replaced by 𝑔_𝜇𝜈(𝜔), the induced field will be 𝑔_𝜇𝜈(𝜔+𝜎). 𝑔_𝜇𝜈(𝜎) is the unique function trajectory that does not depend on the 𝑋^∗ chosen.

We can compare it with the previously mentioned noncritical string coupling with 2𝑑 gravity: 𝑍𝐹;𝑔=^∞_𝜑=0𝐷_(𝑒𝜑𝑔₎𝜑𝐷_(𝑒𝜑𝑔₎𝑋volconf₂𝑒^{−𝑆(𝐹,𝑒𝜑}𝑔^{,𝑋)−𝑆}_𝐿^(𝜑,𝑔⁾, (233) where 𝐹(𝑋) is the 𝑑 dimensional background field. 𝑔 is the cutoff metric. With 𝑔 replaced by 𝑔^𝜔=𝑒^𝜔𝑔, 𝑍𝐹;𝑒^𝜔𝑔=^∞_𝜑=0𝐷_(𝑒𝜑𝑔^𝜔₎𝜑𝐷_(𝑒𝜑𝑔^𝜔₎𝑋volconf₂𝑒^{−𝑆(𝐹,𝑒𝜑}𝑔^𝜔_{,𝑋)−𝑆𝐿(𝜑,}𝑔^𝜔₎=^∞_𝜑=𝜔𝐷_(𝑒𝜑𝑔₎𝜑𝐷_(𝑒𝜑𝑔₎𝑋volconf₂𝑒^{−𝑆(𝐹,𝑒𝜑}𝑔^{,𝑋)−𝑆}_𝐿^(𝜑,𝑔⁾. (234)𝑍(𝐹;𝑒^𝜔𝑔)≠𝑍(𝐹;𝑔). 𝑔→𝑒^𝜔𝑔  is a conformal transformation, which should be accompanied by 𝐹→𝐹(𝜔) to make the partition function invariant. 𝑍(𝐹;𝑔)=𝑍(𝐹(𝜔);𝑒^𝜔𝑔). 𝐹(𝜔) is a function trajectory that does not depend on the 𝑔 chosen. A further 𝑒^𝜑𝑔^𝜔→𝑔^𝜔 transformation gives 𝑍𝐹(𝜔);𝑒^𝜔𝑔=^∞_𝜑=0𝐷𝑔^𝜔𝜑𝐷𝑔^𝜔𝑋volconf₂𝑒⁻𝑆_{(𝜑,𝑋,𝐹}^𝜔_(𝜑),𝑔^𝜔₎, (235) in which 𝐹^𝜔(𝜑)=𝐹(𝜑+𝜔) is induced.

Just as (224), ln𝑍(𝐹(𝜔);𝑒^𝜔𝑔) cannot be identified with the 𝑑+1 dimensional gravity action with a cutoff. The gravity action interpretation is possible only when 𝑔→0; that is, the cutoff is removed.

Here, 𝜑 is directly related to the worldsheet metric, 𝑔=𝑒^𝜑𝑔. 𝐹(𝜑) gives the RG flow of the 𝑑 dimensional fields. In gauge theory case, 𝜎 represents the radial direction, along which all fields, including the metric 𝑔_𝜇𝜈, will evolve. 𝐹(𝜎) 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 𝑊_ren=𝑆_ren, for which (see, e.g., [60]) 𝑑⁴𝑥√𝑔₀𝜔(𝑥)−2𝑔₀^𝜇𝜈𝛿𝛿𝑔₀^𝜇𝜈+Δ^𝑖−4+𝛽^𝑖𝐹^𝑖₀𝛿𝛿𝐹^𝑖₀𝑊_ren=𝑑⁴𝑥𝜔(𝑥)𝐴+𝜕_𝜇𝜔(𝑥)𝑍^𝜇, (236) where 𝐴 represents the local anomalies. For the renormalization scale 𝜇, 𝜇𝜕𝑊_ren𝜕𝜇+𝑑⁴𝑥√𝑔₀×2𝑔₀^𝜇𝜈𝛿𝛿𝑔₀^𝜇𝜈−Δ^𝑖−4𝐹^𝑖₀𝛿𝛿𝐹^𝑖₀𝑊_ren𝜇=0,𝜕𝑊_ren+𝑑𝜕𝜇⁴𝑥√𝑔₀𝛽^𝑖𝛿𝑊_ren𝛿𝐹^𝑖₀=𝑑⁴𝑥𝐴. (237)𝛽^𝑖 is the anomalous dimension. 𝛽^𝑖=𝛽^𝑖𝐹^𝑖₀ is the 𝛽 function.

5. Conclusion

With 𝑁→∞being fixed and𝑅 varying, the free energy of the Matrix theory on a 𝑥⁻-translation invariant 11𝑑 supergravity background 𝐹 is the functional of 𝑅 and 𝐹; that is, 𝑊=𝑊(𝑅,𝐹). Under the coordinate transformation 𝑥⁻=𝛼𝑥^′−,𝑥⁺=𝑓𝑥^′+,𝑥=𝑔𝑥^′, (238) with 𝑥=(𝑥¹,…,𝑥⁹), if 𝑅→𝑅^′, 𝐹→𝐹^′, 𝑊(𝑅,𝐹)=𝑊(𝑅^′,𝐹^′). 𝑊(𝑅,𝐹) preserves part of the 11𝑑 diffeomorphism invariance. 𝑊(𝑅,𝐹) can be compared with 𝑆_eff(𝑅,𝐹), the effective action of the supergravity for the same field 𝐹. On field theory side, naively, 𝑆_eff(𝐹) is calculated from 𝑒^{𝑆eff(𝐹)}=_1PI𝑑𝐹𝑒^{−𝑆cla(𝐹+}𝐹⁾ (239) with 𝑆_cla being the classical action of the 11𝑑 supergravity. 𝑒^{𝑆eff(𝐹)} can be taken as the sum of the 1PI graphs of the vacuum-vacuum amplitude for supergravitons living on background 𝐹.(Of course, (239) is not well-defined.) In 𝑀 theory, the basic objects are membranes other than particles. 𝑒^𝑊(𝐹) is the sum of the membrane configurations on background 𝐹. In some sense, 𝑊(𝐹) gives an 𝑀 theory refined version of the 𝑆_eff(𝐹) in (239).

In Matrix theory, the problem is that we only consider the 𝑀 theory sector with the definite light-cone momentum 𝑝⁺ and the background 𝐹 that is translation invariant along 𝑥⁻. For the same 𝐹 but different 𝑝⁺, 𝑊(𝑅,𝐹) also has the 𝑅 or, equivalently, 𝑝⁺ dependence. Nevertheless, 𝑊(𝑅,𝐹)=𝑊(𝐹₀,𝑅𝐹₋,𝑅𝐹₋₋), with 𝑅 being the radius of the 𝑥⁻. In fact, for 𝑆_eff on the same 11𝑑 background, there is also 𝑆_eff(𝑅,𝐹)=𝑆_eff(𝐹₀,𝑅𝐹₋,𝑅𝐹₋₋). In 𝑊(𝑝⁺,𝐹), 𝑝⁺ is only a parameter encoding the scale of 𝑥⁻, so it is enough to consider the sector with the definite 𝑝⁺. 𝑁→∞ in Matrix theory is kept fixed in order to maintain the consistency with membrane theory, which is only characterized by one parameter 𝑝⁺.

On field theory side, to describe the supergravity interactions among 𝑀 theory objects with no light-cone momentum exchange, one may consider Γ_𝑔(𝑌) given by 𝑒^Γ𝑔(𝑌)=[𝑑𝐹]𝑒^{𝑆𝑅,𝐹(𝑌)−𝑆cla(𝑅,𝐹)}, (240)where𝐹 is the zero mode of the 11𝑑 supergravity along 𝑥⁻. The integration out of 𝐹 gives the effective action for the 𝑀 theory object 𝑌 with the supergravity interactions (without transferring the light-cone momentum) all taken into account. Under the Legendre transformation, we may get Γ_𝑔(𝑌)=𝑆_eff(𝑅,𝐹)+𝑆_𝑅,𝐹(𝑌), where 𝑆_eff is given by (239). 𝛿[𝑆_eff(𝑅,𝐹)+𝑆_𝑅,𝐹(𝑌)]/𝛿𝐹=0. With 𝑆_eff(𝑅,𝐹) being replaced by 𝑊(𝑅,𝐹), one can define another effective action Γ(𝑌)=𝑊(𝑅,𝐹)+𝑆_𝑅,𝐹(𝑌), which is totally constructed from the Matrix theory with no input like 𝑆_cla(𝑅,𝐹) added. Γ(𝑌) is 𝑝⁺ independent, which is expected, since the scattering amplitudes with the zero light-cone momentum transfer should not depend on 𝑝⁺ due to the Lorentz invariance [61]. When 𝛿𝑆₀(𝑌)/𝛿𝑌=0, at the one-loop order, Γ(𝑌) and the standard Matrix theory effective action Γ_eff(𝑌) 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], SYM_𝑅×𝑆², SYM_𝑅×𝑆³_/𝑍𝑘, and SYM_𝑅×𝑆³ are all obtained by expanding the PWMM around the particular 1/2 BPS states, while the backreaction of the 1/2 BPS states on pp-wave after the 𝑥⁻ reduction produces the dual 10𝑑 geometry [19]. A special type IIA limit is taken in both gauge theory and gravity side. In this limit, the backreacted 11𝑑 geometry approaches the pp-wave, so the Matrix theory dual is still PWMM. On the other hand, the 𝑥⁻ reduced 10𝑑 geometry (122) is a 𝐷0-type solution. The action of the PWMM equals the action of the SYM_𝑅 on background (122). To study the fluctuations around the 1/2 BPS states, the 𝜂→𝜂+𝑛 and the 𝑌→𝑌+𝑌 redefinition should be made on gravity and the Matrix theory side. The 10𝑑 geometry then becomes (133), a 𝐷2-type solution. Correspondingly, 𝑆^PW(𝑌+𝑌) becomes the action of SYM_𝑅×𝑆², which, with the background (133) plugged in, remains invariant. With the SYM_𝑅×𝑆² and the gravity dual at hand, a further 𝑇-duality-like transformation gives SYM_𝑅×𝑆³ and AdS₅×𝑆⁵ [19, 21].

SYM₄ is the nonpertubative definition of the type IIB string theory on AdS₅×𝑆⁵. It is unlikely such gauge/string correspondence will suddenly vanish just because the metric in SYM₄ deviates 𝜂_𝜇𝜈 a little. If the correspondence still exists, the gravity dual will be the type IIB string theory on AdS₅×𝑆⁵ with a little background perturbation turned on. Then a one-to-one correspondence should exist between the 4𝑑 field 𝐹₀(𝑥) in SYM₄ and the 5𝑑 field 𝐅(𝑥,𝜎) on AdS₅. The natural candidate of 𝐅(𝑥,𝜎) is the gravity solution on AdS₅ with 𝐹₀(𝑥) the boundary condition. For the correspondence to be valid, it is necessary to derive 𝐅(𝑥,𝜎) merely from SYM₄. In the simplest situation, when the background metric in gauge theory is the standard 𝜂_𝜇𝜈, the near-horizon geometry of 𝐷3, 𝑀2, and 𝑀5 could be induced from SYM₄, SCFT₃, and SCFT₆ by a 𝑋→𝑋 transformation. It is expected that the same method, when applied to SYM₄ with the arbitrary 𝐹₀(𝑥) turned on, will give the corresponding 𝐅(𝑥,𝜎). With 𝑋→𝑋, 𝐹₀(𝑥)→𝐹(𝑥,𝜎), SYM₄ becomes a gauge theory living in 5𝑑 background 𝐅(𝑥,𝜎) times the transverse 𝑆⁵. The free energy can be expressed as the functional of 𝐅(𝑥,𝜎); that is, 𝑊₀(𝐹₀)=𝑊(𝐅). 𝛿𝑊(𝐅)/𝛿𝐅(𝑥,𝜎)=0, so if 𝑊(𝐅) is the 10𝑑 gravity action as is in Matrix theory case, 𝐅(𝑥,𝜎) will be the on-shell solution.

Appendices

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

The action of the (𝑝+1)-dimensional SYM theory on 𝑅^9,1 background could be written as 𝐼₀1=−𝑔²_𝑝𝑑^𝑝+1𝑥−𝑔1×Tr2𝑔^𝜇𝜆𝑔^𝜈𝜎𝐹_𝜇𝜈𝐹_𝜆𝜎+𝑔^𝜇𝜈𝑔_𝑚𝑛𝐷_𝜇𝑋^𝑚𝐷_𝜈𝑋^𝑛−𝑖𝜆Γ^𝜇𝐷_𝜇𝜆−𝜆Γ^𝑚𝑋_𝑚−1,𝜆2𝑔_𝑚𝑝𝑔_𝑛𝑞𝑋^𝑚,𝑋^𝑛𝑋^𝑝,𝑋^𝑞, (A.1) where 𝑔_𝜇𝜈=𝜂_𝜇𝜈, 𝑔_𝑚𝑛=𝛿_𝑚𝑛, 𝑔=det𝑔_𝜇𝜈. The near-horizon geometry of the 𝐷𝑝 branes is 𝑔_𝜇𝜈=𝑟𝑟_𝑝^(7−𝑝)/2𝜂_𝜇𝜈,𝑔_𝑚𝑛=𝑟𝑟_𝑝^{−(7−𝑝)/2}𝛿_𝑚𝑛,𝑒^Φ=𝑔_𝑠𝑟𝑟_𝑝^{(7−𝑝)(𝑝−3)/4},𝐴_01⋯𝑝=𝑟𝑟_𝑝^7−𝑝𝜀_01⋯𝑝. (A.2) With 𝑟 replaced by the matrix 𝑅=(𝑋^𝑚𝑋^𝑚)^1/2, the gauge theory on the near-horizon geometry background has the action 𝐼_𝐹1=−4𝜋𝑑^𝑝+1𝑥Tr√−𝑔𝑒^−Φ×12𝑔^𝜇𝜆𝑔^𝜈𝜎𝐹_𝜇𝜈𝐹_𝜆𝜎+𝑔^𝜇𝜈𝑔_𝑚𝑛𝐷_𝜇𝑋^𝑚𝐷_𝜈𝑋^𝑛−𝑖𝜆Γ^𝜇𝐷_𝜇𝜆−𝜆Γ^𝑚𝑋_𝑚−1,𝜆2𝑔_𝑚𝑝𝑔_𝑛𝑞×𝑋^𝑚,𝑋^𝑛𝑋^𝑝,𝑋^𝑞, (A.3) where 𝑔=det𝑔_𝜇𝜈. With 4𝜋𝑔_𝑠=1, except for a rescaling of 𝜆, 𝐼₀=𝐼_𝐹. The SYM action in 𝑅^9,1 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 𝑀2 and 𝑀5 branes are 𝑔_𝜇𝜈=𝑟𝑟₂⁴𝜂_𝜇𝜈,𝑔_𝑚𝑛=𝑟𝑟₂⁻²𝛿_𝑚𝑛,𝐴₀₁₂=𝑟𝑟₂⁶𝜀₀₁₂,𝑔_𝜇𝜈=𝑟𝑟₅𝜂_𝜇𝜈,𝑔_𝑚𝑛=𝑟𝑟₅⁻²𝛿_𝑚𝑛,𝐴₀₁₂=𝑟𝑟₅³𝜀₀₁₂, (A.4) respectively. For both 𝑀2 and 𝑀5, √−𝑔𝑔^𝜇𝜈𝑔_𝑚𝑛𝐷_𝜇𝑋^𝑚𝐷_𝜈𝑋^𝑛=−𝑔𝑔^𝜇𝜈𝑔_𝑚𝑛𝐷_𝜇𝑋^𝑚𝐷_𝜈𝑋^𝑛. (A.5) But √−𝑔𝑔_𝑚𝑝𝑔_𝑛𝑞𝑋^𝑚,𝑋^𝑛𝑋^𝑝,𝑋^𝑞≠−𝑔𝑔_𝑚𝑝𝑔_𝑛𝑞𝑋^𝑚,𝑋^𝑛𝑋^𝑝,𝑋^𝑞. (A.6) Instead, for 𝑀2, (the 3-algebra was proposed in [62–64] to construct the model for two coincident 𝑀2 branes)√−𝑔𝑔_𝑙𝑝𝑔_𝑚𝑞𝑔_𝑛𝑟𝑋^𝑙,𝑋^𝑚,𝑋^𝑛𝑋^𝑝,𝑋^𝑞,𝑋^𝑟=−𝑔𝑔_𝑙𝑝𝑔_𝑚𝑞𝑔_𝑛𝑟𝑋^𝑙,𝑋^𝑚,𝑋^𝑛𝑋^𝑝,𝑋^𝑞,𝑋^𝑟. (A.7) We see another necessity of the sextic potential. For 3𝑑 SCFT, the dimension of the scalar field is 1/2, so the potential term should contain six scalars. For 𝑀5, (3-algebra like this appeared in [65])√−𝑔𝑔_𝜇𝜈𝑔_𝑚𝑞𝑔_𝑛𝑟𝐶^𝜇,𝑋^𝑚,𝑋^𝑛𝐶^𝜈,𝑋^𝑞,𝑋^𝑟=−𝑔𝑔_𝜇𝜈𝑔_𝑚𝑞𝑔_𝑛𝑟𝐶^𝜇,𝑋^𝑚,𝑋^𝑛𝐶^𝜈,𝑋^𝑞,𝑋^𝑟. (A.8) The scalar dimension is 2, so, effectively, the potential should contain two-dimension 2 scalars and two-dimension 1 scalars.

B. Path Integral Measure and the External Fields

For simplicity, consider the scalar field with the action 𝐼𝑔_𝜇𝜈0,𝜙₀=𝑑,𝑋₄𝑥√−𝑔₀Tr𝑔₀^𝜇𝜈𝜕_𝜇𝑋^𝑚𝜕_𝜈𝑋^𝑚+16𝑅₀𝑋²_𝑚+𝜙₀𝑂, (B.1) where 𝑂=𝐶_{𝑎1⋯𝑎𝑛}(𝑋^𝑎1⋯𝑋^𝑎𝑛). 𝐼[𝑒^2𝜔𝑔_𝜇𝜈0,𝑒^{(𝑛−4)𝜔}𝜙₀,𝑒^−𝜔𝑋]=𝐼[𝑔_𝜇𝜈0,𝜙₀,𝑋]. Following [66], we will define the scalar density field 𝑋 to eliminate the dependence of the path integral measure on 𝑔_𝜇𝜈0: 𝑋^𝑚=−𝑔₀^1/4𝑋^𝑚. (B.2) In terms of 𝑋, 𝐼𝑔_𝜇𝜈0,𝜙₀,𝑋=𝐼𝑔_𝜇𝜈0,𝜙₀,𝑋=𝑑⁴𝑥Tr𝑔₀^𝜇𝜈𝜕_𝜇𝑋^𝑚𝜕_𝜈𝑋^𝑚+16𝑅₀𝑋²_𝑚+−𝑔₀^(2−𝑛)/4𝜙₀𝑂𝐼𝑒^2𝜔𝑔_𝜇𝜈0,𝑒^{(𝑛−4)𝜔}𝜙₀,𝑒^𝜔𝑋=𝐼𝑔_𝜇𝜈0,𝜙₀,𝑋. (B.3) The partition function is then 𝑍𝑔_𝜇𝜈0,𝜙₀=𝐷𝑋exp−𝐼𝑔_𝜇𝜈0,𝜙₀,𝑋, (B.4) with no 𝑔_𝜇𝜈0 entering into the path integral measure. 𝑍𝑒^2𝜔𝑔_𝜇𝜈0,𝑒^{(𝑛−4)𝜔}𝜙₀=𝐷𝑋exp−𝐼𝑒^2𝜔𝑔_𝜇𝜈0,𝑒^{(𝑛−4)𝜔}𝜙₀,𝑋=𝐷𝑋exp−𝐼𝑔_𝜇𝜈0,𝜙₀,𝑒^−𝜔𝑋=𝐷𝑒^𝜔𝑋^′exp−𝐼𝑔_𝜇𝜈0,𝜙₀,𝑋^′, (B.5) where 𝑋^′=𝑒^−𝜔𝑋. If 𝐷(𝑒^𝜔𝑋^′)=𝑒^𝐴𝜔𝐷𝑋^′; then, ln𝑍𝑒^2𝜔𝑔_𝜇𝜈0,𝑒^{(𝑛−4)𝜔}𝜙₀=𝐴_𝜔+ln𝑍𝑔_𝜇𝜈0,𝜙₀ (B.6) with 𝐴_𝜔 being the conformal anomaly. Note that to arrive at (B.6), we potentially assumed 𝐷𝑋exp−𝐼𝑔_𝜇𝜈0,𝜙₀,𝑋=𝐷𝑒^−𝜔𝑋exp−𝐼𝑔_𝜇𝜈0,𝜙₀,𝑒^−𝜔𝑋. (B.7) On gravity side, this is equivalent to assume that lim_𝜖→0𝑆(𝑔_𝜇𝜈0,𝜙₀,𝜖)=lim_𝜖→0𝑆(𝑔_𝜇𝜈0,𝜙₀,𝑒^2𝜔𝜖), so 𝐴_𝜔=lim_𝜖→0𝑆𝑒^2𝜔𝑔_𝜇𝜈0,𝑒^{(𝑛−4)𝜔}𝜙₀,𝜖−lim_𝜖→0𝑆𝑔_𝜇𝜈0,𝜙₀,𝜖. (B.8)𝐴_𝜔 in (B.8) could be compared with (227). From (B.5), 𝛿𝐴_𝜔𝛿𝜔_𝜔=0=1𝑍𝐷𝑋exp−𝐼𝑔_𝜇𝜈0,𝜙₀,𝑋𝛿𝐼𝛿𝑋𝑋=𝛿𝐼𝛿𝑋𝑋, (B.9) indicating that 𝐴_𝜔 will depend on both 𝑔_𝜇𝜈0 and 𝜙₀. Indeed, the direct calculation of the conformal anomaly gives 𝐴_𝜔=𝐴_𝜔(𝑔_𝜇𝜈0,𝜙₀) composed by the gravity part (𝑔_𝜇𝜈0) and the matter part (𝜙₀). Correspondingly, the transformation of the path integral measure will also depend on 𝑔_𝜇𝜈0 and 𝜙₀, although neither of them enters into 𝐷𝑋 explicitly.

Similarly, when the theory is coupled to the external 𝑆𝑈(4)_𝑅 gauge field 𝐴^𝑎_𝜇, although 𝐴^𝑎_𝜇 does not enter into the path integral measure, the Jacobian of the path integral measure under the 𝑅-symmetry transformation gives the 𝑅-symmetry anomaly which is the function of 𝐴^𝑎_𝜇.

Finally, on gravity side, one can read the 𝑅-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 𝑅-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.