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

Authors: Shan Hu; Dimitri Nanopoulos
Journal: Advances in High Energy Physics
 (2013)

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## Abstract

WithN→∞  being fixed, R→∞, the free energy of the Matrix theory on a supergravity background F is a functional of F, W=W(R,F). We try to relate this functional with Seff(R,F), the effective action of F, where F is translation invariant along x-. The vertex function is then associated with the connected correlation function of the current densities. 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)=Seff(R,F), Γ(Y) will give the supergravity interactions among M theory objects with no light-cone momentum exchange. We then discuss the Matrix theory dual of the 11d background generated by branes with the definite p+ as well as the gauge theory dual of the 10d background arising from the x- reduction. Finally, for SYM4 with the 4d background field F0, we give a possible way to induce the radial dependent 5d field F(σ).


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## Body

## 1. Introduction

Up to present, two kinds of nonperturbative formulation ofM/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 R11-n×Tn, the dual gauge theory will become SYMn+1, which is Matrix theory compactified on Tn [10–12].As the nonperturbative description ofM/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 theM/string theory. For SYM4, we have [8, 9]
(1)ZSYM4(β,ϕ0)=ZAdS5×S5(β,ϕ0),
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]
(2)ZSYM4(ϕ0)=eW[ϕ(ϕ0)].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 ZSYM4(ϕ0)=eW[ϕ(ϕ0)].Except for (2), ZSYM4(ϕ0) also has another expression on AdS5×S5. Let Ws be the free energy of the type IIB string theory on AdS5×S5 and ZSYM4 the partition function of SYM4, it is expected that
(3)ZSYM4=eWs,
where W=∑nh=0‍Wnh. 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:
(4)ZSYM4(ϕ0)=eWs[ϕ(ϕ0)].
On AdS5 side, the string free energy Ws[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,
(5)W[ϕ(ϕ0)]=Ws[ϕ(ϕ0)].
The free energy of the string theory on a given background equals the effective action of the background fields.For Matrix theory, similarly,(6)ZMatrix(β,p+,F)=ZM(β,p+,F).F is the 11d supergravity background and is the translation invariant along the x- direction. ZM(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=∞,
(7)ZMatrix(p+,F)=eW(p+,F).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 backgroundF:
(8)Ws(F)=∑m=0∞‍eϕ(2-2m)∫M2-2m‍[dX]e-SF.
In [17, 18], it was shown that, for F satisfying the free field equation, Ws(F) could be taken as the effective action of the renormalized background field F~(F); that is, Ws(F)=Seff[F~(F)].It is tempting to establish a relation betweenW(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 Seff(R,F) is the effective action of F, and there is also Seff(R,F)=Seff(F0,RF-,R2F--). The R dependence of W(R,F) is consistent with Seff(R,F).One may want to consider the complete partition function with all light-cone momentum taken into account, which is roughlyeW(F)=∫‍dp+eW(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 Seff(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 configurationY, we may define Γ(Y) via
(9)eΓ(Y)=∫[dY~]e-SR,F(Y~)+SR,F(Y)=eW(R,F)+SR,F(Y),
with F(x) solved from
(10)-δW(R,F)δF(x)|F=δSR,F(Y)δF(x)|F.
It is easy to see that Γ(Y) is R independent.To describe the supergravity interactions amongM theory objects with no light-cone momentum exchange, and one may define Γg(Y) through
(11)eΓg(Y)=∫[dF]eSR,F(Y)-Scla(R,F),
with F being the zero mode of the 11d supergravity along x- and Scla 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. 						(12)Γg(Y)=∑n‍1n!∫d10x1⋯∫d10xnGFc(x1,…,xn)pppppppppppppppp×VF(Y)(x1)⋯VF(Y)(xn),
where GFc(x1,…,xn) is the connected Green’s function of supergravity in light-cone gauge with the zero light-cone momentum and VF(Y)(x) is the current density of the configuration Y coupling with the supergravity field F(x). Under a Legendre transformation, Γg(Y) could be written as
(13)Γg(Y)=Seff(R,F)+∫d10xF(x)VR,F(Y)(x)~Seff(R,F)+SR,F(Y),
with F(x) solved from
(14)-δSeff(R,F)δF(x)|F=VR,F(Y)(x)=δSR,F(Y)δF(x)|F.
So, if W(R,F)=Seff(R,F), Γ(Y)=Γg(Y).AlthoughM 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 SYMR×S2, SYMR×S3/Zk, and SYMR×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 Seff(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 XI. Suppose the coordinate of AdS5×S5 is (x,r,Ω), for a 10d scalar Hn(x,r,Ω)=hn(x,r)Yn(Ω) with Yn(Ω)=Ca1⋯an(ya1⋯yan)/|y|n the spherical harmonic of S5; the operator counterpart is
(15)Hn(x,r,X^)=hn(x,r)Ca1⋯an(X^a1⋯X^an).hn(x,r) can be the arbitrary 5d function. In SYM4, we only have hn(x)Ca1⋯an(Xa1⋯Xan) 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 eW[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 Section2, 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, SYMR×S2, SYMR×S3/Zk, and SYMR×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 considerW(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 SR,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 SR,F(Y) is SR,F(Y)=SF0,RF-,R2F--(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 Seff(R,F)=Seff(F0,RF-,R2F--). From W(R,F), we can define Γ(Y)=SR,F(Y)+W(R,F), with F solved through δ[SR,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)=Seff(R,F), we will have Γ(Y)=Γg(Y) with Γg(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(16)SR=R∫dx+×Tr(1lp312R2D+XID+XI+14lp6[XI,XJ]2pppppppppp-iRθD+θ+1lp3θγI[XI,θ]),
where XI, θ, and A0 are N×N hermitian matrices with N→∞, and I=1⋯9. R→∞, p+=N/R,  lp is the Planck length.X
I, x+, and R have the dimension of length, so each commutator is multiplied by a factor 1/lp3 to make the action dimensionless. With the replacement XI→lpXI, x+→lpx+, R→lpR, we get the action
(17)SR=R∫dx+×Tr(12R2D+XID+XIpppppppppp+14[XI,XJ]2-iRθD+θ+θγI[XI,θ]),
in which lp is cancelled and XI, θ, x+, and R are all dimensionless. In the following, we will still adopt this convention, so lp will not appear explicitly.The11d supergravity field, after the gauge fixing, has the nonzero components (g+-, g++, gI+, gIJ), (CIJ+, CIJK), and (ψ+, ψI) [22]. Based on the Hamiltonian in [22], one can write down the action of the bosonic membrane on such supergravity background:
(18)SF0b=∫dx+d2σg+-P+×[P+2D+XID+XJ2g+-2gIJ+P+D+XIg+-(CJ+P+g+Jg+-)gIJ+P+2g++2g+-2-{XI,XJ}{XK,XL}gIKgJL+P+C+g+-],
where CI={XJ,XK}CJKI and C+={XJ,XK}C+JK. Note that it is g+-/P+ that appears. The Matrix theory version is
(19)SF0b=∫dx+×Tr(Rg+-{D+XID+XJ2(Rg+-)2gIJppppppppppppppp+D+XIRg+-(CJ+g+JRg+-)gIJppppppppppppppp+14[XI,XJ][XK,XL]gIKgJLppppppppppppppp+g++2(Rg+-)2+C+Rg+-{D+XID+XJ2(Rg+-)2gIJ}),
where CI=-(i/2)[XJ,XK]CJKI and C+=-(i/2)[XJ,XK]C+JK. Without the gauge fixing, gI-, g--, C-IJ, C+-I, ψ-≠0, so terms involving them can also be added:
(20)SgI-b=∫dx+×Tr(Rg+-{D+XID+XJD+XK2(Rg+-)3gJKpppppppppppppp-D+XI4Rg+-[XK,XL][XM,XN]pppppppppppppp×gKMgLNpppppppppppppp+D+XLRg+-[XJ,XM][XI,XN]pppppppppppppp×gLJgMND+XID+XJD+XK2(Rg+-)3}pppppppp×RgI-D+XID+XJD+XK2(Rg+-)3)
and similarly for Sg--b, 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--, gI-, 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+XID+XI+(1/ν2){XI,XJ}{XI,XJ} and DaX-=D+XIDaXI into the action, which could couple with the x- indexed fields.In all these terms, it isRg+-, RgI-, R2g--, RC-IJ, RC+-I, Rψ- that are involved. (Similarly, in supermembrane action, it is g+-/P+, gI-/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′,
(21)g++′=f′2(x′+)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,
the action is invariant. This is consistent with the coordinate transformation
(22)ds2=g++df(x′+)df(x′+)+2gI+dxIdf(x′+)+gIJdxIdxJ+2αg+-df(x′+)dx′-+2αgI-dxIdx′-+α2g--dx′-dx′-=g++f′2(x′+)dx′+dx′++2f′(x′+)gI+dxIx′++gIJdxIdxJ+2αg+-f′(x′+)dx′+dx′-+2αgI-dxIdx′-+α2g--dx′-dx′-,C=CIJKdxI∧dxJ∧dxK+CIJ+dxI∧dxJ∧df(x′+)+αCIJ-dxI∧dxJ∧dx′-+αCI+-dxI∧df(x′+)∧dx′-=CIJKdxI∧dxJ∧dxK+CIJ+f′(x′+)dxI∧dxJ∧dx′++αCIJ-dxI∧dxJ∧dx-+αCI+-f′(x′+)dxI∧dx′+∧dx-.S
R
,
F, the supersymmetric extension of (19) and (20), is not constructed yet. With SR,F given, the current density for F can be defined as
(23)VF(Y)(x)=δSR,F(Y)δF(x)|F.
In particular, when F is the flat background, for which, the only nonvanishing fields are g-+-=-1 and g-IJ=δIJ, VF(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 SR,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]:
(24)SR,F=SR-R∫dx+pppppppp×STr[VhIJ(X)hIJ(X)+VhI+(X)hI+(X)pppppppppppppp+Vh++(X)h++(X)+VhI-(X)hI-(X)pppppppppppppp+Vh+-(X)h+-(X)+Vh--(X)h--(X)pppppppppppppp+VCIJK(X)CIJK(X)+VC+JK(X)C+JK(X)pppppppppppppp+VCIJ-(X)CIJ-(X)+VC+-K(X)C+-K(X)pppppppppppppp+Vψ(X)ψ(X)+Vψ+(X)ψ+(X)pppppppppppppp+Vψ-(X)ψ-(X)hIJ(X)],
where, for example,
(25)VhI+(X)hI+(X)=(D+XIR-14θγIJθ∂∂XJ)hI+(X)R.hIJ(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 a1d SYM theory, just as that in AdS/CFT. Then, the background fields, like hI+(X), should be expanded as the Taylor series:
(26)hI+(x+,X1,…,X9)=∑n=0∞‍1n!(∂k1⋯∂knhI+)(x+,0,…,0)Xk1⋯Xkn,
with (∂k1⋯∂knhI+)(x+,0,…,0) the derivative of hI+(x+,x→) at (x+,0) [3, 23–25]. 								(27)STr[VhI+(X)hI+(X)]=∑n=0∞‍1n!(∂k1⋯∂knhI+)(x+,0,…,0)×STr[1R(D+XIR-14θγIJθ∂∂XJ)Xk1⋯Xkn]=∑n=0∞‍1n!(∂k1⋯∂knhI+)(x+,0,…,0)IhI+k1⋯kn(x+).
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
(28)Xi⟶Xi-ai,f(x+,0,…,0)⟶f(x+,a1,…,a9),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 Φk1⋯kn in AdS/CFT, the moment operators Ik1⋯kn do not need to be traceless. As a result, {1,Ik1,…,Ik1⋯kn,…} couples with the 10d field, while {1,Φk1,…,Φk1⋯kn,…} only couples with the 9d field.In the second form, the vertex operator is defined in10d spacetime:
(29)δδF(x)F(x+,X1,…,X9)=δ(x0-x+)δ9(xI-XI)=δ10(xμ-Xμ),
where XI are N×N matrices and for uniformity; we have set X0=x+1N×N. This is a matrix generalization of the δ-function. In special situations, when all of the XI are diagonal, that is, XI=diag{x1I,x2I,…,xNI}, (29) becomes
(30)δδF(x)F(x+,X1,…,X9)=diag{δ10(xμ-x1μ),δ10(xμ-x2μ),…,δ10(xμ-xNμ)}.
With the generalized δ-function, it is straightforward to write down the current densities for various fields. For hI+, we have
(31)VhI+(x)=δδhI+(x){-R∫‍dx+STr[VhI+(X)hI+(X)]}=-R∫‍dx+STr[VhI+(X)δ10(x-X)].It is not convenient to deal with theδ-function. One may want to do a Fourier transformation, which gives the third representation of the vertex operator:
(32)VhI+(p)=∫d10xVhI+(x)eipx=-R∫dx+STr[VhI+(X)eipX]=-R∫dx+STr{[-D+XIR+i4θγIJθpJ]eipX}.


### 2.2. Partition Function of Matrix Theory on Curved Background

Suppose the exact form ofSF is given, and the partition function of Matrix theory on supergravity background is
(33)Z(R,F)=eW(R,F)=∫[dXdθdA0]e-SR,F(X,θ,A0)=∫[dY]e-SR,F(Y),
where F is the 11d background field (g+-, g++, gI+, gIJ, g--, gI-), (CIJ+, CIJK, CIJ-, CI+-), and (ψ+, ψI, ψ-) mentioned before, and Y collectively represents (XI,θ,A0).In gauge/string correspondence, partition function is an important quantity, the value of which should be equal on both sides. ForAdS5/CFT4 correspondence, it is expected that
(34)ZSYM4=exp{-F}
should hold, where F=∑nh=0‍Fnh is the free energy of the strings on AdS5×S5, ZSYM4 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).SupposeSP+,FM(Y) is the membrane action on background F. SP+,FM(Y) should be general covariant, so, for P+=P′+/α, x+=f(x′+), and X=g(X′), there is SP+,FM(Y)=SP′+,F′M(Y′), where F′ is the field coming from the coordinate transformation:
(35)x-=αx′-,x+=f(x′+),x=g(x′).
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
(36)∫[dY]e-SP+,FM(Y)=∫[dY′]e-SP′+,F′M(Y′).
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
(37)Z(R,F)=eW(R,F)=∫[dY]e-SR,F(Y)=∫[dY′]e-SR′,F′(Y′)=eW(R′,F′)=Z(R′,F′).W(R,F)=W(R′,F′). At least restricted to (35), W(R,F) is the diffeomorphism invariant functional of F.LetF0, F-, F-- represent fields with zero, one, and two x- indices. Since
(38)SR,F(Y)=SF0,RF-,R2F--(Y),(39)W(R,F)=W(F0,RF-,R2F--).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 Seff(R;F0,F-,F--).  Seff is invariant under the coordinate transformation:
(40)x-⟶x-R,F0⟶F0,F-⟶RF-,F--⟶R2F--,
so
(41)Seff(R;F0,F-,F--)=Seff(1;F0,RF-,R2F--).
The radius of x- is absorbed in F-, F--, as is in (39). In this respect, W is consistent with Seff.LetF=F-+F~, where F- is the flat background with g-+-=-1, g-IJ=δIJ, and the rest fields being zero. W(R,F-)=0:
(42)W(R,F)=∫d10xΓR,Fc(x)F~(x)+12!∫d10x1×∫d10x2ΓR,Fc(x1,x2)F~(x1)F~(x2)+⋯=∑n‍1n!∫d10x1⋯∫d10xnΓR,Fc(x1,…,xn)pppppppppppppppp×F~(x1)⋯F~(xn),
where (F collectively represents supergravity fields. For example, ΓR,g++,g--c(x1,x2)=δ2W(R,F)/δg++(x1)δg--(x2)|F=F-.)(43)ΓR,Fc(x1,…,xn)=δnW(R,F)δF(x1)⋯δF(xn)|F=F-.
For the same F, the 11d supergravity effective action Seff(R,F) is
(44)Seff(R,F)=∑n‍1n!∫02πRdx1-×∫dx1+d9x1⋯∫02πRdxn-∫dxn+d9xn×ΓFc(x1-,x1+,x1;…;xn-,xn+,xn)×F~(x1+,x1)⋯F~(xn+,xn).ΓFc(x1-,x1+,x1;…;xn-,xn+,xn) is the vertex function of the 11d supergravity. If Seff(R,F)=W(R,F), there will be
(45)ΓR,Fc(x1,…,xn)=ΓR,Fc(x1+,x1;…;xn+,xn)=∫02πRdx1-⋯∫02πRdxn-ΓFc(x1-,x1+,x1;…;xn-,xn+,xn).In linear gravity approximation,δVF(x)/δF(y)=0, where VF is the current density coupling with F(x) as is defined in (23):
(46)ΓR,Fc(x1,…,xn)=(-1)n〈VF(x1)⋯VF(xn)〉c.〈VF(x1)⋯VF(xn)〉c is the connected correlation function of the current density, in contrast to
(47)〈VF(x1)⋯VF(xn)〉=∫[dXdθdA0]e-S0VF(x1)⋯VF(xn)=(-1)nδnZ(R,F)δF(x1)⋯δF(xn)|F=F-,
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 ΓR,Fc(x1,…,xn).In terms ofVF(p),
(48)〈VF(x1)⋯VF(xn)〉c=∫d10p1⋯∫d10pne-i(p1x1+⋯+pnxn)pppppppppppppp×〈VF(p1)⋯VF(pn)〉c
Note that VF(X)(x+,xI-aI)=VF(X+a)(x+,xI), S0(X+a)=S0(X), so
(49)〈VF(x1)⋯VF(xn)〉c=〈VF(x1+a1)⋯VF(xn+an)〉c.
The correlation function is translation invariant. 〈VF(p1)⋯VF(pn)〉c=0 if p1+⋯+pn≠0.Let us first consider the one point function(50)〈VF(x)〉=-δW(R,F)δF(x)|F=F-=v-F.v-F is the vacuum expectation value of the current density, when the background field is F-, and is a constant in spacetime. SR is SO(9) invariant, so v-F should also be SO(9) invariant. As a result,
(51)v-gI+=v-gI-=v-CIJK=v-CIJ+=v-CIJ-=v-CI+-=v-ψI=0.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,
(52)〈VF(x)〉F=-δW(R,F)δF(x)|F=F-+F~=vF(x)=∑n‍-1(n-1)!×∫d10x1⋯∫d10xn-1ΓR,Fc(x,x1,…,xn-1)ppppppppppppppp×F~(x1)⋯F~(xn-1)
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 〈VF(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 givenF(x), vF(x) is uniquely determined. Conversely, different F(x) may result in the same vF(x). This is quite like the source-gravity coupled system. For the given gravity field, the density of the source can be obtained through δSg/δgμν=-Tμν. 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 vF(x)=0, F~(x)=0. F~(x) could be interpreted as the field generated by the current density vF(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 betweenvF and F, and so a Legendre transformation is possible. Before that, we will first define Γ(Y):
(53)Γ(Y)=SR,F(Y)+W(R,F),(54)eΓ(Y)=∫[dY~]e-SR,F(Y~)+SR,F(Y).F is solved from the equation
(55)∫[dY~]δδF(x)e-SR,F(Y~)+SR,F(Y)=∫[dY~]e-SR,F(Y~)+SR,F(Y)[-δSR,F(Y~)δF(x)+δSR,F(Y)δF(x)]=0
or equivalently,
(56)∫‍[dY~]e-SR,F(Y~)δSR,F(Y~)δF(x)=δSR,F(Y)δF(x)=-δW(R,F)δF(x).
In some sense, F(x) is the field generated by Y. Take a derivative of (54) with respect to Y; using (55), we get
(57)δΓ(Y)δY=δSR,F(Y)δY,
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 theR dependence of SR,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). SR,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 SP+,FM(Y), the R dependence is canceled, so the obtained Γ(Y) is the same.With these properties collected, we may consider the possible interpretation ofΓ(Y) and W(R,F). If W(R,F) is the effective action of supergravity, since SR,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=δSR,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=δSR,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 ofΓ(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, SF(Y)=S0(Y)+∫‍d10xF~(x)VF(Y)(x), Γ(Y)=S0(Y)+Γ~(vF), where
(58)Γ~(vF)=∫d10xF~(x)vF(x)+W(F-+F~)
is the Legendre transformation of W(F):
(59)-δW(F)δF~(x)|F~=vF=VF(Y)(x),δΓ~(vF)δvF(x)|vF=F~(x).
Let
(60)ΛF(x1,…,xn)=δnΓ~(vF)δvF(x1)⋯δvF(xn)|vF=0,
then
(61)Γ(Y)=S0(Y)+∑n‍1n!ppppppppppp×∫d10x1⋯∫d10xnΛF(x1,…,xn)ppppppppppppppppppppppp×vF(x1)⋯vF(xn)=S0(Y)+∑n‍1n!pppppppppp×∫d10x1⋯∫d10xnΛF(x1,…,xn)pppppppppppppppppppppp×VF(Y)(x1)⋯VF(Y)(xn).
For the given vF(x),
(62)F(x)=∑n‍1(n-1)!×∫d10x1⋯∫d10xn-1ΛF(x,x1,…,xn-1)ppppppppppppppp×vF(x1)⋯vF(xn-1).
Equation (52) gives the current density vF(x) generated by the field F(x); (62) gives the field F(x) generated by the current density vF(x). If ΛF 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 vF. In classical level, F(x) could be calculated by δSg/δF(x)=-vF(x) with Sg the classical action of supergravity:
(63)ΓFc(x-z)=-δvF(x)δF~(z)|F~=0,ΛF(z-y)=δ2Γ~(vF)δvF(z)δvF(y)|vF=0=δF~(z)δvF(y)|vF=0,
so
(64)-∫d10zΓFc(x-z)ΛF(z-y)=δ10(x-y).-ΛF(x1-x2) is the inverse of ΓFc(x1-x2). Subsequently,
(65)ΛF(x1,x2,x3)=∫d10y1∫d10y2×∫d10y3ΛF(y1-x1)ΛF(y2-x2)×ΛF(y3-x3)ΓFc(y1,y2,y3),ΓFc(x1,x2,x3)=-∫d10y1∫d10y2×∫d10y3ΓFc(y1-x1)ΓFc(y2-x2)×ΓFc(y3-x3)ΛF(y1,y2,y3).ΛF(x1,x2,x3,x4)=∫d10y1∫d10y2∫d10y3×∫d10y4ΛF(y1-x1)ΛF(y2-x2)ppppp×ΛF(y3-x3)ΛF(y4-x4)ppppp×[∫d10z1ΓFc(y1,y2,y3,y4)pppppppp+∫d10z1pppppppp×∫d10z2ΛF(z1-z2)ppppppppppp×(ΓFc(z1,y1,y4)ppppppppppppp×ΓFc(z2,y2,y3)ppppppppppppp+ΓFc(z1,y2,y4)ppppppppppppp×ΓFc(z2,y1,y3)ppppppppppppp+ΓFc(z1,y3,y4)ppppppppppppp×ΓFc(z2,y1,y2))∫d10z1],ΓFc(x1,x2,x3,x4)=∫d10y1∫d10y2∫d10y3×∫d10y4ΓFc(y1-x1)ΓFc(y2-x2)×ΓFc(y3-x3)ΓFc(y4-x4)×[∫d10z1ΛF(y1,y2,y3,y4)+∫d10z1×∫d10z2ΓFc(z1-z2)ppppppp×(ΛF(z1,y1,y4)ppppppppp×ΛF(z2,y2,y3)ppppppppp+ΛF(z1,y2,y4)ppppppppp×ΛF(z2,y1,y3)ppppppppp+ΛF(z1,y3,y4)ppppppppp×ΛF(z2,y1,y2))∫d10z1].
The relation between ΛF and ΓFc shows that ΛF and ΓFc are the connected Green’s function and the vertex function of a particular quantum field theory. ΛF(z-y)=(δF~(z)/δvF(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,
(66)eΓ(Y)=∫[dη]e-SF(Y+η)+SF(Y).-SF(Y+η)+SF(Y)=-∫dτδSF(Y)δY(τ)η(τ)-12∫dτdτ′δ2SF(Y)δY(τ)δY(τ′)×η(τ)η(τ′)+⋯=-SF(1)(Y,η)-SF(2)(Y,η)+⋯.SF could be written as (We only write ∂τ2, but one should be aware that the fermionic kinetic term ∂τ also exists.)(67)SF(Y)=∫dτ[12Y∂τ2Y+V0(Y)+V(F,Y)]=S0(Y)+∫dτV(F,Y)
with V(F,Y) taking the form as that in (27). For the given Y, solve F0 from δSF0(Y)/δY=0, and expand SF(n)(Y,η) around F0. F=F0+f. Consider
(68)SF(1)(Y,η)=δSF(1)(Y,η)δF|F0f+⋯SF(2)(Y,η)=SF0(2)(Y,η)+δSF(2)(Y,η)δF|F0f+⋯.f=f(Y) is solved as the functional of Y. Let Γ(Y)=Γ1(Y)+Γ2(Y)+⋯ since
(69)-SF(Y+η)+SF(Y)=-SF0(2)(Y,η)+⋯,Γ1(Y)=ln{∫[dη]e-SF0(2)(Y,η)}=ln(detAF0)1/2
with
(70)AF0(τ,τ′)=[∂τ2+∂2V0(Y)∂Y2+∂2V(F0,Y)∂Y2]δ(τ-τ′).
In particular, when δS0(Y)/δY=0, F0=0, AF0 reduces to
(71)A(τ,τ′)=[∂τ2+∂2V0(Y)∂Y2]δ(τ-τ′).
The corresponding Γ1(Y) is the same as the one-loop contribution Γeff1(Y) of the standard effective action Γeff(Y) in background gauge:
(72)eΓeff(Y)=∫[dη]e-S0(Y+η)+∫‍dτTr[J(τ)η(τ)],
subject to the condition
(73)∫[dη]δδJ(τ)e-S0(Y+η)+∫‍dτTr[J(τ)η(τ)]=0.Γ
1
(
Y
)
=
Γ
eff
1
(
Y
) has been calculated for the arbitrary Y satisfying δS0(Y)/δY=0 [3, 25]:
(74)Γ1(Y)=12∫d10x1×∫d10x2GFc0(x1,x2)VF(Y)(x1)VF(Y)(x2),
with GFc0 being the free supergraviton propagator. Compared with (61), ΛF,1(x1,x2)=GFc0(x1,x2). If the result can be extended to the n-point Green’s functions and to the full-loop calculation, there will be
(75)ΛF(x1,…,xn)=GFc(x1,…,xn),
where GFc is the connected full Green's function of supergravity. ΓFc(x1,…,xn) and W(F) will then become the n-point vertex function and the effective action of supergravity, respectively.In quantum field theory, unlike theS-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 Γeff(Y) differ from a parameterization transformation. Γ(Y)≠Γeff(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
(76)Γg[Y(τ)]=∑n‍1n!∫d10x1⋯∫d10xnGFc(x1,…,xn)pppppppppppppppp×VF(Y)(x1)⋯VF(Y)(xn).
The effective action defined in (61) is also the Taylor series of VF(Y) and thus could be compared with (76) directly. The standard effective action Γeff(Y) is expanded as the Taylor series of Y. A careful reorganization is needed to get VF(Y), but it is unclear whether it is always possible to do so. Moreover, under a Legendre transformation, (76) becomes
(77)Γg[Y(τ)]=Seff(F)+∫d10xF(x)VF(Y)(x)~Seff(F)+SF[Y(τ)]
with
(78)-δSeff(F)δF(x)|F=VF(Y)(x)=δSF(Y)δF(x)|F.Seff(F) is the effective action of supergravity. Equation (77) is almost the same as (53). The only problem is whether W(F)=Seff(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:
(79)Seff[F~(F)]=∑m=0∞‍eϕ(2-2m)∫M2-2m[dX]e-SF=Ws(F),
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 gab=e2σg^ab is always imposed, and so, Ws(F,σ) also has the dependence on σ:
(80)δWs(F,σ)δσ=-δWs(F,σ)δFβ(F).
When β(F)=0, Ws(F,σ)=Ws(F), although β(F)=0 is not the necessary condition.SupposeSeff(F) is the effective action of the string modes:
(81)Seff(F)=-12∫dDxFΔF+Sint(F).
Consider F0 with ΔF0=0. Since β(F0)≠0, F will evolve along the RG flow; that is, Ws(F0,σ)=Ws[F(σ′),σ′]. A special property of F0 is that it will finally reach an IR fixed point F~, Ws(F0,σ)=Ws(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, Ws(F~,∞)=Ws(F~,σ)=Ws(F~)=Ws(F0,σ)=Ws(F0).F
~ is calculated from F0 via
(82)F~=F0+Δ-1δSint(F~)δF~=F0(x)+∫dDyD(x-y,F)δSint(F~)δF~(y).δSeff(F)/δF(x)|F=F~=0. Since δSeff(F)/δFi=κijβ(Fj), β(F~)=0 is equivalent to δSeff(F)/δF(x)|F=F~=0 [18]. In [18], it was shown that Ws(F0,σ)=Seff(F~), which means Ws(F~,σ)=Ws(F~)=Seff(F~). There is a one-to-one correspondence between the solution space of δSeff(F)/δF(x)=0 and the solution space of ΔF=0. So, for all F with δSeff(F)/δF(x)=0=β(F), Ws(F,σ)=Seff(F). In σ-model approach, only for such F, the conformal factor decouples; thus, the calculated string free energy is physical. In this subspace, Ws(F)=Seff(F).W
s
(
F
,
σ
) with β(F)≠0 usually depends on σ. However, the σ-dependence drops out for F0. In fact,(83)Ws(F0,σ)=Ws(F0)=Seff(F~)=Seff[F0+Δ-1δSint(F0)δF0+⋯]
is the S-matrix functional [27]. For the on-shell p1,…,pn,
(84)Ws(F0,σ)=∑n‍1n!∫d9p1⋯∫d9pnSFc(p1,…,pn)ppppppppppppppp×F0(p1)⋯F0(pn),F0(x)=∫d9pF0(p),SFc(p1,…,pn)=1N0∑m=0∞‍eϕ(2-2m)pppppppp×∫M2-2m[dX]e-S0VF(p1)⋯VF(pn)
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, forF satisfying δSeff(F)/δF(x)=0, Ws(F,σ)=Ws(F)=Seff(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, ZSYM4=eWs, where Ws 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 ημν+δgμν(x) other than ημν for very small δgμν(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 δGμν(x,σ) that is entirely determined by δgμν(x). Correspondingly, ZSYM4[δgμν(x)]=eWs[δGμν(x,σ)], where Ws[δGμν(x,σ)] is the free energy of the string theory on AdS5×S5 with the background perturbation δGμν(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 ZSYM4[δgμν(x)]=eSeff[δGμν(x,σ)], where Seff[δGμν(x,σ)] is the supergravity effective action for the modified background fields, so Seff[δGμν(x,σ)]=Ws[δGμν(x,σ)] if ZSYM4=eWs holds.There is a naive way to interpretWs(F)=Seff(F). Suppose the classical supergravity action is Sc(F), from the field theory's point of view:
(85)eSeff(F)=∫1PI[dF~]e-Sc(F+F~),
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 Sc(F+F~). Now, consider the string theory on background F, we may have
(86)eSeff(F)=Zs(F)=eWs(F),
where Ws(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 Sc(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 δSeff(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 Seff(F) should be the effective action with the α′ corrections taken into account.A naturalM-theory extension is
(87)eSeff(F)=ZM(F)=∫[dY]e-SFM(Y),
where SFM(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 eSeff(F) other than Seff(F). For the generic 11d background F, SFM(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+.


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

The Matrix theory action in flat spacetime is(16)SR=R∫dx+×Tr(1lp312R2D+XID+XI+14lp6[XI,XJ]2pppppppppp-iRθD+θ+1lp3θγI[XI,θ]),
where XI, θ, and A0 are N×N hermitian matrices with N→∞, and I=1⋯9. R→∞, p+=N/R,  lp is the Planck length.X
I, x+, and R have the dimension of length, so each commutator is multiplied by a factor 1/lp3 to make the action dimensionless. With the replacement XI→lpXI, x+→lpx+, R→lpR, we get the action
(17)SR=R∫dx+×Tr(12R2D+XID+XIpppppppppp+14[XI,XJ]2-iRθD+θ+θγI[XI,θ]),
in which lp is cancelled and XI, θ, x+, and R are all dimensionless. In the following, we will still adopt this convention, so lp will not appear explicitly.The11d supergravity field, after the gauge fixing, has the nonzero components (g+-, g++, gI+, gIJ), (CIJ+, CIJK), and (ψ+, ψI) [22]. Based on the Hamiltonian in [22], one can write down the action of the bosonic membrane on such supergravity background:
(18)SF0b=∫dx+d2σg+-P+×[P+2D+XID+XJ2g+-2gIJ+P+D+XIg+-(CJ+P+g+Jg+-)gIJ+P+2g++2g+-2-{XI,XJ}{XK,XL}gIKgJL+P+C+g+-],
where CI={XJ,XK}CJKI and C+={XJ,XK}C+JK. Note that it is g+-/P+ that appears. The Matrix theory version is
(19)SF0b=∫dx+×Tr(Rg+-{D+XID+XJ2(Rg+-)2gIJppppppppppppppp+D+XIRg+-(CJ+g+JRg+-)gIJppppppppppppppp+14[XI,XJ][XK,XL]gIKgJLppppppppppppppp+g++2(Rg+-)2+C+Rg+-{D+XID+XJ2(Rg+-)2gIJ}),
where CI=-(i/2)[XJ,XK]CJKI and C+=-(i/2)[XJ,XK]C+JK. Without the gauge fixing, gI-, g--, C-IJ, C+-I, ψ-≠0, so terms involving them can also be added:
(20)SgI-b=∫dx+×Tr(Rg+-{D+XID+XJD+XK2(Rg+-)3gJKpppppppppppppp-D+XI4Rg+-[XK,XL][XM,XN]pppppppppppppp×gKMgLNpppppppppppppp+D+XLRg+-[XJ,XM][XI,XN]pppppppppppppp×gLJgMND+XID+XJD+XK2(Rg+-)3}pppppppp×RgI-D+XID+XJD+XK2(Rg+-)3)
and similarly for Sg--b, 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--, gI-, 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+XID+XI+(1/ν2){XI,XJ}{XI,XJ} and DaX-=D+XIDaXI into the action, which could couple with the x- indexed fields.In all these terms, it isRg+-, RgI-, R2g--, RC-IJ, RC+-I, Rψ- that are involved. (Similarly, in supermembrane action, it is g+-/P+, gI-/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′,
(21)g++′=f′2(x′+)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,
the action is invariant. This is consistent with the coordinate transformation
(22)ds2=g++df(x′+)df(x′+)+2gI+dxIdf(x′+)+gIJdxIdxJ+2αg+-df(x′+)dx′-+2αgI-dxIdx′-+α2g--dx′-dx′-=g++f′2(x′+)dx′+dx′++2f′(x′+)gI+dxIx′++gIJdxIdxJ+2αg+-f′(x′+)dx′+dx′-+2αgI-dxIdx′-+α2g--dx′-dx′-,C=CIJKdxI∧dxJ∧dxK+CIJ+dxI∧dxJ∧df(x′+)+αCIJ-dxI∧dxJ∧dx′-+αCI+-dxI∧df(x′+)∧dx′-=CIJKdxI∧dxJ∧dxK+CIJ+f′(x′+)dxI∧dxJ∧dx′++αCIJ-dxI∧dxJ∧dx-+αCI+-f′(x′+)dxI∧dx′+∧dx-.S
R
,
F, the supersymmetric extension of (19) and (20), is not constructed yet. With SR,F given, the current density for F can be defined as
(23)VF(Y)(x)=δSR,F(Y)δF(x)|F.
In particular, when F is the flat background, for which, the only nonvanishing fields are g-+-=-1 and g-IJ=δIJ, VF(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 SR,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]:
(24)SR,F=SR-R∫dx+pppppppp×STr[VhIJ(X)hIJ(X)+VhI+(X)hI+(X)pppppppppppppp+Vh++(X)h++(X)+VhI-(X)hI-(X)pppppppppppppp+Vh+-(X)h+-(X)+Vh--(X)h--(X)pppppppppppppp+VCIJK(X)CIJK(X)+VC+JK(X)C+JK(X)pppppppppppppp+VCIJ-(X)CIJ-(X)+VC+-K(X)C+-K(X)pppppppppppppp+Vψ(X)ψ(X)+Vψ+(X)ψ+(X)pppppppppppppp+Vψ-(X)ψ-(X)hIJ(X)],
where, for example,
(25)VhI+(X)hI+(X)=(D+XIR-14θγIJθ∂∂XJ)hI+(X)R.hIJ(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 a1d SYM theory, just as that in AdS/CFT. Then, the background fields, like hI+(X), should be expanded as the Taylor series:
(26)hI+(x+,X1,…,X9)=∑n=0∞‍1n!(∂k1⋯∂knhI+)(x+,0,…,0)Xk1⋯Xkn,
with (∂k1⋯∂knhI+)(x+,0,…,0) the derivative of hI+(x+,x→) at (x+,0) [3, 23–25]. 								(27)STr[VhI+(X)hI+(X)]=∑n=0∞‍1n!(∂k1⋯∂knhI+)(x+,0,…,0)×STr[1R(D+XIR-14θγIJθ∂∂XJ)Xk1⋯Xkn]=∑n=0∞‍1n!(∂k1⋯∂knhI+)(x+,0,…,0)IhI+k1⋯kn(x+).
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
(28)Xi⟶Xi-ai,f(x+,0,…,0)⟶f(x+,a1,…,a9),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 Φk1⋯kn in AdS/CFT, the moment operators Ik1⋯kn do not need to be traceless. As a result, {1,Ik1,…,Ik1⋯kn,…} couples with the 10d field, while {1,Φk1,…,Φk1⋯kn,…} only couples with the 9d field.In the second form, the vertex operator is defined in10d spacetime:
(29)δδF(x)F(x+,X1,…,X9)=δ(x0-x+)δ9(xI-XI)=δ10(xμ-Xμ),
where XI are N×N matrices and for uniformity; we have set X0=x+1N×N. This is a matrix generalization of the δ-function. In special situations, when all of the XI are diagonal, that is, XI=diag{x1I,x2I,…,xNI}, (29) becomes
(30)δδF(x)F(x+,X1,…,X9)=diag{δ10(xμ-x1μ),δ10(xμ-x2μ),…,δ10(xμ-xNμ)}.
With the generalized δ-function, it is straightforward to write down the current densities for various fields. For hI+, we have
(31)VhI+(x)=δδhI+(x){-R∫‍dx+STr[VhI+(X)hI+(X)]}=-R∫‍dx+STr[VhI+(X)δ10(x-X)].It is not convenient to deal with theδ-function. One may want to do a Fourier transformation, which gives the third representation of the vertex operator:
(32)VhI+(p)=∫d10xVhI+(x)eipx=-R∫dx+STr[VhI+(X)eipX]=-R∫dx+STr{[-D+XIR+i4θγIJθpJ]eipX}.


## 2.2. Partition Function of Matrix Theory on Curved Background

Suppose the exact form ofSF is given, and the partition function of Matrix theory on supergravity background is
(33)Z(R,F)=eW(R,F)=∫[dXdθdA0]e-SR,F(X,θ,A0)=∫[dY]e-SR,F(Y),
where F is the 11d background field (g+-, g++, gI+, gIJ, g--, gI-), (CIJ+, CIJK, CIJ-, CI+-), and (ψ+, ψI, ψ-) mentioned before, and Y collectively represents (XI,θ,A0).In gauge/string correspondence, partition function is an important quantity, the value of which should be equal on both sides. ForAdS5/CFT4 correspondence, it is expected that
(34)ZSYM4=exp{-F}
should hold, where F=∑nh=0‍Fnh is the free energy of the strings on AdS5×S5, ZSYM4 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).SupposeSP+,FM(Y) is the membrane action on background F. SP+,FM(Y) should be general covariant, so, for P+=P′+/α, x+=f(x′+), and X=g(X′), there is SP+,FM(Y)=SP′+,F′M(Y′), where F′ is the field coming from the coordinate transformation:
(35)x-=αx′-,x+=f(x′+),x=g(x′).
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
(36)∫[dY]e-SP+,FM(Y)=∫[dY′]e-SP′+,F′M(Y′).
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
(37)Z(R,F)=eW(R,F)=∫[dY]e-SR,F(Y)=∫[dY′]e-SR′,F′(Y′)=eW(R′,F′)=Z(R′,F′).W(R,F)=W(R′,F′). At least restricted to (35), W(R,F) is the diffeomorphism invariant functional of F.LetF0, F-, F-- represent fields with zero, one, and two x- indices. Since
(38)SR,F(Y)=SF0,RF-,R2F--(Y),(39)W(R,F)=W(F0,RF-,R2F--).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 Seff(R;F0,F-,F--).  Seff is invariant under the coordinate transformation:
(40)x-⟶x-R,F0⟶F0,F-⟶RF-,F--⟶R2F--,
so
(41)Seff(R;F0,F-,F--)=Seff(1;F0,RF-,R2F--).
The radius of x- is absorbed in F-, F--, as is in (39). In this respect, W is consistent with Seff.LetF=F-+F~, where F- is the flat background with g-+-=-1, g-IJ=δIJ, and the rest fields being zero. W(R,F-)=0:
(42)W(R,F)=∫d10xΓR,Fc(x)F~(x)+12!∫d10x1×∫d10x2ΓR,Fc(x1,x2)F~(x1)F~(x2)+⋯=∑n‍1n!∫d10x1⋯∫d10xnΓR,Fc(x1,…,xn)pppppppppppppppp×F~(x1)⋯F~(xn),
where (F collectively represents supergravity fields. For example, ΓR,g++,g--c(x1,x2)=δ2W(R,F)/δg++(x1)δg--(x2)|F=F-.)(43)ΓR,Fc(x1,…,xn)=δnW(R,F)δF(x1)⋯δF(xn)|F=F-.
For the same F, the 11d supergravity effective action Seff(R,F) is
(44)Seff(R,F)=∑n‍1n!∫02πRdx1-×∫dx1+d9x1⋯∫02πRdxn-∫dxn+d9xn×ΓFc(x1-,x1+,x1;…;xn-,xn+,xn)×F~(x1+,x1)⋯F~(xn+,xn).ΓFc(x1-,x1+,x1;…;xn-,xn+,xn) is the vertex function of the 11d supergravity. If Seff(R,F)=W(R,F), there will be
(45)ΓR,Fc(x1,…,xn)=ΓR,Fc(x1+,x1;…;xn+,xn)=∫02πRdx1-⋯∫02πRdxn-ΓFc(x1-,x1+,x1;…;xn-,xn+,xn).In linear gravity approximation,δVF(x)/δF(y)=0, where VF is the current density coupling with F(x) as is defined in (23):
(46)ΓR,Fc(x1,…,xn)=(-1)n〈VF(x1)⋯VF(xn)〉c.〈VF(x1)⋯VF(xn)〉c is the connected correlation function of the current density, in contrast to
(47)〈VF(x1)⋯VF(xn)〉=∫[dXdθdA0]e-S0VF(x1)⋯VF(xn)=(-1)nδnZ(R,F)δF(x1)⋯δF(xn)|F=F-,
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 ΓR,Fc(x1,…,xn).In terms ofVF(p),
(48)〈VF(x1)⋯VF(xn)〉c=∫d10p1⋯∫d10pne-i(p1x1+⋯+pnxn)pppppppppppppp×〈VF(p1)⋯VF(pn)〉c
Note that VF(X)(x+,xI-aI)=VF(X+a)(x+,xI), S0(X+a)=S0(X), so
(49)〈VF(x1)⋯VF(xn)〉c=〈VF(x1+a1)⋯VF(xn+an)〉c.
The correlation function is translation invariant. 〈VF(p1)⋯VF(pn)〉c=0 if p1+⋯+pn≠0.Let us first consider the one point function(50)〈VF(x)〉=-δW(R,F)δF(x)|F=F-=v-F.v-F is the vacuum expectation value of the current density, when the background field is F-, and is a constant in spacetime. SR is SO(9) invariant, so v-F should also be SO(9) invariant. As a result,
(51)v-gI+=v-gI-=v-CIJK=v-CIJ+=v-CIJ-=v-CI+-=v-ψI=0.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,
(52)〈VF(x)〉F=-δW(R,F)δF(x)|F=F-+F~=vF(x)=∑n‍-1(n-1)!×∫d10x1⋯∫d10xn-1ΓR,Fc(x,x1,…,xn-1)ppppppppppppppp×F~(x1)⋯F~(xn-1)
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 〈VF(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 givenF(x), vF(x) is uniquely determined. Conversely, different F(x) may result in the same vF(x). This is quite like the source-gravity coupled system. For the given gravity field, the density of the source can be obtained through δSg/δgμν=-Tμν. 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 vF(x)=0, F~(x)=0. F~(x) could be interpreted as the field generated by the current density vF(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 betweenvF and F, and so a Legendre transformation is possible. Before that, we will first define Γ(Y):
(53)Γ(Y)=SR,F(Y)+W(R,F),(54)eΓ(Y)=∫[dY~]e-SR,F(Y~)+SR,F(Y).F is solved from the equation
(55)∫[dY~]δδF(x)e-SR,F(Y~)+SR,F(Y)=∫[dY~]e-SR,F(Y~)+SR,F(Y)[-δSR,F(Y~)δF(x)+δSR,F(Y)δF(x)]=0
or equivalently,
(56)∫‍[dY~]e-SR,F(Y~)δSR,F(Y~)δF(x)=δSR,F(Y)δF(x)=-δW(R,F)δF(x).
In some sense, F(x) is the field generated by Y. Take a derivative of (54) with respect to Y; using (55), we get
(57)δΓ(Y)δY=δSR,F(Y)δY,
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 theR dependence of SR,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). SR,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 SP+,FM(Y), the R dependence is canceled, so the obtained Γ(Y) is the same.With these properties collected, we may consider the possible interpretation ofΓ(Y) and W(R,F). If W(R,F) is the effective action of supergravity, since SR,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=δSR,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=δSR,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 ofΓ(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, SF(Y)=S0(Y)+∫‍d10xF~(x)VF(Y)(x), Γ(Y)=S0(Y)+Γ~(vF), where
(58)Γ~(vF)=∫d10xF~(x)vF(x)+W(F-+F~)
is the Legendre transformation of W(F):
(59)-δW(F)δF~(x)|F~=vF=VF(Y)(x),δΓ~(vF)δvF(x)|vF=F~(x).
Let
(60)ΛF(x1,…,xn)=δnΓ~(vF)δvF(x1)⋯δvF(xn)|vF=0,
then
(61)Γ(Y)=S0(Y)+∑n‍1n!ppppppppppp×∫d10x1⋯∫d10xnΛF(x1,…,xn)ppppppppppppppppppppppp×vF(x1)⋯vF(xn)=S0(Y)+∑n‍1n!pppppppppp×∫d10x1⋯∫d10xnΛF(x1,…,xn)pppppppppppppppppppppp×VF(Y)(x1)⋯VF(Y)(xn).
For the given vF(x),
(62)F(x)=∑n‍1(n-1)!×∫d10x1⋯∫d10xn-1ΛF(x,x1,…,xn-1)ppppppppppppppp×vF(x1)⋯vF(xn-1).
Equation (52) gives the current density vF(x) generated by the field F(x); (62) gives the field F(x) generated by the current density vF(x). If ΛF 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 vF. In classical level, F(x) could be calculated by δSg/δF(x)=-vF(x) with Sg the classical action of supergravity:
(63)ΓFc(x-z)=-δvF(x)δF~(z)|F~=0,ΛF(z-y)=δ2Γ~(vF)δvF(z)δvF(y)|vF=0=δF~(z)δvF(y)|vF=0,
so
(64)-∫d10zΓFc(x-z)ΛF(z-y)=δ10(x-y).-ΛF(x1-x2) is the inverse of ΓFc(x1-x2). Subsequently,
(65)ΛF(x1,x2,x3)=∫d10y1∫d10y2×∫d10y3ΛF(y1-x1)ΛF(y2-x2)×ΛF(y3-x3)ΓFc(y1,y2,y3),ΓFc(x1,x2,x3)=-∫d10y1∫d10y2×∫d10y3ΓFc(y1-x1)ΓFc(y2-x2)×ΓFc(y3-x3)ΛF(y1,y2,y3).ΛF(x1,x2,x3,x4)=∫d10y1∫d10y2∫d10y3×∫d10y4ΛF(y1-x1)ΛF(y2-x2)ppppp×ΛF(y3-x3)ΛF(y4-x4)ppppp×[∫d10z1ΓFc(y1,y2,y3,y4)pppppppp+∫d10z1pppppppp×∫d10z2ΛF(z1-z2)ppppppppppp×(ΓFc(z1,y1,y4)ppppppppppppp×ΓFc(z2,y2,y3)ppppppppppppp+ΓFc(z1,y2,y4)ppppppppppppp×ΓFc(z2,y1,y3)ppppppppppppp+ΓFc(z1,y3,y4)ppppppppppppp×ΓFc(z2,y1,y2))∫d10z1],ΓFc(x1,x2,x3,x4)=∫d10y1∫d10y2∫d10y3×∫d10y4ΓFc(y1-x1)ΓFc(y2-x2)×ΓFc(y3-x3)ΓFc(y4-x4)×[∫d10z1ΛF(y1,y2,y3,y4)+∫d10z1×∫d10z2ΓFc(z1-z2)ppppppp×(ΛF(z1,y1,y4)ppppppppp×ΛF(z2,y2,y3)ppppppppp+ΛF(z1,y2,y4)ppppppppp×ΛF(z2,y1,y3)ppppppppp+ΛF(z1,y3,y4)ppppppppp×ΛF(z2,y1,y2))∫d10z1].
The relation between ΛF and ΓFc shows that ΛF and ΓFc are the connected Green’s function and the vertex function of a particular quantum field theory. ΛF(z-y)=(δF~(z)/δvF(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,
(66)eΓ(Y)=∫[dη]e-SF(Y+η)+SF(Y).-SF(Y+η)+SF(Y)=-∫dτδSF(Y)δY(τ)η(τ)-12∫dτdτ′δ2SF(Y)δY(τ)δY(τ′)×η(τ)η(τ′)+⋯=-SF(1)(Y,η)-SF(2)(Y,η)+⋯.SF could be written as (We only write ∂τ2, but one should be aware that the fermionic kinetic term ∂τ also exists.)(67)SF(Y)=∫dτ[12Y∂τ2Y+V0(Y)+V(F,Y)]=S0(Y)+∫dτV(F,Y)
with V(F,Y) taking the form as that in (27). For the given Y, solve F0 from δSF0(Y)/δY=0, and expand SF(n)(Y,η) around F0. F=F0+f. Consider
(68)SF(1)(Y,η)=δSF(1)(Y,η)δF|F0f+⋯SF(2)(Y,η)=SF0(2)(Y,η)+δSF(2)(Y,η)δF|F0f+⋯.f=f(Y) is solved as the functional of Y. Let Γ(Y)=Γ1(Y)+Γ2(Y)+⋯ since
(69)-SF(Y+η)+SF(Y)=-SF0(2)(Y,η)+⋯,Γ1(Y)=ln{∫[dη]e-SF0(2)(Y,η)}=ln(detAF0)1/2
with
(70)AF0(τ,τ′)=[∂τ2+∂2V0(Y)∂Y2+∂2V(F0,Y)∂Y2]δ(τ-τ′).
In particular, when δS0(Y)/δY=0, F0=0, AF0 reduces to
(71)A(τ,τ′)=[∂τ2+∂2V0(Y)∂Y2]δ(τ-τ′).
The corresponding Γ1(Y) is the same as the one-loop contribution Γeff1(Y) of the standard effective action Γeff(Y) in background gauge:
(72)eΓeff(Y)=∫[dη]e-S0(Y+η)+∫‍dτTr[J(τ)η(τ)],
subject to the condition
(73)∫[dη]δδJ(τ)e-S0(Y+η)+∫‍dτTr[J(τ)η(τ)]=0.Γ
1
(
Y
)
=
Γ
eff
1
(
Y
) has been calculated for the arbitrary Y satisfying δS0(Y)/δY=0 [3, 25]:
(74)Γ1(Y)=12∫d10x1×∫d10x2GFc0(x1,x2)VF(Y)(x1)VF(Y)(x2),
with GFc0 being the free supergraviton propagator. Compared with (61), ΛF,1(x1,x2)=GFc0(x1,x2). If the result can be extended to the n-point Green’s functions and to the full-loop calculation, there will be
(75)ΛF(x1,…,xn)=GFc(x1,…,xn),
where GFc is the connected full Green's function of supergravity. ΓFc(x1,…,xn) and W(F) will then become the n-point vertex function and the effective action of supergravity, respectively.In quantum field theory, unlike theS-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 Γeff(Y) differ from a parameterization transformation. Γ(Y)≠Γeff(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
(76)Γg[Y(τ)]=∑n‍1n!∫d10x1⋯∫d10xnGFc(x1,…,xn)pppppppppppppppp×VF(Y)(x1)⋯VF(Y)(xn).
The effective action defined in (61) is also the Taylor series of VF(Y) and thus could be compared with (76) directly. The standard effective action Γeff(Y) is expanded as the Taylor series of Y. A careful reorganization is needed to get VF(Y), but it is unclear whether it is always possible to do so. Moreover, under a Legendre transformation, (76) becomes
(77)Γg[Y(τ)]=Seff(F)+∫d10xF(x)VF(Y)(x)~Seff(F)+SF[Y(τ)]
with
(78)-δSeff(F)δF(x)|F=VF(Y)(x)=δSF(Y)δF(x)|F.Seff(F) is the effective action of supergravity. Equation (77) is almost the same as (53). The only problem is whether W(F)=Seff(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:
(79)Seff[F~(F)]=∑m=0∞‍eϕ(2-2m)∫M2-2m[dX]e-SF=Ws(F),
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 gab=e2σg^ab is always imposed, and so, Ws(F,σ) also has the dependence on σ:
(80)δWs(F,σ)δσ=-δWs(F,σ)δFβ(F).
When β(F)=0, Ws(F,σ)=Ws(F), although β(F)=0 is not the necessary condition.SupposeSeff(F) is the effective action of the string modes:
(81)Seff(F)=-12∫dDxFΔF+Sint(F).
Consider F0 with ΔF0=0. Since β(F0)≠0, F will evolve along the RG flow; that is, Ws(F0,σ)=Ws[F(σ′),σ′]. A special property of F0 is that it will finally reach an IR fixed point F~, Ws(F0,σ)=Ws(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, Ws(F~,∞)=Ws(F~,σ)=Ws(F~)=Ws(F0,σ)=Ws(F0).F
~ is calculated from F0 via
(82)F~=F0+Δ-1δSint(F~)δF~=F0(x)+∫dDyD(x-y,F)δSint(F~)δF~(y).δSeff(F)/δF(x)|F=F~=0. Since δSeff(F)/δFi=κijβ(Fj), β(F~)=0 is equivalent to δSeff(F)/δF(x)|F=F~=0 [18]. In [18], it was shown that Ws(F0,σ)=Seff(F~), which means Ws(F~,σ)=Ws(F~)=Seff(F~). There is a one-to-one correspondence between the solution space of δSeff(F)/δF(x)=0 and the solution space of ΔF=0. So, for all F with δSeff(F)/δF(x)=0=β(F), Ws(F,σ)=Seff(F). In σ-model approach, only for such F, the conformal factor decouples; thus, the calculated string free energy is physical. In this subspace, Ws(F)=Seff(F).W
s
(
F
,
σ
) with β(F)≠0 usually depends on σ. However, the σ-dependence drops out for F0. In fact,(83)Ws(F0,σ)=Ws(F0)=Seff(F~)=Seff[F0+Δ-1δSint(F0)δF0+⋯]
is the S-matrix functional [27]. For the on-shell p1,…,pn,
(84)Ws(F0,σ)=∑n‍1n!∫d9p1⋯∫d9pnSFc(p1,…,pn)ppppppppppppppp×F0(p1)⋯F0(pn),F0(x)=∫d9pF0(p),SFc(p1,…,pn)=1N0∑m=0∞‍eϕ(2-2m)pppppppp×∫M2-2m[dX]e-S0VF(p1)⋯VF(pn)
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, forF satisfying δSeff(F)/δF(x)=0, Ws(F,σ)=Ws(F)=Seff(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, ZSYM4=eWs, where Ws 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 ημν+δgμν(x) other than ημν for very small δgμν(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 δGμν(x,σ) that is entirely determined by δgμν(x). Correspondingly, ZSYM4[δgμν(x)]=eWs[δGμν(x,σ)], where Ws[δGμν(x,σ)] is the free energy of the string theory on AdS5×S5 with the background perturbation δGμν(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 ZSYM4[δgμν(x)]=eSeff[δGμν(x,σ)], where Seff[δGμν(x,σ)] is the supergravity effective action for the modified background fields, so Seff[δGμν(x,σ)]=Ws[δGμν(x,σ)] if ZSYM4=eWs holds.There is a naive way to interpretWs(F)=Seff(F). Suppose the classical supergravity action is Sc(F), from the field theory's point of view:
(85)eSeff(F)=∫1PI[dF~]e-Sc(F+F~),
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 Sc(F+F~). Now, consider the string theory on background F, we may have
(86)eSeff(F)=Zs(F)=eWs(F),
where Ws(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 Sc(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 δSeff(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 Seff(F) should be the effective action with the α′ corrections taken into account.A naturalM-theory extension is
(87)eSeff(F)=ZM(F)=∫[dY]e-SFM(Y),
where SFM(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 eSeff(F) other than Seff(F). For the generic 11d background F, SFM(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

IfW(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)=-vF(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].)
(88)W(R,F)=-∫d10x[v-g++g++(x)+v-g--g--(x)+v-g+-g+-(x)]+12∫d10x1×∫d10x2ΓR,g++,g--c(x1,x2)g++(x1)g--(x2)+12∫d10x1×∫d10x2ΓR,g+-,g+-c(x1,x2)g+-(x1)g+-(x2)+⋯.
With α,β=+,-, one may try to solve gαβ(x) from δW(R,F)/δgαβ(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 gαβ(x) are also constants (although the constant gαβ(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 δSR,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
(89)eΩ(R,Y,F)=∫[dY~]e-SR,F(Y~)+SR,F(Y)=∫[dη]e-SR,F(Y+η)+SR,F(Y)=∫[dη]e-SR,F(Y;η).Ω(R,Y,F)=W(R,F)+SR,F(Y) is the action of the source-gravity coupled system and is not necessarily on shell with respect to gravity. If SR,F(η) gives the Matrix theory description of M theory on background F, SR,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, δSR,F(Y)/δY=0,
(90)SR,F(Y;η)=∫dτδSR,F(Y)δY(τ)η(τ)+12∫dτdτ′δ2SR,F(Y)δY(τ)δY(τ′)η(τ)η(τ′)+⋯=12∫dτdτ′δ2SR,F(Y)δY(τ)δY(τ′)η(τ)η(τ′)+⋯SR,F(Y;η) starts from the quadratic term. For F satisfying δΩ(R,Y,F)/δF(x)=0, SR,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):
(91)0=∫[dη]e-SR,F(Y;η)δSR,F(Y;η)δF(x)=〈δSR,F(Y;η)δF(x)〉.
The expectation values of all current densities vanish. There is no tadpole.TheU(∞) symmetry of Matrix theory is destroyed by Y. In particular, if
(92)Y=diag[M1,…,M1︸N1,…,Mm,…,Mm︸Nm],
the preserved gauge symmetry is U(N1)×⋯×U(Nm). Nevertheless, the original U(∞) symmetry still has its manifestation. For all u∈U(∞),
(93)η⟶uηu++(uYu+-Y),η⟶η+i[α,η]+i[α,Y],
so for u∉U(N1)×⋯×U(Nm), 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:
(94)ds2=-2dx+dx-+∑i=19‍dxidxi-[∑a=13‍μ29(xa)2+∑a′=49‍μ236(xa′)2]dx+dx+F123+=μ.
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:
(95)SRPW=R∫dx+×Tr{14R2(D+Xi)2pppppppppp-iRθD+θ+12[Xi,Xj]2+2θγi[Xi,θ]pppppppppp-(μ6R)2(Xa)2-(μ12R)2(Xa′)2pppppppppp-μ6RiϵabcXaXbXc-μ8Rθγ123θ},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]
(96)[X^i,X^j]=iμ3RϵijkX^k,i,j,k=1,2,3,X^i=0,i=4,…,9,X^˙i=0,i=1,…,9.{X^1,X^2,X^3} form the N dimensional representation of SU(2). Suppose (3R/μ)X^i=Li, i=1,2,3, Li can be decomposed as
(97)Li=diag[Lj1i⋯Lj1i︸N1Lj2i⋯Lj2i︸N2⋯Ljti⋯Ljti︸Nt],
where Ljsi(s=1,…,t) stands for the spin js irreducible representation of SU(2):
(98)(2j1+1)N1+(2j2+1)N2+⋯+(2jt+1)Nt=N.The solution can be taken as a set ofNs coincident fuzzy spheres with the radius (μ/3R)js(js+1) [2]. The U(N) gauge symmetry is broken into U(N1)×U(N2)×⋯×U(Nt). All fuzzy spheres are concentric. When js→∞, the fuzzy spheres become a set of membrane spheres given by ∑i=13‍(xsi)2=rs2 with rs∝(2js+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 sphericalM5 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 Ns are finite, they are fuzzy M5 branes. When Ns→∞, the fuzzy M5 become the spherical M5 wrapping S5 given by ∑i=49‍(xri)2=rr2. In patricular, the trivial vacuum X^i=0 represents a single M5 brane.All of the supersymmetric solutions preserve16 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.WithY^ 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
(99)SRPW(Y^;η)=SRPW(Y^+η)-SRPW(Y^).
Since (δSRPW(Y)/δY)|Y=Y^=0, SRPW(Y^;η) starts from the quadratic term.For simplicity, in the following, we will assumeR 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:
(100)eΓPW(Y^)=∫[dη]e-SF(Y^)PW(η)+SF(Y^)PW(Y^)=eWPW[F(Y^)]+SF(Y^)PW(Y^),
where
(101)δSFPW(Y^)δF(x)|F=F(Y^)=-WPW(F)δF(x)|F=F(Y^).
pp-wave with Y^ added respects the SU(2∣4) symmetry, so the generated F(Y^) as well as the corresponding SF(Y^)PW(η) will have the same symmetry.With the fieldsF(Y^) given, in principle, one can write down SF(Y^)PW(η) explicitly. The classical supersymmetric solutions of SF(Y^)PW(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 δΓPW(Y)/δY|Y=Y^=δSF(Y^)PW(Y)/δY|Y=Y^=0. With the pp-wave background replaced by the backreacted geometry, SRPW(Y^;η) is modified to
(102)SF(Y^)PW(Y^;η)=SF(Y^)PW(Y^+η)-SF(Y^)PW(Y^).SF(Y^)PW(Y^;η) still starts from the quadratic term since δSF(Y^)PW(Y)/δY|Y=Y^=0.Notice that, although any configurationY^′ in (96) may be the classical vacuum of SF(Y^)PW(η), Y^ is special because the vacuum expectation values of current densities vanish only for SF(Y^)PW(Y^;η) but not for the generic SF(Y^)PW(Y^′;η).The11d 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),
(103)SF(Y^)PW(Y^)=0,δSFPW(Y^)δF(x)|F=F(Y^)=0,ΓPW(Y^)=WPW[F(Y^)],δWPW(F)δF(x)|F=F(Y^)=0.eWPW[F(Y^)]=∫[dη]e-SF(Y^)PW(Y^;η).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]
(104)ds112=2dx+dx-+g--dx-dx-+dxidxig--=Qr7,
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=30π3lp9N/R2 because lp is absorbed into R and r.)  With the coordinate transformation x→gs-1/3x, x-→gs2/3x-, (104) becomes
(105)ds112=2gs1/3dx+dx-+gs4/3Qr7dx-dx-+gs-2/3dxidxi,
while the radius of x- is now gs-2/3R.Under thex- reduction, in string frame,
(106)ds2=e-2ϕ/3Gmndxmdxn+e4ϕ/3(dx-+Amdxm)2.gmn=e-2ϕ/3Gmn+e4ϕ/3AmAn,g--=e4ϕ/3,gm-=e4ϕ/3Am.
In particular, for (105),
(107)ds102=-r7/2Q1/2dx+dx++Q1/2r7/2dxIdxI,eϕ=gs(Qr7)3/4,A+=r7gsQ
is the near-horizon geometry of the D0 branes [41].The type IIA solution coming from thex- 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 theM theory on background (104) in presence of the supergraviton with p+=N/R is SR,F(Y;η), where F is the field in (104), Y=0  (if F is the field in (105), SR,F→(1/gs)SR,F.) Cosider
(108)SR,F(0;η)=SR(η)+∫dx+Tr(g--T--)(η).SR 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 SR=1(η). Since (104) becomes (107) under the x- reduction, it is desirable to find a limit to make (108) reduce to SR=1(η).Consider the coordinate transformationxI→xI/R, x-→Rx-, under which
(109)F(x)=(g+-,gIJ,CI+-,CIJK,ψI,g++,gI+,g--,gI-,CIJ+,CIJ-,ψ+,ψ-)⟶F*(R,x)=RqF(xR)=(Rg+-,gIJR2,CI+-,CIJKR3,ψIR,g++,gI+R,R2g--,gI-,CIJ+R2,CIJ-R,ψ+,Rψ-).
Correspondingly, in Matrix theory action SR,F, with a field redefinition XI→XI/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). SR,F→S1,F*(R). With a further rescaling θ→θ/R3/2, SR,F→S1,F*(R)/R3. In S1,F*(R), it is
(110)F*(R,X)=RpF(XR)=(CIJ-Rg+-,gIJ,CI+-,CIJK,ψI,R2g++,RgI+,g--R2,gI-R,RCIJ+,CIJ-R,Rψ+,ψ-R)
that will appear.Return to (108). For any R, SR,F is equivalent to S1,F*(R)/R3, in which g--*(R,x)=g--(r/R)/R2~R3/r7. In R→0 limit, g--*(R,x)→0,
(111)limR→0SR,F⟺limR→01R3SR=1=limR→01R3∫dx+Tr(12D+XID+XI+14[XI,XJ]2pppppppppppppppp-iθD+θ+θγI[XI,θ]12).SYM1 arrives. Note that, for finite x, limR→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 RCIJ+(x/R)=CIJ+(x), all fields are marginal, so, for the arbitrary R [5],
(112)SRPW⟺1R3SR=1PW=1R3∫dx+ppippp×Tr{14(D+Xi)2-iθD+θppppppppppp+12[Xi,Xj]2+2θγi[Xi,θ]ppppppppppp-(μ6)2(Xa)2-(μ12)2(Xa′)2ppppppppppp-μ6iϵabcXaXbXc-μ8θγ123θ14}.Now consider the backgroundF(Y^) coming from the backreaction of the giant graviton Y^ on pp-wave. Suppose Y^ is a vacuum in (96) with
(113)2js+1=n+αs,(114)n⟶∞,αsfinite.
The radii of the spherical membranes are rs~(n+αs)/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]
(115)ds102=(V¨-2V˙-V′′)1/2×{-4V¨V¨-2V˙dx+dx++-2V′′V˙(dρ2+dη2)+4dΩ52+2V′′V˙ΔdΩ22},e4ϕ=4(V¨-2V˙)3-V′′V˙2Δ2,C1=-2V˙′V˙V¨-2V˙dx+,F4=dC3,C3=-4V˙2V′′Δdx+∧d2Ω,H3=dB2,B2=(V˙V˙′Δ+η)d2Ω,Δ=(V¨-2V˙)V′′-(V˙′)2,
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)=Vb(ρ,η)+vη-(ρ,η), where Vb is the background potential and vη- is determined by a configuration of conducting disks. Each SU(2∣4) symmetric theory is specified by Vb; 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,
(116)ds112=22/3(V¨-2V˙)(-V′′V˙2Δ2)1/3dx-dx--25/3V˙′V˙(-V′′V˙2Δ2)1/3dx+dx-+25/3V˙1/3[(V˙′)2-V¨V′′](-V′′Δ2)1/3dx+dx++22/3(V˙Δ)1/3(-V′′)1/3{-V′′V˙(dρ2+dη2)ppppppppppppppp+2dΩ52+V′′V˙ΔdΩ22}.For PWMM,(117)VbPW(ρ,η)=ρ2η-23η3.
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 η-1=(π/2R)(2j1+1),η-2=(π/2R)(2j2+1),…,η-t=(π/2R)(2jt+1). The electric charges on these disks are equal to (π2/8)N1,(π2/8)N2,…,(π2/8)Nt, 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(118)V(ρ,η)=ρ2η-23η3+vη-(ρ,η),
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 transformationx→x/R, x-→Rx-, the disks are now located at η-s=(π/2)(2js+1), while the fields will transform as F(x)→F*(R,x)=RqF(x/R), F10(x)→F*10(R,x). As a result, SR,F→S1,F*(R). F*10(R,x) and F*(R,x) can be obtained by plugging
(119)V(ρR,ηR)=1R3(ρ2η-23η3)+vη-/R(ρR,ηR)
into (115) and (116). Except for S1,F*(R), we also have SR,F~S1,F*(R)/R3 with F*(R,x)=RpF(x/R) given by (110). Plug
(120)R3V(ρR,ηR)=ρ2η-23η3+R3vη-/R(ρR,ηR)
into (115) and (116); one can get F10*(R,x) and the corresponding F*(R,x). In particular, for PWMM, V=VbPW, 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. SR,F⇔S1,F*(R)⇔S1,F*(R)/R3.The limit of interest isn→∞,  R→∞, R3/n=α2 and μ fixed. In this limit, the concentric spherical membranes all have the infinite radii with the difference (αi-αj)/R→0. For the finite ρ, η, ρ/R→0, η/R→0, we need to consider the behavior of vη-(ρ,η) around 0. Since η-s/R→∞, near 0, vη-/R can be approximated as the potential generated by dipoles located at ±η-s/R:
(121)R3vη-/R(ρR,ηR)~∑s=1t‍NsR3{[(η-s-ηR)2+(ρR)2]-1/2ppppppippp-[(η-s+ηR)2+(ρR)2]-1/2}~∑s=1t‍2R4Nsηη-s2⟶0.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, limSR,F⇔limSR=1PW/R3. Even if the background field is F(Y^), in the limit taken, we still get PWMM.On the other hand, plugR3V(ρ/R,η/R) into the reduced 10d solution (115); V¨-2V˙=R3(v¨η-/R-2v˙η-/R); the vη-/R part now matters. Let w(ρ2,η)=R3vη-/R(ρ/R,η/R):
(122)ds102=4{-(∂ρ22wη)-1/2dx+dx++(∂ρ22wη)1/2[dρ2+ρ2dΩ52+dη2+η2dΩ22]},e4ϕ=(∂ρ22wη)3,C1=-2(∂ρ22wη)-1dx+,F4=dC3,C3=-16η3dx+∧d2Ω,H3=dB2,B2=(4η2∂ρ22w-∂ρ2w+η∂ρ2∂ηw)d2Ω.
The 10d solution has the dependence on the disk configuration. Equation (122) is limF10*(R,x). With eϕ→R3eϕ, 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
(123)ds102=k2(-u7/2N1/2du+du++N1/2u7/2duIduI),e4ϕ/3=Nu7,A+=u7N.
This is similar to the limit taken in AdS/CFT [7].Now, consider the exact form ofS1,F*(R). For the generic 11d background F(x), the bosonic part of S1,F*(R) is
(124)S1,F*(R)b=∫dx+×Tr(g*+-{D+XID+XJ2g*+-2g*IJpppppppppppppp+D+XIg*+-(c*J+g*+Jg*+-)g*IJpppppppppppppp+14[XI,XJ][XK,XL]g*IKg*JLpppppppppppppp+g*++2g*+-2+c*+g*+-})+Sg*I-+Sg*--.
The relation between F*(R,x) and the x- reduced 10d background F*10(R,x) is
(125)g*mn=e-2ϕ*/3G*mn+e4ϕ*/3A*mA*n,g*--=e4ϕ*/3,g*m-=e4ϕ*/3A*m.
In the limit with n→∞, R→∞, R3/n fixed, if g--*→0, gI-*=0, the rest F10*(R,x) is finite, there will be
(126)R3SgI-=0,R3Sg--⟶0,G*++⟶-e2ϕ*A*+A*+,G*IJ=e2ϕ*/3g*IJ,-G*++⟶eϕ*A*+.
With these relations,
(127)limS1,F*(R)b=lim{1eϕ*∫dx+Tr-G*++pppppp×(-12G*IJG*++D+XID+XJppppppppp-G*IJG*++D+XIg*+Jppppppppp+(-G*++)1/2G*IJC*KLJppppppppp×D+XI[XK,XL]ppppppppp+14G*IJG*KL[XI,XK][XJ,XL]ppppppppp-12e2ϕ*/3G*++g*++ppppppppp+(-G*++)1/2C*+JK[XJ,XK]12)}=limS1,F*10(R)b.
For g*++=μ++XIXJg*IJ, g*+J=ν+XJ,
(128)e2ϕ*/3G*++g*++=μ++G*++G*IJXIXJ,G*IJG*++D+XIg*+J=ν+G*IJG*++D+XIXJ.
So, in the limit taken, S1,F*(R)b could be identified with S1,F*10(R)b, the action of SYM1 on the 10d background F*10(R,x). The discussion can also be extended to the full S1,F*(R) with the fermionic part included.Specified to the background generated byY^ on pp-wave, (126) is satisfied for the solution in (122), so limS1,F*(R)=limSR=1PW/R3=limS1,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,
(129)∫dx+Tr(12D+XID+XI+14[XI,XJ]2pppppppp-iθD+θ+θγI[XI,θ]12)=∫dx+×Tr{-G++eϕ(-12GIJG++D+XID+XJpppppppppppppppp+14GIJGKL[XI,XK][XJ,XL]pppppppppppppppp-ieϕ(-G++)1/2θD+θpppppppppppppppp+eϕGII1/2θγI[XI,θ]12)-G++eϕ},
where ϕ, G++, and so forth are the near-horizon geometry of the D0 branes in (107) with gs=1.In the limit withn→∞, 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,
(130)1R3VbPW=1R3[ρ2(η+n)-23(η+n)3]⟶nR3(ρ2-2η2)=1α2VbSYMR×S2,
where VbSYMR×S2 is the background potential for SYMR×S2. Alternatively, we may directly plug
(131)V(ρR,η+nR)=1R3[ρ2(η+n)-23(η+n)3]+vη-/R(ρR,η+nR)
into (116) and (115), taking the limit at the end. Let vη-/R(ρ/R,(η+n)/R)=u(ρ2,η); the 11d background F*(R,x) is
(132)ds112=(2α2n2∂ρ22u)2/3×[α4n∂ρ22u4dx-dx--2α2n(1+∂ρ2∂ηu)dx+dx-+n2(4dx+dx+-dΩ22)-4(dη2+dρ2+ρ2dΩ52)α4n∂ρ22u4],
which has the topology of R×R7×S2×R. The 10d background F*10(R,x) is
(133)ds102=(∂ρ22un)-1/2(-4dx+dx++dΩ22)+4(∂ρ22un)1/2(dη2+dρ2+ρ2dΩ52),eϕ=α22(∂ρ22un)1/4,C1=-2(1+∂ρ2∂ηu)α2∂ρ22udx+,F4=dC3,C3=-4nα2∂ρ22udx+∧d2Ω,H3=dB2,B2=(-1+∂ρ2∂ηu4∂ρ22u+η)d2Ω.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 expandSR=1PW/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 SYMR×S2 around each vacuum and (ii) SYMR×S2 around a certain vacuum with a periodicity imposed is equivalent to SYMR×S3/Zk around each vacuum. In particular, SYMR×S3 can be realized as the PWMM around a certain vacuum with a periodicity condition imposed.Concretely,(n/R3)SR=1PW(Y^;η)=(n/R3)SR=1PW(Y^+η)=SR×S2(y^+y), where SR×S2 is the action of the U(N-)  SYMR×S2 with N-=N1+⋯+Nt:
(134)SR×S2=1gR×S22∫dx+dΩ2μ2pppppppp×Tr{-14FabFab-12DaΦDaΦppppppppppppp-μ22Φ2+μF12Φ-12DaXmDaXmppppppppppppp-μ28Xm2+14[Xm,Xn]2+12[Φ,Xm]2ppppppppppppp-i2λ-ΓaDaλ+iμ8λ-Γ123λppppppppppppp-12λ-Γ3[Φ,λ]-12λ-Γm[Xm,λ]},1/μ is the radius of S2 which is parameterized by (θ,ϕ). gR×S22=R3/μ3n. y^  represents the vacuum:
(135)Φ^=μ2diag[α1,…,α1︸N1,α2,…,α2︸N2,…,αt,…,αt︸Nt],A^1=0,A^2={tanθ2Φ^inregionI-cotθ2Φ^inregionII.
Region I and II correspond to 0≤θ<(π/2)+ε and (π/2)-ε<θ≤π, respectively. αs is identified with the αs in (113). The U(N-) gauge group is spontaneously broken to U(N1)×U(N2)×⋯×U(Nt). Equation (135) corresponds to t disks located at η-1=(π/2)α1,η-2=(π/2)α2,…,η-t=(π/2)αt with the electric charges on each equal to (π2/8)N1,(π2/8)N2,…,(π2/8)Nt. The previous η-s=(π/2)(n+αs) in PWMM now becomes η-s=(π/2)αs due to the η→η+n redefinition.The correspondence between the spacetime coordinates and theSYMR×S2 fields is Φ↔η, (X4,…,X9)↔(ρ,Ω5), (x+,Ω2)↔(x+,Ω2). Equation (134) is the SYMR×S2 on flat background. Plug (133) into (134), similar to (127), for bosonic part, we have
(136)SbR×S2=μ3∫dx+dΩ2μ2ppppp×Tr{-Geϕpppppppppp×(-14GacGbdFabFcdppppppppppppp-12GabGηηDaΦDbΦ-μ22GΩ2GηηΦ2ppppppppppppp+μ(-G)-1/2Gηη1/2F12Φppppppppppppp-12GabGmnDaXmDbXnppppppppppppp-μ28GΩ2GmnXmXnppppppppppppp+14GmpGnq[Xm,Xn][Xp,Xq]pppppppppppip+12GηηGmn[Φ,Xm][Φ,Xn]-Geϕ)},
where Gab~Gmn~Gηη~GΩ2~(∂ρ22u/n)1/2, G=detGab~-(∂ρ22u/n)-3/2, eϕ~α2(∂ρ22u/n)1/4. The action remains the same, so (134) is also the action of SYMR×S2 on background (133). The conclusion holds for fermionic part except for a rescaling of λ.Indeed, as we will show in AppendixA, such phenomenon is very common. Generically, the action of SYMp+1 on flat background is equal to the action of the SYMp+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
(
ρ
2
,
η
) is the function of the radial directions. For the given field configuration [Φ,X4,…,X9], let
(137)[Φ^,X^4,…,X^9]=[(∂ρ22un)1/4Φ,(∂ρ22un)1/4X4,…,(∂ρ22un)1/4X9];
then
(138)SbR×S2=μ3∫dx+dΩ2μ2ppppp×Tr{-Geϕppppppppppp×(-14GacGbdFabFcdppppppppppppp-12GabDaΦ^DbΦ^-μ22GΩ2Φ^2ppppppppppppp+μ(-G)-1/2F12Φ^-12GabDaX^mDbX^nppppppppppppp-μ28GΩ2X^mX^nppppppppppppp+14[X^m,X^n][X^p,X^q]ppppppppppipp+12[Φ^,X^m][Φ^,X^n])-Geϕ}.
Compared with (134), with Xm→X^m, Φ→Φ^, the 3d background fields on R×S2 will get the radial dependence:
(139)eϕ=1⟶eϕ=α22(∂ρ22un)1/4(ρ2,η),ds32=-4dx+dx++dΩ22⟶ds32=(∂ρ22un)-1/2(ρ2,η)(-4dx+dx++dΩ22).
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; ds102 in (133) can be written as
(140)ds102=(∂ρ22un)-1/2(-4dx+dx++dΩ22)+4(∂ρ22un)1/2dr2+4r2(∂ρ22un)1/2dΩ62,
with SYMR×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 Xm→X^m, Φ→Φ^ transformation to recover ρη dependence of the 3d background fields in SYMR×S2.For PWMM, we have(141)eWPW(F)=∫[dη]e-SPW(Y^;η).WPW(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,
(142)SPW(Y^;η)⟶SR×S2(y^;y)WPW(F)⟶WR×S2(F),
where WR×S2(F) is the type IIA action for the above-mentioned supergravity solution dual to a vacuum of SYMR×S2. Then,
(143)eWR×S2(F)=∫[dy]e-SR×S2(y^;y).As is demonstrated in [19, 20], from SYMR×S2 and the corresponding type IIA solution, it is also possible to get the SYMR×S3 and the associated type IIB solution. In (135), suppose
(144)Φ^=μ2×diag[…,s-1,…,s-1︸M,s,…,s︸M,s+1,…,s+1︸M,…],
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]
(145)SR×S2(y^+y)|[y(s+1,t+1)=y(s,t)]=SR×S3(z)=1gR×S32∫dtdΩ3(μ/2)3ppppppppp×Tr{-14FabFab-12DaXmDaXm-112R^Xm2pppppppppppppp-i2λ-ΓaDaλ-12λ-Γm[Xm,λ]pppppppppppppp+14[Xm,Xn]2},
where
(146)gR×S32=4πμgR×S22.SR×S3 is the action of the U(M) SYM on R×S3.This is a special example of Taylor’s prescription for the compactification (theT-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 ofSYMR×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 SYMR×S2 in type IIA become SR×S3 in type IIB. Equation (143) turns into
(147)eWR×S3(F)=∫[dz]e-SR×S2(z).
For the trivial vacuum of SR×S3, F represents AdS5×S5 background.Finally, the geometry arising from the backreaction of theD3 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.


## 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:
(94)ds2=-2dx+dx-+∑i=19‍dxidxi-[∑a=13‍μ29(xa)2+∑a′=49‍μ236(xa′)2]dx+dx+F123+=μ.
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:
(95)SRPW=R∫dx+×Tr{14R2(D+Xi)2pppppppppp-iRθD+θ+12[Xi,Xj]2+2θγi[Xi,θ]pppppppppp-(μ6R)2(Xa)2-(μ12R)2(Xa′)2pppppppppp-μ6RiϵabcXaXbXc-μ8Rθγ123θ},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]
(96)[X^i,X^j]=iμ3RϵijkX^k,i,j,k=1,2,3,X^i=0,i=4,…,9,X^˙i=0,i=1,…,9.{X^1,X^2,X^3} form the N dimensional representation of SU(2). Suppose (3R/μ)X^i=Li, i=1,2,3, Li can be decomposed as
(97)Li=diag[Lj1i⋯Lj1i︸N1Lj2i⋯Lj2i︸N2⋯Ljti⋯Ljti︸Nt],
where Ljsi(s=1,…,t) stands for the spin js irreducible representation of SU(2):
(98)(2j1+1)N1+(2j2+1)N2+⋯+(2jt+1)Nt=N.The solution can be taken as a set ofNs coincident fuzzy spheres with the radius (μ/3R)js(js+1) [2]. The U(N) gauge symmetry is broken into U(N1)×U(N2)×⋯×U(Nt). All fuzzy spheres are concentric. When js→∞, the fuzzy spheres become a set of membrane spheres given by ∑i=13‍(xsi)2=rs2 with rs∝(2js+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 sphericalM5 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 Ns are finite, they are fuzzy M5 branes. When Ns→∞, the fuzzy M5 become the spherical M5 wrapping S5 given by ∑i=49‍(xri)2=rr2. In patricular, the trivial vacuum X^i=0 represents a single M5 brane.All of the supersymmetric solutions preserve16 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.WithY^ 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
(99)SRPW(Y^;η)=SRPW(Y^+η)-SRPW(Y^).
Since (δSRPW(Y)/δY)|Y=Y^=0, SRPW(Y^;η) starts from the quadratic term.For simplicity, in the following, we will assumeR 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:
(100)eΓPW(Y^)=∫[dη]e-SF(Y^)PW(η)+SF(Y^)PW(Y^)=eWPW[F(Y^)]+SF(Y^)PW(Y^),
where
(101)δSFPW(Y^)δF(x)|F=F(Y^)=-WPW(F)δF(x)|F=F(Y^).
pp-wave with Y^ added respects the SU(2∣4) symmetry, so the generated F(Y^) as well as the corresponding SF(Y^)PW(η) will have the same symmetry.With the fieldsF(Y^) given, in principle, one can write down SF(Y^)PW(η) explicitly. The classical supersymmetric solutions of SF(Y^)PW(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 δΓPW(Y)/δY|Y=Y^=δSF(Y^)PW(Y)/δY|Y=Y^=0. With the pp-wave background replaced by the backreacted geometry, SRPW(Y^;η) is modified to
(102)SF(Y^)PW(Y^;η)=SF(Y^)PW(Y^+η)-SF(Y^)PW(Y^).SF(Y^)PW(Y^;η) still starts from the quadratic term since δSF(Y^)PW(Y)/δY|Y=Y^=0.Notice that, although any configurationY^′ in (96) may be the classical vacuum of SF(Y^)PW(η), Y^ is special because the vacuum expectation values of current densities vanish only for SF(Y^)PW(Y^;η) but not for the generic SF(Y^)PW(Y^′;η).The11d 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),
(103)SF(Y^)PW(Y^)=0,δSFPW(Y^)δF(x)|F=F(Y^)=0,ΓPW(Y^)=WPW[F(Y^)],δWPW(F)δF(x)|F=F(Y^)=0.eWPW[F(Y^)]=∫[dη]e-SF(Y^)PW(Y^;η).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]
(104)ds112=2dx+dx-+g--dx-dx-+dxidxig--=Qr7,
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=30π3lp9N/R2 because lp is absorbed into R and r.)  With the coordinate transformation x→gs-1/3x, x-→gs2/3x-, (104) becomes
(105)ds112=2gs1/3dx+dx-+gs4/3Qr7dx-dx-+gs-2/3dxidxi,
while the radius of x- is now gs-2/3R.Under thex- reduction, in string frame,
(106)ds2=e-2ϕ/3Gmndxmdxn+e4ϕ/3(dx-+Amdxm)2.gmn=e-2ϕ/3Gmn+e4ϕ/3AmAn,g--=e4ϕ/3,gm-=e4ϕ/3Am.
In particular, for (105),
(107)ds102=-r7/2Q1/2dx+dx++Q1/2r7/2dxIdxI,eϕ=gs(Qr7)3/4,A+=r7gsQ
is the near-horizon geometry of the D0 branes [41].The type IIA solution coming from thex- 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 theM theory on background (104) in presence of the supergraviton with p+=N/R is SR,F(Y;η), where F is the field in (104), Y=0  (if F is the field in (105), SR,F→(1/gs)SR,F.) Cosider
(108)SR,F(0;η)=SR(η)+∫dx+Tr(g--T--)(η).SR 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 SR=1(η). Since (104) becomes (107) under the x- reduction, it is desirable to find a limit to make (108) reduce to SR=1(η).Consider the coordinate transformationxI→xI/R, x-→Rx-, under which
(109)F(x)=(g+-,gIJ,CI+-,CIJK,ψI,g++,gI+,g--,gI-,CIJ+,CIJ-,ψ+,ψ-)⟶F*(R,x)=RqF(xR)=(Rg+-,gIJR2,CI+-,CIJKR3,ψIR,g++,gI+R,R2g--,gI-,CIJ+R2,CIJ-R,ψ+,Rψ-).
Correspondingly, in Matrix theory action SR,F, with a field redefinition XI→XI/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). SR,F→S1,F*(R). With a further rescaling θ→θ/R3/2, SR,F→S1,F*(R)/R3. In S1,F*(R), it is
(110)F*(R,X)=RpF(XR)=(CIJ-Rg+-,gIJ,CI+-,CIJK,ψI,R2g++,RgI+,g--R2,gI-R,RCIJ+,CIJ-R,Rψ+,ψ-R)
that will appear.Return to (108). For any R, SR,F is equivalent to S1,F*(R)/R3, in which g--*(R,x)=g--(r/R)/R2~R3/r7. In R→0 limit, g--*(R,x)→0,
(111)limR→0SR,F⟺limR→01R3SR=1=limR→01R3∫dx+Tr(12D+XID+XI+14[XI,XJ]2pppppppppppppppp-iθD+θ+θγI[XI,θ]12).SYM1 arrives. Note that, for finite x, limR→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 RCIJ+(x/R)=CIJ+(x), all fields are marginal, so, for the arbitrary R [5],
(112)SRPW⟺1R3SR=1PW=1R3∫dx+ppippp×Tr{14(D+Xi)2-iθD+θppppppppppp+12[Xi,Xj]2+2θγi[Xi,θ]ppppppppppp-(μ6)2(Xa)2-(μ12)2(Xa′)2ppppppppppp-μ6iϵabcXaXbXc-μ8θγ123θ14}.Now consider the backgroundF(Y^) coming from the backreaction of the giant graviton Y^ on pp-wave. Suppose Y^ is a vacuum in (96) with
(113)2js+1=n+αs,(114)n⟶∞,αsfinite.
The radii of the spherical membranes are rs~(n+αs)/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]
(115)ds102=(V¨-2V˙-V′′)1/2×{-4V¨V¨-2V˙dx+dx++-2V′′V˙(dρ2+dη2)+4dΩ52+2V′′V˙ΔdΩ22},e4ϕ=4(V¨-2V˙)3-V′′V˙2Δ2,C1=-2V˙′V˙V¨-2V˙dx+,F4=dC3,C3=-4V˙2V′′Δdx+∧d2Ω,H3=dB2,B2=(V˙V˙′Δ+η)d2Ω,Δ=(V¨-2V˙)V′′-(V˙′)2,
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)=Vb(ρ,η)+vη-(ρ,η), where Vb is the background potential and vη- is determined by a configuration of conducting disks. Each SU(2∣4) symmetric theory is specified by Vb; 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,
(116)ds112=22/3(V¨-2V˙)(-V′′V˙2Δ2)1/3dx-dx--25/3V˙′V˙(-V′′V˙2Δ2)1/3dx+dx-+25/3V˙1/3[(V˙′)2-V¨V′′](-V′′Δ2)1/3dx+dx++22/3(V˙Δ)1/3(-V′′)1/3{-V′′V˙(dρ2+dη2)ppppppppppppppp+2dΩ52+V′′V˙ΔdΩ22}.For PWMM,(117)VbPW(ρ,η)=ρ2η-23η3.
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 η-1=(π/2R)(2j1+1),η-2=(π/2R)(2j2+1),…,η-t=(π/2R)(2jt+1). The electric charges on these disks are equal to (π2/8)N1,(π2/8)N2,…,(π2/8)Nt, 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(118)V(ρ,η)=ρ2η-23η3+vη-(ρ,η),
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 transformationx→x/R, x-→Rx-, the disks are now located at η-s=(π/2)(2js+1), while the fields will transform as F(x)→F*(R,x)=RqF(x/R), F10(x)→F*10(R,x). As a result, SR,F→S1,F*(R). F*10(R,x) and F*(R,x) can be obtained by plugging
(119)V(ρR,ηR)=1R3(ρ2η-23η3)+vη-/R(ρR,ηR)
into (115) and (116). Except for S1,F*(R), we also have SR,F~S1,F*(R)/R3 with F*(R,x)=RpF(x/R) given by (110). Plug
(120)R3V(ρR,ηR)=ρ2η-23η3+R3vη-/R(ρR,ηR)
into (115) and (116); one can get F10*(R,x) and the corresponding F*(R,x). In particular, for PWMM, V=VbPW, 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. SR,F⇔S1,F*(R)⇔S1,F*(R)/R3.The limit of interest isn→∞,  R→∞, R3/n=α2 and μ fixed. In this limit, the concentric spherical membranes all have the infinite radii with the difference (αi-αj)/R→0. For the finite ρ, η, ρ/R→0, η/R→0, we need to consider the behavior of vη-(ρ,η) around 0. Since η-s/R→∞, near 0, vη-/R can be approximated as the potential generated by dipoles located at ±η-s/R:
(121)R3vη-/R(ρR,ηR)~∑s=1t‍NsR3{[(η-s-ηR)2+(ρR)2]-1/2ppppppippp-[(η-s+ηR)2+(ρR)2]-1/2}~∑s=1t‍2R4Nsηη-s2⟶0.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, limSR,F⇔limSR=1PW/R3. Even if the background field is F(Y^), in the limit taken, we still get PWMM.On the other hand, plugR3V(ρ/R,η/R) into the reduced 10d solution (115); V¨-2V˙=R3(v¨η-/R-2v˙η-/R); the vη-/R part now matters. Let w(ρ2,η)=R3vη-/R(ρ/R,η/R):
(122)ds102=4{-(∂ρ22wη)-1/2dx+dx++(∂ρ22wη)1/2[dρ2+ρ2dΩ52+dη2+η2dΩ22]},e4ϕ=(∂ρ22wη)3,C1=-2(∂ρ22wη)-1dx+,F4=dC3,C3=-16η3dx+∧d2Ω,H3=dB2,B2=(4η2∂ρ22w-∂ρ2w+η∂ρ2∂ηw)d2Ω.
The 10d solution has the dependence on the disk configuration. Equation (122) is limF10*(R,x). With eϕ→R3eϕ, 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
(123)ds102=k2(-u7/2N1/2du+du++N1/2u7/2duIduI),e4ϕ/3=Nu7,A+=u7N.
This is similar to the limit taken in AdS/CFT [7].Now, consider the exact form ofS1,F*(R). For the generic 11d background F(x), the bosonic part of S1,F*(R) is
(124)S1,F*(R)b=∫dx+×Tr(g*+-{D+XID+XJ2g*+-2g*IJpppppppppppppp+D+XIg*+-(c*J+g*+Jg*+-)g*IJpppppppppppppp+14[XI,XJ][XK,XL]g*IKg*JLpppppppppppppp+g*++2g*+-2+c*+g*+-})+Sg*I-+Sg*--.
The relation between F*(R,x) and the x- reduced 10d background F*10(R,x) is
(125)g*mn=e-2ϕ*/3G*mn+e4ϕ*/3A*mA*n,g*--=e4ϕ*/3,g*m-=e4ϕ*/3A*m.
In the limit with n→∞, R→∞, R3/n fixed, if g--*→0, gI-*=0, the rest F10*(R,x) is finite, there will be
(126)R3SgI-=0,R3Sg--⟶0,G*++⟶-e2ϕ*A*+A*+,G*IJ=e2ϕ*/3g*IJ,-G*++⟶eϕ*A*+.
With these relations,
(127)limS1,F*(R)b=lim{1eϕ*∫dx+Tr-G*++pppppp×(-12G*IJG*++D+XID+XJppppppppp-G*IJG*++D+XIg*+Jppppppppp+(-G*++)1/2G*IJC*KLJppppppppp×D+XI[XK,XL]ppppppppp+14G*IJG*KL[XI,XK][XJ,XL]ppppppppp-12e2ϕ*/3G*++g*++ppppppppp+(-G*++)1/2C*+JK[XJ,XK]12)}=limS1,F*10(R)b.
For g*++=μ++XIXJg*IJ, g*+J=ν+XJ,
(128)e2ϕ*/3G*++g*++=μ++G*++G*IJXIXJ,G*IJG*++D+XIg*+J=ν+G*IJG*++D+XIXJ.
So, in the limit taken, S1,F*(R)b could be identified with S1,F*10(R)b, the action of SYM1 on the 10d background F*10(R,x). The discussion can also be extended to the full S1,F*(R) with the fermionic part included.Specified to the background generated byY^ on pp-wave, (126) is satisfied for the solution in (122), so limS1,F*(R)=limSR=1PW/R3=limS1,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,
(129)∫dx+Tr(12D+XID+XI+14[XI,XJ]2pppppppp-iθD+θ+θγI[XI,θ]12)=∫dx+×Tr{-G++eϕ(-12GIJG++D+XID+XJpppppppppppppppp+14GIJGKL[XI,XK][XJ,XL]pppppppppppppppp-ieϕ(-G++)1/2θD+θpppppppppppppppp+eϕGII1/2θγI[XI,θ]12)-G++eϕ},
where ϕ, G++, and so forth are the near-horizon geometry of the D0 branes in (107) with gs=1.In the limit withn→∞, 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,
(130)1R3VbPW=1R3[ρ2(η+n)-23(η+n)3]⟶nR3(ρ2-2η2)=1α2VbSYMR×S2,
where VbSYMR×S2 is the background potential for SYMR×S2. Alternatively, we may directly plug
(131)V(ρR,η+nR)=1R3[ρ2(η+n)-23(η+n)3]+vη-/R(ρR,η+nR)
into (116) and (115), taking the limit at the end. Let vη-/R(ρ/R,(η+n)/R)=u(ρ2,η); the 11d background F*(R,x) is
(132)ds112=(2α2n2∂ρ22u)2/3×[α4n∂ρ22u4dx-dx--2α2n(1+∂ρ2∂ηu)dx+dx-+n2(4dx+dx+-dΩ22)-4(dη2+dρ2+ρ2dΩ52)α4n∂ρ22u4],
which has the topology of R×R7×S2×R. The 10d background F*10(R,x) is
(133)ds102=(∂ρ22un)-1/2(-4dx+dx++dΩ22)+4(∂ρ22un)1/2(dη2+dρ2+ρ2dΩ52),eϕ=α22(∂ρ22un)1/4,C1=-2(1+∂ρ2∂ηu)α2∂ρ22udx+,F4=dC3,C3=-4nα2∂ρ22udx+∧d2Ω,H3=dB2,B2=(-1+∂ρ2∂ηu4∂ρ22u+η)d2Ω.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 expandSR=1PW/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 SYMR×S2 around each vacuum and (ii) SYMR×S2 around a certain vacuum with a periodicity imposed is equivalent to SYMR×S3/Zk around each vacuum. In particular, SYMR×S3 can be realized as the PWMM around a certain vacuum with a periodicity condition imposed.Concretely,(n/R3)SR=1PW(Y^;η)=(n/R3)SR=1PW(Y^+η)=SR×S2(y^+y), where SR×S2 is the action of the U(N-)  SYMR×S2 with N-=N1+⋯+Nt:
(134)SR×S2=1gR×S22∫dx+dΩ2μ2pppppppp×Tr{-14FabFab-12DaΦDaΦppppppppppppp-μ22Φ2+μF12Φ-12DaXmDaXmppppppppppppp-μ28Xm2+14[Xm,Xn]2+12[Φ,Xm]2ppppppppppppp-i2λ-ΓaDaλ+iμ8λ-Γ123λppppppppppppp-12λ-Γ3[Φ,λ]-12λ-Γm[Xm,λ]},1/μ is the radius of S2 which is parameterized by (θ,ϕ). gR×S22=R3/μ3n. y^  represents the vacuum:
(135)Φ^=μ2diag[α1,…,α1︸N1,α2,…,α2︸N2,…,αt,…,αt︸Nt],A^1=0,A^2={tanθ2Φ^inregionI-cotθ2Φ^inregionII.
Region I and II correspond to 0≤θ<(π/2)+ε and (π/2)-ε<θ≤π, respectively. αs is identified with the αs in (113). The U(N-) gauge group is spontaneously broken to U(N1)×U(N2)×⋯×U(Nt). Equation (135) corresponds to t disks located at η-1=(π/2)α1,η-2=(π/2)α2,…,η-t=(π/2)αt with the electric charges on each equal to (π2/8)N1,(π2/8)N2,…,(π2/8)Nt. The previous η-s=(π/2)(n+αs) in PWMM now becomes η-s=(π/2)αs due to the η→η+n redefinition.The correspondence between the spacetime coordinates and theSYMR×S2 fields is Φ↔η, (X4,…,X9)↔(ρ,Ω5), (x+,Ω2)↔(x+,Ω2). Equation (134) is the SYMR×S2 on flat background. Plug (133) into (134), similar to (127), for bosonic part, we have
(136)SbR×S2=μ3∫dx+dΩ2μ2ppppp×Tr{-Geϕpppppppppp×(-14GacGbdFabFcdppppppppppppp-12GabGηηDaΦDbΦ-μ22GΩ2GηηΦ2ppppppppppppp+μ(-G)-1/2Gηη1/2F12Φppppppppppppp-12GabGmnDaXmDbXnppppppppppppp-μ28GΩ2GmnXmXnppppppppppppp+14GmpGnq[Xm,Xn][Xp,Xq]pppppppppppip+12GηηGmn[Φ,Xm][Φ,Xn]-Geϕ)},
where Gab~Gmn~Gηη~GΩ2~(∂ρ22u/n)1/2, G=detGab~-(∂ρ22u/n)-3/2, eϕ~α2(∂ρ22u/n)1/4. The action remains the same, so (134) is also the action of SYMR×S2 on background (133). The conclusion holds for fermionic part except for a rescaling of λ.Indeed, as we will show in AppendixA, such phenomenon is very common. Generically, the action of SYMp+1 on flat background is equal to the action of the SYMp+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
(
ρ
2
,
η
) is the function of the radial directions. For the given field configuration [Φ,X4,…,X9], let
(137)[Φ^,X^4,…,X^9]=[(∂ρ22un)1/4Φ,(∂ρ22un)1/4X4,…,(∂ρ22un)1/4X9];
then
(138)SbR×S2=μ3∫dx+dΩ2μ2ppppp×Tr{-Geϕppppppppppp×(-14GacGbdFabFcdppppppppppppp-12GabDaΦ^DbΦ^-μ22GΩ2Φ^2ppppppppppppp+μ(-G)-1/2F12Φ^-12GabDaX^mDbX^nppppppppppppp-μ28GΩ2X^mX^nppppppppppppp+14[X^m,X^n][X^p,X^q]ppppppppppipp+12[Φ^,X^m][Φ^,X^n])-Geϕ}.
Compared with (134), with Xm→X^m, Φ→Φ^, the 3d background fields on R×S2 will get the radial dependence:
(139)eϕ=1⟶eϕ=α22(∂ρ22un)1/4(ρ2,η),ds32=-4dx+dx++dΩ22⟶ds32=(∂ρ22un)-1/2(ρ2,η)(-4dx+dx++dΩ22).
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; ds102 in (133) can be written as
(140)ds102=(∂ρ22un)-1/2(-4dx+dx++dΩ22)+4(∂ρ22un)1/2dr2+4r2(∂ρ22un)1/2dΩ62,
with SYMR×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 Xm→X^m, Φ→Φ^ transformation to recover ρη dependence of the 3d background fields in SYMR×S2.For PWMM, we have(141)eWPW(F)=∫[dη]e-SPW(Y^;η).WPW(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,
(142)SPW(Y^;η)⟶SR×S2(y^;y)WPW(F)⟶WR×S2(F),
where WR×S2(F) is the type IIA action for the above-mentioned supergravity solution dual to a vacuum of SYMR×S2. Then,
(143)eWR×S2(F)=∫[dy]e-SR×S2(y^;y).As is demonstrated in [19, 20], from SYMR×S2 and the corresponding type IIA solution, it is also possible to get the SYMR×S3 and the associated type IIB solution. In (135), suppose
(144)Φ^=μ2×diag[…,s-1,…,s-1︸M,s,…,s︸M,s+1,…,s+1︸M,…],
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]
(145)SR×S2(y^+y)|[y(s+1,t+1)=y(s,t)]=SR×S3(z)=1gR×S32∫dtdΩ3(μ/2)3ppppppppp×Tr{-14FabFab-12DaXmDaXm-112R^Xm2pppppppppppppp-i2λ-ΓaDaλ-12λ-Γm[Xm,λ]pppppppppppppp+14[Xm,Xn]2},
where
(146)gR×S32=4πμgR×S22.SR×S3 is the action of the U(M) SYM on R×S3.This is a special example of Taylor’s prescription for the compactification (theT-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 ofSYMR×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 SYMR×S2 in type IIA become SR×S3 in type IIB. Equation (143) turns into
(147)eWR×S3(F)=∫[dz]e-SR×S2(z).
For the trivial vacuum of SR×S3, F represents AdS5×S5 background.Finally, the geometry arising from the backreaction of theD3 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

SYM
4 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 Xi, i=1,…,9 representing nine transverse coordinates, so for a scalar field
(148)ϕ(t,x1,…,x9)=∑n1=0∞‍⋯∑n9=0∞‍1n1!⋯n9!pppppppppp×(∂1n1⋯∂9n9ϕ)(t,0,…,0)pppppppppp×(x1)n1⋯(x9)n9,
we may get the operator realization
(149)Φ(t)=∑n1=0∞‍⋯∑n9=0∞‍1n1!⋯n9!pppppppppp×(∂1n1⋯∂9n9ϕ)(t,0,…,0)pppppppppp×STr[(X1)n1⋯(X9)n9],
with xi replaced by matrix Xi. 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,
(150)h(x,y)=∑n=2∞‍hn(x,|y|)Yn(y|y|),Yn(y|y|)=Ca1⋯an(ya1⋯yan)|y|n,whereYn(y/|y|) is a spherical harmonic of rank n. hn(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 Yn(y/|y|). In SYM4, we have Ca1⋯anTr(Xa1⋯Xan), which, however, is the operator realization of Ca1⋯an(ya1⋯yan).Naively, letR2=e2Σ=XmXm, Σ=(1/2)ln(XmXm); R and Σ may be the operators corresponding to r=|y|=eσ and σ, respectively. For the 5d field F(x,r), for all σ0=lnr0, formally,
(151)F(x,R)=f(x,Σ)=f(0)(x,σ0)+f(1)(x,σ0)(Σ-σ0I)+12f(2)(x,σ0)(Σ-σ0I)2+⋯=f(0)(x,σ0)+f(1)(x,σ0)[12ln(XmXmr02)]+12f(2)(x,σ0)[12ln(XmXmr02)]2+⋯.
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 Σs=Σ-σI:
(152)σ=1NTrΣ=12NTrln(XmXm)=12Nlndet(XmXm),Σs=12ln(XsmXsm),det(XsmXsm)=1,Xm=eσXsm. The configuration (X5,…,X10) is equivalent to (Xs5,…,Xs10,σ). σ→σ+a, Xm→eaXm.One may takeCa1⋯anTr(Xsa1⋯Xsan) as the operator corresponding to Yn(y/|y|) and consider the gauge theory with the vertex operator perturbation realized as (in fact, we will choose X^ instead of Xs with det(X^mX^m)=∞. Ca1⋯anTr(X^a1⋯X^an)~Yn(y^), where |y^|=∞)(153)Hn(x,σ)=hn(x,σ)Ca1⋯anTr(Xsa1⋯Xsan);hn(x,σ) can be the arbitrary 5d function. Gauge theory like this is difficult to approach directly. We still prefer SYM4 with hn(x)Ca1⋯anTr(Xa1⋯Xan) added, which, after a suitable transition, will become a gauge theory with the operator hn(x,σ)Ca1⋯anTr(X^a1⋯X^an). The 5d field hn(x,σ) is not arbitrary anymore but is determined by hn(x).This is quite similar to the noncritical string coupling with2d 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 with2d 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
(154)S(g,X)=14πα′∫d2ξg[gab∂aXμ∂bXνGμν(X)ppppppppppppppp+ϵab∂aXμ∂bXνBμν(X)ppppppppppppppp+α′RΦ(X)+T(X)+⋯],μ,ν=1,…,d. The partition function is defined as
(155)Z=∫DggDgXvol(diff2)e-S(g,X).∫DggDgXvol(diff2)e-S(g,X)=∫Deωg(eωg)DeωgXvol(diff2)e-S(eωg,X).
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=eφg^, φ≥0, g^ is a small metric giving the cut-off scale
(156)Dggvol(diff2)=Dgφvol(conf2)e-SL(φ;g^),
where SL(φ;g^) is the Liouville action. With DgX→Dg^X, Dgφ→Dg^φ, the partition function becomes
(157)Z(g^,0)=∫φ=0∞DgφDgXvol(conf2)e-S(g,X)-SL(φ;g^)=∫φ=0∞Dg^φDg^Xvol(conf2)e-S^(φ,X;g^),
where
(158)S^(φ,X;g^)=14πα′∫d2ξg^[R^Φ^g^ab∂aφ∂bφppppppppppppppp+g^ab∂aXμ∂bXνG^μν(X,φ)ppppppppppppppp+ϵab∂aXμ∂bXνB^μν(X,φ)ppppppppppppppp+α′R^Φ^(X,φ)+T^(X,φ)+⋯].
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=-∞:
(159)Z(g^,0)=∫φ=0∞Dg^φDg^Xvol(conf2)e-S^(φ,X;g^)=∫φ=0∞Dg^(φ-∞)Dg^Xvol(conf2)e-S^(φ-∞+∞,X;g^)=∫φ=-∞∞Dg^φDg^Xvol(conf2)e-S^(φ+∞,X;g^)=∫φ=-∞∞Dg^φDg^Xvol(conf2)e-S^′(φ,X;g^).
In S^′, F^′(φ)=F(φ+∞).Sinceg=eφ+ωe-ωg^, for ω≥0,
(160)Z(g^,0)=∫φ=0∞De-ωg^(φ+ω)De-ωg^Xvol(conf2)e-S^(φ+ω,X;e-ωg^)=∫φ=ω∞De-ωg^φDe-ωg^Xvol(conf2)e-S^(φ,X;e-ωg^)=Z(e-ωg^,ω).
With the cutoff being introduced, the conformal transformation should be accompanied by a change of the boundary:
(161)Z(e-ωg^,0)-Z(g^,0)=∫φ=0ωDe-ωg^φDe-ωg^Xvol(conf2)e-S^(φ,X;e-ωg^)=Z(e-ωg^,0)-Z(e-ωg^,ω).
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, considerSYM4 with the action
(162)I[g^μν,X,A,λ]=-1gYM2∫d4x-g^pppppppp×Tr(12g^μνg^λσFμλFνσppppppppppppp+g^μνDμXmDνXmppppppppppppp+16R^Xm2-iλ-ΓμDμλppppppppppppp-λ-Γm[Xm,λ]ppppppppppppp-12[Xm,Xn]2).
Select X^ so that X can be expressed as X=eσX^ with σ∈(-∞,0]. X^ is the infinite matrix representing the maximum X. The partition function is
(163)Z(g^μν,X^)=∫X^Dg^XDg^ADg^λexp{-I[g^μν,X,A,λ]}=eW(g^μν,X^).Due to the Weyl invariance,(164)I[g^μν,X,A,λ]=I[e2ωg^μν,e-ωX,A,e-(3/2)ωλ].
Suppose
(165)Dg^XDg^ADg^λ=De2ωg^(e-ωX)De2ωg^×ADe2ωg^(e-(3/2)ωλ)eSc[ω;g^],
with no counterterm added,  SYM4 is conformal invariant, so at least for finite ω and g^μν, Sc[ω;g^]=0. Z(g^μν;X^)=Z(e-2ωg^μν;eωX^). However, this may not be the case when ω=∞ or when g^μν→0. Sc[ω;g^] will still be kept for the infinite conformal transformation.Nevertheless, withg^μν=ημν, we always have Sc[σ;g^]=0. With X parameterized as X=eσX^,
(166)Z(g^μν;X^)=∫σ=-∞0Dg^(eσX^)Dg^ADg^λpppppp×exp{-I[g^μν,eσX^,A,λ]}(167)=∫σ=-∞0De2σg^X^De2σg^ADe2σg^λpppppp×exp{-I[e2σg^μν,X^,A,λ]}.
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:
(168)ds2=e2σdx42+r32(dσ2+dΩ52)=(rr3)2dx42+(r3r)2(dr2+r2dΩ52).r=eσr3. 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 (eσX^,g^μν)→(X^,e4σg^μν). The induced metric is
(169)ds2=e4σdx32+r22(dσ2+dΩ72)=(rr2)4dx32+(r2r)2(dr2+r2dΩ72)
with r=eσr2. For SCFT6, X has the weight 2, and the Weyl transformation is (eσX^,g^μν)→(X^,eσg^μν). The induced metric is
(170)ds2=eσdx62+r52(dσ2+dΩ42)=(rr5)dx62+(r5r)2(dr2+r2dΩ42),
with r=eσr5. Equations (169) and (170) are the near-horizon geometries of M2 and M5, respectively.More generically, as we will show in AppendixA, for SYMp+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 dx32→(r/r2)4dx32 and dx62→(r/r5)dx62. For Dp with the near-horizon geometry
(171)ds2=(rrp)(7-p)/2dxp+12+(rrp)-(7-p)/2(dr2+r2dΩ8-p2),eΦ=gs(rrp)(7-p)(p-3)/4,
the X→(rp/r)(7-p)/4X transformation will then make dxp+12→(r/rp)(7-p)/2dxp+12, eΦ=gs→eΦ=gs(r/rp)(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 forX. This is not always the case. Consider the 2dN=(4,4) SCFT and the AdS3×S3×T4 geometry [50, 51]. The scalars Xi, i=1,2,3,4 are related to the transverse T4, for which the metric is dx42. 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 toSYM4, with the operator perturbation added:
(172)I[gμν0,ϕ0,X,A,λ]=I[gμν0,X,A,λ]+∫d4x-gμν0ϕ0(x)O(x),
where O=Ca1⋯anTr(Xa1⋯Xan). gμν0 is the arbitrary 4d metric. The partition function is (here, the path integral measure depends on both gμν0 and ϕ0. However, except for gμν0, 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)(173)Z(gμν0,ϕ0;X^)=∫X^D(g0,ϕ0)XD(g0,ϕ0)AD(g0,ϕ0)λ×exp{-I[gμν0,ϕ0,X,A,λ]}.I[gμν0,ϕ0,X,A,λ]=I[e2ωgμν0,e(n-4)ωϕ0,e-ωX,A,e-(3/2)ωλ].
For finite ω, gμν0, ϕ0,
(174)Z(gμν0,ϕ0;X^)=Z(e2ωgμν0,e(n-4)ωϕ0;e-ωX^).We still want to do aX→X^ transformation:
(175)D(g0,ϕ0)(eσX^)D(g0,ϕ0)AD(g0,ϕ0)λ×exp{-I[gμν0,ϕ0,eσX^,A,λ]}=D(e2σg0,e(n-4)σϕ0)X^D(e2σg0,e(n-4)σϕ0)×AD(e2σg0,e(n-4)σϕ0)(e-(3/2)σλ)×exp{-I[e2σgμν0,e(n-4)σϕ0,X^,A,e-(3/2)σλ]ppppppppp+Sc(σ;gμν0,ϕ0)}.
When σ=-∞, Sc(σ;gμν0,ϕ0) may not be zero. Instead of e2σgμν0 and e(n-4)σϕ0, we can find the suitably adjusted gμν(σ) and ϕ(σ) so that
(176)D(g0,ϕ0)(eσX^)D(g0,ϕ0)AD(g0,ϕ0)λ×exp{-I[gμν0,ϕ0,eσX^,A,λ]}=D[g(σ),ϕ(σ)]X^D[g(σ),ϕ(σ)]AD[g(σ),ϕ(σ)]λ×exp{-I[gμν(σ),ϕ(σ),X^,A,λ]}.
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, gμν0=gμν(0), ϕ0=ϕ(0). We also require
(177)D[g(σ1),ϕ(σ1)](eσ2X^)D[g(σ1),ϕ(σ1)]AD[g(σ1),ϕ(σ1)]λ×exp{-I[gμν(σ1),ϕ(σ1),eσ2X^,A,λ]}=D[g(σ1+σ2),ϕ(σ1+σ2)]X^D[g(σ1+σ2),ϕ(σ1+σ2)]×AD[g(σ1+σ2),ϕ(σ1+σ2)]λ×exp{-I[gμν(σ1+σ2),ϕ(σ1+σ2),X^,A,λ]}.
The 5d functions gμν(x,σ) and ϕ(x,σ) are induced from the 4d fields gμν0(x) and ϕ(x). For finite σ, gμν(σ)=e2σgμν0, ϕ(σ)=e(n-4)σϕ0.More generically, with the4d background field F0 turned on, for the given 4d function σ(x), since σ≤0, eσX^→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. eσX^→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:
(178)Z(gμν0,F0;X^)=∫σ=-∞0D[g(σ),F(σ)]X^D[g(σ),F(σ)]AD[g(σ),F(σ)]λppppppp×exp{-I[gμν(σ),F(σ),X^,A,λ]}=Z^(gμν(σ),F(σ);X^;0).In (178), with F specified to ϕ, the induced action I[gμν(σ),F(σ),X^,A,λ] contains the term ∫‍d4x-g(σ)ϕ(x,σ)O^(x) with O^(x) given by O^=Ca1⋯anTr(X^a1⋯X^an). O^(x)~Ca1⋯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)Yn(Ω) and gμν(x,σ)=|y^|4gμν(x,σ), where Yn(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 gμν(x,σ) turned on.On gravity side, the asymptotic expansion of the gravity solutiong(σ) is [52]
(179)g(σ)=e2σ[g0+e-2σg2+e-4σg4-2σe-4σh4]+O(e-3σ),
where σ∈(-∞,+∞). Let σ→σ+σ∞, with σ∈(-∞,0], σ∞=+∞, and g(σ)=e-2σ∞g(σ).
(180)g(σ)=e2σ[g0+e-2(σ+σ∞)g2+e-4(σ+σ∞)g4ppppp-2(σ+σ∞)e-4(σ+σ∞)h4]+O(e-5σ∞e-3σ).
For finite σ, g(σ)=e2σg0. Similarly, for the solution of the scalar field Φ(σ) [53],
(181)Φ(σ)=e(n-4)σ[ϕ0+e-2σϕ2+⋯+e(4-2n)σϕ2n-4pppppppp-2σe(4-2n)σφ2n-4]+O[e-(n+1)σ]
with σ∈(-∞,+∞). ϕ(σ)=e(4-n)σ∞Φ(σ+σ∞).
(182)ϕ(σ)=e(n-4)σ[ϕ0+e-2(σ+σ∞)ϕ2+⋯pppppppp+e(4-2n)(σ+σ∞)ϕ2n-4pppppppp-2(σ+σ∞)e(4-2n)(σ+σ∞)φ2n-4]pppppppp+O[e(3-2n)σ∞e-(n+1)σ],
with σ∈(-∞,0]. When n>2, ϕ(σ)=e(n-4)σϕ0 for finite σ. For g-Φ coupled system, the subleading terms are determined by both g0 and ϕ0. Φ(x,σ) and gμν(x,σ) are fields on AdS5 with r3 finite. It is expected that the X→X^ transformation will give gμν(x,σ) and ϕ(x,σ) in (180) and (182).From (177),
(183)D[g(σ),ϕ(σ)]X^D[g(σ),ϕ(σ)]AD[g(σ),ϕ(σ)]λ×exp{-I[gμν(σ),ϕ(σ),X^,A,λ]}=D[g(σ+ω),ϕ(σ+ω)](e-ωX^)D[g(σ+ω),ϕ(σ+ω)]×AD[g(σ+ω),ϕ(σ+ω)]λ×exp{-I[gμν(σ+ω),ϕ(σ+ω),e-ωX^,A,λ]}.
As a result,
(184)Z(gμν0,ϕ0;X^)=Z^(gμν(σ),ϕ(σ);X^;0)=∫σ=-∞0D[g(σ+ω),ϕ(σ+ω)](e-ωX^)D[g(σ+ω),ϕ(σ+ω)]pppppppp×AD[g(σ+ω),ϕ(σ+ω)]λpppppppp×exp{-I[gμν(σ+ω),ϕ(σ+ω),e-ωX^,A,λ]}=Z^(gμν(σ+ω),ϕ(σ+ω);e-ωX^;0)=∫σ=-∞ωD[g(σ),ϕ(σ)](e-ωX^)D[g(σ),ϕ(σ)]AD[g(σ),ϕ(σ)]λppppppppp×exp{-I[gμν(σ),ϕ(σ),e-ωX^,A,λ]}=Z^(gμν(σ),ϕ(σ);e-ωX^;ω).
With X^→e-ωX^, the induced fields become gμν(σ+ω) and ϕ(σ+ω). Also, since
(185)Z(gμν0,ϕ0;X^)=Z(e2ωgμν0,e(n-4)ωϕ0;e-ωX^),
we have
(186)Z^(gμν(σ),ϕ(σ);X^;0)=Z^(gμνω(σ),ϕω(σ);e-ωX^;0),
where gμνω(σ) and ϕω(σ) are fields induced from e2ωgμν0 and e(n-4)ωϕ0, respectively. There will be gμνω(σ)=gμν(σ+ω), ϕω(σ)=ϕ(σ+ω). Indeed, in (182) and (180), with ϕ0 and g0 replaced by e(n-4)ωϕ0 and e2ωg0, one may get ϕ(σ+ω) and g(σ+ω). g0→e2ωg0, ϕ0→e(n-4)ωϕ0, σ→σ+ω is a diffeomorphism transformation of the gravity solution gμν(σ) and Φ(σ) [54].Equation (184) is valid for [gμν(x,σ),ϕ(x,σ)] induced from [gμν0(x),ϕ0(x)]. For the generic 5d functions [g~μν(x,σ),ϕ~(x,σ)], we only have
(187)Z^(g~μν(x,σ),ϕ~(x,σ);X^;0)=Z^(g~μνω(x,σ),ϕ~ω(x,σ);e-ωX^;0)
for some 5d functions [g~μνω(x,σ),ϕ~ω(x,σ)]. For constant ω, under the scale transformation x→eωx,
(188)Z^(g~μν(x,σ),ϕ~(x,σ);X^;0)=Z^(e-2ωg~μνω(eωx,σ),ϕ~ω(eωx,σ);e-ωX^;0).
In special cases, if e-2ωg~μνω(σ)=g~μν(σ), ϕ~ω(σ)=ϕ~(σ), the background is “conformal”. For [gμν(x,σ),ϕ(x,σ)], this requires gμν(σ+ω)=e2ωgμν(σ), ϕ(σ+ω)=ϕ(σ) and thus gμν(σ)=e2σgμν0, ϕ(σ)=ϕ0. ϕ is the marginal field.For fields which are “conformal”,(189)Z^(g~μν(x,σ),ϕ~(x,σ);X^;0)=Z^(g~μν(eωx,σ),ϕ~(eωx,σ);e-ωX^;0),
and so
(190)δZ^(g~μν(x,σ),ϕ~(x,σ);X^;0)δϕ~(x0,σ0)=δZ^(g~μν(eωx,σ),ϕ~(eωx,σ);e-ωX^;0)δϕ~(eωx0,σ0).
As a result,
(191)〈-g~(x0,σ0)O^(x0)δ(σ-σ0)〉=∫σ=-∞0D[g~(σ),ϕ~(σ)]X^D[g~(σ),ϕ~(σ)]AD[g~(σ),ϕ~(σ)]λppppppppp×exp{-I[g~μν(x,σ),ϕ~(x,σ),X^,A,λ]}ppppppppp×-g~(x0,σ0)O^(x0)δ(σ-σ0)=〈e4ω-g~(eωx0,σ0)O^ω(eωx0)δ(σ-σ0)〉ω=∫σ=-∞0D[g~(σ),ϕ~(σ)](e-ωX^)D[g~(σ),ϕ~(σ)]AD[g~(σ),ϕ~(σ)]λppppppppp×exp{-I[g~μν(eωx,σ),ϕ~(eωx,σ),e-ωX^,A,λ]}ppppppppp×e(4-n)ω-g~(eωx0,σ0)O^(eωx0)δ(σ-σ0)=∫σ=-∞0D[g~(σ),ϕ~(σ)]X^D[g~(σ),ϕ~(σ)]AD[g~(σ),ϕ~(σ)]λppppppppp×exp{-I[g~μν(x,σ),ϕ~(x,σ),X^,A,λ]}ppppppppp×e(4-n)ω-g~(eωx0,σ0)O^(eωx0)δ(σ-σ0)=e(4-n)ω〈-g~(eωx0,σ0)O^(eωx0)δ(σ-σ0)〉,
and then
(192)0=〈-g~(x0,σ0)O^(x0)δ(σ-σ0)〉=δZ^(g~μν(x,σ),ϕ~(x,σ);X^;0)δϕ~(x0,σ0).ϕ~(x,σ) is on shell.Restricted to[gμν(x,σ),ϕ(x,σ)], usually, [e-2ωgμν(σ+ω),ϕ(σ+ω)]≠[gμν(σ),ϕ(σ)], [gμν(σ),ϕ(σ)] is not “conformal”. Instead of (191), if we require
(193)〈-g(x0,σ0)O^(x0)δ(σ-σ0)〉=∫σ=-∞0D[g(σ),ϕ(σ)]X^D[g(σ),ϕ(σ)]AD[g(σ),ϕ(σ)]λppppppppp×exp{-I[gμν(x,σ),ϕ(x,σ),X^,A,λ]}ppppppppp×-g(x0,σ0)O^(x0)δ(σ-σ0)=〈e(4-n)ω-g(x0,σ0)O^(x0)eip(σ+ω)〉ω=∫σ=-∞0D[g(σ+ω),ϕ(σ+ω)](e-ωX^)D[g(σ+ω),ϕ(σ+ω)]ppppppppp×AD[g(σ+ω),ϕ(σ+ω)]λppppppppp×exp{-I[gμν(σ+ω),ϕ(σ+ω),e-ωX^,A,λ]}ppppppppp×e(4-n)ω-g(x0,σ0)O^(x0)eip(σ+ω)=∫σ=-∞0D[g(σ),ϕ(σ)]X^D[g(σ),ϕ(σ)]AD[g(σ),ϕ(σ)]λppppppppp×exp{-I[gμν(σ),ϕ(σ),X^,A,λ]}ppppppppp×e(4-n)ω-g(x0,σ0)O^(x0)eip(σ+ω)=e(ip-n+4)ω〈-g(x0,σ0)O^(x0)eipσ〉;
still,
(194)0=〈-g(x0,σ0)O^(x0)eipσ〉=〈-g(x0,σ0)O^(x0)δ(σ-σ0)〉=δZ^(gμν(x,σ),ϕ(x,σ);X^;0)δϕ(x0,σ0).ϕ(x,σ) is also on shell.ϕ
(
x
,
σ
) is the functional of ϕ0(x). In general,
(195)δZ^(gμν(x,σ),ϕ(x,σ);X^;0)δϕ0(x0)=δZ(gμν0(x),ϕ0(x);X^)δϕ0(x0)≠0
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
(196)δZ^(gμν(x,σ),ϕ(x,σ);X^;0)δΦ(x0,σ0)=0.The difference betweenI[gμν0(x),ϕ0(x),X,A,λ] and I[gμν(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[gμν(x,σ),ϕ(x,σ),X^,A,λ] has the natural interpretation as a gauge theory on AdS5×S5 with the external field
(197)-g(x,σ)F(x,σ,Ω)=-g(x,σ)F(x,σ)Y^n=enσ∞-g(x,σ)F(x,σ)Yn=-g(x,σ)F(x,σ)Yn
turned on, where F(x,σ) and gμν(x,σ) can be the arbitrary 5d functions. gμν=e4σ∞gμν and F=e(n-4)σ∞F are physical fields on AdS5. According to the previous philosophy,
(198)Z^(gμν(x,σ),ϕ(x,σ);X^;0)=exp{Sg[gμν(x,σ),ϕ(x,σ);X^;0]},Sg[gμν(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),
(199)δZ^(gμν(x,σ),ϕ(x,σ);X^;0)δϕ(x0,σ0)=0,
and then
(200)δSg[gμν(x,σ),ϕ(x,σ);X^;0]δΦ(x0,σ0)=0;
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 δSg/δΦ(x0,σ0)=0, imposing limσ→0e(4-n)σ∞Φ(x,σ)=ϕ0(x) as the boundary condition.Finally, we arrive at(201)W(gμν0,ϕ0;X^)=lnZ(gμν0,ϕ0;X^)=Sg[gμν(x,σ),ϕ(x,σ);X^;0].
Starting from I[gμν0(x),ϕ0(x),X,A,λ], SYM4 with the source term added, after a X→X^ transformation, we get I[gμν(x,σ),ϕ(x,σ),X^,A,λ], the action of the gauge theory on AdS5×S5, whose free energy may be equal to Sg[gμν(x,σ),ϕ(x,σ);X^;0]. During the transition, F(x,σ) are carefully adjusted to make the partition function of I[gμν0(x),ϕ0(x),X,A,λ] and I[gμν(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 Sg.


### 4.3. Imposing the Cutoff

(202)
S
g
[
g
μ
ν
(
x
,
σ
)
,
ϕ
(
x
,
σ
)
;
X
^
;
0
]
=
∫
-
∞
0
d
σ
∫
d
4
x
L
[
g
μ
ν
(
x
,
σ
)
,
ϕ
(
x
,
σ
)
]
,
where  L is the Lagrangian of the supergravity. Sg is divergent. One may impose a cutoff σ-, σ-=-∞,
(203)Sg[gμν(x,σ),ϕ(x,σ);X^;σ-]=∫-∞σ-dσ∫d4xL[gμν(x,σ),ϕ(x,σ)].
On gauge theory side, the one-loop effective action W(gμν0)=-lndet(D)/2 is also divergent. With a cutoff ϵbeing introduced [55],
(204)W(gμν0;ϵ)=12∫ϵ∞dρρTr(e-ρD)=∫-∞σ-dσTr(exp{-e-2(σ+σ∞)D})=∫-∞σ-dσTr(exp{-e-2σD})=W(gμν(0);σ-),
where e-2(σ+σ∞)=ρ, gμν(0)=e2σ∞gμν0, D is the Laplace operator with the metric gμν(0). Obviously, W(e2ωgμν0,e2ωϵ)=W(gμν0,ϵ)=W(e2ωgμν(0);σ--ω)=W(gμν(0);σ-). For small but finite ρ [56],
(205)Tr(e-ρD)=∑n=0∞‍A2nρn-2,A2n=∫‍d4x-ga2n(g),
so
(206)W(gμν0)=limϵ→0W(gμν0,ϵ)=limϵ→0[A04ϵ2+A22ϵ-A4lnϵ2]+Wren(gμν0).
The renormalized Wren(gμν0) is finite. This is the realization of the holographic renormalization [52] on gauge theory side. With gμν0→e2ωgμν0, ϵ→e2ωϵ, [52]
(207)Wren(e2ωgμν0)-Wren(gμν0)=ωA4.A4 is the conformal anomaly [57]. On the other hand,
(208)W(gμν(0);σ-)=A04e-4σ-+A22e-2σ-+A4σ-+Wnonloc(gμν(0)),
so
(209)W(gμν(0);0)=A04+A22+Wnonloc(gμν(0))=Wloc(gμν(0))+Wnonloc(gμν(0)),Wnonloc(gμν(0))-Wren(gμν0)=σ∞A4.Wloc(gμν(0))=A0/4+A2/2 and Wnonloc(gμν(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(210)G(g0,ρ)=g0ρ+g2+ρ(g4+lnρh4)+O(ρ2).
At ρ=ϵ, the metric is G(g0,ϵ). The on-shell gravity action with the cutoff ρ=ϵ is
(211)S(gμν0,ϵ)=S[G(g0,ϵ)]=∫ϵ∞‍dρ2ρL(ρ),
which is entirely determined by the boundary value G(g0,ϵ). 								(212)S(gμν0)=limϵ→0S(gμν0,ϵ)=limϵ→0{a0(g)ϵ2l16πGN∫d4x-gppppppppppppp×[a0(g)ϵ2+a2(g)ϵppppppppppppppp-a4(g)lnϵa0(g)ϵ2]}+Sren(gμν0).
The renormalized Sren(gμν0) is finite [52]. Since G(e2ωg0,e2ωρ)=G(g0,ρ), G(e2ωg0,e2ωϵ)=G(g0,ϵ), S(e2ωgμν0,e2ωϵ)=S(gμν0,ϵ) [52]
(213)Sren(e2ωgμν0)-Sren(gμν0)=l8πGN∫d4x-gωa4(g)=ωA4.
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.TakeX-=eσ-X^ other than X^ as the upper bound of X; the partition function is
(214)Z(gμν0;X-)=∫X-Dg0XDg0ADg0λexp{-I[gμν0,X,A,λ]}=Z^(gμν(σ);X^;σ-).
For finite gμν0 and ω, Z(gμν0;eωX-)=Z(e2ωgμν0;X-). There is also
(215)Z(gμν0;eωX-)=Z^(gμν(σ);X^;σ-+ω)=Z^(gμν(σ+ω);X^;σ-)=Z(e2ωgμν0;X-),
since the field induced from e2ωgμν0 is gμν(σ+ω). From Z(e2ωgμν0;X-) and Z(gμν0;eωX-), the induced fields have the same boundary value gμν(σ-+ω).For the infinitesimal4d function δσ(x),
(216)Z(gμν0;eδσX-)=∫eδσX-Dg0XDg0ADg0λexp{-I[gμν0,X,A,λ]}=exp{∫d4xL(g0;X-)a(x)δσ(x)}Z(gμν0;X-),
where
(217)L(g0;X-)a(x)=limδσ→0[∫eδσ(x)X-Dg0XDg0ADg0λ1δσ(x)ppppppp×(∫eδσ(x)X-Dg0XDg0ADg0λpppppppppppppp×exp{-I[gμν0,X,A,λ]}pppppppppppppp-Z(gμν0;X-)∫eδσ(x)X-Dg0XDg0ADg0λ)]×Z-1(gμν0;X-).
With Sa(g0;X-)=∫‍d4xL(g0;X-)a(x), we may have
(218)W(gμν0)=∫0∞dσSa(g0;eσX-)+W(gμν0;X-)=∫-∞∞dσSa(g0;eσX-)=∫0∞‍dρρSa(g0;X-ρ),
where ρ=e-σ(219)W(gμν0;X-)=∫-∞0dσSa(g0;eσX-)=∫1∞dρρSa(g0;X-ρ)(220)W(gμν0;eδσX-)-W(gμν0;X-)=∫d4xL(g0;X-)a(x)δσ(x).
Also, we may define L(g0;X-)b, with
(221)W(e2δσgμν0;X-)-W(gμν0;X-)=∫d4xL(g0;X-)b(x)δσ(x)if(222)W(e2δσgμν0;X-)-W(gμν0;eδσX-)=∫‍d4xL(g0;X-)c(x)δσ(x),L(g0;X-)c(x)=L(g0;X-)b(x)-L(g0;X-)a(x).
For finite δσ and gμν0, L(g0;X-)c(x)=0. For infinite δσ, for example, W(gμν0;eωX^) and W(e2ωgμν0;X^) with ω=-∞,
(223)W(gμν0;eωX^)=lnZ^(gμν(σ+ω);X^;0),W(e2ωgμν0;X^)=lnZ^(gμνω(σ);X^;0).gμν(ω)≠e2ωgμν0, so gμνω(σ)≠gμν(σ+ω).Although (219) and (204) look similar, the two kinds of cutoffs are different. In particular,
(224)lnZ(gμν0;X-)=lnZ^(gμν(σ);X^;σ-)≠Sg[gμν(x,σ);X^;σ-]=∫-∞σ-dσ∫d4xL[gμν(x,σ)]
because the Z(gμν0;X^)-Z(gμν0;X-) part also has the contribution to Sg[gμν(x,σ);X^;σ-].Let(225)Sb(g0;X-)=∫d4xL(g0;X-)b(x),Sc(g0;X-)=∫d4xL(g0;X-)c(x);
then, Sc(g0;X-)=Sb(g0;X-)-Sa(g0;X-). From (206) or (212), one can see
(226)ϵ∂∂ϵW(gμν0,ϵ)=-A02ϵ2-A22ϵ-A42+O(ϵ)=-gμν0∂∂gμν0W(gμν0,ϵ).
For finite X- and ϵ, (226) cannot be directly identified with Sa and Sb. However, when X-→X^, ϵ→0,
(227)Sb(g0;X^)=limϵ→0[A0ϵ2+A2ϵ+A4]=Sa(g0;X^).A0/ϵ2+A2/ϵ+A4 is the conformal anomaly, which, with the infinite part subtracted, becomes A4.In the previous subsection, the radial dependent functionF(σ) is induced via (176). Nevertheless, it can also be induced from W(gμν0;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=eσX* with σ∈(-∞,+∞):
(228)W(gμν0;X)=W(gμν0;eσX*)=lnZ(gμν0;eσX*).
Starting from σ=0, with σ increasing, W(gμν0;eσX*) will also increase, so gμν0 should change accordingly to make the partition function invariant. Namely, we have
(229)W(gμν0;X*)=W(gμν(σ);eσX*)
with gμν(0)=gμν0(230)δWδgμν(σ)δgμν(σ)δσ+δWδσ=0.
For finite σ, gμν(σ)=e-2σgμν0. When σ→∞, gμν(σ)→0, the simple scaling relation may not be valid. For finite gμν0 and ω, W(gμν0;e-ωX*)=W(e-2ωgμν0;X*). If
(231)W(e-2ωgμν0;X*)=W[gμν(ω+σ);eσX*]=W(gμν(ω);X*),
there will be
(232)W(gμν0;e-ωX*)=W[gμν(σ);eσ(e-ωX*)].
So, in (229), with gμν(0)=gμν0 fixed andX* replaced by e-ωX*, the induced gμν(σ) remains the same. Also, with X* fixed andgμν(0) replaced by gμν(ω), the induced field will be gμν(ω+σ). gμν(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 with2d gravity:
(233)Z(F;g^)=∫φ=0∞D(eφg^)φD(eφg^)Xvol(conf2)e-S(F,eφg^,X)-SL(φ,g^),
where F(X) is the d dimensional background field. g^ is the cutoff metric. With g^ replaced by g^ω=eωg^,
(234)Z(F;eωg^)=∫φ=0∞D(eφg^ω)φD(eφg^ω)Xvol(conf2)e-S(F,eφg^ω,X)-SL(φ,g^ω)=∫φ=ω∞D(eφg^)φD(eφg^)Xvol(conf2)e-S(F,eφg^,X)-SL(φ,g^).Z(F;eωg^)≠Z(F;g^). g^→eωg^  is a conformal transformation, which should be accompanied by F→F(ω) to make the partition function invariant. Z(F;g^)=Z(F(ω);eωg^). F(ω) is a function trajectory that does not depend on the g^ chosen. A further eφg^ω→g^ω transformation gives
(235)Z(F(ω);eωg^)=∫φ=0∞Dg^ωφDg^ωXvol(conf2)e-S^(φ,X,Fω(φ),g^ω),
in which Fω(φ)=F(φ+ω) is induced.Just as (224), lnZ(F(ω);eωg^) 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=eφg^. 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 gμν, 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 Wren=Sren, for which (see, e.g., [60])
(236)∫d4xg0ω(x)[-2g0μνδδg0μν+(Δi-4+β^i)F0iδδF0i]Wren=∫d4x[ω(x)A+∂μω(x)Zμ],
where A represents the local anomalies. For the renormalization scale μ,
(237)μ∂Wren∂μ+∫d4xg0ppppppp×[2g0μνδδg0μν-(Δi-4)F0iδδF0i]Wren=0,μ∂Wren∂μ+∫d4xg0[βiδWrenδF0i]=∫d4xA.β^i is the anomalous dimension. βi=β^iF0i is the β function.


## 4.1. Noncritical String Coupling with2d 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
(154)S(g,X)=14πα′∫d2ξg[gab∂aXμ∂bXνGμν(X)ppppppppppppppp+ϵab∂aXμ∂bXνBμν(X)ppppppppppppppp+α′RΦ(X)+T(X)+⋯],μ,ν=1,…,d. The partition function is defined as
(155)Z=∫DggDgXvol(diff2)e-S(g,X).∫DggDgXvol(diff2)e-S(g,X)=∫Deωg(eωg)DeωgXvol(diff2)e-S(eωg,X).
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=eφg^, φ≥0, g^ is a small metric giving the cut-off scale
(156)Dggvol(diff2)=Dgφvol(conf2)e-SL(φ;g^),
where SL(φ;g^) is the Liouville action. With DgX→Dg^X, Dgφ→Dg^φ, the partition function becomes
(157)Z(g^,0)=∫φ=0∞DgφDgXvol(conf2)e-S(g,X)-SL(φ;g^)=∫φ=0∞Dg^φDg^Xvol(conf2)e-S^(φ,X;g^),
where
(158)S^(φ,X;g^)=14πα′∫d2ξg^[R^Φ^g^ab∂aφ∂bφppppppppppppppp+g^ab∂aXμ∂bXνG^μν(X,φ)ppppppppppppppp+ϵab∂aXμ∂bXνB^μν(X,φ)ppppppppppppppp+α′R^Φ^(X,φ)+T^(X,φ)+⋯].
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=-∞:
(159)Z(g^,0)=∫φ=0∞Dg^φDg^Xvol(conf2)e-S^(φ,X;g^)=∫φ=0∞Dg^(φ-∞)Dg^Xvol(conf2)e-S^(φ-∞+∞,X;g^)=∫φ=-∞∞Dg^φDg^Xvol(conf2)e-S^(φ+∞,X;g^)=∫φ=-∞∞Dg^φDg^Xvol(conf2)e-S^′(φ,X;g^).
In S^′, F^′(φ)=F(φ+∞).Sinceg=eφ+ωe-ωg^, for ω≥0,
(160)Z(g^,0)=∫φ=0∞De-ωg^(φ+ω)De-ωg^Xvol(conf2)e-S^(φ+ω,X;e-ωg^)=∫φ=ω∞De-ωg^φDe-ωg^Xvol(conf2)e-S^(φ,X;e-ωg^)=Z(e-ωg^,ω).
With the cutoff being introduced, the conformal transformation should be accompanied by a change of the boundary:
(161)Z(e-ωg^,0)-Z(g^,0)=∫φ=0ωDe-ωg^φDe-ωg^Xvol(conf2)e-S^(φ,X;e-ωg^)=Z(e-ωg^,0)-Z(e-ωg^,ω).
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, considerSYM4 with the action
(162)I[g^μν,X,A,λ]=-1gYM2∫d4x-g^pppppppp×Tr(12g^μνg^λσFμλFνσppppppppppppp+g^μνDμXmDνXmppppppppppppp+16R^Xm2-iλ-ΓμDμλppppppppppppp-λ-Γm[Xm,λ]ppppppppppppp-12[Xm,Xn]2).
Select X^ so that X can be expressed as X=eσX^ with σ∈(-∞,0]. X^ is the infinite matrix representing the maximum X. The partition function is
(163)Z(g^μν,X^)=∫X^Dg^XDg^ADg^λexp{-I[g^μν,X,A,λ]}=eW(g^μν,X^).Due to the Weyl invariance,(164)I[g^μν,X,A,λ]=I[e2ωg^μν,e-ωX,A,e-(3/2)ωλ].
Suppose
(165)Dg^XDg^ADg^λ=De2ωg^(e-ωX)De2ωg^×ADe2ωg^(e-(3/2)ωλ)eSc[ω;g^],
with no counterterm added,  SYM4 is conformal invariant, so at least for finite ω and g^μν, Sc[ω;g^]=0. Z(g^μν;X^)=Z(e-2ωg^μν;eωX^). However, this may not be the case when ω=∞ or when g^μν→0. Sc[ω;g^] will still be kept for the infinite conformal transformation.Nevertheless, withg^μν=ημν, we always have Sc[σ;g^]=0. With X parameterized as X=eσX^,
(166)Z(g^μν;X^)=∫σ=-∞0Dg^(eσX^)Dg^ADg^λpppppp×exp{-I[g^μν,eσX^,A,λ]}(167)=∫σ=-∞0De2σg^X^De2σg^ADe2σg^λpppppp×exp{-I[e2σg^μν,X^,A,λ]}.
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:
(168)ds2=e2σdx42+r32(dσ2+dΩ52)=(rr3)2dx42+(r3r)2(dr2+r2dΩ52).r=eσr3. 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 (eσX^,g^μν)→(X^,e4σg^μν). The induced metric is
(169)ds2=e4σdx32+r22(dσ2+dΩ72)=(rr2)4dx32+(r2r)2(dr2+r2dΩ72)
with r=eσr2. For SCFT6, X has the weight 2, and the Weyl transformation is (eσX^,g^μν)→(X^,eσg^μν). The induced metric is
(170)ds2=eσdx62+r52(dσ2+dΩ42)=(rr5)dx62+(r5r)2(dr2+r2dΩ42),
with r=eσr5. Equations (169) and (170) are the near-horizon geometries of M2 and M5, respectively.More generically, as we will show in AppendixA, for SYMp+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 dx32→(r/r2)4dx32 and dx62→(r/r5)dx62. For Dp with the near-horizon geometry
(171)ds2=(rrp)(7-p)/2dxp+12+(rrp)-(7-p)/2(dr2+r2dΩ8-p2),eΦ=gs(rrp)(7-p)(p-3)/4,
the X→(rp/r)(7-p)/4X transformation will then make dxp+12→(r/rp)(7-p)/2dxp+12, eΦ=gs→eΦ=gs(r/rp)(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 forX. This is not always the case. Consider the 2dN=(4,4) SCFT and the AdS3×S3×T4 geometry [50, 51]. The scalars Xi, i=1,2,3,4 are related to the transverse T4, for which the metric is dx42. 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 toSYM4, with the operator perturbation added:
(172)I[gμν0,ϕ0,X,A,λ]=I[gμν0,X,A,λ]+∫d4x-gμν0ϕ0(x)O(x),
where O=Ca1⋯anTr(Xa1⋯Xan). gμν0 is the arbitrary 4d metric. The partition function is (here, the path integral measure depends on both gμν0 and ϕ0. However, except for gμν0, 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)(173)Z(gμν0,ϕ0;X^)=∫X^D(g0,ϕ0)XD(g0,ϕ0)AD(g0,ϕ0)λ×exp{-I[gμν0,ϕ0,X,A,λ]}.I[gμν0,ϕ0,X,A,λ]=I[e2ωgμν0,e(n-4)ωϕ0,e-ωX,A,e-(3/2)ωλ].
For finite ω, gμν0, ϕ0,
(174)Z(gμν0,ϕ0;X^)=Z(e2ωgμν0,e(n-4)ωϕ0;e-ωX^).We still want to do aX→X^ transformation:
(175)D(g0,ϕ0)(eσX^)D(g0,ϕ0)AD(g0,ϕ0)λ×exp{-I[gμν0,ϕ0,eσX^,A,λ]}=D(e2σg0,e(n-4)σϕ0)X^D(e2σg0,e(n-4)σϕ0)×AD(e2σg0,e(n-4)σϕ0)(e-(3/2)σλ)×exp{-I[e2σgμν0,e(n-4)σϕ0,X^,A,e-(3/2)σλ]ppppppppp+Sc(σ;gμν0,ϕ0)}.
When σ=-∞, Sc(σ;gμν0,ϕ0) may not be zero. Instead of e2σgμν0 and e(n-4)σϕ0, we can find the suitably adjusted gμν(σ) and ϕ(σ) so that
(176)D(g0,ϕ0)(eσX^)D(g0,ϕ0)AD(g0,ϕ0)λ×exp{-I[gμν0,ϕ0,eσX^,A,λ]}=D[g(σ),ϕ(σ)]X^D[g(σ),ϕ(σ)]AD[g(σ),ϕ(σ)]λ×exp{-I[gμν(σ),ϕ(σ),X^,A,λ]}.
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, gμν0=gμν(0), ϕ0=ϕ(0). We also require
(177)D[g(σ1),ϕ(σ1)](eσ2X^)D[g(σ1),ϕ(σ1)]AD[g(σ1),ϕ(σ1)]λ×exp{-I[gμν(σ1),ϕ(σ1),eσ2X^,A,λ]}=D[g(σ1+σ2),ϕ(σ1+σ2)]X^D[g(σ1+σ2),ϕ(σ1+σ2)]×AD[g(σ1+σ2),ϕ(σ1+σ2)]λ×exp{-I[gμν(σ1+σ2),ϕ(σ1+σ2),X^,A,λ]}.
The 5d functions gμν(x,σ) and ϕ(x,σ) are induced from the 4d fields gμν0(x) and ϕ(x). For finite σ, gμν(σ)=e2σgμν0, ϕ(σ)=e(n-4)σϕ0.More generically, with the4d background field F0 turned on, for the given 4d function σ(x), since σ≤0, eσX^→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. eσX^→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:
(178)Z(gμν0,F0;X^)=∫σ=-∞0D[g(σ),F(σ)]X^D[g(σ),F(σ)]AD[g(σ),F(σ)]λppppppp×exp{-I[gμν(σ),F(σ),X^,A,λ]}=Z^(gμν(σ),F(σ);X^;0).In (178), with F specified to ϕ, the induced action I[gμν(σ),F(σ),X^,A,λ] contains the term ∫‍d4x-g(σ)ϕ(x,σ)O^(x) with O^(x) given by O^=Ca1⋯anTr(X^a1⋯X^an). O^(x)~Ca1⋯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)Yn(Ω) and gμν(x,σ)=|y^|4gμν(x,σ), where Yn(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 gμν(x,σ) turned on.On gravity side, the asymptotic expansion of the gravity solutiong(σ) is [52]
(179)g(σ)=e2σ[g0+e-2σg2+e-4σg4-2σe-4σh4]+O(e-3σ),
where σ∈(-∞,+∞). Let σ→σ+σ∞, with σ∈(-∞,0], σ∞=+∞, and g(σ)=e-2σ∞g(σ).
(180)g(σ)=e2σ[g0+e-2(σ+σ∞)g2+e-4(σ+σ∞)g4ppppp-2(σ+σ∞)e-4(σ+σ∞)h4]+O(e-5σ∞e-3σ).
For finite σ, g(σ)=e2σg0. Similarly, for the solution of the scalar field Φ(σ) [53],
(181)Φ(σ)=e(n-4)σ[ϕ0+e-2σϕ2+⋯+e(4-2n)σϕ2n-4pppppppp-2σe(4-2n)σφ2n-4]+O[e-(n+1)σ]
with σ∈(-∞,+∞). ϕ(σ)=e(4-n)σ∞Φ(σ+σ∞).
(182)ϕ(σ)=e(n-4)σ[ϕ0+e-2(σ+σ∞)ϕ2+⋯pppppppp+e(4-2n)(σ+σ∞)ϕ2n-4pppppppp-2(σ+σ∞)e(4-2n)(σ+σ∞)φ2n-4]pppppppp+O[e(3-2n)σ∞e-(n+1)σ],
with σ∈(-∞,0]. When n>2, ϕ(σ)=e(n-4)σϕ0 for finite σ. For g-Φ coupled system, the subleading terms are determined by both g0 and ϕ0. Φ(x,σ) and gμν(x,σ) are fields on AdS5 with r3 finite. It is expected that the X→X^ transformation will give gμν(x,σ) and ϕ(x,σ) in (180) and (182).From (177),
(183)D[g(σ),ϕ(σ)]X^D[g(σ),ϕ(σ)]AD[g(σ),ϕ(σ)]λ×exp{-I[gμν(σ),ϕ(σ),X^,A,λ]}=D[g(σ+ω),ϕ(σ+ω)](e-ωX^)D[g(σ+ω),ϕ(σ+ω)]×AD[g(σ+ω),ϕ(σ+ω)]λ×exp{-I[gμν(σ+ω),ϕ(σ+ω),e-ωX^,A,λ]}.
As a result,
(184)Z(gμν0,ϕ0;X^)=Z^(gμν(σ),ϕ(σ);X^;0)=∫σ=-∞0D[g(σ+ω),ϕ(σ+ω)](e-ωX^)D[g(σ+ω),ϕ(σ+ω)]pppppppp×AD[g(σ+ω),ϕ(σ+ω)]λpppppppp×exp{-I[gμν(σ+ω),ϕ(σ+ω),e-ωX^,A,λ]}=Z^(gμν(σ+ω),ϕ(σ+ω);e-ωX^;0)=∫σ=-∞ωD[g(σ),ϕ(σ)](e-ωX^)D[g(σ),ϕ(σ)]AD[g(σ),ϕ(σ)]λppppppppp×exp{-I[gμν(σ),ϕ(σ),e-ωX^,A,λ]}=Z^(gμν(σ),ϕ(σ);e-ωX^;ω).
With X^→e-ωX^, the induced fields become gμν(σ+ω) and ϕ(σ+ω). Also, since
(185)Z(gμν0,ϕ0;X^)=Z(e2ωgμν0,e(n-4)ωϕ0;e-ωX^),
we have
(186)Z^(gμν(σ),ϕ(σ);X^;0)=Z^(gμνω(σ),ϕω(σ);e-ωX^;0),
where gμνω(σ) and ϕω(σ) are fields induced from e2ωgμν0 and e(n-4)ωϕ0, respectively. There will be gμνω(σ)=gμν(σ+ω), ϕω(σ)=ϕ(σ+ω). Indeed, in (182) and (180), with ϕ0 and g0 replaced by e(n-4)ωϕ0 and e2ωg0, one may get ϕ(σ+ω) and g(σ+ω). g0→e2ωg0, ϕ0→e(n-4)ωϕ0, σ→σ+ω is a diffeomorphism transformation of the gravity solution gμν(σ) and Φ(σ) [54].Equation (184) is valid for [gμν(x,σ),ϕ(x,σ)] induced from [gμν0(x),ϕ0(x)]. For the generic 5d functions [g~μν(x,σ),ϕ~(x,σ)], we only have
(187)Z^(g~μν(x,σ),ϕ~(x,σ);X^;0)=Z^(g~μνω(x,σ),ϕ~ω(x,σ);e-ωX^;0)
for some 5d functions [g~μνω(x,σ),ϕ~ω(x,σ)]. For constant ω, under the scale transformation x→eωx,
(188)Z^(g~μν(x,σ),ϕ~(x,σ);X^;0)=Z^(e-2ωg~μνω(eωx,σ),ϕ~ω(eωx,σ);e-ωX^;0).
In special cases, if e-2ωg~μνω(σ)=g~μν(σ), ϕ~ω(σ)=ϕ~(σ), the background is “conformal”. For [gμν(x,σ),ϕ(x,σ)], this requires gμν(σ+ω)=e2ωgμν(σ), ϕ(σ+ω)=ϕ(σ) and thus gμν(σ)=e2σgμν0, ϕ(σ)=ϕ0. ϕ is the marginal field.For fields which are “conformal”,(189)Z^(g~μν(x,σ),ϕ~(x,σ);X^;0)=Z^(g~μν(eωx,σ),ϕ~(eωx,σ);e-ωX^;0),
and so
(190)δZ^(g~μν(x,σ),ϕ~(x,σ);X^;0)δϕ~(x0,σ0)=δZ^(g~μν(eωx,σ),ϕ~(eωx,σ);e-ωX^;0)δϕ~(eωx0,σ0).
As a result,
(191)〈-g~(x0,σ0)O^(x0)δ(σ-σ0)〉=∫σ=-∞0D[g~(σ),ϕ~(σ)]X^D[g~(σ),ϕ~(σ)]AD[g~(σ),ϕ~(σ)]λppppppppp×exp{-I[g~μν(x,σ),ϕ~(x,σ),X^,A,λ]}ppppppppp×-g~(x0,σ0)O^(x0)δ(σ-σ0)=〈e4ω-g~(eωx0,σ0)O^ω(eωx0)δ(σ-σ0)〉ω=∫σ=-∞0D[g~(σ),ϕ~(σ)](e-ωX^)D[g~(σ),ϕ~(σ)]AD[g~(σ),ϕ~(σ)]λppppppppp×exp{-I[g~μν(eωx,σ),ϕ~(eωx,σ),e-ωX^,A,λ]}ppppppppp×e(4-n)ω-g~(eωx0,σ0)O^(eωx0)δ(σ-σ0)=∫σ=-∞0D[g~(σ),ϕ~(σ)]X^D[g~(σ),ϕ~(σ)]AD[g~(σ),ϕ~(σ)]λppppppppp×exp{-I[g~μν(x,σ),ϕ~(x,σ),X^,A,λ]}ppppppppp×e(4-n)ω-g~(eωx0,σ0)O^(eωx0)δ(σ-σ0)=e(4-n)ω〈-g~(eωx0,σ0)O^(eωx0)δ(σ-σ0)〉,
and then
(192)0=〈-g~(x0,σ0)O^(x0)δ(σ-σ0)〉=δZ^(g~μν(x,σ),ϕ~(x,σ);X^;0)δϕ~(x0,σ0).ϕ~(x,σ) is on shell.Restricted to[gμν(x,σ),ϕ(x,σ)], usually, [e-2ωgμν(σ+ω),ϕ(σ+ω)]≠[gμν(σ),ϕ(σ)], [gμν(σ),ϕ(σ)] is not “conformal”. Instead of (191), if we require
(193)〈-g(x0,σ0)O^(x0)δ(σ-σ0)〉=∫σ=-∞0D[g(σ),ϕ(σ)]X^D[g(σ),ϕ(σ)]AD[g(σ),ϕ(σ)]λppppppppp×exp{-I[gμν(x,σ),ϕ(x,σ),X^,A,λ]}ppppppppp×-g(x0,σ0)O^(x0)δ(σ-σ0)=〈e(4-n)ω-g(x0,σ0)O^(x0)eip(σ+ω)〉ω=∫σ=-∞0D[g(σ+ω),ϕ(σ+ω)](e-ωX^)D[g(σ+ω),ϕ(σ+ω)]ppppppppp×AD[g(σ+ω),ϕ(σ+ω)]λppppppppp×exp{-I[gμν(σ+ω),ϕ(σ+ω),e-ωX^,A,λ]}ppppppppp×e(4-n)ω-g(x0,σ0)O^(x0)eip(σ+ω)=∫σ=-∞0D[g(σ),ϕ(σ)]X^D[g(σ),ϕ(σ)]AD[g(σ),ϕ(σ)]λppppppppp×exp{-I[gμν(σ),ϕ(σ),X^,A,λ]}ppppppppp×e(4-n)ω-g(x0,σ0)O^(x0)eip(σ+ω)=e(ip-n+4)ω〈-g(x0,σ0)O^(x0)eipσ〉;
still,
(194)0=〈-g(x0,σ0)O^(x0)eipσ〉=〈-g(x0,σ0)O^(x0)δ(σ-σ0)〉=δZ^(gμν(x,σ),ϕ(x,σ);X^;0)δϕ(x0,σ0).ϕ(x,σ) is also on shell.ϕ
(
x
,
σ
) is the functional of ϕ0(x). In general,
(195)δZ^(gμν(x,σ),ϕ(x,σ);X^;0)δϕ0(x0)=δZ(gμν0(x),ϕ0(x);X^)δϕ0(x0)≠0
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
(196)δZ^(gμν(x,σ),ϕ(x,σ);X^;0)δΦ(x0,σ0)=0.The difference betweenI[gμν0(x),ϕ0(x),X,A,λ] and I[gμν(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[gμν(x,σ),ϕ(x,σ),X^,A,λ] has the natural interpretation as a gauge theory on AdS5×S5 with the external field
(197)-g(x,σ)F(x,σ,Ω)=-g(x,σ)F(x,σ)Y^n=enσ∞-g(x,σ)F(x,σ)Yn=-g(x,σ)F(x,σ)Yn
turned on, where F(x,σ) and gμν(x,σ) can be the arbitrary 5d functions. gμν=e4σ∞gμν and F=e(n-4)σ∞F are physical fields on AdS5. According to the previous philosophy,
(198)Z^(gμν(x,σ),ϕ(x,σ);X^;0)=exp{Sg[gμν(x,σ),ϕ(x,σ);X^;0]},Sg[gμν(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),
(199)δZ^(gμν(x,σ),ϕ(x,σ);X^;0)δϕ(x0,σ0)=0,
and then
(200)δSg[gμν(x,σ),ϕ(x,σ);X^;0]δΦ(x0,σ0)=0;
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 δSg/δΦ(x0,σ0)=0, imposing limσ→0e(4-n)σ∞Φ(x,σ)=ϕ0(x) as the boundary condition.Finally, we arrive at(201)W(gμν0,ϕ0;X^)=lnZ(gμν0,ϕ0;X^)=Sg[gμν(x,σ),ϕ(x,σ);X^;0].
Starting from I[gμν0(x),ϕ0(x),X,A,λ], SYM4 with the source term added, after a X→X^ transformation, we get I[gμν(x,σ),ϕ(x,σ),X^,A,λ], the action of the gauge theory on AdS5×S5, whose free energy may be equal to Sg[gμν(x,σ),ϕ(x,σ);X^;0]. During the transition, F(x,σ) are carefully adjusted to make the partition function of I[gμν0(x),ϕ0(x),X,A,λ] and I[gμν(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 Sg.


## 4.3. Imposing the Cutoff

(202)
S
g
[
g
μ
ν
(
x
,
σ
)
,
ϕ
(
x
,
σ
)
;
X
^
;
0
]
=
∫
-
∞
0
d
σ
∫
d
4
x
L
[
g
μ
ν
(
x
,
σ
)
,
ϕ
(
x
,
σ
)
]
,
where  L is the Lagrangian of the supergravity. Sg is divergent. One may impose a cutoff σ-, σ-=-∞,
(203)Sg[gμν(x,σ),ϕ(x,σ);X^;σ-]=∫-∞σ-dσ∫d4xL[gμν(x,σ),ϕ(x,σ)].
On gauge theory side, the one-loop effective action W(gμν0)=-lndet(D)/2 is also divergent. With a cutoff ϵbeing introduced [55],
(204)W(gμν0;ϵ)=12∫ϵ∞dρρTr(e-ρD)=∫-∞σ-dσTr(exp{-e-2(σ+σ∞)D})=∫-∞σ-dσTr(exp{-e-2σD})=W(gμν(0);σ-),
where e-2(σ+σ∞)=ρ, gμν(0)=e2σ∞gμν0, D is the Laplace operator with the metric gμν(0). Obviously, W(e2ωgμν0,e2ωϵ)=W(gμν0,ϵ)=W(e2ωgμν(0);σ--ω)=W(gμν(0);σ-). For small but finite ρ [56],
(205)Tr(e-ρD)=∑n=0∞‍A2nρn-2,A2n=∫‍d4x-ga2n(g),
so
(206)W(gμν0)=limϵ→0W(gμν0,ϵ)=limϵ→0[A04ϵ2+A22ϵ-A4lnϵ2]+Wren(gμν0).
The renormalized Wren(gμν0) is finite. This is the realization of the holographic renormalization [52] on gauge theory side. With gμν0→e2ωgμν0, ϵ→e2ωϵ, [52]
(207)Wren(e2ωgμν0)-Wren(gμν0)=ωA4.A4 is the conformal anomaly [57]. On the other hand,
(208)W(gμν(0);σ-)=A04e-4σ-+A22e-2σ-+A4σ-+Wnonloc(gμν(0)),
so
(209)W(gμν(0);0)=A04+A22+Wnonloc(gμν(0))=Wloc(gμν(0))+Wnonloc(gμν(0)),Wnonloc(gμν(0))-Wren(gμν0)=σ∞A4.Wloc(gμν(0))=A0/4+A2/2 and Wnonloc(gμν(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(210)G(g0,ρ)=g0ρ+g2+ρ(g4+lnρh4)+O(ρ2).
At ρ=ϵ, the metric is G(g0,ϵ). The on-shell gravity action with the cutoff ρ=ϵ is
(211)S(gμν0,ϵ)=S[G(g0,ϵ)]=∫ϵ∞‍dρ2ρL(ρ),
which is entirely determined by the boundary value G(g0,ϵ). 								(212)S(gμν0)=limϵ→0S(gμν0,ϵ)=limϵ→0{a0(g)ϵ2l16πGN∫d4x-gppppppppppppp×[a0(g)ϵ2+a2(g)ϵppppppppppppppp-a4(g)lnϵa0(g)ϵ2]}+Sren(gμν0).
The renormalized Sren(gμν0) is finite [52]. Since G(e2ωg0,e2ωρ)=G(g0,ρ), G(e2ωg0,e2ωϵ)=G(g0,ϵ), S(e2ωgμν0,e2ωϵ)=S(gμν0,ϵ) [52]
(213)Sren(e2ωgμν0)-Sren(gμν0)=l8πGN∫d4x-gωa4(g)=ωA4.
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.TakeX-=eσ-X^ other than X^ as the upper bound of X; the partition function is
(214)Z(gμν0;X-)=∫X-Dg0XDg0ADg0λexp{-I[gμν0,X,A,λ]}=Z^(gμν(σ);X^;σ-).
For finite gμν0 and ω, Z(gμν0;eωX-)=Z(e2ωgμν0;X-). There is also
(215)Z(gμν0;eωX-)=Z^(gμν(σ);X^;σ-+ω)=Z^(gμν(σ+ω);X^;σ-)=Z(e2ωgμν0;X-),
since the field induced from e2ωgμν0 is gμν(σ+ω). From Z(e2ωgμν0;X-) and Z(gμν0;eωX-), the induced fields have the same boundary value gμν(σ-+ω).For the infinitesimal4d function δσ(x),
(216)Z(gμν0;eδσX-)=∫eδσX-Dg0XDg0ADg0λexp{-I[gμν0,X,A,λ]}=exp{∫d4xL(g0;X-)a(x)δσ(x)}Z(gμν0;X-),
where
(217)L(g0;X-)a(x)=limδσ→0[∫eδσ(x)X-Dg0XDg0ADg0λ1δσ(x)ppppppp×(∫eδσ(x)X-Dg0XDg0ADg0λpppppppppppppp×exp{-I[gμν0,X,A,λ]}pppppppppppppp-Z(gμν0;X-)∫eδσ(x)X-Dg0XDg0ADg0λ)]×Z-1(gμν0;X-).
With Sa(g0;X-)=∫‍d4xL(g0;X-)a(x), we may have
(218)W(gμν0)=∫0∞dσSa(g0;eσX-)+W(gμν0;X-)=∫-∞∞dσSa(g0;eσX-)=∫0∞‍dρρSa(g0;X-ρ),
where ρ=e-σ(219)W(gμν0;X-)=∫-∞0dσSa(g0;eσX-)=∫1∞dρρSa(g0;X-ρ)(220)W(gμν0;eδσX-)-W(gμν0;X-)=∫d4xL(g0;X-)a(x)δσ(x).
Also, we may define L(g0;X-)b, with
(221)W(e2δσgμν0;X-)-W(gμν0;X-)=∫d4xL(g0;X-)b(x)δσ(x)if(222)W(e2δσgμν0;X-)-W(gμν0;eδσX-)=∫‍d4xL(g0;X-)c(x)δσ(x),L(g0;X-)c(x)=L(g0;X-)b(x)-L(g0;X-)a(x).
For finite δσ and gμν0, L(g0;X-)c(x)=0. For infinite δσ, for example, W(gμν0;eωX^) and W(e2ωgμν0;X^) with ω=-∞,
(223)W(gμν0;eωX^)=lnZ^(gμν(σ+ω);X^;0),W(e2ωgμν0;X^)=lnZ^(gμνω(σ);X^;0).gμν(ω)≠e2ωgμν0, so gμνω(σ)≠gμν(σ+ω).Although (219) and (204) look similar, the two kinds of cutoffs are different. In particular,
(224)lnZ(gμν0;X-)=lnZ^(gμν(σ);X^;σ-)≠Sg[gμν(x,σ);X^;σ-]=∫-∞σ-dσ∫d4xL[gμν(x,σ)]
because the Z(gμν0;X^)-Z(gμν0;X-) part also has the contribution to Sg[gμν(x,σ);X^;σ-].Let(225)Sb(g0;X-)=∫d4xL(g0;X-)b(x),Sc(g0;X-)=∫d4xL(g0;X-)c(x);
then, Sc(g0;X-)=Sb(g0;X-)-Sa(g0;X-). From (206) or (212), one can see
(226)ϵ∂∂ϵW(gμν0,ϵ)=-A02ϵ2-A22ϵ-A42+O(ϵ)=-gμν0∂∂gμν0W(gμν0,ϵ).
For finite X- and ϵ, (226) cannot be directly identified with Sa and Sb. However, when X-→X^, ϵ→0,
(227)Sb(g0;X^)=limϵ→0[A0ϵ2+A2ϵ+A4]=Sa(g0;X^).A0/ϵ2+A2/ϵ+A4 is the conformal anomaly, which, with the infinite part subtracted, becomes A4.In the previous subsection, the radial dependent functionF(σ) is induced via (176). Nevertheless, it can also be induced from W(gμν0;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=eσX* with σ∈(-∞,+∞):
(228)W(gμν0;X)=W(gμν0;eσX*)=lnZ(gμν0;eσX*).
Starting from σ=0, with σ increasing, W(gμν0;eσX*) will also increase, so gμν0 should change accordingly to make the partition function invariant. Namely, we have
(229)W(gμν0;X*)=W(gμν(σ);eσX*)
with gμν(0)=gμν0(230)δWδgμν(σ)δgμν(σ)δσ+δWδσ=0.
For finite σ, gμν(σ)=e-2σgμν0. When σ→∞, gμν(σ)→0, the simple scaling relation may not be valid. For finite gμν0 and ω, W(gμν0;e-ωX*)=W(e-2ωgμν0;X*). If
(231)W(e-2ωgμν0;X*)=W[gμν(ω+σ);eσX*]=W(gμν(ω);X*),
there will be
(232)W(gμν0;e-ωX*)=W[gμν(σ);eσ(e-ωX*)].
So, in (229), with gμν(0)=gμν0 fixed andX* replaced by e-ωX*, the induced gμν(σ) remains the same. Also, with X* fixed andgμν(0) replaced by gμν(ω), the induced field will be gμν(ω+σ). gμν(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 with2d gravity:
(233)Z(F;g^)=∫φ=0∞D(eφg^)φD(eφg^)Xvol(conf2)e-S(F,eφg^,X)-SL(φ,g^),
where F(X) is the d dimensional background field. g^ is the cutoff metric. With g^ replaced by g^ω=eωg^,
(234)Z(F;eωg^)=∫φ=0∞D(eφg^ω)φD(eφg^ω)Xvol(conf2)e-S(F,eφg^ω,X)-SL(φ,g^ω)=∫φ=ω∞D(eφg^)φD(eφg^)Xvol(conf2)e-S(F,eφg^,X)-SL(φ,g^).Z(F;eωg^)≠Z(F;g^). g^→eωg^  is a conformal transformation, which should be accompanied by F→F(ω) to make the partition function invariant. Z(F;g^)=Z(F(ω);eωg^). F(ω) is a function trajectory that does not depend on the g^ chosen. A further eφg^ω→g^ω transformation gives
(235)Z(F(ω);eωg^)=∫φ=0∞Dg^ωφDg^ωXvol(conf2)e-S^(φ,X,Fω(φ),g^ω),
in which Fω(φ)=F(φ+ω) is induced.Just as (224), lnZ(F(ω);eωg^) 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=eφg^. 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 gμν, 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 Wren=Sren, for which (see, e.g., [60])
(236)∫d4xg0ω(x)[-2g0μνδδg0μν+(Δi-4+β^i)F0iδδF0i]Wren=∫d4x[ω(x)A+∂μω(x)Zμ],
where A represents the local anomalies. For the renormalization scale μ,
(237)μ∂Wren∂μ+∫d4xg0ppppppp×[2g0μνδδg0μν-(Δi-4)F0iδδF0i]Wren=0,μ∂Wren∂μ+∫d4xg0[βiδWrenδF0i]=∫d4xA.β^i is the anomalous dimension. βi=β^iF0i is the β function.


## 5. Conclusion

WithN→∞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
(238)x-=αx′-,x+=f(x′+),x=g(x′),
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 Seff(R,F), the effective action of the supergravity for the same field F. On field theory side, naively, Seff(F) is calculated from
(239)eSeff(F)=∫1PI[dF~]e-Scla(F+F~)
with Scla being the classical action of the 11d supergravity. eSeff(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. eW(F) is the sum of the membrane configurations on background F. In some sense, W(F) gives an M theory refined version of the Seff(F) in (239).In Matrix theory, the problem is that we only consider theM 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 Seff on the same 11d background, there is also Seff(R,F)=Seff(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 amongM theory objects with no light-cone momentum exchange, one may consider Γg(Y) given by
(240)eΓg(Y)=∫[dF]eSR,F(Y)-Scla(R,F),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 Γg(Y)=Seff(R,F)+SR,F(Y), where Seff is given by (239). δ[Seff(R,F)+SR,F(Y)]/δF=0. With Seff(R,F) being replaced by W(R,F), one can define another effective action Γ(Y)=W(R,F)+SR,F(Y), which is totally constructed from the Matrix theory with no input like Scla(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 Γeff(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], SYMR×S2, SYMR×S3/Zk, and SYMR×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 SYMR 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, SPW(Y+Y^) becomes the action of SYMR×S2, which, with the background (133) plugged in, remains invariant. With the SYMR×S2 and the gravity dual at hand, a further T-duality-like transformation gives SYMR×S3 and AdS5×S5 [19, 21].SYM
4 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 ημν 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 ημν, 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.



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*Source: 604637-2013-11-27.xml*