Pis'ma v ZhETF, vol.93, iss.9, pp.603-607
© 2011 May 10
Hints on integrability in the Wilsonian/holographic renormalization
group
E. T. Akhmedov*V1), I. B. Gahramanovt ^, E. T. Musaev*v ^ * Moscow Institute of Physics and Technology, Dolgoprudny, Moscow reg., Russia t National University of Science and Technology "MISIS", Moscow, Russia v Institute for Theoretical and Experimental Physics, 117218 Moscow, Russia Submitted 9 March 2011
The Polchinski equations for the Wilsonian renormalization group in the D-dimensional matrix scalar field theory can be written at large A" in a Hamiltonian form. The Hamiltonian defines evolution along one extra holographic dimension (energy scale) and can be found exactly for the subsector of Tr<pn (for all n) operators. We show that at low energies independently of the dimensionality D the Hamiltonian system in question reduces to the integrable effective theory. The obtained Hamiltonian system describes large wavelength KdV type (Burger-Hopf) equation with an external potential and is related to the effective theory obtained by Das and Jevicki for the matrix quantum mechanics.
Introduction. One of the greatest achievements of the modern fundamental physics is the holographic duality between D dimensioned gauge and (D + 1)-dimensional gravity theories. The seminal example of this duality is the equality between the quantum generating functional of the correlation functions for the flat space gauge theory and the classical wave functional for the AdS gravity theory [1]. The latter is known as AdS/CFT-correspondence.
The extra dimension in the gravity theory has a natural interpretation as the energy scale in the gauge theory [1] (see e.g. [2] for a review). Moreover, (D + 1)-dimensional gravity equations of motion can be related to the renormalization group (RG) equations on the gauge theory side [3] (see as well [4-11]).
To understand deeper such a holographic duality, we would like to address the following more general question: what kind of the (D + l)-dimensional theories govern RG flows of the large N D dimensioned field theories? In [12] matrix scalar field theory was considered. The Polchinski [13] equations for the Wilsonian RG for this theory were formulated in the subsector of the Tr [(f>n] (for all n) operators. At large N the Polchinski equations reduce to the Hamiltonian ones. The latter Hamiltonian system is rather artificial and contains non-local terms. However, in this note we show that in the infrared (IR) limit the RG dynamics is governed by the Hamiltonian2) [12]:
e-mail: akhmedov0itep.ru, ilmar.gh8gmail.com, musaevOitep.ru
Below in this paper we correct the important mistake made by two of us in the paper [12].
H = J+ da J dDx n2 J'. (1)
Here J' = dJ/da, J(T,a,x) = £kakJk(T,x), Jk(T,x) are sources for the operators Tr[0fe(ar)]; U(T,a,x) = Efc^(fe+1) nk(T,x), Uk(T,x) are vaguely speaking vacuum expectation values (VEV) of the operators Tr[(f>h(x)] (see below). The sources and VEVs are conjugate to each other via the Poisson bracket: {Tl(T,a,x), J(T,a',x')} oc 5(a - a')5(x - x'). On the D dimensioned matrix scalar field theory side this can be traced from the Legendre (functional Fourier) relation between the effective actions for the sources and VEVs [12]. The role of the time T for the Hamiltonian system in question is played by the energy scale in the scalar field theory. Note that we observe here the appearance of the one extra (on top of the energy scale) dimension a conjugate to the number k enumerating the operators Tr[^>fe(a;)].
Usually in the holographic duality the (D + 1)-dimensional theory contains gravity. The Hamiltonian system (1) does not contain the symmetric tensor particle because among the Tr4>l{x) operators there is no energy-momentum tensor of the matrix field theory. Hence, naively there is no gravity in the above theory. However, as we discuss at the end of the paper the Hamiltonian system in question might be related to the effective string field theory.
In general, however, it is not clear for us so far under what circumstances and/or how the Hamiltonian equations in question are converted into the Hamiltonian constraint equations of the generally covariant theory. It is not clear for us whether the theory which governs the
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E. T. Akhmedov, I. B. Gahramanov, E. T. Musaev
RG flow of field theory on the full OPE basis should always (for all large N field theories) be generally co-variant or not. This remains to be a challenge for the future work. However, we refer to the RG in question as holographic because, unlike the standard definition of the RG, one can find the theory at any energy scale (high or low one) once he specified this theory at some scale. As well we believe that AdS/CFT-correspondence is of the same origin.
The Hamiltonian flow along the original D directions, denoted in (1) as x, is trivial. I.e. basically we deal with continuous collection of non-linear mechanics enumerated by the index x. So one can just forget this index and study the dynamics of the system with the Hamiltonian H = f+7T daIl2J'.
J —7T
One of the goals of this paper is to show that the Hamiltonian system in question is integrable. And to show that it is equivalent to the effective field theory derived by Das and Jevicki for the matrix quantum mechanics [14] (see as well [15, 16]).
Holographic formulation of the large N Wilsonian RG. We consider the I?-dimensional Euclidian matrix scalar field theory whose action in the Fourier transformed form can be written as:
m = ~ x
x ITr [cf>(p) (p2 + to2) K^(p2)d>(^p)] dDp +
oo „
+ N^2 dDk1...dDklTv^¡){k1) ...¿(fc,)] x
i=0 '
X Jj(-fci
h).
(2)
Here Ji are the sources. We assume that there is some momentum cut-off imposed, i.e. KA(p2) ~ 1 as p2 -C A2, while KA(p2) 0 as p2 > A2. As well we assume that Ji(k) = 0 for all I for |fc| > A, where A is some low energy scale (where we measure our physics).
The Polchinski equations for the theory in question follow from the RG invariance of the functional integral Z [13]:
A
x
dSr[4>] dA
N-1
M
dDp _AdKA(p2)
p¿
s2 s![<!>}
m" dA
Sftii-pjSfi'íjp)
oo „ 1=0 J
x Ji{-k-
Sfti(p) 5<j>ii(-p)_ dDki... dDk Tr [^(fei) ... x
- h), (3)
i.e. this equation is supposed to specify the scale A dependence of the sources J; to fulfill the equation dZ/dA = 0.
To proceed, we verify the following relations: 52Si
Tr
oo a —1 «
Tr
_5<j)(p)5<j)(-p) _ (a + l)Tr[0(fc!) ...d>(ks)]x
a=l s=0 *'fe(>i-i)
X Tr ^>(ka+1) ...0(fca)] Ja+l(-fc(a-l));
£/ ("-Ox
SSr SSr
Hip) H(-p)
x Tr [<¡>(qi) ... ¿(qj^) <j>(h) ... Hh-i)] x
x Jj(-P~Qi
Qj-i) Jiip-h
■ki-i); (4)
Here we use the same notations as in [12]: Ji(-k^) :=
Ji(-ki-----h) and fh := f dDh ... dDkh The first
among these two equations is obtained after the regrouping summands under the sum.
If one substitutes the obtained expressions for the variations of <5/ into the RG equation (3), he has to make two observations. First, the operators containing derivatives of ^>'s do not appear on the RHS of the RG equation. That is true simply because operators containing powers the momenta k and q are not generated from the functional derivatives of <5/ from (3). Second, one encounters higher trace operators on the RHS of the RG equation (3). It seems that to close the obtained system one has to add to (2) sources for higher trace operators as well (and of cause in general one has to add operators containing derivatives). That is the standard approach in the Wilsonian RG: at the end one has to use Operator Product Expansion and the completeness of the basis of operators.
However, in the large N limit one can take a different way of addressing the problem [11]. In this limit one has the factorization property, (J|n On) = = Un(On) + O (l/iV2), which is valid for any choice of the bare action used for performing the quantum average (...). Due to the factorization property at large N one can express any operator of the complete OPE algebra basis as a polynomial in the single trace operators. I.e. in the limit in question single trace operators form a basis through which any operator in the complete OPE algebra can be expressed algebraically. In this note we of cause consider only a subspace of the single trace operators, which contains only the operators without derivatives.
Thus, our idea is that the first step to obtain the closed system of equations for the single trace operators is to take the quantum average of (3) [12]. To do that we separate the field 4> into two contributions 4> = (f>o + <p -
the low energy one 4>o, which solves the equations of motion following from (2)3), and the high energy ones <p, which contain harmonics between A and A. In the quantum average we take the functional integral over the <p. In [12] we have used the averaging with respect to the Gaussian theory. However, all our formulas remain valid even if we average with the use of an interacting action. The reason for that is the validity of the factorization property at large N.
Thus, substituting Si to the equation (3), then taking its quantum average and using the factorization property, one obtains:
oo „
i_i J km
1=1 Jkd) 1 f Ka(P
fD .
N~
2 Jp p* +m- i —i~oJha-D
oo a —1 « __i —n J ki
(a+ 1) x
02 + TO2
X ^^({ka^i^T^k^Ja+i^k^) +
(l-m+i-2({kl-l}Ali-l}) X
• £ /
X 1) -p)jj(-<lu-1) +p)
(5)
where overdot means Ad/dA and Ta({ka}) := (1\[<i>(h)...<i>(ka)}).
It is probably worth stressing now that after the quantum averaging the operators containing differentials of 4> do appear in the RG evolution. However, they contribute to the RG flow of J's under consideration within Ti = (Tr^>') = Tr^+ higher derivative (d(f>o) terms. Note that eq. (5) contains only J's and T's.
The equation (5) still is not closed since the RG dynamics for the sources J depends on the VEVs T. However, one can close the system by deriving the RG equations for the VEVs T as well. To derive the above Polchinski equation, we have used the fact that
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