Abstract
We continue the development of a systematic procedure for deriving closed string pp wave string field theory from the large N Berenstein-Maldacena-Nastase limit. In the present paper the effects of the Yang-Mills interaction are considered in detail for general BMN states. The SFT interaction with the appropriate operator insertion at the interaction point is demonstrated.
Introduction
The Berenstein-Maldacena-Nastase correspondence outlines a precise relationship between large N N = 4 super Yang-Mills theory and closed string theory in the ppwave background [1] . This limit simplifies and extends the AdS/CFT correspondence [2] , [3] , [4] and is a subject of detailed studies. Of particular interest is the derivation of pp wave string field theory from the Yang-Mills/matrix theory.
In the present paper we continue the work begun in [5] (here after referred to as (I)) on developing a direct and systematic approach for mapping large N gauge theory to closed string theory. We have applied in I the methods of collective field theory and adapted them to the large N BMN limit. The approach was described in the context of a quantum mechanical model. The essential feature of the collective field approach is the mechanism of joining and splitting (of loops) represented in an effective hamiltonian. Consequently the simplest example of obtaining string type interactions was seen to appear already from a free matrix theory. It was shown in particular that the SUGRA type amplitudes are correctly reproduced through the above mechanism. Likewise the 3-string overlap vertex was seen to naturally arise. In this approach the effect of Yang-Mills interactions is expected to renormalize the zeroth order expressions and we discuss this in the present work.
The matrix model language was used essentially for notational simplicity. The matrix model contains the effects, which in our framework, are identical in the full gauge theory.
It may also be that there is an effective matrix model description of the full pp theory and our result have shed light on this possibility. (For a most recent discussion of holography in the pp limit see [6] .)
Our plan is as follows: In Sect.2 we summarize the basics of the approach reviewing and extending the methods of I. In Sect.3 we give a simple example of string type amplitudes and exhibit the manner in which Yang-Mills type interactions modify the basic couplings.
We make use of a coherent state picture and describe a map between this and the physical picture of I. In Sect. 4 we then take these interactions into account concentrating on general string-type states. In the continuum BMN limit we derive the emerging 3-string interaction with the appropriate operator prefactor. We reproduce the spectrum of BMN loops in an appendix.
Collective field theory
In this section, we give a general overview of collective field theory as applied to the large N limit of Berenstein, Maldacena and Nastase. This section defines the formalism and gives a short summary of the results reached in I. The collective method in general provides a systematic formalism for describing the dynamics of observable, physical degrees of freedom in a theory with large N symmetry. In gauge or matrix theory the physical observables are given by loops or traces of matrix products (words). The method then provides a direct change of variables to the invariant observables. The resulting effective or collective Hamiltonian describes the full dynamics of these invariants. The essential two terms that define the effective hamiltonian are the interaction terms containing joining and splitting of the loops. In N =4 SUSY Yang-Mills gauge theory we follow the degrees of freedom consisting of the Higgs fields: φ 1 φ 2 · · · φ 5 , φ 6 . In the proposal of Berenstein, Maldacena and Nastase two of the Higgs matrices (φ 5 , φ 6 ) are chosen to play a special role , they are selected to define the light cone coordinates Z = φ 5 + iφ 6 . The above matrix variables define a sector of the full theory that we are interested in. In addition we consider these as functions of time only, so that the dynamics reduces to matrix quantum mechanics. This gives a minimal set of fields that are capable of capturing the essential features of the BMN correspondence.
We therefore concentrate on a complex multi-matrix system with a Hamiltonian that is invariant under
The basic equal time variables are given by single trace operators
In the large N limit, one then has a change of variables from the original Yang-Mills fields to the invariant loop variables denoted collectively by φ C , where C stands for a loop or word index. This index also includes complex conjugate loopsφ C . One then considers a change of variables to this new set in the canonical operator formalism. Concentrating on the kinetic term, there follows
with
and
These two operations define the processes of joining and splitting. In particular Ω(C, C ′ )
"joins" loops, or words. As an example, if φ C = Tr(Z
). In general, one has schematically
where C + C ′ is obtained by adding the two words C and C ′ . Similarly, ω "splits" loops.
Again,
represents the processes of splitting the word C into C ′ and C ′′ . The hermiticity of the new collective description is assured through a field transformation and this completes the hamiltonian [7] .
The non-triviality in applying collective field theory to multimatrix models comes from the enormous set of loops (words) that can be generated. For the present problem 
with T l the l ordering operator -it orders the φ(l) factors so that l i increases from left to right. We also haveΦ
T l is a second l ordering operator -it orders theφ(l) factors so that l i decreases from left to right.
The joining and splitting processes applied to these lattice string fields proceed as follows. One first has the open loop defined by
where we have l i − a mod J so that 0
In terms of these split (open) loops we generate through the joining operation the composite, joined loop. In particular these are contained in the interaction of the collective hamiltonian through the composite trace Ω
This composite trace involves mixed combinations of Z andZ which do not survive as excitations in the BMN correspondence. In I we have introduced a mechanism of factorization based on which we replace such new, unwanted loops by a sum of physical, BMN approved loops. Concretely,
The coefficients (structure constants) in this relation can be generated by Schwinger-Dyson equations (as we have discussed in (I)). A more complete procedure for finding these coefficients is through consistency conditions obtained by taking expectation values of both sides of the equation. In this way they can be read of from the correlator
is again most simply deduced from the correlator
The above sums appearing in C and G are equal and represent a form of a (lattice) threestring vertex |V . In order to demonstrate that, let us consider the string states
these are in direct correspondence with our matrix theory fields (states)Φ J ({m}), Φ J 1 ({l})
The sums appearing in the matrix theory result correspond to reparametrizations that occur in the calculation of
The sum in C would appear in the above calculation if we fix one of the delta functions between string (2) and (3) and one of the delta functions between (1) and (3). This allowed us to write the part of our collective Hamiltonian coming from H 0 as
where
and |ψ i are the lattice string states associated with Φ J i ({l i }). The characteristic feature of this interaction is that it appears proportional to 1/N and exhibits the prefactor (E
). The occurrence of the energy prefactor was an early conjecture of [8] . It was explained in I in a specific example as resulting from a projection to light cone fields. It is expected that this form will be modified once Yang-Mills interactions are taken into account.
We note that this interaction can be transformed away through a (nonlinear) field redefinition. The Hamiltonian H 2 + H 3 can be reduced (in leading order of 1/N ) to H 2 .
Another way to understand this fact is to realize that in the creation-annihilation (coherent state) basis the free oscillator hamiltonian is first order in the derivatives and its collective representation in this basis is still a quadratic Hamiltonian
The nonlinear transformation
from coherent state fields A({l}) to the physical collective fields relates the two representations. The coherent state picture will in the next section provide the simplest framework for incorporating the effect of Yang-Mills interactions.
Examples
We will start by considering the g 2 Y M effects of the dimensionally reduced YangMills system and discuss simple, illustrative examples of corrections that this term gives.
Consider the Hamiltonian
With
the interaction term in the above Hamiltonian is equivalently written as
showing the usual split into D and F terms respectively. Having in mind the passage to the infinite momentum frame, we work in a coherent state basis and project
We don't consider B † quanta. These quanta correspond, in the pp-wave string field theory, to modes with p + < 0. In the infinite momentum frame these modes decouple.
For the complex impurity fields ψ and φ we project
This last projection ensures that our hamiltonian keeps us within the subspace of loops that are near to chiral primary operators. This truncation could be justified by appealing to supersymmetry. Here we follow a more pedestrian approach and justify the truncation by showing that the resulting hamiltonian reproduces the impurity number conserving amplitudes of the light cone string field theory vertex. We have also checked that the correct string mass spectrum (i.e. anomalous dimensions) is obtained to order 1/N 2 .
An immediate consequence of the above reduction of degrees of freedom is that the D terms in (3.1) are trivial as a result of the commutator
We therefore arrive at the Hamiltonian
which can be recognized as the (dimensionally reduced) operator∆ −Ĵ in a coherent state basis. In obtaining this expression, we have subtracted the free terms in the Hamiltonian, which in the large J limit simply contribute an additive term equal toĴ.
In I, we have already demonstrated agreement for a class of sugra states given by the loop variables
The sum is over all possible permutations of the φ and ψ fields, that is, the above loop is a chiral primary operator. Based on the free part of the above hamiltonian the following interacting cubic hamiltonian was shown to arise
The sugra amplitudes are special in that they do not seem to receive corrections from the
By changing the order of the double summations, expressing O l = n e − 2πinl J +1Õ J n /(J + 1), performing the intermediate sums, taking into account the normalization of the states and Carrying out steps similar to those leading to equations (3.8) and (3.9), we obtain
Combining the results (3.8), (3.9) and (3.10) we obtain the following Hamiltonian acting on loop spaceĤ
The coefficients can be read by inspection from equations (3.8), (3.9) and (3.10). In this coherent representation, we in general reach a Hamiltonian of the form
which is not hermitean. The transformation from coherent to physical fields given in Section 2 reads
It leads to an interacting Hamiltonian of the form
(3.14)
For the example of three states denoted as (in what follows y ≡ Some care must be exercised when computing these correlators, since the use of coherent states implies that one has a non-trivial inner product. These overlaps can equivalently be computed as correlators in the free matrix model in accord with the state operator map.
As discussed in section 2, our transformation makes use of these correlators. The cubic contribution from (3.14) becomes
We then evaluate the coefficient in the respective couplings
Collecting these couplings, we obtain
n,my =Γ
(1)
n,y =Γ
(1)
These are indeed the cubic couplings generated from the pp wave SFT [9] , [10] , [11] , [12] .
Lattice Strings and The Vertex
We have in sect.2 and in (I) given the collective Hamiltonian corresponding to H 0 and we now concentrate on H 1 . We start with the contributions to the cubic interaction.
In what follows we will consider loops of the form
The subscript l on b is understood as l mod J, so that we can assume this index is in the range 0 ≤ l ≤ J − 1. The symbol T l orders the b l so that l increases from left to right.
These loops correspond to the string states
In the continuum limit the sparse occupation of lattice sites becomes trivial.
For simplicity, we focus on a single complex impurity. The cubic interaction is represented as
where, because we have a single impurity, joining is generated bŷ
Towards this end, consider the following derivatives
which produce
for the loop joining terms.
In I the free matrix model was seen to generate loop joinings that were reproducing the string field theory interaction vertex with the trivial energy prefactor (zeroth order in µ). These free matrix model loop joining operations were associated with
and
Now from the action of (4.1) we will find another sequence of joining contributions to the full cubic interaction. As before we may again use our correspondence but this time with a different state from |V 0 3 . However, in the continuum limit (large J) the new state will reduce to be |V 0 3 multiplied by an insertion which will be shown to be, in combination to the effect of the field redefinition, precisely the large µ limit of the string field theory prefactor. Consider the term
2)
The summation in P 1 reflects the averaging in order to have the physical state |ψ 1 , a part of the non-triviality of the above comes from Q 2 . We would expect the same to happen with the second string variables after seen in the third string in the case of |V gives an insertion b
2) we can easily read off the appropriate state for this term to be,
In exactly the same way we can argue for the following identifications
The remaining terms follow
Next we have the four remaining contributions. Consider
After employing once more our correspondence we have
where
We are now ready to collect the contributions to the three string vertex and take the continuum limit. Let us recall the contribution from the free part of the Hamiltonian
This is essentially the result of I, but now we have the improved O(λ) corrected energies E i in the prefactor multiplying V 0 3 . This form is deduced from our free coherent Hamiltonian after performing the coherent physical space map. The relevant transformation given generally in sect.3 by eqs. (3.11)-(3.14). From the action of H 1 we have generated eight terms. In the continuum limit, the difference between V 0 3,1 and V 0 3 is negligible so the eight terms appear with the same 3-vertex:
Altogether we therefore find the total matrix theory vertex comes with the prefactor
where ∆b stands for a finite difference. We now clearly see the leading effect of the Y-M interaction: first it induces a correction to the energies(dimensions) in the prefactor renormalizing the free matrix theory result ,in addition there are novel contributions to the prefactor .
In the continuum limit these become operator insertions at the interaction point, in particular we get the derivative b + (0) 1 of the string creation coordinate at the interaction point. In order to demonstrate agreement with SFT now show that the SFT prefactor can be written in an identical form andP M atrix →P SF T .
To perform the comparison consider the SFT prefactor [13] 
We can "separate" out an energy contribution to write it as
where
Denoting the first term containing positive modes only by P + we have the 3-vertex
since we may use the expansion
and
, where r = {1, 2}
For all three strings we have:
a form predicted in our matrix model calculation.
Conclusions
We have in the present paper constructed the pp wave cubic SFT interaction from large N matrix theory using the Berenstein-Maldacena-Nastase limit. Even though this construction was performed at leading order in the Yang-Mills theory coupling constant, the method that we employ is generally not limited to weak coupling. We have concentrated on the sector of the theory generated by the Higgs (scalar field) degrees of freedom. This and the use of a matrix model language was for the purpose of notational simplicity. The construction can be extended to include fermionic fields of SUSY Yang-Mills theory and also states generated by the (covariant) derivatives of the Higgs fields. In the basic scheme that we presented the closed string theory cubic interactions are seen to be correctly generated. Most importantly we have seen how the correct prefactor or operator insertion at the interaction point is obtained from the large N matrix theory construction. There has been some debate on the form of the prefactor, our direct calculations provide a unique and well defined form. Apart from the extension of the present approach to include the fermionic and derivative degrees of freedom of the full Yang-Mills theory the most important future goal is that of presenting a derivation of the full nonperturbative SFT T r T l θ(n j+1 − 1)

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