"Non-renormalization"without supersymmetry

The g_{YM} perturbed, non supersymmetric extension of the dual single matrix description of 1/2 BPS states, within the Hilbert space reduction to the oscillator subsector associated with chiral primaries is considered. This matrix model is described in terms of a single hermitean matrix. It is found that, apart from a trivial shift in the energy, the large N background, spectrum and interaction of invariant states are independent of g_{YM}. This property applies to more general D terms.


Background and Motivation
Recent studies of giant gravitons in AdS backgrounds [1][2][3][4][5][6][7][8] have led to a dual matrix model description of 1/2 BPS states in terms of the large N limit of a single complex matrix [8] in a harmonic potential [9].
This can be motivated as follows: starting with the leading Kaluza Klein compactification of the bosonic sector of N = 4 SYM on S 3 × R, one chooses the plane defined by two adjoint scalars (N × N hermitean matrices) X 1 and X 2 . The corresponding bosonic sector of the Hamiltonian iŝ H ≡ 1 2 Tr(P 2 1 ) + 1 2 Tr(P 2 2 ) + w 2 2 Tr(X 2 1 ) + (1) with P 1 (P 2 ) canonical conjugate to X 1 (X 2 , respectively). The harmonic potential arises as a result of the coupling to the curvature of the manifold.
In terms of the complex matrix Z = X 1 + iX 2 and canonical momentum Π = 1/2(P 1 − iP 2 ), Introducing matrix valued creation and annhilation operators motion in this plane is then characterized by the energy E and the two dimensional angular momentum: A(and B) quanta carry well defined charge 1 (−1, respectively). When imbedded in N = 4 SYM, the g Y M dependent interaction in (2) is one of the so called D terms, and is subject to non-renormalization theorems. One may then consider the free theory. In this case, the (invariant) eigenstates are Tr((A † ) n )Tr((B † ) m )|0 >.
1/2 BPS states then correspond to a restriction of the theory to the (chiral) sector with no B excitations, for which E = E 0 = wJ (E 0 is the free theory energy) 1 .
When restricted to correlators of these chiral primary operators, the dynamics of the system is fully described by free fermions in the harmonic oscillator potential. It actually turns out, as shown in [11], that the gravity description of 1/2 BPS states is completely determined by a phase space density function associated with a general fermionic droplet configuration.
As suggested in [12], the dynamics of the A, A † system can be described in terms of an hermitean matrix M while retaining B, B † creation and annhilation operators. The change of variables is: As shown in [12], in the 1/2 BPS sector, the gravitational degrees of freedom emerge in a natural way from the density of eigenvalues description of the large N dynamics of the hermitean matrix M. The energy and flux of the 1/2 BPS states obtained in [11] are exactly those of the leading large N configuration of the hermitean matrix M in a harmonic potential, in a bosonic phase space density description 2 . This will be briefly reviewed in the next section. Fluctuations about the large N matrix configuration were shown to agree with fluctuations [12], [14] about the gravity background of [11]. The purpose of this note is to investigate, in the absence of supersymmetry, the non-perturbative consequences of the quartic g Y M interaction in (1) while still restricting the theory to the sector with no B excitations. In other words, the system of interest iŝ 1 The B sector can also be projected out by taking a pp wave limit [10]. This requires J to become large.
In order to study its large N limit, I will use the variables (5), in terms of which the hamiltonian (6) becomes: In the absence of supersymmetry, the argument for the decoupling of B excitations is potentially weakened. It should always be a good approximation for external states with large J charges [13] and for large w. However, the hamiltonian (6) (or (7)) is of great interest per se as it contains an interaction with the structure typical of a Yang-Mills interaction.
Remarkably, we will find that, apart from a trivial shift in the energy, the large N background, spectrum and interaction of gauge invariant states are independent of g Y M , even in the absence of supersymmetry.
This letter is organized as follows: within the collective field theory approach [15], a general argument is presented in Section 2 for the "nonrenormalization' properties of the theory, and an explicit non-perturbative argument is developed in Section 3. A simple diagrammatic check is carried out Section 4, and a generalization presented in Section 5. Section 6 is reserved for a brief conclusion.

A general argument
In order to obtain the large N limit of (7), we will make use of collective field theory approach [15]. The starting point of this approach is to consider the action of the Hamiltonian on wave functionals of gauge invariant operators, i.e., operators invariant under the transformation: For a single matrix model such as (7), these can be chosen as the density of eigenvalues λ i , i = 1, ..., N of the matrix M: or its fourier tranform Let us analyse the structure of the g Y M dependent operator in (7) in more detail. It can be written as We recognize the generators of the transformation (8). Therefore, when restricted to the gauge invariant subspace, the g Y M term in (7) does not contribute, except for the trivial constant shift in energy in (11) 3 .

Explicit non-perturbative argument
Because, as a result of the use of the chain rule in the kinetic energy operator, the interactions in the theory organize themselves with different powers of N, it is important to provide an explicit verification of the general argument presented in the previous section. What is diferent from previous applications of the collective field theory to the large N limit of the single matrix hamiltonian (7), is the sigma model nature of the kinetic energy operator and the presence of terms linear in momentum: Denoting by φ α a generic gauge invariant variable, the kinetic energy operator (13) takes the form, when acting on functionals of φ α , For Ω α,β , I obtain Due to the antisymmetric nature of the Yang-Mills interaction reflected in the sigma model nature of the kinetic energy (13), the result above for Ω k,k ′ is independent of g Y M and is the same as that for the standard kinetic energy. Forω α , I obtain 2w 2 x dyφ(y) − N Interpreting N = Tr(1) = dxφ(x), one would immediatly conclude that both (15) and (16) are independent of g Y M , proving our result. However, we follow the more rigorous approach of imposing this constraint via a Lagrange multiplier, and choose to enforce the constraint after variation. Due to the change of variables X A → φ α , one performs the similarity transformation [15] induced by the Jacobian J: where only the leading (in N) expression for ln J is described. One obtains the form of the collective field hamiltonian sufficient for the description of the leading large N background and fluctuations: The lagrange multiplier µ enforces the contraint To exhibit explicitly the N dependence, we rescale Using the identity we obtain where λ = g 2 Y M N is 't Hooft's coupling. The large N configuration is the semiclassical background corresponding to the minimum of the effective potential in (22), and satisfies where the constraint dxφ 0 (x) = 1 has been applied after variation. This constraint fixes µ = w + λ 4w 2 , and we arrive at the Wigner distribution background This large N background is independent of λ.
It is useful to review here the emergence of the droplet picture when λ = 0. In this case, if we let [16] p ± ≡ ∂ x Π(x)/N 2 ± πφ(x), then (22) has a very natural phase space representation as As N → ∞, the boundary of the droplet is given by p 2 ± + x 2 = 2µ = 2w, since p ± → ±πφ 0 (x) = ± 2µ − w 2 x 2 . This is in agreement with the energy of the gravity solutions considered in [11], with x 1 → x, x 2 → p 4 .
When λ = 0, all possible λ dependence is absorbed in the definition of µ, with the result that the large N background has a similar λ independent fermionic description.
For the small fluctuation spectrum, one shifts to find the quadratic operator The variable with a gravity interpretation as the angular variable in the plane of the droplet [11] is the classical "time" of flight φ in terms of which the quadratic Hamiltonian takes the form: Except for the last two terms, one recognizes the Hamiltonian of a 1 + 1 massless boson. For consistency of the time evolution of the contraint (19), we impose Dirichelet boundary conditions at the classical turning points: Since then the last two terms in (27) vanish, and we obtain a 1 + 1 massless boson hamiltonian with spectrum ǫ j = wj , j = 1, 2, 3, ...
again independent of λ. It is not dificult to see that as a result of both (19) and (29) the collective field cubic interaction is independent of λ.
In addition to (18), there is in the collective field theory an extra subleading contribution proportional to dx∂w(x)/∂φ(x), responsible for the one loop contribution to the energy and a tadpole interaction term [17]. The additional λ dependent contribution is proportonial to a total derivative of xφ(x).

A simple diagrammatic test
Let us consider, for instance, the g 2 Y M contribution to the two point function 1 resulting from the quartic interaction in (6). This can be done using the propagator Consider first the Tr(A † A † AA) interaction. There are two types of connected planar diagrams, one the usual connected diagram without self contractions and the other involving the "dressing" of an internal leg. For the first, one obtains For the other type of diagram, one obtains Turning now to the Tr(A † AA † A) interaction, one establishes that the only type of planar diagram that is generated is the diagram (34) with a selfcontraction associated with the "dressing" of an internal leg. This diagram now has a symmetry factor of 12, and since the interaction has the opposite sign, it exactly cancels the sum of (33) and (34).
The arguments of the previous sections show that this generalizes to general invariant external states and to arbitrary orders of perturbation theory.
Other families of states, that are not protected by BPS arguments but with energies independent of g Y M for different diagrammatic reasons, have also been reported in the literature [18].

General D terms
written in terms of complex fields Z = X 1 + iX 2 , φ = X 3 + iX 4 and ψ = X 5 + iX 6 . The D terms take the form They can be equivalently written as One introduces, as before, matrix valued creation and annihilation operators If one is interested only in correlators of chiral primaries such as Tr(Z † m 1 φ † q 1 ψ † s 1 ....), these can be excited from the vacuum as Tr(A † m 1 C † q 1 E † s 1 ....) and described in terms of hermitean matrices M,Q and S. Using an argument similar to the the general argument of Section 2, and taking into account the special form of (37), we conclude that the terms in (36) involving only A, C and E oscillators (and their conjugates) do not contribute to these amplitudes.

Conclusion
In this letter, the large N limit of the system when restricted to the subsector of chiral primary states Tr(A † n ) has been shown to be independent of g Y M . In order to explicitly confirm this result, the collective field theory method has been generalized to include sigma model type kinetic energy operators. This "non-renormalization" result applies to more general D-terms.

Acknowledgment
I wish to thank Robert de Mello Koch for reading the manuscript and for useful comments.