###### Abstract

Motivated by recent disagreements in the context of AdS/CFT, we study the non-planar sector of the BMN correspondence. In particular, we reconsider the energy shift of states with two stringy excitations in light-cone string field theory and explicitly determine its complete perturbative contribution from the impurity-conserving channel. Surprisingly, our result neither agrees with earlier leading order computations, nor reproduces the gauge theory prediction. More than that, it features half-integer powers of the effective gauge coupling representing a qualitative difference to gauge theory. Based on supersymmetry we argue that the above truncation is not suited for conclusive tests of the BMN duality.

hep-th/0407098

AEI-2004-052

UW/PT-04-08

New aspects of the BMN correspondence

beyond the planar limit

Petra Gutjahr^{*}^{*}*E-mail address:
and Ari Pankiewicz^{†}^{†}†E-mail address:

Max-Planck-Institut für Gravitationsphysik, Albert-Einstein Institut

Am Mühlenberg 1, D-14476 Golm GERMANY

Department of Physics, University of Washington

P.O. Box 351560, Seattle WA 98195 USA

July 2004

## 1 Introduction

In the last two years a lot of progress has been made in understanding the ideas of the AdS/CFT correspondence. The starting point of a long chain of
developments marks the BMN duality [1], which connects super Yang-Mills (SYM) in the sector of operators with large charge
and type IIB string theory on the maximally supersymmetric plane-wave background [2]. In contrast to ,
here the worldsheet -model reduces to a free theory in the light-cone gauge and thus, can easily be quantized [3]. Moreover, string interactions can
be treated in the context of light-cone string field theory [4, 5, 6].

The parameters of the two sides of the duality are linked
via [1, 7]^{1}^{1}1Here denotes the curvature scale of the plane-wave
geometry and is the light-cone momentum.

(1.1) |

and therefore free string theory corresponds to the planar () sector, while string interactions are identified with non-planar, interacting gauge theory. This statement has been subject to a wide range of tests based on the key relation

(1.2) |

where is the light-cone energy of string states and denotes the conformal dimension of the dual SYM-operators. In particular for the planar case, could explicitly be verified for so-called two-impurity operators (i.e. two stringy excitations) up to two loops [1, 8, 9]; for all-loop arguments see [10, 11]. The extension to the interacting part proved to be much more involved. On the field theory side, the anomalous dimension of two-impurity operators is known up to [12, 13, 9]

(1.3) |

whereas in light-cone SFT only the leading order energy shift has been computed [14]. Over and above, matrix elements of and the dilatation
operator as well as decay widths of one-string states and single-trace operators have been successfully compared to leading order, see e.g. [15].
For more details we refer to the reviews [16] and references therein.

It was subsequently realized, that the BMN correspondence can be generalized in essentially two directions: Curvature corrections to the plane-wave background
can be taken into account [17] having their counterpart in corrections of planar BMN gauge theory [18, 19].
On the other hand, a completely new field was established by considering semiclassical string states in [20] and
the insight that the planar spectrum of SYM operators is governed by an integrable Hamiltonian (long-range) spin chain [21], leading to a vast class of
tests, see the review [22] and references therein.
Both cases show perfect agreement up to two loops [23], see also [24], but – despite of this tremendous progress – several open
questions and puzzles gradually emerged at three loops:
In the latter approach, string and gauge theory continue to exhibit qualitatively similar structures but start to differ in detail [25].
Somewhat more disturbing results have been found in [17]: degeneracies present in the gauge theory and crucial for
integrability [9, 19, 26] are lifted in the near plane-wave background.

These recent developments raise the question whether disagreements do already occur in the BMN duality itself, namely in the non-planar, interacting sector.
As was pointed out in [27], certain matrix elements of and seem to mismatch starting at . This however need not necessarily
imply that physical quantities show disagreement as well and deserves a more careful investigation, which we initiate in the present paper.
Eventually one would like to compute independently the energy of states with two (stringy) excitations on the string theory side and hopefully reproduce (1.3).

Therefore, in section 2, we briefly introduce some well-known facts about the free theory and explicitly construct the supermultiplet for states with two stringy
excitations. Especially we find that states consisting of two fermionic/bosonic oscillators in general mix with each other.

The evaluation of the energy shift demands the knowledge of cubic and quartic terms in the Hamiltonian. We review (section 3) the
formulation of the three-string vertex restricted by the superalgebra at order and comment on further constraints on the quartic interaction. In particular, we notice
that a term induced by the second order dynamical supercharges cannot a priori be excluded.

It is a known fact, that all members of a supermultiplet receive the same energy corrections. Furthermore (section 4), one can show by using the states in the
supermultiplet, degenerate perturbation theory becomes redundant. Note, that both statements are only valid when including impurity-conserving and
-non-conserving intermediate states.
Here, we calculate as a first step the complete perturbative (in ) impurity-conserving contribution for one particular representation. Quite surprisingly our result
disagrees with that given in the literature [14] and also fails to reproduce the gauge theory prediction (1.3) at two loops.
Above all as a qualitative difference to gauge theory the series features not only integer, but also half-integer powers of .
It seems to be apparent that this truncated analysis does not reveal the whole story.
We conclude with a discussion.

## 2 The free theory

In this section, we introduce the free theory and its symmetries and analyze the underlying supermultiplet structure. Type IIB string theory on the plane-wave space-time can be quantized in light-cone gauge [3] resulting in the Hamiltonian

(2.1) |

where is the light-cone momentum, the frequencies are given by and indicates the number operator

(2.2) |

Compared to [4] we performed a redefinition of the oscillator basis to have the
standard level-matching condition (cf. Appendix A), i.e. the operator
has to vanish on the space of physical states.
Here the bosonic oscillators transform as and under the transverse
isometry of the plane-wave background, whereas the
fermionic oscillators give the representations and .
Both obey the standard (anti)-commutation relations.

Due to the effective harmonic oscillator potential of the background geometry, the theory possesses an essentially unique singlet ground state
(labeled by its light-cone momentum since is a central element of the plane-wave superalgebra) defined by

(2.3) |

Excited states are obtained by acting with the creation oscillators on subject to the level-matching condition and can be organized into
multiplets of the plane-wave superalgebra. Its bosonic generators are , , , () and the angular momentum
generators of the transverse and , while
the 32 supersymmetries are generated by , (both transforming as and )
and , (transforming as and ).

On a general eigenstate of in a given irreducible representation of the nontrivial action of generators is as follows:
Certain combinations of and add or remove a bosonic zero-mode excitation; this raises or lowers the energy of the state by and is the
discretized analog of giving a state transverse momentum. Note, that is not a quantum number in the plane-wave space-time since it does not commute with .
Similarly and add or remove a fermionic zero-mode excitation.
This is why not the energy but , which only counts the non-zero-mode (‘stringy’) excitations, is a Casimir of
the superalgebra.
Finally, the most interesting generators are and ; these do not change the number of excitations, commute with and, therefore transform states
of the same energy but different representations into each other. This action of the generators is schematically depicted in Fig. 1.

In the following we will call a multiplet containing states with stringy excitations a ‘–impurity’ multiplet. The simplest example is the ’zero-impurity’, i.e. supergravity multiplet [28]. In this case the highest weight state is the ground state annihilated by all , and, therefore, this multiplet is short and protected against quantum corrections through string interactions.

### 2.1 The two-impurity supermultiplet

In contrast to the supergravity multiplet, where the state of lowest energy is unique, for the two-impurity multiplet the states of lowest energy are those with two stringy oscillators, schematically

All of these 256 states are linked to each other by acting with half of the dynamical supercharges on a highest weight state which we will explicitly determine below, see also [17] for the discussion in the case of the near plane-wave background. States with two bosonic oscillators are decomposed into irreducible representations of , namely

(2.4) | ||||

(2.5) | ||||

(2.6) | ||||

(2.7) |

and analogously for . Here indicate the (anti)-selfdual representations. Under worldsheet-parity (i.e. )
the singlets and the symmetric-traceless representations are even, whereas the (anti)-selfdual are odd.
The (-function)-normalization^{2}^{2}2We will always suppress the -function normalization factor , where and denote the
light-cone momenta of the in-/out-states, respectively.
of is , all other states are normalized to one.

Two fermionic oscillators lead to the states (cf. Appendix A for our conventions for the -matrices)

(2.8) | ||||

(2.9) | ||||

(2.10) | ||||

(2.11) | ||||

(2.12) |

and similarly for the remaining representations. Here, the singlets and have odd, whereas the (anti)-selfdual representations have even
worldsheet-parity. Again we have normalized the states to one.
Notice that only , and are uniquely constructed of
bosonic or fermionic oscillators, respectively.

The remaining representations are realized both with bosonic and fermionic oscillators and, therefore potentially mix with each
other^{3}^{3}3In this case it is sometimes convenient to introduce and
. Then e.g.
..
For completeness we mention that fermionic two-impurity states transform as , ,
and the same representations with the two ’s exchanged. These are e.g.

(2.13) | ||||

(2.14) | ||||

(2.15) |

To construct the two-impurity multiplet explicitly we determine the highest weight state that is annihilated by , , and (no zero-mode excitations), and (a singlet) and by half of the dynamical supercharges which we choose to be and (cf. Appendix A for their explicit oscillator expressions). The latter requirements impose three conditions on the most general ansatz (with to normalize it to one)

(2.16) |

with the (up to a phase in ) unique solution

(2.17) |

As contains states of opposite worldsheet parity, is not a quantum number to label states in the supermultiplet; in particular for large and we have

(2.18) |

reflecting that in the BMN gauge theory the singlet operator built out of scalar impurities starts to mix with covariant derivatives and fermions at higher
(than one) loops.
By applying and successively to we generate all states with two stringy
excitations. For example, acting twice with we find
^{4}^{4}4We define

(2.19) |

Here

(2.20) | ||||

(2.21) | ||||

(2.22) |

and analogously for , while the mixing-coefficients are

(2.23) |

In the large limit and this yields e.g. , so the mixing of states is again a effect. For the mixing is maximal in agreement with the gauge theory result. Up to irrelevant phases and an overall factor of (which can be absorbed in the definition of , ) the leading order SUSY variation (2.1) precisely agrees with [18].

## 3 Turning on Interactions

String interactions in the plane-wave background have been treated within the framework of light-cone string field theory [4, 5, 6]. Its guiding principles are worldsheet continuity and the realization of the superalgebra in the full interacting theory: the superalgebra gives rise to two types of constraints – kinematical and dynamical – depending on whether the participating generators receive corrections (, and ) or not. Kinematical constraints lead to the continuity conditions in superspace, whereas dynamical constraints require the insertion of interaction point operators [29]. In practice these constraints will be solved perturbatively, for example , the full Hamiltonian of the interacting theory, has an expansion in

(3.1) |

Here the operator represents a three-string interaction, but it is more convenient to express it as a state in the multi-string Hilbert space and work in the number basis [30]. Then the dynamical generators are of the form , where are the prefactors determined by the dynamical constraints (i.e. the oscillator expressions of the interaction point operators mentioned above) and the kinematical part of the vertex common to all the dynamical generators implements the continuity conditions. These follow for example from , so the interaction vertex is translationally invariant and conserves transverse momentum. In the number basis the bosonic part of has the form

(3.2) |

where is the tensor product of three (bosonic) vacuum states and
are known as Neumann matrices, see [31] for explicit expressions as functions of , .

To fulfil the dynamical constraints we define the linear combinations of the free supercharges and
() which satisfy on the space of physical states e.g.

(3.3) |

and similar relations for and . Since and are not corrected by the interaction, it follows that at order the dynamical generators have to obey

(3.4) | ||||

(3.5) | ||||

(3.6) |

Substituting the most general ansatz for, say , compatible with the requirement that the Hamiltonian prefactor in
its functional form is quadratic in derivatives, into (3.4) and demanding that the result only involves the tensor
fixes and consequently also up to their normalization. The same procedure applies to and requires that its normalization is the
same as of .

In short, the three-string vertex and dynamical supercharges are

(3.7) | ||||

(3.8) | ||||

(3.9) |

with similar expressions for . Here , for the incoming and for the outgoing strings and
is defined in (A.26). Further we list
the relevant parts of , and in the next section, for complete expressions see e.g. [5].
In equations (3.7)-(3.9) we suppressed the integrals over light-cone momenta
and the -function normalization factor .

More importantly, as alluded to above, the normalization of the dynamical generators is
not fixed by the superalgebra at order and can be an arbitrary (dimensionless) function due to the fact that is a central element of the algebra.
Indeed, it does not seem that further consistency conditions at higher orders in would allow to fix .

Now consider the constraints at order . These are e.g.
of the light-cone
momenta and

(3.10) | ||||

(3.11) | ||||

(3.12) |

and have been analyzed in some detail in [32]. In particular, it was found that and diverge in the - strings channel leading in the latter case to the introduction of a non-vanishing (and ). The new supercharges – together with – generate a contact term needed for finite scattering amplitudes. An analogue argumentation presumably holds for the - string channel. We have checked that for our calculations the above conditions are not violated if is set to zero. Still, this constitutes only a necessary but clearly not sufficient requirement.

## 4 Computing energy shifts in light-cone SFT

To compute the energy shift of two-impurity states to leading order in , the relevant part of the Hamiltonian is

(4.1) |

where the contact term acting in the single-string Hilbert space is induced by the cubic supercharges^{5}^{5}5At this point we neglect possible contributions of .

(4.2) |

As we have seen in section 2, there are generically several two-impurity eigenstates transforming in the same irreducible representation of ; hence these will mix with each other and we have to use degenerate perturbation theory to compute their energy shift. The required formula for the energy shift is standard and reads

(4.3) |

Here and label the degenerate one-string eigenstates, is the two-string projector and . Thus, we essentially have to diagonalize the mixing matrix

(4.4) |

Since the theory is supersymmetric, this can be achieved by constructing supermultiplets: Suppose we have constructed the complete supermultiplet by acting with eight supercharges, say and on a highest weight state that is annihilated by the remaining charges. Now consider two states and carrying the same quantum numbers, related by, say

(4.5) |

Therefore we have to show that the off-diagonal matrix elements of vanish

(4.6) |

where we used that . As a matter of fact, equation (4.6) is a consequence of supersymmetry. Recall , so in particular to order we have the conditions

(4.7) | ||||

(4.8) | ||||