String Theory and M-Theory

680 Gauge theory/string theory dualities
The mass terms in the world-sheet action mix left-movers and right-movers.
Therefore, it is convenient to allow mode numbers to run over all integers
rather than to treat left-movers and right-movers separately. In the limit
µ → 0, left-movers and right-movers would decouple and correspond to
positive and negative indices. Note also that the zero modes are described
by harmonic oscillators, rather than continuous momenta pI. This reflects
the fact that g++ acts like a confining quadratic potential restricting motion
into the transverse directions.
The frequency of the nth oscillator is
ωn = 1 + (n/µα) 2 , (12.162)
where α = α ′ P−. Then the light-cone Hamiltonian, which describes evolution
in τ (and hence x + ) is
Hℓc = µ
∞
8
n=−∞ I=1
ωna I†
n a I n + fermions. (12.163)
The eigenvalues of this Hamiltonian give the allowed values of P+. The
zero-point energies of the bosons and fermions cancel, so no regularization
is required.
The Fock space is constrained by
∞
8
n=−∞ I=1
n a I†
n a I n + fermions = 0, (12.164)
which generalizes the usual level-matching condition. This constraint arises
as a consequence of translation symmetry of the spatial world-sheet coordinate.
The dual gauge theory limit
Let us now consider the implications for the dual gauge theory. The Penrose
limit R → ∞ corresponds to J, N → ∞ with finite
λ ′ = g 2 YMN/J 2 , (12.165)
which is the loop expansion parameter introduced by Berenstein, Maldacena,
and Nastase (BMN). By definition, BMN operators are the class of gaugeinvariant
operators of the gauge theory that survive, with finite anomalous
dimension, in the Penrose/BMN limit.
The key duality formula relates the anomalous-dimension operator ∆a of