JHEP07(2018)146
2 Free CFT
Our basic goal is to organize and study excitations of an LLM background, using the
dual N = 4 super Yang-Mills theory. Any LLM geometry is specified by a boundary
condition, given by coloring the bubbling plane into black and white regions [9]. The LLM
backgrounds we consider have boundary conditions given by concentric annuli, possibly
with a central black disk. The LLM geometry is described by a CFT operator with a
bare dimension of order N
2
. Concretely, it is a Schur polynomial [13] labeled by a Young
diagram with O(N
2
) boxes and O(1) corners. Large N correlators of these operators are not
captured by summing only planar diagrams, so we talk about the large N but non-planar
limit of the theory. The excitation is described by adding J boxes to the background, with
J
2
N. Consequently, we can ignore back reaction of the excitation on the LLM geometry.
The CFT operators corresponding to the background and excitation are given by
restricted Schur polynomials [15, 16]. Construction of these operators and their correlators
becomes an exercise in group representation theory. In section 2.1 we discuss elements of
this description, placing an emphasis on if the quantity being considered depends on or is
independent of the collection of branes being excited. This distinction will clarify general
patterns in the CFT computations that follow.
We begin our study in the free field theory. The Hilbert space of possible excita-
tions can be written as a direct sum of subspaces. There are subspaces that collect the
excitations localized at the outer or inner edge of a given annulus, or at the outer edge
of the central disk. The excitations are obtained by adding boxes to the Young diagram
describing the background, at a specific location. They are also localized in the dual grav-
itational description, at a specific radius on the bubbling plane [21, 22]. Each localized
Hilbert space is labeled by the edge at which it is localized. There are also delocalized
excitations, where the description of the excitation involves adding boxes at different lo-
cations on the background Young diagram [21, 22]. We will not have much to say about
delocalized excitations.
The excitations belonging to the localized Hilbert spaces play a central role in our
study. These are the Hilbert spaces of the emergent gauge theories. We give a bijection
between the states belonging to the planar Hilbert space of an emergent gauge theory, and
the states of the planar limit of the original CFT without background. To show that the
bijection takes on a physical meaning, we argue that correlation functions of operators that
are in bijection are related in a particularly simple way, in the large N limit. This result
is significant because the basic observables of any quantum field theory are its correlation
functions and many properties of the theory can be phrased as statements about correlation
functions. Thanks to the map between correlation functions, any statement about the
planar limit that can be phrased in terms of correlators, immediately becomes a statement
about the planar emergent gauge theories that arise in the large N but non-planar limits
we consider.
2.1 Background dependence
Irreducible representations of the symmetric group S
n
are labeled by Young diagrams with
n boxes. States in the carrier space of the representation are labeled by standard tableau,
– 6 –
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