String Theory and M-Theory

656 Gauge theory/string theory dualities
In the string description short multiplets arise as the five-dimensional
supergravity multiplet and all of its Kaluza–Klein excitations on the fivesphere.
The harmonics on the five-sphere give SU(4) irreducible representations
denoted (0, n, 0) in Dynkin notation. In SO(6) language, these correspond
to rank-n tensors that are totally symmetric and traceless. Clearly,
the helicities range from −2 to +2 for these multiplets, since the five-sphere
harmonic does not contribute to the helicity. All of the excited string states
belong to long multiplets, which are much more difficult to analyze. However,
it is possible to say something about a certain class of them in the
plane-wave limit, as is done in Section 12.5.
Let us now consider some local operators in the gauge theory that belong
to short multiplets. The SU(N) super Yang–Mills fields are described
as traceless N × N hermitian matrices. The way to form gauge-invariant
combinations is to consider traces of various products. The quantities that
are allowed inside the traces are the six scalars, four spinors, and Yang–
Mills field strength, as well as arbitrary covariant derivatives of these fields.
One can also consider products of such traces. However, it turns out that
single-trace operators correspond to single-particle states and multi-trace
operators correspond to multi-particle states in leading order. At higher
orders in λ and 1/N, there can be mixing between operators with differing
numbers of traces.
A convenient way of characterizing a supermultiplet is by finding the primary
operator of lowest dimension. By definition, this operator is annihilated
by all of the conformal symmetries Sα and Kµ. The other operators in
the supermultiplet are reached by commuting or anticommuting the primary
operator with the super-Poincaré generators Qα and Pµ. These operators
are called descendants and are characterized by the fact that they can be
expressed as Q acting on some operator. In the case of short multiplets,
the primary operator is also annihilated by half of the Q supersymmetry
generators. Such operators are called chiral primary operators.
As an example, consider the trace of a product of n scalar fields
O I1I2···In
= Tr φ I1
I2 In φ · · · φ . (12.122)
It turns out that if any of the indices are antisymmetrized this operator is a
descendant. A commutator [φI, φJ] is a descendant field because it appears
in the supersymmetry transformation of fermion fields. To understand this,
recall that in ten dimensions δψ ∼ FµνΓ µν ε. On reduction to four dimensions
FIJ → [φI, φJ].
The way to make a primary operator is to totally symmetrize all n indices