160
7 CFT Systems
7.2
First-Order bc Ghost System
First-order systems describe two free fields called ghosts which have a first-order
action and whose conformal weights sum to 1. Commuting (resp. anti-commuting)
fields are often denoted by β and γ (resp. b and c) and correspondingly first-order
systems are also called βγ or bc systems. We will introduce a sign = ±1 to
denote the Grassmann parity of the fields and always write them as b and c. In
string theory, first-order systems describe the Faddeev–Popov ghosts associated with
reparametrizations and supersymmetries (Sects. 2.4 and 17.1).
7.2.1 Covariant Action
A first-order system is defined by two symmetric and traceless fields b μ 1 ···μ λ and
c μ 1 ···μ λ−1 called ghosts. For fields of integer spins, the dynamics is governed by the
first-order action
S =
1
4π
d
2 x
√ g g
μν b μμ 1 ···μ λ−1 ∇ ν c
μ 1 ···μ λ−1
(7.93)
after taking into account the symmetries of the field indices. Obviously, for λ =
2, one recovers the reparametrization ghost action (2.145). The action (7.93) is
invariant under Weyl transformations (the fields and covariant derivatives are inert)
such that it describes a CFT on flat space.
When the fields have half-integer spins (and often denoted as β and γ in this
case), they carry a spinor index. In this case, the action contains a Dirac matrix, and
the covariant derivative a spin connection.
The ghost action (7.93) is invariant under a global U(1) symmetry
b μ 1 ···μ n −→ e
−iθ b μ 1 ···μ n ,
c
μ 1 ···μ n−1 −→ e
iθ c
μ 1 ···μ n−1 .
(7.94)
7.2.2 Action on the Complex Plane
The simplest description of the system is on the complex plane. Due to the
conditions imposed on the fields, they have only two independent components for
all n, and the equations of motion imply that one is holomorphic, and the other
anti-holomorphic:
b(z) := b z···z (z),
¯
b(¯ z) := b ¯
z···¯ z (¯ z),
c(z) := c
z···z (¯ z),
¯
c(¯ z) := c
¯
z···¯ z (z).
(7.95)
7 CFT Systems
7.2
First-Order bc Ghost System
First-order systems describe two free fields called ghosts which have a first-order
action and whose conformal weights sum to 1. Commuting (resp. anti-commuting)
fields are often denoted by β and γ (resp. b and c) and correspondingly first-order
systems are also called βγ or bc systems. We will introduce a sign = ±1 to
denote the Grassmann parity of the fields and always write them as b and c. In
string theory, first-order systems describe the Faddeev–Popov ghosts associated with
reparametrizations and supersymmetries (Sects. 2.4 and 17.1).
7.2.1 Covariant Action
A first-order system is defined by two symmetric and traceless fields b μ 1 ···μ λ and
c μ 1 ···μ λ−1 called ghosts. For fields of integer spins, the dynamics is governed by the
first-order action
S =
1
4π
d
2 x
√ g g
μν b μμ 1 ···μ λ−1 ∇ ν c
μ 1 ···μ λ−1
(7.93)
after taking into account the symmetries of the field indices. Obviously, for λ =
2, one recovers the reparametrization ghost action (2.145). The action (7.93) is
invariant under Weyl transformations (the fields and covariant derivatives are inert)
such that it describes a CFT on flat space.
When the fields have half-integer spins (and often denoted as β and γ in this
case), they carry a spinor index. In this case, the action contains a Dirac matrix, and
the covariant derivative a spin connection.
The ghost action (7.93) is invariant under a global U(1) symmetry
b μ 1 ···μ n −→ e
−iθ b μ 1 ···μ n ,
c
μ 1 ···μ n−1 −→ e
iθ c
μ 1 ···μ n−1 .
(7.94)
7.2.2 Action on the Complex Plane
The simplest description of the system is on the complex plane. Due to the
conditions imposed on the fields, they have only two independent components for
all n, and the equations of motion imply that one is holomorphic, and the other
anti-holomorphic:
b(z) := b z···z (z),
¯
b(¯ z) := b ¯
z···¯ z (¯ z),
c(z) := c
z···z (¯ z),
¯
c(¯ z) := c
¯
z···¯ z (z).
(7.95)