10
Free BRST String Field Theory
Abstract
In this chapter, we construct the BRST (or covariant) free bosonic string field
theories. It is useful to first ignore the interactions in order to introduce some
general tools and structures in a simpler setting. Moreover, the free SFT is easily
constructed and does not require as much input as the interactions. In this chapter,
we discuss mostly the open string, keeping the closed string for the last section.
We start by describing the classical theory: equations of motion, action, gauge
invariance and gauge fixing. Then, we perform the path integral quantization and
compute the action in terms of spacetime fields for the first two levels (tachyon
and gauge field).
10.1 Classical Action for the Open String
While the rest of this book discusses the closed string only, the ideas of this chapter
are simpler to introduce for the open string. For this reason, we make an exception
and start this chapter with the open string. There are also a few subtleties which
can be more easily explained once the general structure is understood. Everything
needed for the open string for this chapter can be found in Chap. 8; in fact, describing
the open string (at this level) is equivalent to considering only the holomorphic
sector of the CFT and setting p L = p (instead of p/2). We consider a generic
matter CFT in addition to the ghost system, and we denote as H the space of states.
The open and closed string fields are denoted, respectively, by and , such that it
is clear which theory is studied.
An action can be either constructed from first principles or derived from the
equations of motion. Since the fundamental structure of string field theory is not
(really) known, one needs to rely on the second approach. But do we already know
the (free) equations of motion for the string field? The answer is yes. But, before
© Springer Nature Switzerland AG 2021
H. Erbin, String Field Theory, Lecture Notes in Physics 980,
https://doi.org/10.1007/978-3-030-65321-7_10
211
Free BRST String Field Theory
Abstract
In this chapter, we construct the BRST (or covariant) free bosonic string field
theories. It is useful to first ignore the interactions in order to introduce some
general tools and structures in a simpler setting. Moreover, the free SFT is easily
constructed and does not require as much input as the interactions. In this chapter,
we discuss mostly the open string, keeping the closed string for the last section.
We start by describing the classical theory: equations of motion, action, gauge
invariance and gauge fixing. Then, we perform the path integral quantization and
compute the action in terms of spacetime fields for the first two levels (tachyon
and gauge field).
10.1 Classical Action for the Open String
While the rest of this book discusses the closed string only, the ideas of this chapter
are simpler to introduce for the open string. For this reason, we make an exception
and start this chapter with the open string. There are also a few subtleties which
can be more easily explained once the general structure is understood. Everything
needed for the open string for this chapter can be found in Chap. 8; in fact, describing
the open string (at this level) is equivalent to considering only the holomorphic
sector of the CFT and setting p L = p (instead of p/2). We consider a generic
matter CFT in addition to the ghost system, and we denote as H the space of states.
The open and closed string fields are denoted, respectively, by and , such that it
is clear which theory is studied.
An action can be either constructed from first principles or derived from the
equations of motion. Since the fundamental structure of string field theory is not
(really) known, one needs to rely on the second approach. But do we already know
the (free) equations of motion for the string field? The answer is yes. But, before
© Springer Nature Switzerland AG 2021
H. Erbin, String Field Theory, Lecture Notes in Physics 980,
https://doi.org/10.1007/978-3-030-65321-7_10
211