Approaches to Quantum Gravity

214 W. Taylorraising operators α−n. μ For the bosonic open string, the string field can then beexpanded as∫ = d 26 p [ ϕ(p) |p〉+A μ (p) α μ |p〉+···] −1. (12.3)The leading fields in this expansion are a space-time tachyon field ϕ(p) and amassless space-time vector field A μ (p).12.2.1 Witten’s cubic OSFT actionThe action proposed by Witten for the open bosonic string field theory takes thesimple cubic formS =− 1 ∫⋆Q − g ∫⋆⋆. (12.4)23In this action, g is the (open) string coupling constant. The field is the openstring field. Abstractly, this field can be considered to take values in an algebra A.Associated with the algebra A there is a star product⋆ : A ⊗ A → A, (12.5)The algebra A is graded, such that the open string field has degree G = 1, and thedegree G is additive under the star product (G ⋆ = G + G ). There is also anoperatorQ : A → A, (12.6)called the BRST operator, which is of degree one (G Q = 1 + G ). String fieldscan be integrated using∫: A → C . (12.7)This integral vanishes for all with degree G ̸= 3. Thus, the action (12.4) isonly nonvanishing for a string field of degree 1. The action (12.4) thus has thegeneral form of a Chern–Simons theory on a 3-manifold, although for string fieldtheory there is no explicit interpretation of the integration in terms of a concrete3-manifold.The elements Q,⋆, ∫ that define the string field theory are assumed to satisfythe following axioms.(a) Nilpotency of Q: Q 2 = 0, ∀ ∈ A.(b) ∫ Q = 0, ∀ ∈ A.(c) Derivation property of Q:Q( ⋆ ) = (Q) ⋆ + (−1) G ⋆(Q), ∀, ∈ A.