Approaches to Quantum Gravity

(d) Cyclicity: ∫ ⋆= (−1) G G ∫⋆, ∀, ∈ A.String field theory 215(e) Associativity: ( ⋆ ) ⋆ = ⋆ ( ⋆ ), ∀, , ∈ A.When these axioms are satisfied, the action (12.4) is invariant under the gaugetransformationsδ = Q + ⋆− ⋆, (12.8)for any gauge parameter ∈ A with degree 0.When the string coupling g is taken to vanish, the equation of motion for thetheory defined by (12.4) simply becomes Q = 0, and the gauge transformations(12.8) simply becomeδ = Q. (12.9)This structure at g = 0 is precisely what is needed to describe a free bosonic stringin the BRST formalism, where physical states live in the cohomology of the BRSToperator Q, which acts on the string Fock space. 2 The motivation for introducingthe extra structure in (12.4) was to find a simple interacting extension of the freetheory, consistent with the perturbative expansion of open bosonic string theory.Witten presented this formal structure and argued that all the needed axioms aresatisfied when A is taken to be the space of string fields of the form (12.3). In thisrealization, the star product ⋆ acts on a pair of functionals , by gluing the righthalf of one string to the left half of the other using a delta function interaction.Similarly, the integral over a string field corresponds to gluing the left and righthalves of the string together with a delta function interaction.Combining these pictures, the three-string vertex ∫ 1 ⋆ 2 ⋆ 3 corresponds toa three-string overlap.2 For a detailed introduction to BRST string quantization, see [26]