Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1149terms of a noncommutative extension of differential geometry, where stringfields, the BRST operator Q and the integration over the string configurations∫ in string field theory are analogs of differential forms, the exteriorderivative d and the integration over the manifold ∫ in the differentialMgeometry, respectively. The ghost number assigned to the string field correspondsto the degree of the differential form. Also the (noncommutative)product between the string fields ∗ is interpreted as an analog of the wedgeproduct ∧ between the differential forms.The axioms obeyed by the system of ∫ , ∗ and Q are∫QA = 0, Q(A ∗ B) = (QA) ∗ B + (−1) n AA ∗ (QB),∫∫(A ∗ B) ∗ C = A ∗ (B ∗ C), A ∗ B = (−1) n An BB ∗ A,where A, B and C are arbitrary string fields, whose ghost number is half–integer valued: The ghost number of A is defined by the integer n A asn A + 1 2 .Then Witten discussed the following string–field–theory actionS = 1 G s∫ ( 12 ψ ∗ Qψ + 1 3 ψ ∗ ψ ∗ ψ ), (6.236)where G s is the open string coupling constant and ψ is the string field withthe ghost number - 1 2. The action is invariant under the gauge transformationδψ = QΛ + ψ ∗ Λ − Λ ∗ ψ,with the gauge parameter Λ of the ghost number - 3 2 .6.5.9.1 Operator Formulation of String Field TheoryThe objects defined above can be explicitly constructed by using the operatorformulation, where the string field is represented as a Fock space, andthe integration ∫ as an inner product on the Fock space. It was consideredby [Gross and Jevicki(1987a); Gross and Jevicki(1987b)] in the case of theNeumann boundary condition. We will heavily use the notation of [Grossand Jevicki(1987a); Gross and Jevicki(1987b)]. In the operator formulation,the action (6.236) is described asS = 1 ( 1〈V 2 ||ψ〉 1 Q|ψ〉 2 + 1 )〈V 3 ||ψ〉 1 |ψ〉 2 |ψ〉 3 , (6.237)G s 2 123 123