Pure spinor formalism as anN= 2 topological string

Abstract

Following suggestions of Nekrasov and Siegel, a non-minimal set of fields are added to the pure spinor formalism for the superstring. Twistedĉ = 3 N=2 generators are then constructed where the pure spinor BRST operator is the fermionic spin-one generator, and the formalism is interpreted as a critical topological string. Three applications of this topological string theory include the super-Poincaré covariant computation of multiloop superstring amplitudes without picture-changing operators, the construction of a cubic open superstring field theory without contact-term problems, and a new four-dimensional version of the pure spinor formalism which computes F-terms in the spacetime action.

Theta expansion of first massive vertex operator in pure spinor

“…The correlation functions of pure spinor fields are computed using the OPEs given in equation (2.8) and the identities of appendix C. 4 After this brief recollection of general pure spinor results, we now turn to unintegrated massive vertex operator.…”

Twistor origin of the superstring

“…In section 5, the "minimal" version of the pure spinor formalism will be derived using this procedure, and in section 6, the "nonminimal" version of the pure spinor formalism [47,48] will be derived by upgrading λ α from a constant pure spinor to a worldsheet variable.…”

“…To obtain the non-minimal pure spinor formalism [47,48] from gauge fixing, one upgrades the constant pure spinor λ α to a worldsheet variable and constrains its conjugate momentum w α to vanish by adding the term…”

“…Equations (6.4) and (6.5) are the gauge-fixed action and BRST operator of the non-minimal pure spinor formalism [47,48] where the term e − d 2 z (λ α λα+θ α rα) in e −S plays the role of a BRST-invariant regulator for integration over the zero modes of the pure spinors.…”

Covariant map between Ramond-Neveu-Schwarz and pure spinor formalisms for the superstring

“…Functional integration over the zero modes of the pure spinors (λ α , w α ) and (λ α , w α ) can be performed using the standard pure spinor regulator [31] N = e −{Q,θ α λα+ g where B is the pure spinor b ghost of (2.41). Since ( β, γ) have the same conformal weight as the (b, c) Virasoro ghosts, they have the same number of zero modes on the worldsheet.…”