Explaining the pure spinor formalism for the superstring

Abstract

After adding a pair of non-minimal fields and performing a similarity transformation, the BRST operator in the pure spinor formalism is expressed as a conventionallooking BRST operator involving the Virasoro constraint and (b, c) ghosts, together with 12 fermionic constraints. This BRST operator can be obtained by gauge-fixing the Green-Schwarz superstring where the 8 first-class and 8 second-class Green-Schwarz constraints are combined into 12 first-class constraints. Alternatively, the pure spinor BRST operator can be obtained from the RNS formalism by twisting the ten spin-half RNS fermions into five spin-one and five spin-zero fermions, and using the SO(10)/U (5) pure spinor variables to parameterize the different ways of twisting. GSO(−) vertex operators in the pure spinor formalism are constructed using spin fields and picture-changing operators in a manner analogous to Ramond vertex operators in the RNS formalism.

Twistor origin of the superstring

“…But despite several attempts [15][16][17][18], this pure spinor BRST operator was not obtained in a simple manner by gauge-fixing a worldsheet reparameterization invariant action.…”

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

“…In [8], a new approach to relating the RNS and pure spinor variables was proposed in which bosonization of the RNS ghost and matter fields is unnecessary. In this approach, one simply rescales the U(5) components (ψ A , ψ A ) of ψ m in opposite directions using the γ ghost as…”

“…Note that Γ m and Γ m each have five independent components since they satisfy Γ m (γ m λ) α = Γ m (γ m λ) α = 0, and were related in [8] to five components of θ α and p α . Although both the bosonization formulas of (1.3) and the twisting formulas of (1.4) and (1.5) map RNS spin-half fermions into GS-like spin-zero and spin-one fermions, the relation of the two maps is unclear.…”

“…This map manifestly preserves an SU(4) subgroup of the SO(8) light-cone symmetry and transforms the eight light-cone RNS vector variables ψ j into the eight lightcone GS spinor variables θ a . Splitting the SO(8) vector ψ j and SO (8) spinor θ a as (ψ J , ψ J ) and (θ J , θ J ) where J = 1 to 4 is an SU(4) index, the map of [1] is obtained by bosonizing ψ J = e iσ J , ψ J = e −iσ J (1.1)…”

Spinor and Hertzian Differential Forms in Electromagnetism

“…Writing P = ßB, M = ßE in agreement with (32) the relations (49) become the Post constitutive relations in a isotropic chiral medium [34], ß being the chirality parameter…”