[Paper Review] Emergence of Superstring from Pure Spinor

Ichiro Oda|arXiv (Cornell University)|Jun 17, 2011

Black Holes and Theoretical Physics17 references3 citations

TL;DR

This paper proposes that the superstring in the pure spinor formalism emerges from a topological field theory where only the pure spinor λα is a fundamental dynamical variable. Through BRST quantization of a topological symmetry and its reducible structure, the spacetime coordinates xm and fermionic θα variables arise as Faddeev-Popov ghosts, with the BRST cohomology reproducing the physical states of the Type II superstring, including supergravity in the massless sector.

ABSTRACT

Starting with a classical action where a pure spinor $λ^α$ is only a fundamental and dynamical variable, the pure spinor formalism for superparticle and superstring is derived by following the BRST formalism. In this formalism, not only the string variable $x^m$ but also the space-time spinor $θ^α$ are emerged as the Faddeev-Popov (FP) ghosts of a topological symmetry and its reducible symmetry. This study suggests that the fundamental theory behind the pure spinor formalism of the superstring might be a topological field theory.

Motivation & Objective

• To explore the origin of the pure spinor formalism of the superstring by treating it as emerging from a more fundamental theory.
• To investigate whether the superstring coordinates (xm, θα) can be derived as Faddeev-Popov ghosts via BRST quantization of a topological symmetry.
• To establish a connection between the pure spinor formalism and topological field theories by comparing degrees of freedom and conformal anomaly.
• To demonstrate that the BRST cohomology of the derived theory reproduces the physical spectrum of the Type II superstring, including supergravity.
• To suggest that the pure spinor formalism may originate from a topological quantum field theory, with the classical action being topological and invariant under a topological symmetry.

Proposed method

• Starting from a classical action with only the pure spinor λα as a dynamical variable, defined by Sc = ∫ dτ(ωα ˙λα + fαλα), where λα satisfies the pure spinor constraint λαγmαβλβ = 0.
• Performing canonical analysis to identify first-class constraints, including the secondary constraint λα ≈ 0, which renders the classical action topological.
• Identifying a topological symmetry generated by G = −εαπα + εαλα, with εα a bosonic parameter, and deriving its BRST transformation by replacing εα with a fermionic ghost pα.
• Gauging the topological symmetry and its reducible structure leads to the emergence of the spacetime bosonic coordinates xm and fermionic θα as Faddeev-Popov ghosts.
• Constructing a BRST-invariant action by introducing ghost fields pα and θα, and deriving the BRST charge QB from the nilpotent transformation rules.
• Analyzing the BRST cohomology to show that physical states are described by the cohomology of the pure spinor formalism, with massless states corresponding to Type II supergravity.

Experimental results

Research questions

• RQ1 Can the superstring in the pure spinor formalism be derived from a topological field theory where only the pure spinor λα is fundamental?
• RQ2 How do the spacetime coordinates xm and θα emerge as Faddeev-Popov ghosts through BRST quantization of a topological symmetry?
• RQ3 What is the role of reducible symmetries in the emergence of the superstring coordinates from a topological action?
• RQ4 Does the BRST cohomology of the derived theory reproduce the known physical spectrum of the Type II superstring?
• RQ5 Is there a correspondence in degrees of freedom and conformal anomaly between the topological field theory and the pure spinor superstring?

Key findings

• The classical action with only the pure spinor λα as a dynamical variable is topological, as the secondary constraint λα ≈ 0 renders the action vanishing, consistent with Witten-type topological field theories.
• The spacetime bosonic coordinates xm and fermionic coordinates θα emerge as Faddeev-Popov ghosts through the BRST quantization of a topological symmetry and its reducible structure.
• The BRST cohomology of the derived theory reproduces the physical states of the Type II superstring, with the massless physical state condition yielding the Type II supergravity equations.
• The theory has 32 bosonic and 32 fermionic degrees of freedom, resulting in a c = 0 conformal field theory, consistent with a topological field theory.
• The BRST charge QB is nilpotent, and the cohomology of QB corresponds to the cohomology of the pure spinor formalism, confirming the consistency of the construction.
• The world-line action of the superparticle is structurally similar to the Chern-Simons and BF theory world-line actions, suggesting a deeper connection between pure spinor theories and topological field theories.