Abstract
Pure spinor formalism implies that supergravity equations in space-time are equivalent to the requirement that the worldsheet sigma-model satisfies certain properties. Here we point out that one of these properties has a particularly transparent geometrical interpretation. Namely, there exists an odd nilpotent vector field on some singular supermanifold, naturally associated to space-time. Is it true that all supergravity fields are encoded in this vector field, as coefficients in its normal form, and the nilpotence is equivalent to the target space equations of motion? We show that this is approximately correct. The normal form is parametrized by some tensor fields, which satisfy hyperbolic equations. These equations are slightly weaker than the full supergravity equations.
Highlights
In the low energy limit of superstring theory, spacetime fields satisfy supergravity (SUGRA)equations of motion, which are super-analogues of the Einstein equations
Pure spinor formalism implies that supergravity equations in space-time are equivalent to the requirement that the worldsheet sigma-model satisfies certain properties
There exists an odd nilpotent vector field on some singular supermanifold, naturally associated to space-time. Is it true that all supergravity fields are encoded in this vector field, as coefficients in its normal form, and the nilpotence is equivalent to the target space equations of motion? We show that this is approximately correct
Summary
In the low energy limit of superstring theory, spacetime fields satisfy supergravity (SUGRA). In the case of pure spinor string, the action of the BRST operator on matter fields and pure spinor ghosts does not contain worldsheet derivatives.. The space of fields in BV formalism is a Q-manifold. We point out that to a pure spinor sigma-model corresponds a finite-dimensional Q-manifold (its target space). An odd nilpotent vector field can be “simplified” by a clever choice of coordinates. ∂ ∂θ where θ is one of fermionic coordinates If it vanishes at some point, the normal form would be (in the notations of [2]). The vector Q vanishes precisely at the singular locus, and the problem of classification of normal forms is a nontrivial cohomological computation
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