Homological Algebra and Divergent Series

“…As explained in [EHS22], this result can be thought of as a version of Kapranov's formulation of Koszul duality [Kap91], which relates D-modules and Ω • -modules on the same space, performed for the translation-invariant objects on a super Lie group; it is also related to Koszul duality for the graded Lie algebra t, followed by an associated-bundle construction. (Yet another relation of pure spinors to Koszul duality occurs when viewing the functions on Ŷ as a commutative algebra and considering the Koszul dual graded Lie algebra, as done in [MS04; GS09;Gál+16]; further work in this direction will appear in [CPS22]. )…”

Section: Introductionmentioning

confidence: 99%

Six-dimensional supermultiplets from bundles on projective spaces

Hahner¹,

Noja²,

Saberi³

et al. 2022

Preprint

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The projective variety of square-zero elements in the six-dimensional minimal supersymmetry algebra is isomorphic to P 1 × P 3 . We use this fact, together with the pure spinor superfield formalism, to study supermultiplets in six dimensions, starting from vector bundles on projective spaces. We classify all multiplets whose derived invariants for the supertranslation algebra form a line bundle over the nilpotence variety; one can think of such multiplets as being those whose holomorphic twists have rank one over Dolbeault forms on spacetime. In addition, we explicitly construct multiplets associated to natural higher-rank equivariant vector bundles, including the tangent and normal bundles as well as their duals.Among the multiplets constructed are the vector multiplet and hypermultiplet, the family of O(n)-multiplets, and the supergravity and gravitino multiplets. Along the way, we tackle various theoretical problems within the pure spinor superfield formalism. In particular, we give some general discussion about the relation of the projective nilpotence variety to multiplets and prove general results on short exact sequences and dualities of sheaves in the context of the pure spinor superfield formalism.

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“…If, however, A is not a complete intersection, then any resolving algebra R is infinitely generated in which case defining a vertex algebroid V(R) becomes problematic because of various divergencies. A regularization procedure for some of these divergencies was suggested in [4] and elaborated on in [17].…”

Section: Introductionmentioning

confidence: 99%