Aspects of Vertex Algebras in Geometry and Physics

• Author / Creator

  Riedler, Wolfgang

• This thesis studies various aspects of the theory of vertex algebras.
  It has been shown that the moonshine module for Conway’s group C0 has close ties to the equivariant elliptic genera of sigma models with a K3 surface as target space. This is taken as a motivation to investigate conditions under which a self-dual vertex operator superalgebra and the bulk Hilbert space of a superconformal field theory may be identified. To that end a classification of self-dual vertex operator superalgebras with central charge less than or equal to 12 is given and several examples of how these vertex algebras can be related to bulk superconformal field theories are provided. This includes field theories which arise from sigma models where the target space is a torus or a K3 surface.
  Following this, we study orbifolds and cosets of the small N = 4 super-conformal algebra. Minimal strong generators for generic and specific levels are found and as a corollary we obtain the vertex algebra of global sections of the chiral de Rham complex on any complex Enriques surface. The com- mutant Com(V l(sl2), V l+1(sl2) ⊗ W−5/2(sl4, frect)) is identified with orbifolds of cosets of the small N = 4 superconformal algebra which, in addition, can be identified with Grassmannian cosets and principal W-algebras of type A at special levels. We conclude by proving a new level-rank duality which includes Grassmannian supercosets.
  Furthermore, we provide a constructive proof of existence of an embedding of the Odake vertex algebra into a lattice vertex algebra in any dimension. In addition, we show that the elliptic genus of this family of lattice vertex algebras at hand is non-vanishing if and only if the dimension does not equal 1.
  Finally, we investigate conformal embeddings of maximal affine vertex algebras into rectangular W-algebras at admissible levels. We prove that such W-algebras are conditionally isomorphic to affine vertex algebras at boundary admissible levels for cases of type A, B, C and D.

• Subjects / Keywords

  • moonshine
  • chiral de rham
  • string theory
  • vertex algebra

• Graduation date

  Spring 2020

• Type of Item

  Thesis

• Degree

  Doctor of Philosophy

• DOI

  https://doi.org/10.7939/r3-d3x4-nt13

• License

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