Highest weight and boundary states of vertex operator algebra modules from Airy ideals

• Author / Creator

  Joshi, Aniket

• In this thesis we study some aspects of `Airy structures' first proposed in \cite{ks}, as an algebraic reformulation of the Chekhov-Eynard-Orantin (CEO) topological recursion initiated in \cite{eo} and \cite{eo1} in order to study the large $N$ expansion of matrix models.

  Our primary goal is to engineer new examples of Airy structures taking inspiration from the representation theory of vertex operator algebras. In particular, we construct a highest weight state of a $\mathcal{W}^{-N-1/2}(\mathfrak{sp}_{2N})$-algebra module as a partition function of an Airy structure, following the approach developed in \cite{Airy}. We do this with the help of an orbifold construction from symplectic fermions that was developed in \cite{cl}. Our second key result is the construction of certain Ishibashi boundary states related to affine vertex algebra modules from partition functions of Airy structures. We make use of the Wakimoto free field realizations of affine Lie algebras for this purpose. A novel aspect of both of these examples is that zero modes of the Heisenberg algebra are realized as derivatives instead of variables, and hence the partition functions are vectors that lie in infinite indecomposable extensions of Fock modules of free field algebras.

  On the other hand, we also give an alternate formulation of Airy structures as left ideals of $\hbar$-adic completions of Rees Weyl algebras. In particular, we obtain a different proof of the existence and uniqueness of partition functions of Airy structures by realizing Airy ideals' as homomorphic images of canonical left ideals generated by derivatives obtained via certain automorphisms of the Rees Weyl algebra calledtransvections'.

• Subjects / Keywords

  • vertex algebras
  • representation theory
  • airy ideals
  • enumerative geometry
  • string theory
  • airy structures
  • topological recursion
  • abstract algebra

• Graduation date

  Fall 2022

• Type of Item

  Thesis

• Degree

  Doctor of Philosophy

• DOI

  https://doi.org/10.7939/r3-d2p7-nk61

• License

  This thesis is made available by the University of Alberta Library with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.