𝒲-Algebra constraints and topological recursion for AN-singularity (with an Appendix by Danilo Lewanski)

We derive a Bouchard–Eynard type topological recursion for the total descendant potential of [Formula: see text]-singularity. Our argument relies on a certain twisted representation of a Heisenberg Vertex Operator Algebra (VOA) constructed via the periods of [Formula: see text]-singularity. In particular, our approach allows us to prove that the topological recursion for the total descendant potential is equivalent to a certain generating set of [Formula: see text]-algebra constraints.

-constraints for the total descendant potential of a simple singularity

Simple, or Kleinian, singularities are classified by Dynkin diagrams of type $ADE$. Let $\mathfrak {g}$ be the corresponding finite-dimensional Lie algebra, and $W$ its Weyl group. The set of $\mathfrak {g}$-invariants in the basic representation of the affine Kac–Moody algebra $\hat {\mathfrak {g}}$ is known as a $\mathcal {W}$-algebra and is a subalgebra of the Heisenberg vertex algebra $\mathcal {F}$. Using period integrals, we construct an analytic continuation of the twisted representation of $\mathcal {F}$. Our construction yields a global object, which may be called a $W$-twisted representation of $\mathcal {F}$. Our main result is that the total descendant potential of the singularity, introduced by Givental, is a highest-weight vector for the $\mathcal {W}$-algebra.

The Inkas as Tyrants: The Construction of a Twisted Representation

The chronicler Pedro Sarmiento de Gamboa wrote a history of the Inkas in 1571 proving that they had been tyrants. This was necessary to convince the King that he had every right to appropriate their lands, riches, and peoples. It is an excellent example of how a representation does not necessarily have to match what actually happened when political issues are at stake. My article analyzes that twisted representation and explores the motives Sarmiento had for kuraka to authenticate his version.