On Generalizing the Partial Semi-Group of Spheres with Tubes in Huang’s Geometric Vertex Operator Algebra and the Algebraic Structures That Arise as a Consequence

We construct a geometric analogue to the sphere with tubes model where there is one incoming and one outgoing tube in Yi-Zhi Huang’s notion of a geometric vertex operator algebra (GVOA) in a generalized setting that resolves the multivaluedness resulting from a generalization of the grading axiom in a GVOA such that it is no longer semisimple; we call the objects in our geometric structure the unfurled worldsheets. We establish a partial semi-group structure on the unfurled worldsheets, and show that a certain associated tangent space forms an algebra that is not Leibniz. Then, we introduce a new piece of data to our notion of unfurled worldsheet, which results in a new collection of objects, called the unfurled worldsheets with dilation. We find that the unfurled worldsheets with dilation are not closed under our analogue of sewing. However, we can close the sewing by creating an extension of the Virasoro algebra. This extension scaffolds our formal identification of the moduli space of the unfurled worldsheets with dilation.