Abstract
We show how to map Grothendieck’s dessins d’enfants to algebraic curves as Seiberg-Witten curves, then use the mirror map and the AGT map to obtain the corresponding 4d mathcal{N} = 2 supersymmetric instanton partition functions and 2d Virasoro conformal blocks. We explicitly demonstrate the 6 trivalent dessins with 4 punctures on the sphere. We find that the parametrizations obtained from a dessin should be related by certain duality for gauge theories. Then we will discuss that some dessins could correspond to conformal blocks satisfying certain rules in different minimal models.
Highlights
ArXiv ePrint: 2101.08843 Dedicated to the memory of our dear friend,Professor Omar Foda, a gentleman and a scholar
We show how to map Grothendieck’s dessins d’enfants to algebraic curves as Seiberg-Witten curves, use the mirror map and the AGT map to obtain the corresponding 4d N = 2 supersymmetric instanton partition functions and 2d Virasoro conformal blocks
We focus on six specific trivalent dessins with 4 punctures on the sphere, which, as we will see, are related to a simple and important class of 4d N = 2 SYM theories and conformal blocks in 2d conformal field theory
Summary
Conformal blocks form a basis of the vertex operator (VO) algebra, used when performing a particular operator product expansion (OPE) of a correlation function. The AGT correspondence makes a connection between the conformal dimensions of the fields in the correlator, and the coordinate ζ, with parameters arising in Nekrasov instanton partition functions, as subsequently described. The instanton partition function is equal to Bαint(αi|ζ), where the conformal block from Vα1 Vα2 Vα3 Vα4 as in (2.1) can be written as B = ζ∆αint −∆α1 −∆α2 Bαint (αi|ζ) and “int” stands for (the VO in) the intermediate channel One may check this perturbatively, and at level |Y |max, Bαint and Zinst should agree up to O ζ|Y |max+1 [1, 21]. Under the 4d limit R → 0, the 4d topological A-model partition function would give the 4d instanton partition function
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
