This summer, for 8 weeks, the undergraduate student James Goodwin (year 2->3) has worked on the Nuffield Foundation funded project "The Yangian on a distinguished Dynkin diagram for the AdS/CFT correspondence", funded by the Nuffield Foundation. This project involved the study of quantum group symmetries and integrable structures in the context of the AdS/CFT correspondence, in a team composed of himself and Alessandro Torrielli.   

The AdS/CFT correspondence states the equivalence between two different models of Mathematical Physics. One model involves strings, the other involves a generalization of the electromagnetic forces. While the consequences of this equivalence can be staggering for our understanding of the fundamental interactions of nature, its mathematical proof is still lacking. A crucial role is likely to be played by certain exactly-solvable quantum-mechanical systems, effectively describing specific sectors of the two models. These systems are characterized by their symmetries. This project studied the infinite dimensional algebra of symmetries of these systems (the so-called Yangian algebra), trying to identify the most suitable basis for its description (bases are associated to the so-called Dynkin diagrams). James focused on a particular class of diagrams (dubbed ‘distinguished’) which still lack a full Yangian generalization, and compared their features with those explored in the literature so far.

To attack this problem it has been necessary for James to take a significant digression into Lie algebras and Lie superalgebras (which are ultimately the algebras one needs) and their associated quantum groups, and to familiarize both with mathematical concepts like the Chevalley-Serre presentation and Drinfeld’s second realization of Yangians, and with tools of (super-) matrix representation theory. James' brilliant work brought the team to provide by the end of week 8 a tentative answer, by guessing a Yangian representation with 6 free parameters and reducing them down to one. The team is thinking of possibly refining the results and collecting them in a preprint in the near future. If you would like to know more, please contact Alessandro.