The Algebra & Geometry of Modern Physics

In two dimensions there is a remarkable correspondence between certain quantum bosonic and fermionic theories. Originally it has been found in physics as an equivalence of a theory of a (massive) Thirring model (i.e. an interacting Dirac fermion) and the so-called bosonic sine-Gordon model, and it provides a prototype example of dualities in quantum field theories and string theory. Mathematically this correspondence gives rise to an isomorphism between certain vector spaces (i.e. bosonic and ferminic Fock spaces), which respectively form representations of infinite-dimensional Heisenberg and Clifford algebras; also beautiful and surprising links with representation theory, soliton equations, integrable hierarchies, etc. arise from this correspondence. In this talk this correspondence and some of its consequences will be reviewed.