Gerbal representations of double loop groups (Q2917605)

For loop groups of reductive Lie groups and their Lie algebras, their nontrivial second cohomology classes give rise to their central extensions, which was proved to be very important in representation theory of loop groups and algebras. Loop groups embed into the group \(\mathrm{GL}_\infty\) of continuous automorphisms of \(\mathbb{C}((t))\), and their nontrivial second cohomology classes come from that of \(\mathrm{GL}_\infty\). The paper first gives a review of the central extensions of \(\mathrm{GL}_\infty\) and \(\mathfrak{gl}_\infty\) from different points of view -- the homological and geometric constructions, and describes a natural representation on a Fock module over an infinite-dimensional Clifford algebra. Then the authors consider the analog for double loop groups, which can be embedded into \(\mathrm{GL}_{\infty,\infty}\). Different from the case of \(\mathfrak{gl}_\infty\), the second cohomology of \(\mathfrak{gl}_{\infty,\infty}\) vanishes and the first nontrivial cohomology class occurs in the third cohomology, which, as the authors state, can be interpreted by the gerbal representations, i.e., the group actions on categories. The authors construct a gerbal representation of \(\mathrm{GL}_{\infty,\infty}\) on a category of modules over a Clifford algebra, discuss the relevant cohomology groups and show that there exists a particular nonzero cohomology class in the third cohomology corresponding to the gerbal representation.