Invariant differential operators. Volume 4: AdS/CFT, (super-)Virasoro, affine (super-)algebras (Q1717079)

This fourth volume continues in the style of the first three volumes (see Zbl 1418.17001 for Vol. 3), that is mainly listing various facts and examples without providing details of proofs. The volume mainly treats various infinite dimensional algebras and their representations. It consists of four chapters, an epilogue, bibliography and index. The first chapter addresses relativistic and non-relativistic holography. It starts by presenting the intertwining operator realization of the Euclidean AdS/CFT correspondence. Then it gives an exposition of the anti de Sitter holography on the example of the anti de Sitter group \(\mathrm{SO}(3,2)\). The last section describes nonrelativistic holography using the Schrödinger group. The second chapter describes nonrelativistic invariant differential operators and equations. It gives a general expression for the invariant differential equations in the case of \((n+1)\)-dimensional space-time. The case of the \((3+1)\)-dimensional space-time is considered in greater detail. The end of the chapter is devoted to the q-deformed situation, in particular, providing a construction of the \(q\)-Schrödinger algebras and describing the corresponding differential operators and equations. A difference analogues of the free Schrödinger equation is also presented. Chapter~3 discusses Virasoro and super-Virasoro algebra and some of their representation theory. The chapter discusses Verma modules and their analogues, Fock modules, unitarizable highest weight modules and various character formulae. Additionally, a number of sections touch upon properties of modular invariants for theta-functions, in particular, classification of modular invariant partition functions for the twisted \(N=2\) superconformal algebra and others. The final, fourth chapter is about affine Lie algebras and affine Lie superalgebras. It mainly discusses various properties of Verma modules and characters of singular highest weight modules. The epilogue presents some references for \(W\)-algebras, Yangians and cluster algebras.