zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Infinite-dimensional Lie algebras in 4D conformal quantum field theory. (English) Zbl 1139.81052
Summary: The concept of global conformal invariance (GCI) opens the way of applying algebraic techniques, developed in the context of two-dimensional chiral conformal field theory, to a higher (even) dimensional spacetime. In particular, a system of GCI scalar fields of conformal dimension two gives rise to a Lie algebra of harmonic bilocal fields, $V_{M}(x, y)$, where the $M$ span a finite dimensional real matrix algebra $\cal M $ closed under transposition. The associative algebra $\cal M $ is irreducible iff its commutant $\cal M^{\prime} $ coincides with one of the three real division rings. The Lie algebra of (the modes of) the bilocal fields is in each case an infinite-dimensional Lie algebra: a central extension of $\text{sp}(\infty,{\mathbb R}) $ corresponding to the field ${{\mathbb R}} $ of reals, of $u(\infty , \infty )$ associated with the field ${{\mathbb C}} $ of complex numbers, and of $so^*(4\infty )$ related to the algebra ${{\mathbb H}} $ of quaternions. They give rise to quantum field theory models with superselection sectors governed by the (global) gauge groups $O(N), U(N)$ and $U(N,{{\mathbb H}})= \text{Sp}(2N) $, respectively.

81T40Two-dimensional field theories, conformal field theories, etc.
81R10Infinite-dimensional groups and algebras motivated by physics
22E70Applications of Lie groups to physics; explicit representations
81R12Relations of groups and algebras in quantum theory with integrable systems
Full Text: DOI