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Abelian Extensions, Fractional Loop Group and Quantum Fields
KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
2010 (engelsk)Doktoravhandling, med artikler (Annet vitenskapelig)
Abstract [en]

This thesis deals with the theory of Lie group extensions, Lie conformal algebras and twisted K-theory, in the context of quantum physics. These structures allow for a mathematically precise description of certain aspects of interacting quantum field theories. We review three concrete examples, namely symmetry breaking (or anomalies) in gauge theory, classification of D-brane charges in string theory and the formulation of integrable hierarchies in the language of Poisson vertex algebras. The main results are presented in three appended scientific papers.

In the first paper we establish, by construction, a criterion for when an infinite dimensional abelian Lie algebra extension corresponds to a Lie group extension.

In the second paper we introduce the fractional loop group LqG, that is the group of maps from a circle to a compact Lie group G, with only a small degree of differentiability q ε R+ in the Sobolev sense. We construct abelian extensions and highest weight modules for the Lie algebra Lqg, and discuss an application to equivariant twisted K-theory on G.

In the third paper, we construct a structure of calculus algebra on the Lie conformal algebra complex and provide a more detailed description in the special case of the complex of variational calculus.

sted, utgiver, år, opplag, sider
Stockholm: KTH , 2010. , s. x, 95
Serie
Trita-FYS, ISSN 0280-316X ; 2010:13
Emneord [en]
Lie group extensions, Lie conformal algebras, gerbes, twisted K-theory, renormalization, anomalies, D-brane charges, variational complex
HSV kategori
Identifikatorer
URN: urn:nbn:se:kth:diva-12155ISBN: 978-91-7415-592-1 (tryckt)OAI: oai:DiVA.org:kth-12155DiVA, id: diva2:303818
Disputas
2010-03-26, FA32, Roslagstullsbacken 21, Albanova universitetscentrum, Stockholm, 13:00 (engelsk)
Opponent
Veileder
Merknad
QC 20100915Tilgjengelig fra: 2010-03-16 Laget: 2010-03-15 Sist oppdatert: 2010-09-15bibliografisk kontrollert
Delarbeid
1. Integrability Criterion for Abelian Extensions of Lie Groups
Åpne denne publikasjonen i ny fane eller vindu >>Integrability Criterion for Abelian Extensions of Lie Groups
2010 (engelsk)Inngår i: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 138, nr 11, s. 4137-4148Artikkel i tidsskrift (Fagfellevurdert) Published
Abstract [en]

We establish a criterion for when an abelian extension of infinite-dimensional Lie algebras (g) over cap = g circle plus(omega) a integrates to a corresponding Lie group extension A (sic) (G) over cap (sic) G, where G is connected, simply connected and A congruent to a/Gamma for some discrete subgroup Gamma subset of a. When pi(1) (G) not equal 0, the kernel A is replaced by a central extension (A) over cap of pi(1) (G) by A.

Emneord
Infinite-dimensional Lie theory, abelian extensions
HSV kategori
Identifikatorer
urn:nbn:se:kth:diva-8043 (URN)10.1090/S0002-9939-2010-10423-9 (DOI)000284022300037 ()2-s2.0-78149258029 (Scopus ID)
Merknad
QC 20100915. Uppdaterad från accepted till published (20101213).Tilgjengelig fra: 2008-02-27 Laget: 2008-02-27 Sist oppdatert: 2017-12-14bibliografisk kontrollert
2. Fractional Loop Group and Twisted K-theory
Åpne denne publikasjonen i ny fane eller vindu >>Fractional Loop Group and Twisted K-theory
2010 (engelsk)Inngår i: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 299, nr 3, s. 741-763Artikkel i tidsskrift (Fagfellevurdert) Published
Abstract [en]

We study the structure of abelian extensions of the group L (q) G of q-differentiable loops (in the Sobolev sense), generalizing from the case of the central extension of the smooth loop group. This is motivated by the aim of understanding the problems with current algebras in higher dimensions. Highest weight modules are constructed for the Lie algebra. The construction is extended to the current algebra of the supersymmetric Wess-Zumino-Witten model. An application to the twisted K-theory on G is discussed.

Emneord
unitary representations, quantization
HSV kategori
Identifikatorer
urn:nbn:se:kth:diva-8044 (URN)10.1007/s00220-010-1108-6 (DOI)000281711100007 ()2-s2.0-84755161215 (Scopus ID)
Merknad
QC 20100915 Uppdaterad från submitted till published (20101130).Tilgjengelig fra: 2008-02-27 Laget: 2008-02-27 Sist oppdatert: 2017-12-14bibliografisk kontrollert
3. Calculus structure on the Lie conformal algebra complex and the variational complex
Åpne denne publikasjonen i ny fane eller vindu >>Calculus structure on the Lie conformal algebra complex and the variational complex
2011 (engelsk)Inngår i: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 52, nr 5Artikkel i tidsskrift (Fagfellevurdert) Published
Abstract [en]

We construct a calculus structure on the Lie conformal algebra cochain complex. By restricting to degree one chains, we recover the structure of a g-complex introduced in [A. De Sole and V. G. Kac, Commun. Math. Phys. 292, 667 (2009)]. A special case of this construction is the variational calculus, for which we provide explicit formulas.

HSV kategori
Identifikatorer
urn:nbn:se:kth:diva-24542 (URN)10.1063/1.3580676 (DOI)000291106000035 ()
Merknad
QC 20100915. Updated from submitted to published 20120326. Previous title: Calculus structure on the Lie conformal algebra complexTilgjengelig fra: 2010-09-15 Laget: 2010-09-15 Sist oppdatert: 2017-12-12bibliografisk kontrollert

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