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Abelian Extensions, Fractional Loop Group and Quantum Fields
KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
2010 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
Stockholm: KTH , 2010. , s. x, 95
Serie
Trita-FYS, ISSN 0280-316X ; 2010:13
Nyckelord [en]
Lie group extensions, Lie conformal algebras, gerbes, twisted K-theory, renormalization, anomalies, D-brane charges, variational complex
Nationell ämneskategori
Annan fysik
Identifikatorer
URN: urn:nbn:se:kth:diva-12155ISBN: 978-91-7415-592-1 (tryckt)OAI: oai:DiVA.org:kth-12155DiVA, id: diva2:303818
Disputation
2010-03-26, FA32, Roslagstullsbacken 21, Albanova universitetscentrum, Stockholm, 13:00 (Engelska)
Opponent
Handledare
Anmärkning
QC 20100915Tillgänglig från: 2010-03-16 Skapad: 2010-03-15 Senast uppdaterad: 2010-09-15Bibliografiskt granskad
Delarbeten
1. Integrability Criterion for Abelian Extensions of Lie Groups
Öppna denna publikation i ny flik eller fönster >>Integrability Criterion for Abelian Extensions of Lie Groups
2010 (Engelska)Ingår i: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 138, nr 11, s. 4137-4148Artikel i tidskrift (Refereegranskat) 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.

Nyckelord
Infinite-dimensional Lie theory, abelian extensions
Nationell ämneskategori
Matematik
Identifikatorer
urn:nbn:se:kth:diva-8043 (URN)10.1090/S0002-9939-2010-10423-9 (DOI)000284022300037 ()2-s2.0-78149258029 (Scopus ID)
Anmärkning
QC 20100915. Uppdaterad från accepted till published (20101213).Tillgänglig från: 2008-02-27 Skapad: 2008-02-27 Senast uppdaterad: 2017-12-14Bibliografiskt granskad
2. Fractional Loop Group and Twisted K-theory
Öppna denna publikation i ny flik eller fönster >>Fractional Loop Group and Twisted K-theory
2010 (Engelska)Ingår i: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 299, nr 3, s. 741-763Artikel i tidskrift (Refereegranskat) 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.

Nyckelord
unitary representations, quantization
Nationell ämneskategori
Annan fysik
Identifikatorer
urn:nbn:se:kth:diva-8044 (URN)10.1007/s00220-010-1108-6 (DOI)000281711100007 ()2-s2.0-84755161215 (Scopus ID)
Anmärkning
QC 20100915 Uppdaterad från submitted till published (20101130).Tillgänglig från: 2008-02-27 Skapad: 2008-02-27 Senast uppdaterad: 2017-12-14Bibliografiskt granskad
3. Calculus structure on the Lie conformal algebra complex and the variational complex
Öppna denna publikation i ny flik eller fönster >>Calculus structure on the Lie conformal algebra complex and the variational complex
2011 (Engelska)Ingår i: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 52, nr 5Artikel i tidskrift (Refereegranskat) 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.

Nationell ämneskategori
Matematik
Identifikatorer
urn:nbn:se:kth:diva-24542 (URN)10.1063/1.3580676 (DOI)000291106000035 ()
Anmärkning
QC 20100915. Updated from submitted to published 20120326. Previous title: Calculus structure on the Lie conformal algebra complexTillgänglig från: 2010-09-15 Skapad: 2010-09-15 Senast uppdaterad: 2017-12-12Bibliografiskt granskad

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