Abstract
In this paper, we identify the Ad-equivariant twisted $K$-theory of a compact Lie group $G$ with the “Verlinde group” of isomorphism classes of admissible representations of its loop groups. Our identification preserves natural module structures over the representation ring $R(G)$ and a natural duality pairing. Two earlier papers in the series covered foundations of twisted equivariant $K$-theory, introduced distinguished families of Dirac operators and discussed the special case of connected groups with free $\pi_1$. Here, we recall the earlier material as needed to make the paper self-contained. Going further, we discuss the relation to semi-infinite cohomology, the fusion product of conformal field theory, the rôle of energy and a topological Peter-Weyl theorem.
-
[atsing]
M. F. Atiyah and I. M. Singer, "Index theory for skew-adjoint Fredholm operators," Inst. Hautes Études Sci. Publ. Math., iss. 37, pp. 5-26, 1969.
@article {atsing, MRKEY = {0285033},
AUTHOR = {Atiyah, M. F. and Singer, I. M.},
TITLE = {Index theory for skew-adjoint {F}redholm operators},
JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
NUMBER = {37},
YEAR = {1969},
PAGES = {5--26},
ISSN = {0073-8301},
MRCLASS = {57.50},
MRNUMBER = {0285033},
MRREVIEWER = {F. Hirzebruch},
URL = {http://www.numdam.org/item?id=PMIHES_1969__37__5_0},
ZBLNUMBER = {0194.55503},
} -
[abs]
M. F. Atiyah, R. Bott, and A. Shapiro, "Clifford modules," Topology, vol. 3, iss. suppl. 1, pp. 3-38, 1964.
@article {abs, MRKEY = {0167985},
AUTHOR = {Atiyah, M. F. and Bott, R. and Shapiro, A.},
TITLE = {Clifford modules},
JOURNAL = {Topology},
FJOURNAL = {Topology. An International Journal of Mathematics},
VOLUME = {3},
YEAR = {1964},
NUMBER = {suppl. 1},
PAGES = {3--38},
ISSN = {0040-9383},
MRCLASS = {57.30 (55.30)},
MRNUMBER = {0167985},
MRREVIEWER = {J. F. Adams},
DOI = {10.1016/0040-9383(64)90003-5},
ZBLNUMBER = {0146.19001},
} -
[atpatsing]
M. F. Atiyah, V. K. Patodi, and I. M. Singer, "Spectral asymmetry and Riemannian geometry. I," Math. Proc. Cambridge Philos. Soc., vol. 77, pp. 43-69, 1975.
@article {atpatsing, MRKEY = {0397797},
AUTHOR = {Atiyah, M. F. and Patodi, V. K. and Singer, I. M.},
TITLE = {Spectral asymmetry and {R}iemannian geometry. {I}},
JOURNAL = {Math. Proc. Cambridge Philos. Soc.},
FJOURNAL = {Mathematical Proceedings of the Cambridge Philosophical Society},
VOLUME = {77},
YEAR = {1975},
PAGES = {43--69},
ISSN = {0305-0041},
MRCLASS = {58G10 (57E15 57D85)},
MRNUMBER = {0397797},
MRREVIEWER = {Kh. Knapp},
DOI = {10.1017/S0305004100049410},
ZBLNUMBER = {0297.58008},
} -
[black] B. Blackadar, $K$-Theory for Operator Algebras, New York: Springer-Verlag, 1986, vol. 5.
@book {black, MRKEY = {0859867},
AUTHOR = {Blackadar, Bruce},
TITLE = {{$K$}-Theory for Operator Algebras},
SERIES = {Math. Sci. Research Inst. Publ.},
VOLUME = {5},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1986},
PAGES = {viii+338},
ISBN = {0-387-96391-X},
MRCLASS = {46L80 (18F25 19Kxx 46M20 58G12)},
MRNUMBER = {0859867},
MRREVIEWER = {Vern Paulsen},
ZBLNUMBER = {0597.46072},
} -
[borel]
A. Borel, "Sous-groupes commutatifs et torsion des groupes de Lie compacts connexes," Tôhoku Math. J., vol. 13, pp. 216-240, 1961.
@article {borel, MRKEY = {0147579},
AUTHOR = {Borel, Armand},
TITLE = {Sous-groupes commutatifs et torsion des groupes de {L}ie compacts connexes},
JOURNAL = {Tôhoku Math. J.},
FJOURNAL = {The Tohoku Mathematical Journal. Second Series},
VOLUME = {13},
YEAR = {1961},
PAGES = {216--240},
ISSN = {0040-8735},
MRCLASS = {22.50 (57.40)},
MRNUMBER = {0147579},
MRREVIEWER = {B. Harris},
DOI = {10.2748/tmj/1178244298},
ZBLNUMBER = {0109.26101},
} -
@article {bott, MRKEY = {0089473},
AUTHOR = {Bott, Raoul},
TITLE = {Homogeneous vector bundles},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {66},
YEAR = {1957},
PAGES = {203--248},
ISSN = {0003-486X},
MRCLASS = {53.3X},
MRNUMBER = {0089473},
MRREVIEWER = {M. F. Atiyah},
DOI = {10.2307/1969996},
ZBLNUMBER = {0094.35701},
} -
[td] T. Bröcker and T. tom Dieck, Representations of Compact Lie Groups, New York: Springer-Verlag, 1995, vol. 98.
@book {td, MRKEY = {1410059},
AUTHOR = {Br{ö}cker, Theodor and tom Dieck, Tammo},
TITLE = {Representations of Compact {L}ie Groups},
SERIES = {Grad. Texts in Math.},
VOLUME = {98},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1995},
PAGES = {x+313},
ISBN = {0-387-13678-9},
MRCLASS = {22E45 (22-01 57-01)},
MRNUMBER = {1410059},
ZBLNUMBER = {0874.22001},
} -
[bunk] U. Bunke and I. Schröder, "Twisted $K$-theory and TQFT," in Mathematisches Institut, Georg-August-Universität Göttingen: Seminars Winter Term 2004/2005, Universitätsdrucke Göttingen, Göttingen, 2005, pp. 33-80.
@incollection {bunk, MRKEY = {2206878},
AUTHOR = {Bunke, U. and Schr{ö}der, I.},
TITLE = {Twisted {$K$}-theory and {TQFT}},
BOOKTITLE = {Mathematisches {I}nstitut, {G}eorg-{A}ugust-{U}niversität {G}öttingen: {S}eminars {W}inter {T}erm 2004/2005},
PAGES = {33--80},
PUBLISHER = {Universitätsdrucke Göttingen, Göttingen},
YEAR = {2005},
MRCLASS = {19L47 (57R56 81T45)},
MRNUMBER = {2206878},
MRREVIEWER = {Jean-Louis Tu},
ZBLNUMBER = {1162.19300},
} -
[turkfreed] D. S. Freed, "The Verlinde algebra is twisted equivariant $K$-theory," Turkish J. Math., vol. 25, iss. 1, pp. 159-167, 2001.
@article {turkfreed, MRKEY = {1829086},
AUTHOR = {Freed, Daniel S.},
TITLE = {The {V}erlinde algebra is twisted equivariant {$K$}-theory},
JOURNAL = {Turkish J. Math.},
FJOURNAL = {Turkish Journal of Mathematics},
VOLUME = {25},
YEAR = {2001},
NUMBER = {1},
PAGES = {159--167},
ISSN = {1300-0098},
MRCLASS = {19L47 (55N91 57R56)},
MRNUMBER = {1829086},
MRREVIEWER = {Stratos Prassidis},
ZBLNUMBER = {0971.55006},
} -
[freedicm] D. S. Freed, "Twisted $K$-theory and loop groups," in Proceedings of the International Congress of Mathematicians, Vol. III, Beijing, 2002, pp. 419-430.
@inproceedings {freedicm, MRKEY = {1957550},
AUTHOR = {Freed, Daniel S.},
TITLE = {Twisted {$K$}-theory and loop groups},
BOOKTITLE = {Proceedings of the {I}nternational {C}ongress of {M}athematicians, {V}ol. {\rm III}},
VENUE={{B}eijing, 2002},
PAGES = {419--430},
PUBLISHER = {Higher Ed. Press},
ADDRESS = {Beijing},
YEAR = {2002},
MRCLASS = {19L47 (22E67 81T45)},
MRNUMBER = {1957550},
MRREVIEWER = {Adrian Clingher},
ZBLNUMBER = {0997.19004},
} -
[ff]
B. L. Feuigin and E. V. Frenkel, "Affine Kac-Moody algebras and semi-infinite flag manifolds," Comm. Math. Phys., vol. 128, iss. 1, pp. 161-189, 1990.
@article {ff, MRKEY = {1042449},
AUTHOR = {Fe{\u\i}gin, Boris L. and Frenkel, Edward V.},
TITLE = {Affine {K}ac-{M}oody algebras and semi-infinite flag manifolds},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {128},
YEAR = {1990},
NUMBER = {1},
PAGES = {161--189},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {17B67 (14M15 17B55)},
MRNUMBER = {1042449},
MRREVIEWER = {Chong Ying Dong},
DOI={10.1007/BF02097051},
ZBLNUMBER = {0722.17019},
} -
[fgz] I. B. Frenkel, H. Garland, and G. J. Zuckerman, "Semi-infinite cohomology and string theory," Proc. Nat. Acad. Sci. U.S.A., vol. 83, iss. 22, pp. 8442-8446, 1986.
@article {fgz, MRKEY = {0865483},
AUTHOR = {Frenkel, I. B. and Garland, H. and Zuckerman, G. J.},
TITLE = {Semi-infinite cohomology and string theory},
JOURNAL = {Proc. Nat. Acad. Sci. U.S.A.},
FJOURNAL = {Proceedings of the National Academy of Sciences of the United States of America},
VOLUME = {83},
YEAR = {1986},
NUMBER = {22},
PAGES = {8442--8446},
ISSN = {0027-8424},
CODEN = {PNASA6},
MRCLASS = {17B65 (17B55 17B67 81E30 81E99)},
MRNUMBER = {0865483},
MRREVIEWER = {J. Stasheff},
ZBLNUMBER = {0607.17007},
} -
[fht1] D. S. Freed, M. J. Hopkins, and C. Teleman, Loop groups and twisted ${K}$-theory I.
@misc{fht1,
author={Freed, Daniel S. and Hopkins, Michael J. and Teleman, Constantin},
TITLE = {Loop groups and twisted ${K}$-theory {I}},
NOTE={preprint},
ARXIV={0711.1906},
} -
[fht2] D. S. Freed, M. J. Hopkins, and C. Teleman, Loop groups and twisted ${K}$-theory II.
@misc{fht2,
author={Freed, Daniel S. and Hopkins, Michael J. and Teleman, Constantin},
TITLE = {Loop groups and twisted ${K}$-theory {II}},
NOTE={preprint},
ARXIV={math.AT/0511232},
} -
[fht3]
D. S. Freed, M. J. Hopkins, and C. Teleman, "Twisted equivariant $K$-theory with complex coefficients," J. Topol., vol. 1, iss. 1, pp. 16-44, 2008.
@article {fht3, MRKEY = {2365650},
AUTHOR = {Freed, Daniel S. and Hopkins, Michael J. and Teleman, Constantin},
TITLE = {Twisted equivariant {$K$}-theory with complex coefficients},
JOURNAL = {J. Topol.},
FJOURNAL = {Journal of Topology},
VOLUME = {1},
YEAR = {2008},
NUMBER = {1},
PAGES = {16--44},
ISSN = {1753-8416},
MRCLASS = {19L10 (19L47 55N15 55N25 57T10)},
MRNUMBER = {2365650},
MRREVIEWER = {Francis Clarke},
DOI = {10.1112/jtopol/jtm001},
ZBLNUMBER = {1188.19005},
} -
[fht4]
D. S. Freed, M. J. Hopkins, and C. Teleman, "Consistent orientation of moduli spaces," in The Many Facets of Geometry, Oxford: Oxford Univ. Press, 2010, pp. 395-419.
@incollection {fht4, MRKEY = {2681705},
AUTHOR = {Freed, Daniel S. and Hopkins, Michael J. and Teleman, Constantin},
TITLE = {Consistent orientation of moduli spaces},
BOOKTITLE = {The Many Facets of Geometry},
PAGES = {395--419},
PUBLISHER = {Oxford Univ. Press},
ADDRESS = {Oxford},
YEAR = {2010},
MRCLASS = {57R56 (19L50 58D29)},
MRNUMBER = {2681705},
MRREVIEWER = {Andrey Yu. Lazarev},
DOI = {10.1093/acprof:oso/9780199534920.003.0019},
ZBLNUMBER = {05853120},
} -
[gar]
H. Garland and J. Lepowsky, "Lie algebra homology and the Macdonald-Kac formulas," Invent. Math., vol. 34, iss. 1, pp. 37-76, 1976.
@article {gar, MRKEY = {0414645},
AUTHOR = {Garland, Howard and Lepowsky, James},
TITLE = {Lie algebra homology and the {M}acdonald-{K}ac formulas},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {34},
YEAR = {1976},
NUMBER = {1},
PAGES = {37--76},
ISSN = {0020-9910},
MRCLASS = {17B55},
MRNUMBER = {0414645},
MRREVIEWER = {S. I. Gelfand},
DOI = {10.1007/BF01418970},
ZBLNUMBER = {0358.17015},
} -
[guilstern]
V. Guillemin and S. Sternberg, "Geometric quantization and multiplicities of group representations," Invent. Math., vol. 67, iss. 3, pp. 515-538, 1982.
@article {guilstern, MRKEY = {0664118},
AUTHOR = {Guillemin, V. and Sternberg, S.},
TITLE = {Geometric quantization and multiplicities of group representations},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {67},
YEAR = {1982},
NUMBER = {3},
PAGES = {515--538},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {58F06 (22E99 81D99)},
MRNUMBER = {0664118},
MRREVIEWER = {I. Vaisman},
DOI = {10.1007/BF01398934},
ZBLNUMBER = {0503.58018},
} -
[kac]
V. G. Kac, Infinite-Dimensional Lie Algebras, Third ed., Cambridge: Cambridge Univ. Press, 1990.
@book {kac, MRKEY = {1104219},
AUTHOR = {Kac, Victor G.},
TITLE = {Infinite-Dimensional {L}ie Algebras},
EDITION = {Third},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {1990},
PAGES = {xxii+400},
ISBN = {0-521-37215-1; 0-521-46693-8},
MRCLASS = {17B65 (17B67 17B68 58F07)},
MRNUMBER = {1104219},
DOI = {10.1017/CBO9780511626234},
ZBLNUMBER = {0716.17022},
} -
[kost1]
B. Kostant, "A cubic Dirac operator and the emergence of Euler number multiplets of representations for equal rank subgroups," Duke Math. J., vol. 100, iss. 3, pp. 447-501, 1999.
@article {kost1, MRKEY = {1719734},
AUTHOR = {Kostant, Bertram},
TITLE = {A cubic {D}irac operator and the emergence of {E}uler number multiplets of representations for equal rank subgroups},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {100},
YEAR = {1999},
NUMBER = {3},
PAGES = {447--501},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {22E46 (19L64 20C35 58J05)},
MRNUMBER = {1719734},
MRREVIEWER = {A. L. Onishchik},
DOI = {10.1215/S0012-7094-99-10016-0},
ZBLNUMBER = {0952.17005},
} -
[kost2]
B. Kostant, "A generalization of the Bott-Borel-Weil theorem and Euler number multiplets of representations. Conference Moshé Flato 1999 (Dijon)," Lett. Math. Phys., vol. 52, iss. 1, pp. 61-78, 2000.
@article {kost2, MRKEY = {1800491},
AUTHOR = {Kostant, Bertram},
TITLE = {A generalization of the {B}ott-{B}orel-{W}eil theorem and {E}uler number multiplets of representations. {C}onference {M}osh{é} {F}lato 1999 ({D}ijon)},
JOURNAL = {Lett. Math. Phys.},
FJOURNAL = {Letters in Mathematical Physics. A Journal for the Rapid Dissemination of Short Contributions in the Field of Mathematical Physics},
VOLUME = {52},
YEAR = {2000},
NUMBER = {1},
PAGES = {61--78},
ISSN = {0377-9017},
CODEN = {LMPHDY},
MRCLASS = {22E46},
MRNUMBER = {1800491},
MRREVIEWER = {Stephen Slebarski},
DOI = {10.1023/A:1007653819322},
} -
[koststern] B. Kostant and S. Sternberg, "Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras," Ann. Physics, vol. 176, iss. 1, pp. 49-113, 1987.
@article {koststern, MRKEY = {0893479},
AUTHOR = {Kostant, Bertram and Sternberg, Shlomo},
TITLE = {Symplectic reduction, {BRS} cohomology, and infinite-dimensional {C}lifford algebras},
JOURNAL = {Ann. Physics},
FJOURNAL = {Annals of Physics},
VOLUME = {176},
YEAR = {1987},
NUMBER = {1},
PAGES = {49--113},
ISSN = {0003-4916},
CODEN = {ADNYA6},
MRCLASS = {58F05 (15A66 17B56 58F06 81D07 81E99)},
MRNUMBER = {0893479},
MRREVIEWER = {Andrew Pressley},
ZBLNUMBER = {0642.17003},
} -
[kumar]
S. Kumar, "Demazure character formula in arbitrary Kac-Moody setting," Invent. Math., vol. 89, iss. 2, pp. 395-423, 1987.
@article {kumar, MRKEY = {0894387},
AUTHOR = {Kumar, Shrawan},
TITLE = {Demazure character formula in arbitrary {K}ac-{M}oody setting},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {89},
YEAR = {1987},
NUMBER = {2},
PAGES = {395--423},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {17B67 (14M15 17B10 20G05)},
MRNUMBER = {0894387},
MRREVIEWER = {H. H. Andersen},
DOI = {10.1007/BF01389086},
ZBLNUMBER = {0635.14023},
} -
[land]
G. D. Landweber, "Multiplets of representations and Kostant’s Dirac operator for equal rank loop groups," Duke Math. J., vol. 110, iss. 1, pp. 121-160, 2001.
@article {land, MRKEY = {1861090},
AUTHOR = {Landweber, Gregory D.},
TITLE = {Multiplets of representations and {K}ostant's {D}irac operator for equal rank loop groups},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {110},
YEAR = {2001},
NUMBER = {1},
PAGES = {121--160},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {17B67 (22E46 22E67 58J20 81R10)},
MRNUMBER = {1861090},
MRREVIEWER = {Stanislav Z. Pakuliak},
DOI = {10.1215/S0012-7094-01-11014-4},
ZBLNUMBER = {1018.17016},
} -
[mich] B. H. Lawson Jr. and M. Michelsohn, Spin Seometry, Princeton, NJ: Princeton Univ. Press, 1989, vol. 38.
@book {mich, MRKEY = {1031992},
AUTHOR = {Lawson, Jr., H. Blaine and Michelsohn, Marie-Louise},
TITLE = {Spin Seometry},
SERIES = {Princeton Math. Ser.},
VOLUME = {38},
PUBLISHER = {Princeton Univ. Press},
ADDRESS = {Princeton, NJ},
YEAR = {1989},
PAGES = {xii+427},
ISBN = {0-691-08542-0},
MRCLASS = {53-02 (53A50 53C20 57R75 58G10)},
MRNUMBER = {1031992},
MRREVIEWER = {N. J. Hitchin},
ZBLNUMBER = {0688.57001},
} -
[mick] J. Mickelsson, "Gerbes, (twisted) $K$-theory, and the supersymmetric WZW model," in Infinite Dimensional Groups and Manifolds, Berlin: de Gruyter, 2004, vol. 5, pp. 93-107.
@incollection {mick, MRKEY = {2104355},
AUTHOR = {Mickelsson, Jouko},
TITLE = {Gerbes, (twisted) {$K$}-theory, and the supersymmetric {WZW} model},
BOOKTITLE = {Infinite Dimensional Groups and Manifolds},
SERIES = {IRMA Lect. Math. Theor. Phys.},
VOLUME = {5},
PAGES = {93--107},
PUBLISHER = {de Gruyter},
ADDRESS = {Berlin},
YEAR = {2004},
MRCLASS = {58J90 (19K56 55R65 58J52 81T50)},
MRNUMBER = {2104355},
MRREVIEWER = {Hisham A. Sati},
ZBLNUMBER = {1058.81067},
} -
[mat] O. Mathieu, "Formules de caractères pour les algèbres de Kac-Moody générales," in Astérisque, , 1988, vol. 159-160, p. 267.
@incollection{mat, MRKEY = {0980506},
AUTHOR = {Mathieu, Olivier},
TITLE = {Formules de caractères pour les algèbres de {K}ac-{M}oody générales},
BOOKTITLE = {Astérisque},
FJOURNAL = {Astérisque},
VOLUME = {159-160},
YEAR = {1988},
PAGES = {267},
ISSN = {0303-1179},
MRCLASS = {17B67 (14M15 17B10 20G05)},
MRNUMBER = {0980506},
MRREVIEWER = {Guy Rousseau},
ZBLNUMBER = {0683.17010},
} -
[psloop] A. Pressley and G. Segal, Loop Groups, New York: The Clarendon Press Oxford Univ. Press, 1986.
@book {psloop, MRKEY = {0900587},
AUTHOR = {Pressley, Andrew and Segal, Graeme},
TITLE = {Loop Groups},
SERIES = {Oxford Math. Monogr.},
PUBLISHER = {The Clarendon Press Oxford Univ. Press},
ADDRESS = {New York},
YEAR = {1986},
PAGES = {viii+318},
ISBN = {0-19-853535-X},
MRCLASS = {22E65 (58D15 81D15)},
MRNUMBER = {0900587},
MRREVIEWER = {Jouko Mickelsson},
ZBLNUMBER = {0618.22011},
ZBLNUMBER = {0603.17012},
} -
[seg] G. Segal, "The representation ring of a compact Lie group," Inst. Hautes Études Sci. Publ. Math., iss. 34, pp. 113-128, 1968.
@article {seg, MRKEY = {0248277},
AUTHOR = {Segal, Graeme},
TITLE = {The representation ring of a compact {L}ie group},
JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
NUMBER = {34},
YEAR = {1968},
PAGES = {113--128},
ISSN = {0073-8301},
MRCLASS = {22.50},
MRNUMBER = {0248277},
MRREVIEWER = {M. L. Laplaza},
ZBLNUMBER = {0209.06203},
} -
[taub] C. H. Taubes, Notes on the Dirac operator on loop space, 1989.
@misc{taub,
author={Taubes, C. H.},
TITLE={Notes on the {D}irac operator on loop space},
NOTE={unpublished manuscript},
YEAR={1989},
} -
[tel]
C. Teleman, "Borel-Weil-Bott theory on the moduli stack of $G$-bundles over a curve," Invent. Math., vol. 134, iss. 1, pp. 1-57, 1998.
@article {tel, MRKEY = {1646586},
AUTHOR = {Teleman, Constantin},
TITLE = {Borel-{W}eil-{B}ott theory on the moduli stack of {$G$}-bundles over a curve},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {134},
YEAR = {1998},
NUMBER = {1},
PAGES = {1--57},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {14D20 (14F20 14H60)},
MRNUMBER = {1646586},
MRREVIEWER = {P. E. Newstead},
DOI = {10.1007/s002220050257},
ZBLNUMBER = {0980.14025},
} -
[tel2] C. Teleman, "${K}$-theory of the moduli of principal bundles on a surface and deformations of the Verlinde algebra," in Proceedings of the 2002 Oxford Symposium in Honour of the 60th Birthday of Graeme Segal, CUP, 2004.
@incollection{tel2,
author={Teleman, Constantin},
TITLE = {${K}$-theory of the moduli of principal bundles on a surface and deformations of the {V}erlinde algebra},
BOOKTITLE={Proceedings of the 2002 Oxford Symposium in Honour of the 60th Birthday of Graeme Segal},
PUBLISHER={CUP},
YEAR={2004},
} -
[val]
V. Toledano Laredo, "Positive energy representations of the loop groups of non-simply connected Lie groups," Comm. Math. Phys., vol. 207, iss. 2, pp. 307-339, 1999.
@article {val, MRKEY = {1724846},
AUTHOR = {Toledano Laredo, Valerio},
TITLE = {Positive energy representations of the loop groups of non-simply connected {L}ie groups},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {207},
YEAR = {1999},
NUMBER = {2},
PAGES = {307--339},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {22E67 (22E65)},
MRNUMBER = {1724846},
MRREVIEWER = {Vladimir Pestov},
DOI = {10.1007/s002200050728},
ZBLNUMBER = {0969.22010},
} -
[wend]
R. Wendt, "A character formula for representations of loop groups based on non-simply connected Lie groups," Math. Z., vol. 247, iss. 3, pp. 549-580, 2004.
@article {wend, MRKEY = {2114428},
AUTHOR = {Wendt, Robert},
TITLE = {A character formula for representations of loop groups based on non-simply connected {L}ie groups},
JOURNAL = {Math. Z.},
FJOURNAL = {Mathematische Zeitschrift},
VOLUME = {247},
YEAR = {2004},
NUMBER = {3},
PAGES = {549--580},
ISSN = {0025-5874},
CODEN = {MAZEAX},
MRCLASS = {22E67 (17B67)},
MRNUMBER = {2114428},
MRREVIEWER = {Alessandro Mani{à}},
DOI = {10.1007/s00209-003-0629-5},
ZBLNUMBER = {1063.22022},
}