%CURRENT ALGEBRAS AND GROUPS, by Jouko Mickelsson, e-mail jouko@theophys.kth.se %

%publ. by Plenum Press, 1989 %

% Back to Jouko Mickelsson's Home Page

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\topmatter\title CONTENTS \endtitle\endtopmatter \pageno=-13
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\line{CHAPTER 1\phantom{A} SEMISIMPLE LIE ALGEBRAS\dotfill\ 1}
\line{\HH 1.1 Lie algebras and homomorphisms\dotfill\ 1}
\line{\HH 1.2 Semisimple Lie algebras\dotfill\ 3}
\line{\HH 1.3 Dynkin diagrams\dotfill\ 10}
\line{\HH 1.4 The enveloping algebra. Linear representations\dotfill\ 12}
\line{\HH 1.5 Highest weight representations of semisimple Lie
          algebras \dotfill\ 17}
\noindent CHAPTER 2\phantom{A} REPRESENTATIONS OF AFFINE\newline
\line{\phantom{AAAAA}KAC-MOODY ALGEBRAS\dotfill\ 21}

\line{\HH 2.1 Affine  Kac-Moody algebras from generalized Cartan \newline}
\line{\phantom{AAAAAAA}  matrices\dotfill\ 21}
\line{\HH 2.2 Affine Lie algebras as central extensions of loop algebras:\newline}
\line{\phantom{AAAAAAA} the untwisted case\dotfill\ 23}
\line{\HH 2.3 Affine Lie algebras as central extensions of loop algebras:\newline}
\line{\phantom{AAAAAAA} the twisted case\dotfill\ 28}
\line{\HH 2.4 The highest weight representations of affine Lie algebras
\dotfill\ 30}
\line{\HH 2.5 The character formula\dotfill\ 37}

\line{CHAPTER 3\phantom{A} PRINCIPAL BUNDLES\dotfill\ 43}
\line{\HH 3.0 A short introduction to calculus of differential forms
\dotfill\ 43}
\line{\HP-Algebra of differential forms\dotfill\ 43}
\line{\HP-The exterior derivative\dotfill\ 45}
\line{\HP-Vector fields and differential forms\dotfill\ 47}
\line{\HP-Differential forms on Lie groups\dotfill\ 49}
\line{\HP-Stokes's theorem\dotfill\ 50}
\line{\HH 3.1 Definition of a principal bundle and examples\dotfill\ 52}
\line{\HH 3.2 Connection and curvature in a principal bundle\dotfill\ 56}
\line{\HH 3.3 Parallel transport\dotfill\ 60}
\line{\HH 3.4 Covariant differentiation in vector bundles\dotfill\ 61}
\line{\HH 3.5 An example: The monopole line bundle\dotfill\ 63}
\line{\HP-Construction of the basic monopole bundle\dotfill\ 63}
\line{\HP-The first Chern class\dotfill\ 66}
\line{\HP-Holomorphic sections\dotfill\ 67}
\line{\HH 3.6 Invariant connections\dotfill\ 69}
\line{\HH 3.7 The Levi-Civit\`a connection\dotfill\ 73}

\noindent CHAPTER 4\phantom{A} EXTENSIONS OF GROUPS OF GAUGE\newline
\line{\phantom{AAAAA}  TRANSFORMATIONS\dotfill\ 75}

\line{\HH 4.0 Introduction\dotfill\ 75}
\line{\HH 4.1 Some group cohomology\dotfill\ 77}
\line{\HP-Cocycles and coboundaries\dotfill\ 77}
\line{\HP-The descent equations\dotfill\ 80}
\line{\HP-Lie algebra cohomology\dotfill\ 82}
\line{\HH 4.2 Groups associated to affine Kac-Moody algebras\dotfill\ 83}
\line{\HP-The central extension $\widehat{LG}$\dotfill\ 83}
\line{\HP-Connection in $\widehat{LG}$\dotfill\ 86}
\line{\HH 4.3 Extensions of $Map(S^3,G)$\dotfill\ 88}
\line{\HP-The Abelian extension $\widehat{S^3}G$\dotfill\ 88}
\line{\HP-Geometry of the extension $\widehat{S^3}G$\dotfill\ 92}
\line{\HH 4.4 Spin and statistics from group extensions\dotfill\ 94}
\line{\HP-The Lagrangian of the Wess-Zumino-Witten model\dotfill\ 94}
\line{\HP-Rotating the soliton\dotfill\ 96}
\line{\HP-Interchange of two solitons\dotfill\ 98}
\line{\HH 4.5 Chern classes\dotfill\ 100}

\line{CHAPTER 5\phantom{A}THE CHIRAL ANOMALY\dotfill\ 105}
\line{\HH 5.0 Introduction\dotfill\ 105}
\line{\HH 5.1 The Clifford algebra\dotfill\ 107}
\line{\HH 5.2 The Dirac operator\dotfill\ 109}
\line{\HH 5.3 The determinant of a Dirac operator\dotfill\ 113}
\line{\HP-The massive Dirac operator\dotfill\ 113}
\line{\HP-The chiral case\dotfill\ 115}
\line{\HP-The Dirac determinant bundle\dotfill\ 117}
\line{\HH 5.4 On the geometry of the Dirac determinant bundle\dotfill\ 120}
\line{\HP-Curvature and anomalies\dotfill\ 120}
\line{\HP-The commutator anomaly\dotfill\ 124}

\noindent CHAPTER 6\phantom{A} DETERMINANT BUNDLES OVER\newline
\line{\phantom{AAAAA}GRASSMANNIANS\dotfill\ 127}

\line{\HH 6.0 Introduction\dotfill\ 127}
\line{\HH 6.1 Embedding $S^d G$ in the general linear group $GL_p$\newline}
\line{\hphantom{AAAAAAA} modelled by Schatten ideals\dotfill\ 129}
\line{\HH 6.2 The determinant bundle over $Gr_p$\dotfill\ 134}
\line{\HP-The Grassmannian $Gr_p$ and the Stiefel manifold $St_p$
\dotfill\ 134}
\line{\HP-Generalized Fredholm determinants\dotfill\ 136}
\line{\HP-The determinant bundle $DET_p$\dotfill\ 138}
\line{\HP-Holomorphic sections\dotfill\ 139}
\line{\HP-The complexification $\Bbb CGr_p$\dotfill\ 141}
\line{\HH 6.3 Lifting the action of $GL_p$ in $Gr_p$ to an action of\newline}
\line{\phantom{AAAAAAA} the extension $\widehat{GL}_p$ in $DET_p$\dotfill\ 142}
\line{\HP-The extension of $GL_p^0$\dotfill\ 142}
\line{\HP-The extension of the Lie algebra $\bold{gl}_p$\dotfill\ 145}
\line{\HP-The extension of $GL_p$\dotfill\ 148}
\line{\HH 6.4 The Dirac field on $Gr_1$\dotfill\ 150}
\line{\HP-A finite-dimensional example: $\Bbb CP^2$\dotfill\ 150}
\line{\HP-The central extension of the orthogonal group $O_1$\dotfill\ 152}
\line{\HP-The spin representation of $\hat O_1$\dotfill\ 155}
\line{\HP-The spin bundle, Clifford algebra, and the Dirac \newline}
\line{\HP\phantom{AAA} operator\dotfill\ 158}
\line{\HH 6.5 The Pl\"ucker embedding and a spherical function\dotfill\ 161}
\line{\HP-The case $p=1$\dotfill\ 161}
\line{\HP-The case $p=2$\dotfill\ 164}
\line{\HP-Spherical function for a highest weight representation \newline}
\line{\HP\phantom{AAA} of $\widehat{GL}_1$\dotfill\ 165}
\line{\HP-The spherical function for $\widehat{GL}_2$\dotfill\ 167}

\line{CHAPTER 7\phantom{A} THE VIRASORO ALGEBRA\dotfill\ 171}
\line{\HH 7.0 Introduction \dotfill\ 171}
\line{\HH 7.1 The Sugawara construction\dotfill\ 173}
\line{\HH 7.2 Embedding \it Diff\rm$\,S^1/S^1$ in $Gr_1$\dotfill\ 176}
\line{\HH 7.3 Semi-infinite forms and representations of\newline}
\line{\phantom{AAAAAAA} the Virasoro algebra\dotfill\ 178}
\line{\HH 7.4 Representations of the Virasoro algebra with\newline}
\line{\phantom{AAAAAAA} central charge $c<1$\dotfill\ 181}
\line{\HH 7.5 Riemann surfaces and generalizations of Virasoro \newline}
\line{\phantom{AAAAAAA} algebras\dotfill\ 185}
\line{\HH 7.6 Extensions of \it Diff\rm$\,S^n$ and diffeomorphism anomalies
\dotfill\ 188}

\noindent CHAPTER 8\phantom{A} THE BOSON FERMION \newline
\line{\phantom{AAAAA}CORRESPONDENCE\dotfill\ 193}

\line{\HH 8.0 Introduction\dotfill\ 193}
\line{\HH 8.1 Representations of $\bold{gl}(\infty)$\dotfill\ 196}
\line{\HH 8.2 The principal Heisenberg subalgebra\dotfill\ 197}
\line{\HH 8.3 Properties of the Schur polynomials\dotfill\ 201}

\noindent CHAPTER 9\phantom{A} HOLOMORPHIC ASPECTS OF\newline
\line{\phantom{AAAAA} STRING THEORY\dotfill\ 203}

\line{\HH 9.0 Introduction\dotfill\ 203}
\line{\HH 9.1 The K\"ahler structure of \it Diff\rm$\,S^1/S^1$\dotfill\ 205}
\line{\HH 9.2 The Fock space of the bosonic string\dotfill\ 209}
\line{\HH 9.3 Reparametrization invariance in string theory\dotfill\ 213}
\line{\HH 9.4 The BRST complex\dotfill\ 218}
\line{\HH 9.5 Strings on a group manifold\dotfill\ 224}

\line{CHAPTER 10\phantom{A} THE NONLINEAR $\sigma$ MODEL\dotfill\ 235}
\line{\HH 10.0 Introduction\dotfill\ 235}
\line{\HH 10.1 The two-dimensional $\sigma$ model\dotfill\ 236}
\line{\HH 10.2 The $\sigma$ model vacua in two dimensions\dotfill\ 238}
\line{\HH 10.3 The $\sigma$ model in four dimensions\dotfill\ 241}

\line{CHAPTER 11\phantom{A} THE KP HIERARCHY\dotfill\ 245}
\line{\HH 11.0 Introduction\dotfill\ 245}
\line{\HH 11.1 The Pl\"ucker relations and the Hirota bilinear \newline}
\line{\phantom{AAAAAAA} equation\dotfill\ 246}
\line{\HH 11.2 Soliton solutions of the KP hierarchy\dotfill\ 250}
\line{\HH 11.3 The Lax formulation of the KP hierarchy\dotfill\ 252}
\line{\HH 11.4 The KdV equation and the reduced KP hierarchy\dotfill\ 260}
\line{\HH 11.5 Vertex operators and kinks\dotfill\ 261}

\noindent CHAPTER 12\phantom{A} THE FOCK BUNDLE OF A DIRAC \newline

\line{\HH 12.0 Introduction\dotfill\ 267}
\line{\HH 12.1 A two-dimensional example: Fermions coupled to \newline}
\line{\phantom{AAAAAAA} a non-Abelian electric field\dotfill\ 268}
\line{\HP-The Fock bundle\dotfill\ 268}
\line{\HP-Quantization of the vector potential\dotfill\ 272}
\line{\HP-The Hamiltonian of the Yang-Mills field\dotfill\ 274}
\line{\HP-The complete Hamiltonian\dotfill\ 276}
\line{\HH 12.2 Dirac operator on a Riemann surface\dotfill\ 277}
\line{\HH 12.3 Dirac operator in $3+1$ space-time dimensions\dotfill\ 280}
\line{\HP-Construction of the Fock bundle\dotfill\ 280}
\line{\HP-Group actions in the Fock bundle\dotfill\ 285}
\line{\HP-The space of holomorphic sections\dotfill\ 286}
\line{\HP-Group actions in $\Cal F_{hol}$\dotfill\ 288}
\line{\HP-The standard Fock bundle $\Cal F^{(2)}$\dotfill\ 290}
\line{\HP-The CAR algebra in $\Cal F^{(2)}$\dotfill\ 291}
\line{\HP-Concluding remarks\dotfill\ 293}
\line{\HH 12.4 A universal Yang-Mills theory\dotfill\ 294}

\line{REFERENCES\dotfill\ 299}

\line{INDEX\dotfill\ 309}