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Loop groups and twisted \(K\)-theory. II. (English) Zbl 1273.22015

The paper is the second one in a series of papers investigating the relationship between the twisted equivariant \(K\)-theory of compact Lie groups and the Verlinde ring of its loop group. The twisted loop groups is the group of gauge transformations of the principal \(G\)-bundle \(P \to \mathbb S^1\). The central extension \((L_PG)^\tau\) of \(L_PG\) by the circle \(\mathbb T\) is called admissible if it extends over this larger group - and so rotation-invariant – and if there is a bilinear form pairing the Lie algebra of the center \(\mathbb T\) and the rotations \(\mathbb T_{rot}\) (Definition 2.10). The adjoint spin representation \(S\) of the loop group determines a distinguished central extension \((L_PG)^\sigma\) and any admissible graded central extension \((L_PG)^\tau\) defines a finite set of isomorphism classes of irreducibles. They generate a free abelian group \(R^\tau_P G)\).
In this paper, the authors introduce the Dirac family of Fredholm operators associated to a positive energy representation of a loop group (§3) by tensoring the positive energy representation of \((L_PG)^{r-\sigma}\) with spinors. This gives a family of Fredholm operators parametrized by the space \(\mathcal A_P\) of connections, equivariant for the central extension \((L_PG)^\tau\). This Fredholm family represents an element of the \(K\)-theory in the model, developed in Part I [J. Topol. 4, No. 4, 737–798 (2011; Zbl 1241.19002)], and so the Dirac operator construction induces a homomorphism \[ \Phi: R^{\tau-\sigma}(L_PG) \to K^{\tau+\dim G}(G[P]), \] where \(G[P]\) is the union of components of \(G\) consisting of all holonomies of connections on \(P\to \mathbb S^1\).
The main result is Theorem 3.44 stating that \(\Phi\) is an isomorphism. In §4, the authors compute the both-side groups and show that \(\Phi\) is an isomorphism for the case when \(G\) is connected with torsion-free fundamental group.
In Part III [Ann. Math. (2) 174, No. 2, 947–1007 (2011; Zbl 1239.19002)], the authors prove the main Theorem 3.44 for any compact Lie group.

MSC:

22E67 Loop groups and related constructions, group-theoretic treatment
57R56 Topological quantum field theories (aspects of differential topology)
19L50 Twisted \(K\)-theory; differential \(K\)-theory
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[1] J. Frank Adams, Lectures on Lie groups, W. A. Benjamin, Inc., New York-Amsterdam, 1969. · Zbl 0206.31604
[2] A. Alekseev and E. Meinrenken, The non-commutative Weil algebra, Invent. Math. 139 (2000), no. 1, 135 – 172. · Zbl 0945.57017
[3] M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes. II. Applications, Ann. of Math. (2) 88 (1968), 451 – 491. · Zbl 0167.21703
[4] M. F. Atiyah, R. Bott, and A. Shapiro, Clifford modules, Topology 3 (1964), no. suppl. 1, 3 – 38. · Zbl 0146.19001
[5] Michael Atiyah and Graeme Segal, Twisted \?-theory, Ukr. Mat. Visn. 1 (2004), no. 3, 287 – 330; English transl., Ukr. Math. Bull. 1 (2004), no. 3, 291 – 334. · Zbl 1151.55301
[6] A. Borel and A. Weil, Representations lineaires et espaces homogenes Kählerians des groupes de Lie compactes, Séminaire Bourbaki, May 1954, Exposé par J.-P. Serre.
[7] Raoul Bott, Homogeneous vector bundles, Ann. of Math. (2) 66 (1957), 203 – 248. · Zbl 0094.35701
[8] J. J. Duistermaat and J. A. C. Kolk, Lie groups, Universitext, Springer-Verlag, Berlin, 2000. · Zbl 0955.22001
[9] Daniel S. Freed, The geometry of loop groups, J. Differential Geom. 28 (1988), no. 2, 223 – 276. · Zbl 0619.58003
[10] Daniel S. Freed, Michael J. Hopkins, and Constantin Teleman, Loop groups and twisted \?-theory I, J. Topol. 4 (2011), no. 4, 737 – 798. · Zbl 1241.19002
[11] Daniel S. Freed, Michael J. Hopkins, and Constantin Teleman, Twisted equivariant \?-theory with complex coefficients, J. Topol. 1 (2008), no. 1, 16 – 44. · Zbl 1188.19005
[12] Daniel S. Freed, Michael J. Hopkins, and Constantin Teleman, Loop groups and twisted \?-theory III, Ann. of Math. (2) 174 (2011), no. 2, 947 – 1007. · Zbl 1239.19002
[13] Daniel S. Freed, Michael J. Hopkins, and Constantin Teleman, Consistent orientation of moduli spaces, The many facets of geometry, Oxford Univ. Press, Oxford, 2010, pp. 395 – 419. · Zbl 1257.19004
[14] Sebastian Goette, Equivariant \?-invariants on homogeneous spaces, Math. Z. 232 (1999), no. 1, 1 – 42. · Zbl 0941.58016
[15] Nigel Hitchin, Generalized geometry — an introduction, Handbook of pseudo-Riemannian geometry and supersymmetry, IRMA Lect. Math. Theor. Phys., vol. 16, Eur. Math. Soc., Zürich, 2010, pp. 185 – 208. · Zbl 1248.53001
[16] Einar Hille, On roots and logarithms of elements of a complex Banach algebra, Math. Ann. 136 (1958), 46 – 57. · Zbl 0081.11203
[17] Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. · Zbl 0716.17022
[18] A. A. Kirillov, Lectures on the orbit method, Graduate Studies in Mathematics, vol. 64, American Mathematical Society, Providence, RI, 2004. · Zbl 1229.22003
[19] Bertram Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. (2) 74 (1961), 329 – 387. · Zbl 0134.03501
[20] Bertram Kostant, A cubic Dirac operator and the emergence of Euler number multiplets of representations for equal rank subgroups, Duke Math. J. 100 (1999), no. 3, 447 – 501. · Zbl 0952.17005
[21] Bertram Kostant and Shlomo Sternberg, Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras, Ann. Physics 176 (1987), no. 1, 49 – 113. · Zbl 0642.17003
[22] Gregory D. Landweber, Multiplets of representations and Kostant’s Dirac operator for equal rank loop groups, Duke Math. J. 110 (2001), no. 1, 121 – 160. · Zbl 1018.17016
[23] Jouko Mickelsson, Gerbes, (twisted) \?-theory, and the supersymmetric WZW model, Infinite dimensional groups and manifolds, IRMA Lect. Math. Theor. Phys., vol. 5, de Gruyter, Berlin, 2004, pp. 93 – 107. · Zbl 1058.81067
[24] Andrew Pressley and Graeme Segal, Loop groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1986. Oxford Science Publications. · Zbl 0618.22011
[25] Alvany Rocha-Caridi and Nolan R. Wallach, Projective modules over graded Lie algebras. I, Math. Z. 180 (1982), no. 2, 151 – 177. · Zbl 0467.17006
[26] Graeme Segal, Cohomology of topological groups, Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69) Academic Press, London, 1970, pp. 377 – 387.
[27] Stephen Slebarski, Dirac operators on a compact Lie group, Bull. London Math. Soc. 17 (1985), no. 6, 579 – 583. · Zbl 0559.58030
[28] C. Taubes, Notes on the Dirac operator on loop space unpublished manuscript (1989).
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