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The chiral anomaly of the free fermion in functorial field theory. (English) Zbl 1436.53032

Summary: When trying to cast the free fermion in the framework of functorial field theory, its chiral anomaly manifests in the fact that it assigns the determinant of the Dirac operator to a top-dimensional closed spin manifold, which is not a number as expected, but an element of a complex line. In functorial field theory language, this means that the theory is twisted, which gives rise to an anomaly theory. In this paper, we give a detailed construction of this anomaly theory, as a functor that sends manifolds to infinite-dimensional Clifford algebras and bordisms to bimodules.

MSC:

53C27 Spin and Spin\({}^c\) geometry
58J52 Determinants and determinant bundles, analytic torsion
53C80 Applications of global differential geometry to the sciences
57N99 Topological manifolds
81T99 Quantum field theory; related classical field theories
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[1] Atiyah, M., Topological quantum field theories, Inst. Hautes Études Sci. Publ. Math., 68, 1989, 175-186 (1988) · Zbl 0692.53053
[2] Atiyah, MF; Patodi, VK; Singer, IM, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Camb. Philos. Soc., 77, 43-69 (1975) · Zbl 0297.58008
[3] Ayala, D.: Geometric Cobordism Categories. ProQuest LLC, Ann Arbor, MI, 2009. Thesis (Ph.D.)—Stanford University
[4] Bär, C., On nodal sets for Dirac and Laplace operators, Commun. Math. Phys., 188, 3, 709-721 (1997) · Zbl 0888.47028
[5] Berline, N., Getzler, E., Vergne, M.: Heat Kernels and Dirac Operators, Volume 298 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1992) · Zbl 0744.58001
[6] Booß-Bavnbek, B.; Lesch, M., The invertible double of elliptic operators, Lett. Math. Phys., 87, 1-2, 19-46 (2009) · Zbl 1161.58312
[7] Booß-Bavnbek, B.; Wojciechowski, KP, Elliptic Boundary Problems for Dirac Operators. Mathematics: Theory & Applications (1993), Boston: Birkhäuser Boston, Inc.,, Boston
[8] Dai, X.; Freed, DS, \( \eta \)-invariants and determinant lines, J. Math. Phys., 35, 10, 5155-5194 (1994) · Zbl 0822.58048
[9] Freed, D.S., Hopkins, M.J.: Reflection positivity and invertible topological phases. ArXiv e-prints (2016) · Zbl 1508.81953
[10] Hassell, A.; Mazzeo, R.; Melrose, RB, A signature formula for manifolds with corners of codimension two, Topology, 36, 5, 1055-1075 (1997) · Zbl 0883.58038
[11] Huang, R-T; Lee, Y., The gluing formula of the zeta-determinants of Dirac Laplacians for certain boundary conditions, Ill. J. Math., 58, 2, 537-560 (2014) · Zbl 1408.58026
[12] Kandel, S., Functorial quantum field theory in the Riemannian setting, Adv. Theor. Math. Phys., 20, 6, 1443-1471 (2016) · Zbl 1431.81153
[13] Kriz, I., On spin and modularity in conformal field theory, Ann. Sci. École Norm. Sup. (4), 36, 1, 57-112 (2003) · Zbl 1028.81050
[14] Mickelsson, J.; Scott, S., Functorial QFT, gauge anomalies and the Dirac determinant bundle, Commun. Math. Phys., 219, 3, 567-605 (2001) · Zbl 0989.81059
[15] Müller, L.; Szabo, RJ, Extended quantum field theory, index theory, and the parity anomaly, Commun. Math. Phys., 362, 3, 1049-1109 (2018) · Zbl 1401.81075
[16] Palais, R.S.: Seminar on the Atiyah-Singer Index Theorem. With contributions by M. F. Atiyah, A. Borel, E. E. Floyd, R. T. Seeley, W. Shih and R. Solovay. Annals of Mathematics Studies, No. 57. Princeton University Press, Princeton (1965)
[17] Ponto, K.; Shulman, M., Traces in symmetric monoidal categories, Expos. Math., 32, 3, 248-273 (2014) · Zbl 1308.18008
[18] Prat Waldron, A.F.: Pfaffian Line Bundles Over Loop Spaces, Spin Structures and the Index Theorem. ProQuest LLC, Ann Arbor, MI, Thesis (Ph.D.)—University of California, Berkeley (2009)
[19] Schochetman, IE; Smith, RL; Tsui, S-K, On the closure of the sum of closed subspaces, Int. J. Math. Math. Sci., 26, 5, 257-267 (2001) · Zbl 1007.46026
[20] Segal, G.B.: The definition of conformal field theory. In: Differential Geometrical Methods in Theoretical Physics (Como, 1987), Volume 250 of NATO Advanced Science Institutes Series C Mathematical and Physical Sciences, pp. 165-171. Kluwer, Dordrecht (1988)
[21] Stolz, S., Teichner, P.: What is an elliptic object? In Topology, Geometry and Quantum Field Theory, Volume 308 of London Mathematical Society. Lecture Note Series, pp. 247-343. Cambridge University Press, Cambridge (2004) · Zbl 1107.55004
[22] Stolz, S., Teichner, P.: Supersymmetric field theories and generalized cohomology. In: Mathematical Foundations of Quantum Field Theory and Perturbative String Theory, Volume 83 of Proceedings of Symposia in Pure Mathematics, pp. 279-340. American Mathematical Society, Providence (2011) · Zbl 1257.55003
[23] Tener, J.E.: Construction of the Unitary Free Fermion Segal Conformal Field Theory. ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)—University of California, Berkeley (2014)
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