One-dimensional Chern–Simons theory and the  genus

Gwilliam, Owen¹ ; Grady, Ryan²

¹Department of Mathematics, University of California, Berkeley, 970 Evans Hall, Berkeley, CA 94720, USA
²Department of Mathematics and Statistics, Boston University, 111 Cummington Mall, Boston, MA 02215, USA

Algebraic and Geometric Topology, Tome 14 (2014) no. 4, pp. 2299-2377

Résumé

We construct a Chern–Simons gauge theory for dg Lie and L–infinity algebras on any one-dimensional manifold and quantize this theory using the Batalin–Vilkovisky formalism and Costello’s renormalization techniques. Koszul duality and derived geometry allow us to encode topological quantum mechanics, a nonlinear sigma model of maps from a 1–manifold into a cotangent bundle T∗X, as such a Chern–Simons theory. Our main result is that the effective action of this theory is naturally identified with the  class of X. From the perspective of derived geometry, our quantization constructs a projective volume form on the derived loop space ℒX that can be identified with the  class.

Détail

DOI : 10.2140/agt.2014.14.2299

Classification : 57R56, 18G55, 58J20
Keywords: $\hat{A}$ genus, BV formalism, Chern–Simons theory, topological quantum mechanics

Bibliographie

[1] M Alexandrov, A Schwarz, O Zaboronsky, M Kontsevich, The geometry of the master equation and topological quantum field theory, Internat. J. Modern Phys. A 12 (1997) 1405

[2] M F Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957) 181

[3] D Ben-Zvi, D Nadler, Loop spaces and connections, (2011)

[4] N Berline, E Getzler, M Vergne, Heat kernels and Dirac operators, Grundl. Math. Wissen. 298, Springer (1992)

[5] I N Bernšteĭn, B I Rosenfel’D, Homogeneous spaces of infinite-dimensional Lie algebras and the characteristic classes of foliations, Uspehi Mat. Nauk 28 (1973) 103

[6] R Bezrukavnikov, D Kaledin, Fedosov quantization in algebraic context, Mosc. Math. J. 4 (2004) 559, 782

[7] P Bressler, R Nest, B Tsygan, Riemann–Roch theorems via deformation quantization, I, II, Adv. Math. 167 (2002) 1, 26

[8] D Calaque, M Van Den Bergh, Hochschild cohomology and Atiyah classes, Adv. Math. 224 (2010) 1839

[9] D Calaque, C A Rossi, Lectures on Duflo isomorphisms in Lie algebra and complex geometry, Eur. Math. Soc. (2011)

[10] A Căldăraru, The Mukai pairing, II: The Hochschild–Kostant–Rosenberg isomorphism, Adv. Math. 194 (2005) 34

[11] A Căldăraru, S Willerton, The Mukai pairing, I: A categorical approach, New York J. Math. 16 (2010) 61

[12] A S Cattaneo, G Felder, L Tomassini, From local to global deformation quantization of Poisson manifolds, Duke Math. J. 115 (2002) 329

[13] Z Chen, M Stiénon, P Xu, From Atiyah classes to homotopy Leibniz algebras,

[14] K Costello, A geometric construction of the Witten genus, I,

[15] K Costello, A geometric construction of the Witten genus, II,

[16] K Costello, Renormalization and effective field theory, Mathematical Surveys and Monographs 170, Amer. Math. Soc. (2011)

[17] K Costello, O Gwilliam, Factorization algebras in quantum field theory, (2014)

[18] P Deligne, P Etingof, D S Freed, L C Jeffrey, D A Každan, J W Morgan, D R Morrison, E Witten, editors, Quantum fields and strings: A course for mathematicians. Vol. 1, 2, Amer. Math. Soc. (1999)

[19] V Dolgushev, A formality theorem for Hochschild chains, Adv. Math. 200 (2006) 51

[20] B V Fedosov, A simple geometrical construction of deformation quantization, J. Differential Geom. 40 (1994) 213

[21] B V Fedosov, Deformation quantization and index theory, Mathematical Topics 9, Akademie Verlag (1996) 325

[22] J Francis, The tangent complex and Hochschild cohomology of $\mathscr{E}_n$–rings, Compos. Math. 149 (2013) 430

[23] E Frenkel, D Ben-Zvi, Vertex algebras and algebraic curves, Mathematical Surveys and Monographs 88, Amer. Math. Soc. (2004)

[24] I M Gel’Fand, D A Každan, Some problems of differential geometry and the calculation of cohomologies of Lie algebras of vector fields, Sov. Math., Dokl. 12 (1971) 1367

[25] I M Gel’Fand, D A Každan, D B Fuks, Actions of infinite-dimensional Lie algebras, Funkcional. Anal. i Priložen. 6 (1972) 10

[26] E Getzler, Pseudodifferential operators on supermanifolds and the Atiyah–Singer index theorem, Comm. Math. Phys. 92 (1983) 163

[27] P B Gilkey, Invariance theory, the heat equation, and the Atiyah–Singer index theorem, CRC Press (1995)

[28] R Grady, The $\hat{A}$–genus as a projective volume form on the derived loop space,

[29] R Grady, On geometric aspects of topological quantum mechanics, PhD thesis, University of Notre Dame (2012)

[30] R Grady, O Gwilliam, ${L}_{\infty}$ spaces and derived loop spaces,

[31] F Hirzebruch, Topological methods in algebraic geometry, Classics in Mathematics, Springer (1995)

[32] F Hirzebruch, T Berger, R Jung, Manifolds and modular forms, Aspects of Mathematics E20, Friedr. Vieweg Sohn (1992)

[33] L Illusie, Complexe cotangent et déformations, I, Lecture Notes in Mathematics 239, Springer (1971)

[34] C Itzykson, J M Luck, P Moussa, M Waldschmidt, editors, From number theory to physics, Springer (1992)

[35] M Kapranov, Rozansky–Witten invariants via Atiyah classes, Compositio Math. 115 (1999) 71

[36] N Markarian, The Atiyah class, Hochschild cohomology and the Riemann–Roch theorem, J. Lond. Math. Soc. 79 (2009) 129

[37] M J Pflaum, H Posthuma, X Tang, Cyclic cocycles on deformation quantizations and higher index theorems, Adv. Math. 223 (2010) 1958

[38] A C Ramadoss, The big Chern classes and the Chern character, Internat. J. Math. 19 (2008) 699

[39] M Schlessinger, J Stasheff, The Lie algebra structure of tangent cohomology and deformation theory, J. Pure Appl. Algebra 38 (1985) 313

[40] B Toën, G Vezzosi, Chern character, loop spaces and derived algebraic geometry, from: "Algebraic topology" (editors N A Baas, E M Friedlander, B Jahren, P A Østvær), Abel Symp. 4, Springer (2009) 331

[41] B Tsygan, Formality conjectures for chains, from: "Differential topology, infinite-dimensional Lie algebras, and applications" (editors A Astashkevich, S Tabachnikov), Amer. Math. Soc. Transl. Ser. 2 194, Amer. Math. Soc. (1999) 261

[42] T Willwacher, Formality of cyclic chains, Int. Math. Res. Not. 2011 (2011) 3939

[43] P Windey, Supersymmetric quantum mechanics and the Atiyah–Singer index theorem, Acta Phys. Polon. B 15 (1984) 435

[44] E Witten, Constraints on supersymmetry breaking, Nuclear Phys. B 202 (1982) 253