Supersymmetric field theories and the elliptic index theorem with complex coefficients

Berwick-Evans, Daniel¹

¹ Department of Mathematics, University of Illinois at Urbana–Champaign, Urbana, IL, United States

Geometry & topology, Tome 25 (2021) no. 5, pp. 2287-2384.

Résumé

We present a cocycle model for elliptic cohomology with complex coefficients in which methods from $2$–dimensional quantum field theory can be used to rigorously construct cocycles. For example, quantizing a theory of vector-bundle-valued fermions yields a cocycle representative of the elliptic Thom class. This constructs the complexified string orientation of elliptic cohomology, which determines a pushforward for families of rational string manifolds. A second pushforward is constructed from quantizing a supersymmetric $\sigma$–model. These two pushforwards agree, giving a precise physical interpretation for the elliptic index theorem with complex coefficients. This both refines and supplies further evidence for the long-conjectured relationship between elliptic cohomology and $2$–dimensional quantum field theory. Analogous methods in supersymmetric mechanics recover path integral constructions of the Mathai–Quillen Thom form in complexified $KO$–theory and a cocycle representative of the $Â$–class for a family of oriented manifolds.

Détail

DOI : 10.2140/gt.2021.25.2287

Classification : 55N34, 81T60
Keywords: elliptic cohomology, topological modular forms, supersymmetric field theories, Witten genus, Mathai-Quillen forms

Bibliographie

[1] L Alvarez-Gaumé, Supersymmetry and the Atiyah–Singer index theorem, Comm. Math. Phys. 90 (1983) 161 | DOI

[2] O Alvarez, T P Killingback, M Mangano, P Windey, The Dirac–Ramond operator in string theory and loop space index theorems, Nuclear Phys. B Proc. Suppl. 1A (1987) 189 | DOI

[3] M Ando, M J Hopkins, C Rezk, Multiplicative orientations of KO–theory and of the spectrum of topological modular forms, preprint (2010)

[4] M Ando, M J Hopkins, N P Strickland, Elliptic spectra, the Witten genus and the theorem of the cube, Invent. Math. 146 (2001) 595 | DOI

[5] M F Atiyah, Circular symmetry and stationary-phase approximation, from: "Colloquium in honor of Laurent Schwartz, I", Astérisque 131, Soc. Math. France (1985) 43

[6] M F Atiyah, R Bott, A Shapiro, Clifford modules, Topology 3 (1964) 3 | DOI

[7] M F Atiyah, L Jeffrey, Topological Lagrangians and cohomology, J. Geom. Phys. 7 (1990) 119 | DOI

[8] M F Atiyah, I M Singer, The index of elliptic operators, V, Ann. of Math. 93 (1971) 139 | DOI

[9] A Bartels, C L Douglas, A Henriques, Conformal nets, I : Coordinate-free nets, Int. Math. Res. Not. 2015 (2015) 4975 | DOI

[10] A Bartels, C L Douglas, A Henriques, Conformal nets, II : Conformal blocks, Comm. Math. Phys. 354 (2017) 393 | DOI

[11] A Bartels, C L Douglas, A Henriques, Conformal nets, IV : The 3–category, Algebr. Geom. Topol. 18 (2018) 897 | DOI

[12] A Bartels, C L Douglas, A Henriques, Fusion of defects, 1237, Amer. Math. Soc. (2019) | DOI

[13] M Batchelor, The structure of supermanifolds, Trans. Amer. Math. Soc. 253 (1979) 329 | DOI

[14] N Berline, E Getzler, M Vergne, Heat kernels and Dirac operators, 298, Springer (1992)

[15] D Berwick-Evans, Supersymmetric localization, modularity and the Witten genus, preprint (2019)

[16] D Berwick-Evans, A Tripathy, A model for complex analytic equivariant elliptic cohomology from quantum field theory, preprint (2018)

[17] D Berwick-Evans, A Tripathy, A de Rham model for complex analytic equivariant elliptic cohomology, preprint (2019)

[18] J M Bismut, The Atiyah–Singer index theorem for families of Dirac operators : two heat equation proofs, Invent. Math. 83 (1986) 91 | DOI

[19] J M Bismut, D S Freed, The analysis of elliptic families, II : Dirac operators, eta invariants, and the holonomy theorem, Comm. Math. Phys. 107 (1986) 103 | DOI

[20] M Blau, The Mathai–Quillen formalism and topological field theory, J. Geom. Phys. 11 (1993) 95 | DOI

[21] M Blau, G Thompson, Aspects of NT ≥ 2 topological gauge theories and D-branes, Nuclear Phys. B 492 (1997) 545 | DOI

[22] C Blohmann, Stacky Lie groups, Int. Math. Res. Not. 2008 (2008) | DOI

[23] P M Cheung, Supersymmetric field theories and generalized cohomology, PhD thesis, Stanford University (2007)

[24] K Costello, A geometric construction of the Witten genus, I, from: "Proceedings of the International Congress of Mathematicians, II" (editors R Bhatia, A Pal, G Rangarajan, V Srinivas, M Vanninathan), Hindustan (2010) 942

[25] K J Costello, A geometric construction of the Witten genus, II, preprint (2011)

[26] P Deligne, P Etingof, D S Freed, L C Jeffrey, D Kazhdan, J W Morgan, D R Morrison, E Witten, editors, Quantum fields and strings: a course for mathematicians, I, Amer. Math. Soc. (1999)

[27] P Deligne, D S Freed, Supersolutions, from: "Quantum fields and strings: a course for mathematicians, I" (editors P Deligne, P Etingof, D S Freed, L C Jeffrey, D Kazhdan, J W Morgan, D R Morrison, E Witten), Amer. Math. Soc. (1999) 227

[28] P Deligne, J Morgan, Notes on supersymmetry, from: "Quantum fields and strings: a course for mathematicians, I" (editors P Deligne, P Etingof, D S Freed, L C Jeffrey, D Kazhdan, J W Morgan, D R Morrison, E Witten), Amer. Math. Soc. (1999) 41

[29] C L Douglas, A G Henriques, Topological modular forms and conformal nets, from: "Mathematical foundations of quantum field theory and perturbative string theory" (editors H Sati, U Schreiber), Proc. Sympos. Pure Math. 83, Amer. Math. Soc. (2011) 341

[30] J J Duistermaat, G J Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math. 69 (1982) 259 | DOI

[31] P Etingof, Mathematical ideas and notions of quantum field theory, lecture notes (2002)

[32] D S Freed, On determinant line bundles, from: "Mathematical aspects of string theory" (editor S T Yau), Adv. Ser. Math. Phys. 1, World Sci. (1987) 189 | DOI

[33] D S Freed, Five lectures on supersymmetry, Amer. Math. Soc. (1999)

[34] D Friedan, P Windey, Supersymmetric derivation of the Atiyah–Singer index and the chiral anomaly, Nuclear Phys. B 235 (1984) 395 | DOI

[35] D Gaiotto, T Johnson-Freyd, Holomorphic SCFTs with small index, preprint (2018)

[36] E Getzler, A short proof of the local Atiyah–Singer index theorem, Topology 25 (1986) 111 | DOI

[37] P G Goerss, Topological modular forms (after Hopkins, Miller and Lurie), from: "Séminaire Bourbaki, 2008/2009", Astérisque 332, Soc. Math. France (2010) 221

[38] I Grojnowski, Delocalised equivariant elliptic cohomology, from: "Elliptic cohomology" (editors H R Miller, D C Ravenel), Lond. Math. Soc. Lect. Note Ser. 342, Cambridge Univ. Press (2007) 114 | DOI

[39] S Gukov, D Pei, P Putrov, C Vafa, 4–manifolds and topological modular forms, preprint (2018)

[40] H Hohnhold, M Kreck, S Stolz, P Teichner, Differential forms and 0–dimensional supersymmetric field theories, Quantum Topol. 2 (2011) 1 | DOI

[41] H Hohnhold, S Stolz, P Teichner, From minimal geodesics to supersymmetric field theories, from: "A celebration of the mathematical legacy of Raoul Bott" (editor P R Kotiuga), CRM Proc. Lect. Notes 50, Amer. Math. Soc. (2010) 207 | DOI

[42] S Hollander, A homotopy theory for stacks, Israel J. Math. 163 (2008) 93 | DOI

[43] M J Hopkins, Topological modular forms, the Witten genus, and the theorem of the cube, from: "Proceedings of the International Congress of Mathematicians, I" (editor S D Chatterji), Birkhäuser (1995) 554

[44] M J Hopkins, Algebraic topology and modular forms, from: "Proceedings of the International Congress of Mathematicians, I" (editor T Li), Higher Ed. (2002) 291

[45] M J Hopkins, I M Singer, Quadratic functions in geometry, topology, and M–theory, J. Differential Geom. 70 (2005) 329 | DOI

[46] P Hu, I Kriz, Conformal field theory and elliptic cohomology, Adv. Math. 189 (2004) 325 | DOI

[47] A Kitaev, Periodic table for topological insulators and superconductors, from: "Advances in theoretical physics" (editors V Lebedev, M Feigelman), AIP Conf. Proc. 1134, Amer. Inst. Phys. (2009) 22 | DOI

[48] M Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003) 157 | DOI

[49] E Lerman, Orbifolds as stacks?, Enseign. Math. 56 (2010) 315 | DOI

[50] J Lurie, On the classification of topological field theories, from: "Current developments in mathematics" (editors D Jerison, B Mazur, T Mrowka, W Schmid, R Stanley, S T Yau), International (2009) 129

[51] J Lurie, A survey of elliptic cohomology, from: "Algebraic topology" (editors N A Baas, E M Friedlander, B Jahren, P A Østvær), Abel Symp. 4, Springer (2009) 219 | DOI

[52] J Lurie, Elliptic cohomology, I: Spectral abelian varieties, preprint (2018)

[53] J Lurie, Elliptic cohomology, II: Orientations, preprint (2018)

[54] V Mathai, D Quillen, Superconnections, Thom classes, and equivariant differential forms, Topology 25 (1986) 85 | DOI

[55] H P Mckean Jr., I M Singer, Curvature and the eigenvalues of the Laplacian, J. Differential Geom. 1 (1967) 43

[56] K Pilch, A N Schellekens, N P Warner, Path integral calculation of string anomalies, Nuclear Phys. B 287 (1987) 362 | DOI

[57] J R Quine, S H Heydari, R Y Song, Zeta regularized products, Trans. Amer. Math. Soc. 338 (1993) 213 | DOI

[58] G Segal, Elliptic cohomology, from: "Séminaire Bourbaki, 1987/1988", Astérisque 161–162, Soc. Math. France (1988) 187

[59] G Segal, The definition of conformal field theory, from: "Topology, geometry and quantum field theory" (editor U Tillmann), Lond. Math. Soc. Lect. Note Ser. 308, Cambridge Univ. Press (2004) 421 | DOI

[60] S Stolz, P Teichner, What is an elliptic object?, from: "Topology, geometry and quantum field theory" (editor U Tillman), Lond. Math. Soc. Lect. Note Ser. 308, Cambridge Univ. Press (2004) 247 | DOI

[61] S Stolz, P Teichner, Supersymmetric field theories and generalized cohomology, from: "Mathematical foundations of quantum field theory and perturbative string theory" (editors H Sati, U Schreiber), Proc. Sympos. Pure Math. 83, Amer. Math. Soc. (2011) 279

[62] Z D Wang, Mathai–Quillen formalism : from the point of view of path integrals, Lett. Math. Phys. 39 (1997) 331 | DOI

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

[64] E Witten, Supersymmetry and Morse theory, J. Differential Geom. 17 (1982) 661

[65] E Witten, Elliptic genera and quantum field theory, Comm. Math. Phys. 109 (1987) 525 | DOI

[66] E Witten, The index of the Dirac operator in loop space, from: "Elliptic curves and modular forms in algebraic topology" (editor P S Landweber), Lecture Notes in Math. 1326, Springer (1988) 161 | DOI

[67] E Witten, Index of Dirac operators, from: "Quantum fields and strings: a course for mathematicians, I" (editors P Deligne, P Etingof, D S Freed, L C Jeffrey, D Kazhdan, J W Morgan, D R Morrison, E Witten), Amer. Math. Soc. (1999) 475

[68] E Witten, Notes on supermanifolds and integration, Pure Appl. Math. Q. 15 (2019) 3 | DOI

[69] S Wu, Mathai–Quillen formalism, from: "Encyclopedia of mathematical physics, III" (editors J P Francoise, G L Naber, T S Tsun), Elsevier (2006) 390 | DOI

[70] D Zagier, Note on the Landweber–Stong elliptic genus, from: "Elliptic curves and modular forms in algebraic topology" (editor P S Landweber), Lecture Notes in Math. 1326, Springer (1988) 216 | DOI

[71] D Zagier, Elliptic modular forms and their applications, from: "The -- of modular forms" (editor K Ranestad), Springer (2008) 1 | DOI