The concept of elliptic cohomology originates around:
Peter Landweber, Elliptic Cohomology and Modular Forms, in: Elliptic Curves and Modular Forms in Algebraic Topology, Lecture Notes in Mathematics 1326 (1988) 55-68 [doi:10.1007/BFb0078038]
Peter Landweber, Douglas Ravenel, Robert Stong, Periodic cohomology theories defined by elliptic curves, in Haynes Miller et. al. (eds.), The Cech centennial: A conference on homotopy theory, June 1993, AMS (1995) (pdf)
and in the universal guise of topological modular forms in:
Surveys:
Matthew Greenberg, Constructing elliptic cohomology, McGill University 2002 (oclc:898194373, pdf)
Paul Goerss, Topological modular forms (after Hopkins, Miller, and Lurie), Séminaire Bourbaki : volume 2008/2009 exposés 997-1011 (arXiv:0910.5130, numdam:AST_2010__332__221_0)
Jacob Lurie, A Survey of Elliptic Cohomology, in: Algebraic Topology, Abel Symposia Volume 4, 2009, pp 219-277 (pdf, doi:10.1007/978-3-642-01200-6_9)
Charles Rezk, Elliptic cohomology and elliptic curves, Felix Klein Lectures, Bonn (2015) web, pdf, pdf, video rec: part 1, 2, 3, 4
Textbook accounts:
Charles Thomas, Elliptic cohomology, Kluwer Academic, 2002 (doi:10.1007/b115001, pdf)
Christopher Douglas, John Francis, André Henriques, Michael Hill (eds.), Topological Modular Forms, Mathematical Surveys and Monographs Volume 201, AMS 2014 (ISBN:978-1-4704-1884-7)
On equivariant elliptic cohomology and positive energy representations of loop groups:
Eduard Looijenga, Root systems and elliptic curves, Invent. Math. 38 1 (1976/77) 17-32 [doi:10.1007/BF01390167]
Ian Grojnowski, Delocalised equivariant elliptic cohomology (1994), in: Elliptic cohomology, Volume 342 of London Math. Soc. Lecture Note Ser., pages 114–121. Cambridge Univ. Press, Cambridge, 2007 (pdf, doi:10.1017/CBO9780511721489.007)
Victor Ginzburg, Mikhail Kapranov, Eric Vasserot, Elliptic Algebras and Equivariant Elliptic Cohomology (arXiv:q-alg/9505012)
Matthew Ando, Power operations in elliptic cohomology and representations of loop groups, Transactions of the American Mathematical Society 352, 2000, pp. 5619-5666. (jstor:221905, pdf)
David Gepner, Homotopy topoi and equivariant elliptic cohomology, University of Illinois at Urbana-Champaign, 2005 (pdf)
David Gepner, Lennart Meier, On equivariant topological modular forms, (arXiv:2004.10254)
Michèle Vergne, Bouquets revisited and equivariant elliptic cohomology, International Journal of Mathematics 2021 (arXiv:2005.00312, doi:10.1142/S0129167X21400127)
Relation to Kac-Weyl characters of loop group representations
Jean-Luc Brylinski, Representations of loop groups, Dirac operators on loop space, and modular forms, Topology, 29(4):461–480, 1990 (doi:10.1016/0040-9383(90)90016-D)
Nora Ganter, The elliptic Weyl character formula, Compositio Mathematica, Vol 150, Issue 7 (2014), pp 1196-1234 (arXiv:1206.0528)
The case of twisted ad-equivariant Tate K-theory:
Nora Ganter, Section 3.1 in: Stringy power operations in Tate K-theory (arXiv:math/0701565)
Nora Ganter, Power operations in orbifold Tate K-theory, Homology Homotopy Appl. Volume 15, Number 1 (2013), 313-342. (arXiv:1301.2754, euclid:hha/1383943680)
Zhen Huan, Quasi-Elliptic Cohomology I, Advances in Mathematics, Volume 337, 15 October 2018, Pages 107-138 (arXiv:1805.06305, doi:10.1016/j.aim.2018.08.007)
Zhen Huan, Quasi-theories (arXiv:1809.06651)
Kiran Luecke, Completed K-theory and Equivariant Elliptic Cohomology, Advances in Mathematics 410 B (2022) 108754 [arXiv:1904.00085, doi:10.1016/j.aim.2022.108754]
Thomas Dove, Twisted Equivariant Tate K-Theory (arXiv:1912.02374)
See also:
Daniel Berwick-Evans, Arnav Tripathy, A geometric model for complex analytic equivariant elliptic cohomology, (arXiv:1805.04146)
Nicolò Sibilla, Paolo Tomasini: Equivariant Elliptic Cohomology and Mapping Stacks I [arXiv:2303.10146]
Formulation of (equivariant) elliptic cohomology in derived algebraic geometry/E-∞ geometry (derived elliptic curves):
Paul Goerss, Michael Hopkins, Moduli spaces of commutative ring spectra, in Structured ring spectra, London
Math. Soc. Lecture Note Ser., vol. 315, Cambridge Univ. Press, Cambridge, 2004, pp. 151-200. (pdf, doi:10.1017/CBO9780511529955.009)
Paul Goerss, Michael Hopkins, Moduli problems for structured ring spectra (pdf)
Jacob Lurie, Elliptic Cohomology I: Spectral abelian varieties, 2018. 141pp (pdf)
Jacob Lurie, Elliptic Cohomology II: Orientations, 2018. 288pp (pdf)
Jacob Lurie, Elliptic Cohomology III: Tempered Cohomology, 2019. 286pp (pdf)
Jacob Lurie, Elliptic Cohomology IV: Equivariant elliptic cohomology, to appear.
The general concept of elliptic genus originates with:
Early development:
Don Zagier, Note on the Landweber-Stong elliptic genus 1986 (pdf, edoc:744944)
Friedrich Hirzebruch, Thomas Berger, Rainer Jung, chapter 2 of: Manifolds and Modular Forms, Aspects of Mathematics 20, Viehweg (1992), Springer (1994) [doi:10.1007/978-3-663-10726-2, pdf]
D.V. Chudnovsky, G.V. Chudnovsky, Elliptic modular functions and elliptic genera, Topology, Volume 27, Issue 2, 1988, Pages 163–170 (doi:10.1016/0040-9383(88)90035-3)
Mark Hovey, Spin Bordism and Elliptic Homology, Math Z 219, 163–170 (1995) (doi:10.1007/BF02572356)
Matthias Kreck, Stephan Stolz, -bundles and elliptic homology, Acta Math, 171 (1993) 231-261 (pdf, euclid:acta/1485890737)
Review:
Peter Landweber, Elliptic genera: An introductory overview In: P. Landweber (eds.) Elliptic Curves and Modular Forms in Algebraic Topology, Lecture Notes in Mathematics, vol 1326. Springer (1988) (doi:10.1007/BFb0078036)
Kefeng Liu, Modular forms and topology, Proc. of the AMS Conference on the Monster and Related Topics, Contemporary Math. (1996) (pdf, pdf, doi:10.1090/conm/193)
Serge Ochanine, What is… an elliptic genus?, Notices of the AMS, volume 56, number 6 (2009) (pdf)
The Stolz conjecture on the Witten genus:
Stephan Stolz, A conjecture concerning positive Ricci curvature and the Witten genus, Mathematische Annalen Volume 304, Number 1 (1996) (doi:10.1007/BF01446319)
Anand Dessai, Some geometric properties of the Witten genus, in: Christian Ausoni, Kathryn Hess, Jérôme Scherer (eds.) Alpine Perspectives on Algebraic Topology, Contemporary Mathematics 504 (2009) (pdf, pdf,
The Jacobi form-property of the Witten genus:
The identification of elliptic genera, via fiber integration/Pontrjagin-Thom collapse, as complex orientations of elliptic cohomology (sigma-orientation/string-orientation of tmf/spin-orientation of Tate K-theory):
Michael Hopkins, Topological modular forms, the Witten genus, and the theorem of the cube, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) (Basel), Birkhäuser, 1995, 554–565. MR 97i:11043 (pdf)
Matthew Ando, Michael Hopkins, Neil Strickland, Elliptic spectra, the Witten genus and the theorem of the cube, Invent. Math. 146 (2001) 595–687 MR1869850 (doi:10.1007/s002220100175, pdf)
Matthew Ando, Michael Hopkins, Neil Strickland, The sigma orientation is an H-infinity map, American Journal of Mathematics Vol. 126, No. 2 (Apr., 2004), pp. 247-334 (arXiv:math/0204053, doi:10.1353/ajm.2004.0008)
Matthew Ando, Michael Hopkins, Charles Rezk, Multiplicative orientations of KO-theory and the spectrum of topological modular forms, 2010 (pdf, pdf)
For the Ochanine genus:
Genera in equivariant elliptic cohomology and the rigidity theorem for equivariant elliptic genera:
The statement, with a string theory-motivated plausibility argument, is due to Witten 87.
The first proof was given in:
Clifford Taubes, -actions and elliptic genera, Comm. Math. Phys. Volume 122, Number 3 (1989), 455-526 (euclid:cmp/1104178471)
Raoul Bott, Clifford Taubes, On the Rigidity Theorems of Witten, Journal of the American Mathematical Society Vol. 2, No. 1 (Jan., 1989), pp. 137-186 (doi:10.2307/1990915)
Reviewed in:
Further proofs and constructions:
Friedrich Hirzebruch, Elliptic Genera of Level for Complex Manifolds, In: Bleuler K., Werner M. (eds) Differential Geometrical Methods in Theoretical Physics NATO ASI Series (Series C: Mathematical and Physical Sciences), vol 250. Springer (1988) (doi:10.1007/978-94-015-7809-7_3)
I. M. Krichever, Generalized elliptic genera and Baker-Akhiezer functions, Mathematical Notes of the Academy of Sciences of the USSR 47, 132–142 (1990) (doi:10.1007/BF01156822)
Kefeng Liu, On modular invariance and rigidity theorems, J. Differential Geom. Volume 41, Number 2 (1995), 343-396 (euclid:jdg/1214456221)
Kefeng Liu, On elliptic genera and theta-functions, Topology Volume 35, Issue 3, July 1996, Pages 617-640 (doi:10.1016/0040-9383(95)00042-9)
Anand Dessai, Rainer Jung, On the Rigidity Theorem for Elliptic Genera, Transactions of the American Mathematical Society Vol. 350, No. 10 (Oct., 1998), pp. 4195-4220 (26 pages) (jstor:117694)
Ioanid Rosu, Equivariant Elliptic Cohomology and Rigidity, American Journal of Mathematics 123 (2001), 647-677 (arXiv:math/9912089)
Matthew Ando, John Greenlees, Circle-equivariant classifying spaces and the rational equivariant sigma genus, Math. Z. 269, 1021–1104 (2011) (arXiv:0705.2687, doi:10.1007/s00209-010-0773-7)
On manifolds with SU(2)-action:
Discussion of elliptic genera twisted by a gauge bundle, i.e. for string^c structure):
Matthew Ando, Maria Basterra, The Witten genus and equivariant elliptic cohomology, Math Z 240, 787–822 (2002) (arXiv:math/0008192, doi:10.1007/s002090100399)
Matthew Ando, The sigma orientation for analytic circle equivariant elliptic cohomology, Geom. Topol., 7:91–153, 2003 (arXiv:math/0201092, euclid:gt/1513883094)
Matthew Ando, Andrew Blumberg, David Gepner, Section 11 of Twists of K-theory and TMF, in Robert S. Doran, Greg Friedman, Jonathan Rosenberg, Superstrings, Geometry, Topology, and -algebras, Proceedings of Symposia in Pure Mathematics, vol 81, American Mathematical Society, 2010 (arXiv:1002.3004)
Qingtao Chen, Fei Han, Weiping Zhang, Generalized Witten Genus and Vanishing Theorems, Journal of Differential Geometry 88.1 (2011): 1-39. (arXiv:1003.2325)
Jianqing Yu, Bo Liu, On the Witten Rigidity Theorem for Manifolds, Pacific Journal of Mathematics 266.2 (2013): 477-508. (arXiv:1206.5955)
Fei Han, Varghese Mathai, Projective elliptic genera and elliptic pseudodifferential genera, Adv. Math. 358 (2019) 106860 (arXiv:1903.07035)
Haibao Duan, Fei Han, Ruizhi Huang, Structures and Modular Invariants, Trans. AMS 2020 (arXiv:1905.02093)
The interpretation of elliptic genera (especially the Witten genus) as the partition function of a 2d superconformal field theory (or Landau-Ginzburg model) – and especially of the heterotic string (“H-string”) or type II superstring worldsheet theory has precursors in
A. N. Schellekens, Nicholas P. Warner, Anomalies and modular invariance in string theory, Physics Letters B 177 (3-4), 317-323, 1986 (doi:10.1016/0370-2693(86)90760-4)
A. N. Schellekens, Nicholas P. Warner, Anomalies, characters and strings, Nuclear Physics B Volume 287, 1987, Pages 317-361 (doi:10.1016/0550-3213(87)90108-8)
Wolfgang Lerche, Bengt Nilsson, A. N. Schellekens, Nicholas P. Warner, Anomaly cancelling terms from the elliptic genus, Nuclear Physics B Volume 299, Issue 1, 28 March 1988, Pages 91-116 (doi:10.1016/0550-3213(88)90468-3)
and then strictly originates with:
Edward Witten, Elliptic genera and quantum field theory, Comm. Math. Phys. Volume 109, Number 4 (1987), 525-536. (euclid:cmp/1104117076)
Edward Witten, On the Landau-Ginzburg Description of Minimal Models, Int. J. Mod. Phys.A9:4783-4800,1994 (arXiv:hep-th/9304026)
Toshiya Kawai, Yasuhiko Yamada, Sung-Kil Yang, Elliptic Genera and Superconformal Field Theory, Nucl. Phys. B414:191-212, 1994 (arXiv:hep-th/9306096, doi:10.1016/0550-3213(94)90428-6)
Sujay K. Ashok, Jan Troost, A Twisted Non-compact Elliptic Genus, JHEP 1103:067, 2011 (arXiv:1101.1059)
Matthew Ando, Eric Sharpe, Elliptic genera of Landau-Ginzburg models over nontrivial spaces, Adv. Theor. Math. Phys. 16 (2012) 1087-1144 (arXiv:0905.1285)
Review in:
Miranda Cheng, (Mock) Modular Forms in String Theory and Moonshine, lecture notes 2016 (pdf)
Katrin Wendland, Section 2.4 in: Snapshots of Conformal Field Theory, in: Mathematical Aspects of Quantum Field Theories Mathematical Physics Studies. Springer 2015 (arXiv:1404.3108, doi:10.1007/978-3-319-09949-1_4)
With emphasis on orbifold CFTs:
Formulation via super vertex operator algebras:
Hirotaka Tamanoi, Elliptic Genera and Vertex Operator Super-Algebras, Springer 1999 (doi:10.1007/BFb0092541)
Chongying Dong, Kefeng Liu, Xiaonan Ma, Elliptic genus and vertex operator algebras, Algebr. Geom. Topol. 1 (2001) 743-762 (arXiv:math/0201135, doi:10.2140/agt.2001.1.743)
and for the topologically twisted 2d (2,0)-superconformal QFT (the heterotic string with enhanced supersymmetry) via sheaves of vertex operator algebras in
based on chiral differential operators:
In relation to error-correcting codes:
Tentative interpretation as indices of Dirac-Ramond operators as would-be Dirac operators on smooth loop space:
Edward Witten, The Index Of The Dirac Operator In Loop Space, in: Elliptic Curves and Modular Forms in Algebraic Topology, Lecture Notes in Mathematics 1326, Springer (1988) 161-181 [doi:10.1007/BFb0078045, spire]
originating from:
Edward Witten, p. 92-94 in: Global anomalies in string theory, in: W. Bardeen and A. White (eds.) Symposium on Anomalies, Geometry, Topology, World Scientific (1985) 61-99 [pdf, spire:214913]
Orlando Alvarez, T. P. Killingback, Michelangelo Mangano, Paul Windey, The Dirac-Ramond operator in string theory and loop space index theorems, Nuclear Phys. B Proc. Suppl., 1A:189–215, 1987, in: Nonperturbative methods in field theory, 1987 (doi"10.1016/0920-5632(87)90110-1)
Orlando Alvarez, T. P. Killingback, Michelangelo Mangano, Paul Windey, String theory and loop space index theorems, Comm. Math. Phys., 111(1):1–10, 1987 (euclid:cmp/1104159462)
Gregory Landweber, Dirac operators on loop space, PhD thesis (Harvard 1999) (pdf)
Orlando Alvarez, Paul Windey, Analytic index for a family of Dirac-Ramond operators, Proc. Natl. Acad. Sci. USA, 107(11):4845–4850, 2010 (arXiv:0904.4748)
Tentative formulation via conformal nets:
The resulting suggestion that, roughly, deformation-classes (concordance classes) of 2d SCFTs with target space are the generalized cohomology of with coefficients in the spectrum of topological modular forms (tmf):
and the more explicit suggestion that, under this identification, the Chern-Dold character from tmf to modular forms, sends a 2d SCFT to its partition function/elliptic genus/supersymmetric index:
This perspective is also picked up in Gukov, Pei, Putrov & Vafa 18.
Discussion of the 2d SCFTs (namely supersymmetric SU(2)-WZW-models) conjecturally corresponding, under this conjectural identification, to the elements of (the third stable homotopy group of spheres):
Davide Gaiotto, Theo Johnson-Freyd, Edward Witten, p. 17 of: A Note On Some Minimally Supersymmetric Models In Two Dimensions, (arXiv:1902.10249) in S. Novikov et al. Integrability, Quantization, and Geometry: II. Quantum Theories and Algebraic Geometry, Proc. Symposia Pure Math., 103(2), 2021 (ISBN: 978-1-4704-5592-7)
Davide Gaiotto, Theo Johnson-Freyd, Mock modularity and a secondary elliptic genus (arXiv:1904.05788)
Theo Johnson-Freyd, Topological Mathieu Moonshine (arXiv:2006.02922)
Discussion properly via (2,1)-dimensional Euclidean field theory:
Daniel Berwick-Evans, How do field theories detect the torsion in topological modular forms? arXiv:2303.09138
Daniel Berwick-Evans, How do field theories detect the torsion in topological modular forms?, talk at QFT and Cobordism, CQTS (Mar 2023) web, video:YT
See also:
Ying-Hsuan Lin, Du Pei, Holomorphic CFTs and topological modular forms [arXiv:2112.10724]
Jan Albert, Justin Kaidi, Ying-Hsuan Lin, Topological modularity of Supermoonshine arXiv:2210.14923
Yuji Tachikawa, Mayuko Yamashita, Kazuya Yonekura, Remarks on mod-2 elliptic genus arXiv:2302.07548
Yuji Tachikawa, Hao Y. Zhang, On a -valued discrete topological term in 10d heterotic string theories [arXiv:2403.08861]
Theo Johnson-Freyd, Mayuko Yamashita, On the 576-fold periodicity of the spectrum SQFT: The proof of the lower bound via the Anderson duality pairing [arXiv:2404.06333]
Vivek Saxena, A T-Duality of Non-Supersymmetric Heterotic Strings and an implication for Topological Modular Forms [arXiv:2405.19409]
Further on the elliptic genus of the heterotic string being the Witten genus:
The interpretation of equivariant elliptic genera as partition functions of parametrized WZW models in heterotic string theory:
Jacques Distler, Eric Sharpe, section 8.5 of Heterotic compactifications with principal bundles for general groups and general levels, Adv. Theor. Math. Phys. 14:335-398, 2010 (arXiv:hep-th/0701244)
Matthew Ando, Equivariant elliptic cohomology and the Fibered WZW models of Distler and Sharpe, talk 2007 (lecture notes pdf)
Proposals on physics aspects of lifting the Witten genus to topological modular forms:
Yuji Tachikawa, Topological modular forms and the absence of a heterotic global anomaly, Progress of Theoretical and Experimental Physics, 2022 4 (2022) 04A107 arXiv:2103.12211, doi:10.1093/ptep/ptab060
Yuji Tachikawa, Mayuko Yamashita, Topological modular forms and the absence of all heterotic global anomalies, Comm. Math. Phys. 402 (2023) 1585-1620 arXiv:2108.13542, doi:10.1007/s00220-023-04761-2
Yuji Tachikawa, Mayuko Yamashita, Anderson self-duality of topological modular forms, its differential-geometric manifestations, and vertex operator algebras arXiv:2305.06196
On the M5-brane elliptic genus:
A 2d SCFT argued to describe the KK-compactification of the M5-brane on a 4-manifold (specifically: a complex surface) originates with
Discussion of the resulting elliptic genus (2d SCFT partition function) originates with:
Davide Gaiotto, Andrew Strominger, Xi Yin, The M5-Brane Elliptic Genus: Modularity and BPS States, JHEP 0708:070, 2007 (hep-th/0607010)
Davide Gaiotto, Xi Yin, Examples of M5-Brane Elliptic Genera, JHEP 0711:004, 2007 (arXiv:hep-th/0702012)
Further discussion in:
Murad Alim, Babak Haghighat, Michael Hecht, Albrecht Klemm, Marco Rauch, Thomas Wotschke, Wall-crossing holomorphic anomaly and mock modularity of multiple M5-branes, Comm. Math. Phys. 339 (2015) 773–814 arXiv:1012.1608, doi:10.1007/s00220-015-2436-3
Sergei Gukov, Du Pei, Pavel Putrov, Cumrun Vafa, 4-manifolds and topological modular forms, J. High Energ. Phys. 2021 84 (2021) arXiv:1811.07884, doi:10.1007/JHEP05(2021)084, spire:1704312
On the elliptic genus of M-strings inside M5-branes:
Stefan Hohenegger, Amer Iqbal, M-strings, Elliptic Genera and String Amplitudes, Fortschritte der PhysikVolume 62, Issue 3 (arXiv:1310.1325)
Stefan Hohenegger, Amer Iqbal, Soo-Jong Rey, M String, Monopole String and Modular Forms, Phys. Rev. D 92, 066005 (2015) (arXiv:1503.06983)
M. Nouman Muteeb, Domain walls and M2-branes partition functions: M-theory and ABJM Theory (arXiv:2010.04233)
Kimyeong Lee, Kaiwen Sun, Xin Wang, Twisted Elliptic Genera [arXiv:2212.07341]
On the elliptic genus of E-strings as wrapped M5-branes:
J. A. Minahan, D. Nemeschansky, Cumrun Vafa, N. P. Warner, E-Strings and Topological Yang-Mills Theories, Nucl. Phys. B527 (1998) 581-623 (arXiv:hep-th/9802168)
Wenhe Cai, Min-xin Huang, Kaiwen Sun, On the Elliptic Genus of Three E-strings and Heterotic Strings, J. High Energ. Phys. 2015, 79 (2015). (arXiv:1411.2801, doi:10.1007/JHEP01(2015)079)
On the elliptic genus of E-strings as M2-branes ending on M5-branes:
Last revised on July 12, 2024 at 07:14:14. See the history of this page for a list of all contributions to it.