superalgebra and (synthetic ) supergeometry
Super-conformal field theory in dimension , locally given by a super vertex operator algebra.
For central charge 15 this is the worldsheet theory of the superstring.
May be regarded as a “2-spectral triple” (see there for more), the 2-dimensional generalization of spectral triples describing the quantum mechanics of spinning particles (super-particles).
See at supersymmetry – Classification – Superconformal algebra – In dimension 2.
A basic but detailed exposition focusing on the super-WZW model (and the perspective of 2-spectral triples) is in Fröhlich & Gawedzki (1993).
Textbook account:
Other accounts:
Lance Dixon, Paul Ginsparg, Jeffrey Harvey, Superconformal field theory (pdf)
Yasuyuki Kawahigashi, Superconformal Field Theory and Operator Algebras (pdf)
Constructing D=2 SCFTs from error-correcting codes and a hint for a relation to tmf, vaguely in line with the lift of the Witten genus to the string orientation of tmf:
further on the resulting elliptic genera:
Discussion of D=2 SCFTs as a higher analog of spectral triples (“2-spectral triples”, see there for more) is in terms of vertex operator algebras in
Jürg Fröhlich, Krzysztof Gawędzki, Conformal Field Theory and Geometry of Strings, extended lecture notes for lecture given at the Mathematical Quantum Theory Conference, Vancouver, Canada, August 4-8 (arXiv:hep-th/9310187)
Yan Soibelman, Collapsing CFTs, spaces with non-negative Ricci curvature and nc-geometry , in Hisham Sati, Urs Schreiber (eds.), Mathematical Foundations of Quantum Field and Perturbative String Theory, Proceedings of Symposia in Pure Mathematics, AMS (2001)
and in terms of conformal nets in
Discussion of D=2 conformal field theory as a functorial field theory, namely as a monoidal functor from a 2d conformal cobordism category to Hilbert spaces:
and including discussion of modular functors:
Graeme Segal, Two-dimensional conformal field theories and modular functors, in: Proceedings of the IXth International Congress on Mathematical Physics, Swansea, 1988, Hilger, Bristol (1989) 22-37.
Graeme Segal, The definition of conformal field theory, in: Ulrike Tillmann (ed.), Topology, geometry and quantum field theory , London Math. Soc. Lect. Note Ser. 308, Cambridge University Press (2004) 421-577 doi:10.1017/CBO9780511526398.019, pdf, pdf
General construction for the case of rational 2d conformal field theory is given by the
See also:
Greg Moore, Graeme Segal, D-branes and K-theory in 2D topological field theory (arXiv:hep-th/0609042)
Richard Blute, Prakash Panangaden, Dorette Pronk, Conformal field theory as a nuclear functor, Electronic Notes in Theoretical Computer Science Volume 172, 1 April 2007, Pages 101-132 GDP Festschrift (pdf, doi:10.1016/j.entcs.2007.02.005)
A different but closely analogous development for chiral 2d CFT (vertex operator algebras, see there for more):
Discussion of the case of Liouville theory:
Early suggestions to refine this to an extended 2-functorial construction:
A step towards generalization to 2d super-conformal field theory:
Discussion of 2-functorial chiral 2d CFT:
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 18, 2024 at 11:44:05. See the history of this page for a list of all contributions to it.