nLab
Dirac-Ramond operator

Idea

The 0-mode of the supercharge in a 2d SCFT (an operator in a (sheaf of) super vertex operator algebra) behaves like a higher dimensional analog of the operator $D$ in a spectral triple, hence like the supercharge in supersymmetric quantum mechanics (see the references there).

Specifically for a sigma-model 2d SCFT induced from some target space geometry – such as the worldsheet-quantum field theory of a superstring propagating on that target spacetimes – the Dirac-Ramond operator is a higher analogue of a Dirac operator on that target spacetime (roughly like what one would expect of a Dirac operator on a smooth loop space). This is called the Dirac-Ramond operator (Ramond 71).

The index of the large volume limit of the Dirac-Ramond operator is what is now known as the Witten genus (but in fact the original article Alvarez-Killingback-Mangano-Windey 87 appeared independently and almost in parallel of Witten’s discussion).

Related concepts

• 2-spectral triple

• string structure, twisted string structure

• Dirac operator on smooth loop space

partition functions in quantum field theory as indices/genera/orientations in generalized cohomology theory:

$d$	partition function in $d$-dimensional QFT	supercharge	index in cohomology theory	genus	logarithmic coefficients of Hirzebruch series
0			push-forward in ordinary cohomology: integration of differential forms			orientation
1	spinning particle	Dirac operator	KO-theory index	A-hat genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin \to KO$
	endpoint of 2d Poisson-Chern-Simons theory string	Spin^c Dirac operator twisted by prequantum line bundle	space of quantum states of boundary phase space/Poisson manifold	Todd genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin^c \to KU$
	endpoint of type II superstring	Spin^c Dirac operator twisted by Chan-Paton gauge field	D-brane charge	Todd genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin^c \to KU$
2	type II superstring	Dirac-Ramond operator	superstring partition function in NS-R sector	Ochanine elliptic genus		SO orientation of elliptic cohomology
	heterotic superstring	Dirac-Ramond operator	superstring partition function	Witten genus	Eisenstein series	string orientation of tmf
	self-dual string		M5-brane charge
3						w4-orientation of EO(2)-theory

References

General

The original article, in the context of the superstring of string theory“

• Pierre Ramond, Dual theory for Free Fermions, Phys Rev. D3 2415 (1971) (doi:10.1103/PhysRevD.3.2415)

The Dirac-Ramond operator originates with the early beginning of superstring models, when they were still called spinning strings – see there for more references.

The concept gained more attention in pure mathematics when it was found that the large volume limit of its index, when properly construed, is a universal elliptic genus, now known as the Witten genus. See there for more references.

Articles that explicitly consider the Dirac-Ramond operator in this context:

• Orlando Alvarez, T. P. Killingback, Michelangelo Mangano, and Paul Windey, The Dirac-Ramond operator in string theory and loop space index theorems, Nuclear Phys. B Proc. Suppl., 1A:189–215, 1987. Nonperturbative methods in field theory (Irvine, CA, 1987)., also: Comm. Math. Phys. Volume 111, Number 1 (1987), 1-160 (Ecudid)

• 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

Elliptic genera as super $p$-brane partition functions

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 – 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 $N=2$ Minimal Models, Int. J. Mod. Phys.A9:4783-4800,1994 (arXiv:hep-th/9304026)

• Toshiya Kawai, Yasuhiko Yamada, Sung-Kil Yang, Elliptic Genera and $N=2$ 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)

Via super vertex operator algebra

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

• Pokman Cheung, The Witten genus and vertex algebras (arXiv:0811.1418)

based on chiral differential operators:

• Vassily Gorbounov, Fyodor Malikov, Vadim Schechtman, Gerbes of chiral differential operators, Math. Res. Lett. 7(1), 55–66 (2000) (arXiv:math/9906117, arXiv:math/0003170, arXiv:math/0005201)

Via Dirac-Ramond operators on free loop space

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, Proc. of Conf. on Elliptic Curves and Modular Forms in Algebraic Topology Princeton (1986) (spire:245523, doi:10.1007/BFb0078045)

• 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)

Via functorial QFT

Tentative formulation via functorial quantum field theory ((2,1)-dimensional Euclidean field theories and tmf):

• Stephan Stolz, Peter Teichner, Supersymmetric field theories and generalized cohomology, in Hisham Sati, Urs Schreiber (eds.), Mathematical foundations of Quantum field theory and String theory, Proceedings of Symposia in Pure Mathematics, Volume 83, AMS 2011 (arXiv:1108.0189)

Via conformal nets

Tentative formulation via conformal nets:

• Chris Douglas, André Henriques, Topological modular forms and conformal nets, in Hisham Sati, Urs Schreiber (eds.), Mathematical Foundations of Quantum Field and Perturbative String Theory, Proceedings of Symposia in Pure Mathematics, AMS (2011) (arXiv:1103.4187, doi:10.1090/pspum/083)

Occurrences in string theory

H-string elliptic 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)

M5-brane elliptic genus

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

• Juan Maldacena, Andrew Strominger, Edward Witten, Black Hole Entropy in M-Theory, JHEP 9712:002, 1997 (arXiv:hep-th/9711053)

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 (arXiv:1012.1608)

• Sergei Gukov, Du Pei, Pavel Putrov, Cumrun Vafa, 4-manifolds and topological modular forms (arXiv:1811.07884, spire:1704312)

M-string elliptic genus

On the elliptic genus of M-strings inside M5-branes:

• Stefan Hohenegger, Amer Iqbal, M-strings, Elliptic Genera and $\mathcal{N}=4$ 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)

E-string elliptic genus

On the elliptic genus of E-strings as wrapped M5-branes:

• J. A. Minahan, D. Nemeschansky, Cumrun Vafa, N. P. Warner, E-Strings and $N=4$ 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:

• Joonho Kim, Seok Kim, Kimyeong Lee, Jaemo Park, Cumrun Vafa, Elliptic Genus of E-strings, JHEP 1709 (2017) 098 (arXiv:1411.2324)