nLab Witten genus

Idea

The Witten genus is a genus with coefficients in power series in one variable, playing the role of a universal elliptic genus. This arises (Witten 87) as the large volume limit of the partition function of the superstring (hence in the string worldsheet perturbation theory about constant worldsheet configurations). Specifically, for the type II superstring this reproduces the universal elliptic genus as previously introduced by Serge Ochanine, while for the heterotic string it yields what is now called the Witten genus proper. Concretely, as Witten argued, this is a formal power series in string oscillation modes of the A-hat genus of the symmetric tensor powers of the tangent bundle that these modes take values in.

In (Witten 86) it is suggested, by regarding the superstring sigma-model as quantum mechanics on the smooth loop space of its target space, that the Witten genus may be thought of as the large volume limit of an $S^1$-equivariant A-hat genus on smooth loop space, hence the index of the Dirac-Ramond operator in that limit. (Ever since this suggestion people have tried to make precise the concept of Dirac operator on a smooth loop space (e.g. Alvarez-Killingback-Mangano-Windey 87). But notice that, by the above, only the formal loop space and the Dirac-Ramond operator really appears in the definition of the Witten genus.)

A priori the coefficients of the Witten genus as a genus on oriented manifolds are formal power series over the rational numbers

$$w \;\colon\; M SO_\bullet \longrightarrow \mathbb{Q}[ [ q ] ] \,.$$

In the construction from string physics this map is interpreted as sending a target spacetime $X$ of the superstring to the function $w_X(q) = w_X(e^{2 \pi i \tau})$ which to each modulus $\tau \in \mathbb{C}$ characterizing a toroidal Riemann surface assigns the partition function of the superstring with worldsheet the torus $\mathbb{C}/(\mathbb{Z} + \mathbb{Z}\tau)$ and propagating on target space $X$.

On manifolds with spin structure the genus refines to integral power series (via the integrality of the A-hat genus (Chudnovsky-Chudnovsky 88, Kreck-Stolz 93, Hovey 91). Moreover on manifolds with rational string structure it takes values in modular forms (Zagier 86) and crucially, on manifolds with string structure it takes values in topological modular forms

$$\array{ M String_\bullet &\longrightarrow& tmf_\bullet \\ \downarrow && \downarrow \\ \Omega_\bullet^{String, rat} &\longrightarrow& MF_\bullet \\ \downarrow && \downarrow \\ M Spin_\bullet &\longrightarrow& \mathbb{Z}[[ q ] ] \\ \downarrow && \downarrow \\ M SO_\bullet &\stackrel{w}{\longrightarrow}& \mathbb{Q}[ [ q ] ] } \,.$$

(On the left is the image under forming Thom spectra/cobordism rings of the first stages in the Whitehead tower of $BO$, see also at higher spin structure.)

Observe here that topological modular forms are the coefficient ring of the E-∞ ring spectrum known as tmf. By the general way in which genera (see there) tend to appear as decategorifications of homomorphisms of E-∞ rings out of a Thom spectrum, this suggests that the Witten genus is the value on homotopy groups of a homomorphism of E-∞ rings of the form

$$\sigma \colon M String \longrightarrow tmf$$

from the Thom spectrum of String bordism to the tmf-spectrum. This lift of the Witten genus to a universal orientation in universal elliptic cohomology indeed exists and is called the sigma-orientation, or the string orientation of tmf.

This construction has been the central motivation behind the search for and construction of tmf (Hopkins 94). A construction of the string orientation of tmf is given in (Ando-Hopkins-Rezk 10) and it is shown that indeed it refines the Witten genus (Ando-Hopkins-Rezk 10, prop. 15.3).

It is maybe noteworthy that tmf (and hence its universal string orientation) also arises canonically from just studying chromatic homotopy theory (see Mazel-Gee 13 for a nice survey of this) a fundamental topic in stable homotopy theory, hence a fundamental topic in mathematics. Therefore in the Witten genus some very fundamental pure mathematics happens to equivalently incarnate as some conjecturally very fundamental physics (string theory).

Properties

Characteristic series

The characteristic series of the Witten genus as a power series in $z$ with coefficients in formal power series in $q$ over $\mathbb{Q}$ is

$$\begin{aligned} K_w(z)(q) & = \frac{z}{\exp_w(z)(q)} \\ & = \frac{z}{\sigma_L(z)(q)} \\ & = \frac{z/2}{sinh(z/2)} \prod_{n \geq 1} \frac{(1-q^n)^2}{(1-q^n e^z)(1-q^n e^{-z})} \\ & = \exp\left( \sum_{k \geq 2} 2G_k(q) \frac{z^k}{k!} \right) \end{aligned} \,,$$

where

• $\sigma_L$ is the Weierstrass sigma-function (see e.g. Ando Basterra 00, section 5.1);

• $G_k$ are the Eisenstein series (Zagier 86, equation (14), Ando-Hopkins-Rezk 10, prop. 10.9).

This is a modular form with respect to the variable $q$, see also the the discussion below at Integrality and modularity . Such functions which are power series of two variables $z$ and $q$ with elliptic nature in $z$ and modular nature in $q$ are called Jacobi forms (Zagier 86, p. 8, Ando-French-Ganter 08).

There are various further ways to equivalently re-express the above in terms of other special modular forms. Here are some:

In terms of Kac-Weyl characters

The Witten genus has a close relation to the Kac-Weyl character of loop group representations.

Consider of four irreducible level-1 positive energy Spin$(2k)$-loop group representation the one denoted

$$\tilde S_+ - \tilde S_- \in Rep(\tilde L Spin(2k))$$

and write its Kac-Weyl character as

$$\chi(\tilde S_+ - \tilde S_-) \in Rep(Spin(2k))[ [ q^{1/12} ] ] \,.$$

Under passing to group characters this is (Brylinski 90, p. 7(467), reviewed in KL 96, section 1.2) equivalently

$$\chi(\tilde S_+ - \tilde S_-) = \frac{\prod_{1}^k \theta}{\eta^k} \,,$$

where on the right we have the Jacobi theta-function $\theta$ divided by the Dedekind eta-function $\eta$.

Comparison shows that in terms of this the exponential series of the Witten genus is equivalently (by the splitting principle the $k$-fold products are left implicit):

$$\exp_w = z/K_w = \eta^2 \, \chi(\tilde S_+ - \tilde S_-) \,.$$

Notice that, by the relation (see here) between equivariant elliptic cohomology and loop group representations, over the complex numbers $\chi(\tilde S_+ - \tilde S_-)$ may be regarded as an element of the $Spin(2k)$-equivariant elliptic cohomology of the point (at the Tate curve, see at twisted ad-equivariant Tate K-theory).

Integrality and modularity

A priori, the Witten genus has coefficients the power series ring $\mathbb{Q}[ [q] ]$ over the rational numbers. But under suitable conditions (quantum anomaly cancellation) it takes values in more interesting subrings.

For the type II superstring

The genus obtained from the type II superstring in the NS-R sector is a modular form for the congruence subgroup $\Gamma_2(2)$. (Witten 87a, below (13)) See at congruence subgroup – Relation to spin structures for more.

Hence, with suitable normalization, the universal Witten-Ochanine genus takes values in the subring $MF_\bullet^{\mathbb{Q}}(\Gamma_0(2)) \hookrightarrow \mathbb{Q}[ [q] ]$ of modular forms for $\Gamma_0(2)\subset SL_2(\mathbb{Z})$ with rational coefficients (Zagier 86, item d) on page 2 based on Chudnovsky-Chudnovsky 88).

For the heterotic superstring

On manifolds with spin structure the heterotic string Witten genus has integral coeffcients, hence in the ring $\mathbb{Z}[ [ q ] ]$ (Chudnovsky-Chudnovsky 88, Landweber 88), see also (Kreck-Stolz 93, Hovey 91).

On manifolds with rational string structure (meaning spin structure and the first fractional Pontryagin class is at most torsion), then the Witten genus takes values in actual modular forms $MF_\bullet$ (Zagier 86, page 6).

On manifolds with actual string structure, finally, the Witten genus factors through topological modular forms (Hopkins 94, Ando-Hopkins-Rezk 10).

Relation to Dirac operators and supersymmetric QM on loop space

Originally in (Witten 87a) the elliptic genus was derived as the large volume limit of the index of the supercharge of the superstring worldsheet 2d SCFT. Here the “large volume limit” is what restricts the oscillations of the string to be “small”. But then in (Witten87b) it was observed that if this supercharge – the Dirac-Ramond operator – would really behave like a Dirac operator on smooth loop space, then the elliptic genus would be the $S^1$-equivariant index of a Dirac operator, where $S^1$ acts by rigid rotationl of the parameterization of the loops, and by analogy standard formulas for equivariant indices in K-theory would imply the localization to the tangent spaces to the space of constant loops.

Notice that the would-be Dirac operator on smooth loop space is what would realize the superstring quantum dynamics as supersymmetric quantum mechanics on smooth loop space. This observation was the original motivation for the study of supersymmetric quantum mechanics in (Witten 82, Witten 85) in the presence of a given Killing vector field (correspinding to the $S^1$-action on loop space ).

Heterotic (twisted) Witten genus, loop group representations and parameterized WZW models

If the superstring in question is the heterotic string then generally there is a “twist” of its background fields by a gauge field, hence by a $G$-principal bundle for $G$ some simply connected compact Lie group (notably E8). The partition function in this case is a “twisted Witten genus” (Witten 87, equations (30), (31), Brylinski 90, KL 95). The modularity condition then is no longer just that the tangent bundle has string structure, but that together with the gauge bundle it has twisted string structure, hence String^c-structure for $c$ the $G$-second Chern class (explicitly identified as such in (Chen-Han-Zhang 10).

An elegant formulation of twisted Witten genera (and proof of their rigidity) in terms of highest weight loop group representations is given in (KL 95) along the lines of (Brylinski 90). In (Distler-Sharpe 07), following suggestions around (Ando 07) this is interpreted geometrically in terms of fiberwise indices of parameterized WZW models associated to the given String-principal 2-bundle.

What should be a concrete computation of the twisted Witten genus specifically for $G =$ E8 is in (Harris 12, section 4).

As the global character of sheaves of vertex operator algebras

For $U \subset \mathbb{C}$ an open subset of the complex plane then the space $\mathcal{D}^{ch}(U)$ of chiral differential operators on $U$ is naturally a super vertex operator algebra. For $X$ a complex manifold such that its first Chern class and second Chern class vanish over the rational numbers, then this assignment gives a sheaf of vertex operator algebras $\mathcal{D}^{ch}_X(-)$ on $X$. Its cochain cohomology $H^\bullet(\mathcal{D}^{ch}_X)$ is itself a super vertex operator algebra and its super-Kac-Weyl character is proportional to the Witten genus $w(X)$ of $X$:

$$char H^\bullet(\mathcal{D}^{ch}_X)\propto w(X) \,.$$

Physically this result is understood by observing that $\mathcal{D}^{ch}_X$is the sheaf of quantum observables of the topologically twisted 2d (2,0)-superconformal QFT (see there for more on this) of which the Witten genus is (the large volume limit of) the partition function.

As highlighted in (Cheung 10, p. 2), there is a resolution by the chiral Dolbeault complex which gives a precise sense in which over a complex manifold the Witten genus is a stringy analog of the Todd genus. See (Cheung 10) for a brief review, where furthermore the problem of generalizing of this construction to sheaves of vertex operator algebras over more general string structure manifolds is addressed.

Stolz conjecture

The Stolz conjecture due to (Stolz 96) asserts that if $X$ is a closed manifold with String structure which furthermore admits a Riemannian metric with positive Ricci curvature, then its Witten genus vanishes.

Relation to BPS state counting on target space

By supersymmetry and by the same argument that controls the expression of the index of a Dirac operator in terms of supersymmetric quantum mechanics, the Witten genus may be thought of as counting those string states on which the left moving supercharge acts trivially. In terms of the target space theory these are the BPS states. (reviews include Dijkgraaf 98).

Therefore the Witten genus may also be used as a generating function for BPS state counting. As such it has for instance been used in the microscopic explanation of Bekenstein-Hawking entropy of black holes, see at black holes in string theory.

Relation to Cayley plane bundles

The rational Witten genus vanishes on total spaces of Cayley plane-fiber bundles, and is indeed characterized by this property (McTague 10, McTague 11).

Related concepts

• Witten index

• Tate K-theory

• elliptic Chern character

• M5-brane elliptic genus

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

Elliptic cohomology

General

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:

• Michael Hopkins, Algebraic topology and modular forms in Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), pages 291–317, Beijing, 2002. Higher Ed. Press (arXiv:math/0212397)

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)

Equivariant elliptic cohomology

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, 2017 (hdl)

• 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]

Via derived $E_\infty$-geometry

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)

  (Goerss-Hopkins-Miller theorem)

• 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.

Elliptic genera

General

The general concept of elliptic genus originates with:

• Serge Ochanine, Sur les genres multiplicatifs definis par des integrales elliptiques, Topology, Vol. 26, No. 2, 1987 (pdf, doi:10.1016/0040-9383(87)90055-3)

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, $HP^2$-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,

  doi:10.1090/conm/504)

The Jacobi form-property of the Witten genus:

• Matthew Ando, Christopher French, Nora Ganter, The Jacobi orientation and the two-variable elliptic genus, Algebraic and Geometric Topology 8 (2008) p. 493-539 (pdf)

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:

• Dylan Wilson, Orientations and Topological Modular Forms with Level Structure (arXiv:1507.05116)

Equivariant elliptic genera

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, $S^1$-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:

• Raoul Bott, On the Fixed Point Formula and the Rigidity Theorems of Witten, Lectures at Cargése 1987. In: ’t Hooft G., Jaffe A., Mack G., Mitter P.K., Stora R. (eds) Nonperturbative Quantum Field Theory. NATO ASI Series (Series B: Physics), vol 185. Springer (1988) (doi:10.1007/978-1-4613-0729-7_2)

Further proofs and constructions:

• Friedrich Hirzebruch, Elliptic Genera of Level $N$ 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:

• Anand Dessai, The Witten genus and $S^3$-actions on manifolds, 1994 (pdf, pdf)

Twisted elliptic genera

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 $C^*$-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 $String^c$ 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, $String^c$ Structures and Modular Invariants, Trans. AMS 2020 (arXiv:1905.02093)

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 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 $N=2$ 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)

With emphasis on orbifold CFTs:

• Sujay K. Ashok, Jan Troost, Orbifolded Elliptic Genera of Non-Compact Models [arXiv:2405.08370]

Formulations

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)

In relation to error-correcting codes:

• Kohki Kawabata, Shinichiro Yahagi, Elliptic genera from classical error-correcting codes $[$arXiv:2308.12592$]$

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, 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)

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)

Conjectural interpretation in tmf-cohomology

The resulting suggestion that, roughly, deformation-classes (concordance classes) of 2d SCFTs with target space $X$ are the generalized cohomology of $X$ with coefficients in the spectrum of topological modular forms (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)

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:

• Davide Gaiotto, Theo Johnson-Freyd, Section 5 of: Holomorphic SCFTs with small index, Canadian Journal of Mathematics (arXiv:1811.00589, doi:10.4153/S0008414X2100002X)

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 $\mathbb{Z}/24$ $\simeq$ $tmf^{-3}(\ast) = \pi_3(tmf)$ $\simeq$ $\pi_3(\mathbb{S})$ (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 $\mathbb{Z}_3$-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]

Occurrences in string theory

H-string elliptic genus

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$]$

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, 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$]$

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)

• Kimyeong Lee, Kaiwen Sun, Xin Wang, Twisted Elliptic Genera [arXiv:2212.07341]

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)