nLab
Witten genus

Context

Index theory

Elliptic cohomology

String theory

Contents

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). 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 1S^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:MSO [[q]]. w \;\colon\; M SO_\bullet \longrightarrow \mathbb{Q}[ [ q ] ] \,.

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

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

MString tmf Ω String,rat MF MSpin [[q]] MSO w [[q]]. \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 BOBO, 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

σ:MStringtmf \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 euivalently incarnate as some conjecturally very fundamental physics (string theory).

Properties

Characteristic series

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

K w(z)(q) =zexp w(z)(q) =zσ L(z)(q) =z/2sinh(z/2) n1(1q n) 2(1q ne z)(1q ne z) =exp( k2G k(q)z kk!), \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} G_k(q) \frac{z^k}{k!} \right) \end{aligned} \,,

where

This is a modular form with respect to the variable qq, see also the the discussion below at Integrality and modularity . Such functions which are power series of two variables zz and qq with elliptic nature in zz and modular nature in qq 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)(2k)-loop group representation the one denoted

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

and write its Kac-Weyl character as

χ(S˜ +S˜ )Rep(Spin(2k))[[q 1/12]]. \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

χ(S˜ +S˜ )= 1 kθη k, \chi(\tilde S_+ - \tilde S_-) = \frac{\prod_{1}^k \theta}{\eta^k} \,,

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

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

exp w=z/K w=η 2χ(S˜ +S˜ ). \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 χ(S˜ +S˜ )\chi(\tilde S_+ - \tilde S_-) may be regarded as an element of the Spin(2k)Spin(2k)-equivariant elliptic cohomology of the point (at the Tate curve).

Integrality and modularity

The Witten genus a priori has coefficients the power series ring [[q]]\mathbb{Q}[ [q] ] over the rational numbers.

But in fact, with suitable normalization, it always takes values in the subring MF (Γ 0(2))[[q]]MF_\bullet^{\mathbb{Q}}(\Gamma_0(2)) \hookrightarrow \mathbb{Q}[ [q] ] of modular forms for Γ 0(2)SL 2()\Gamma_0(2)\subset SL_2(\mathbb{Z}) with rational coefficients (Zagier 86, item d) on page 2 based on Chudnovsky-Chudnovsky 88), but see also (Zagier 86, page 6).

On manifolds with spin structure the Witten genus has integral coeffcients, hence in the ring [[q]]\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 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 1S^1-equivariant index of a Dirac operator, where S 1S^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 1S^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 GG-principal bundle for GG 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 cc the GG-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=G = E8 in in (Harris 12, section 4).

As the global character of sheaves of vertex operator algebras

For UU \subset \mathbb{C} an open subset of the complex plane then the space 𝒟 ch(U)\mathcal{D}^{ch}(U) of chiral differential operators on UU is naturally a super vertex operator algebra. For XX 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 𝒟 X ch()\mathcal{D}^{ch}_X(-) on XX. Its cochain cohomology H (𝒟 X ch)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)w(X) of XX:

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

Physically this result is understood by observing that 𝒟 X ch\mathcal{D}^{ch}_Xis 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 XX is a closed manifold with String structure which furthermore admits a Riemannian metric with positive Ricci curvature, then its Witten genus vanishes.

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

ddpartition function in dd-dimensional QFTsuperchargeindex in cohomology theorygenuslogarithmic coefficients of Hirzebruch series
0push-forward in ordinary cohomology: integration of differential forms
1spinning particleDirac operatorKO-theory indexA-hat genusBernoulli numbers
endpoint of 2d Poisson-Chern-Simons theory stringSpin^c Dirac operator twisted by prequantum line bundlespace of quantum states of boundary phase space/Poisson manifoldTodd genusBernoulli numbers
endpoint of type II superstringSpin^c Dirac operator twisted by Chan-Paton gauge fieldD-brane chargeTodd genusBernoulli numbers
2superstringDirac-Ramond operatorsuperstring partition functionelliptic genus/Witten genusEisenstein series
self-dual stringM5-brane charge

References

General

The original description of the Witten genus from string theory is due to

  • Edward Witten, Elliptic Genera And Quantum Field Theory , Commun.Math.Phys. 109 525 (1987) (Euclid)

  • 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)

based on insights in

  • Peter Landweber, Elliptic Cohomology and Modular Forms, in Elliptic Curves and Modular Forms in Algebraic Topology, Lecture Notes in Mathematics Volume 1326, 1988, pp 55-68 (LandweberEllipticModular.pdf?)

  • 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)

(That the partition function in (Witten 87 (11)) is indeed, after some normalization, an elliptic genus is (Landweber 88, theorem 3)).

Rigorous proofs of the rigidity claims then appeared in

That a spin structure makes the Witten genus take values in integral series is due to

  • D.V. Chudnovsky, G.V. Chudnovsky, Elliptic modular functions and elliptic genera, Topology, Volume 27, Issue 2, 1988, Pages 163–170

  • Matthias Kreck, Stefan Stolz, P 2\mathbb{H}P^2-bundles and elliptic homology, Acta Mathematica 171 (1993), 231–261.

  • Mark Hovey, Spin Bordism and Elliptic Homology (1991) (web)

That it takes a rational string structure to make the elliptic genus land in modular forms was noticed in

  • Don Zagier, Note on the Landweber-Stong elliptic genus 1986 (pdf)

Surveys include

  • Gerald Höhn, Complex elliptic genera and S 1S^1-equivariant cobordism theory (pdf)

Further discussion of the Jacobi form-property of the Witten genus is in

  • 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)

Further discussion of the relation to quantum anomalies and the Green-Schwarz mechanism (string structure, string^c structure) is in

  • Wolfgang Lerche, B. Nilsson, A. Schellekens, N. Warner, Anomaly cancelling terms from the elliptic genus (1987) (pdf)

  • Qingtao Chen, Fei Han, Weiping Zhang, Generalized Witten Genus and Vanishing Theorems, JDG, 88 (2011) 1-39 (arXiv:1003.2325)

Discussion of the Witten genus via a KO-valued Chern-character on elliptic cohomology is in

  • Haynes Miller, The elliptic character and the Witten genus, Contemporary mathematics, volume 96, 1989 (pdf)

Relation to Kac-Weyl characters of loop group representations

The close relation of the Witten genus to Kac-Weyl characters of loop group representations has been highlighted and an elegant proof of rigidity of the Witten genus in these terms in

  • Kefeng Liu, On elliptic genera and Theta functions, Topology, Volume 35, Issue 3, July 1996, Pages 617–640 (pdf)

  • Kefeng Liu, On modular invariance and rigidity theorems, J. Differential Geom. Volume 41, Number 2 (1995), 247-514 (EUCLID, pdf)

along the lines of

  • Jean-Luc Brylinski, Representations of loop groups, Dirac operators on loop space, and modular forms, Topology, 29(4):461–480, 1990.

and further generalized to more general vertex operator algebra representations in (DLM 02).

Review and survey of some of this is in

  • Kefeng Liu, Modular forms and topology, Proc. of the AMS Conference on the Monster and Related Topics, Contemporary Math. (1996) (citeseer)

The Stolz conjecture

The Stolz conjecture on the Witten genus is due to

  • Stephan Stolz, A conjecture concerning positive Ricci curvature and the Witten genus, Mathematische Annalen Volume 304, Number 1 (1996),

Reviews include

Refinement to the string-orientation of tmftmf

The refinement of the Witten genus from values in modular forms to topological modular forms and further to a morphism of E-∞ rings, hence to the string orientation of tmf is due to

see also remark 1.4 of

  • Paul Goerss, Topological modular forms (after Hopkins, Miller and Lurie) (pdf).

and for more on the sigma-orientation see

Via index theory of would-be Dirac operators on loop space

Further literature emphasising the perspective of Dirac-Ramond operators as would-be Dirac operators on smooth loop space includes

  • 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. Nonperturbative methods in field theory (Irvine, CA, 1987).

  • Orlando Alvarez, T. P. Killingback, Michelangelo Mangano,Paul Windey, String theory and loop space index theorems, Comm. Math. Phys., 111(1):1–10, 1987.

  • Jean-Luc Brylinski, Representations of loop groups, Dirac operators on loop space, and modular forms, Topology, 29(4):461–480, 1990.

  • 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

The observation thazt the realization of the Dirac-Ramond operator as a Dirac operator on smooth loop space would realize superstring quantum dynamics as supersymmetric quantum mechanics on smooth loop space is what inspired the observations in

and

  • Edward Witten, Global anomalies in string theory, in W. Bardeen and A. White (eds.) Symposium on Anomalies, Geometry, Topology, pp. 61–99. World Scientific, 1985

Via (sheaves of) super vertex operator algebras

Formalization via super vertex operator algebras is discussed in

  • Hirotaka Tamanoi, Elliptic Genera and Vertex Operator Super-Algebras, 1999

  • Chongying Dong, Kefeng Liu, Xiaonan Ma, Elliptic genus and vertex operator algebras (arXiv:math/0201135)

and for the topologically twisted 2d (2,0)-superconformal QFT (the heterotic string with enhanced supersymmetry) via sheaves of vertex operator algebras in

which is based on the detailed construction via chiral differential operators in

Via other methods

In terms of (2,1)-dimensional Euclidean field theories and tmf:

…and in terms of factorization algebras in

Twisted case

The twisted Witten genus in the present of background gauge field hence for a twisted string structure/string^c structure is discussed in terms of twisted string structures in

  • 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 cString^c Manifolds, Pacific Journal of Mathematics 266.2 (2013): 477-508. (arXiv:1206.5955)

based on formulas from

  • Qingtao Chen, Fei Han, Elliptic Genera, Transgression and Loop Space Chern-Simons Forms (arXiv:math/0611104)

For the moment see the references at string^c structure for more on this.

A geometric interpretation of this in terms of parameterized WZW models is suggested in

and with more emphasis on equivariant elliptic cohomology in

An explicit computation for an E8-gauge bundle is in section 4 of

Revised on March 29, 2014 09:52:33 by Urs Schreiber (82.113.106.247)