nLab
Witten genus

Contents

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). 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 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 interpreted 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 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 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( k22G 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} 2G_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, see at twisted ad-equivariant Tate K-theory).

Integrality and modularity

A priori, the Witten genus has coefficients the power series ring [[q]]\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 Γ 2(2)\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 (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).

For the heterotic superstring

On manifolds with spin structure the heterotic string 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 is 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.

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).

partition functions in quantum field theory as indices/genera/orientations 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 formsorientation
1spinning particleDirac operatorKO-theory indexA-hat genusBernoulli numbersAtiyah-Bott-Shapiro orientation MSpinKOM Spin \to KO
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 numbersAtiyah-Bott-Shapiro orientation MSpin cKUM Spin^c \to KU
endpoint of type II superstringSpin^c Dirac operator twisted by Chan-Paton gauge fieldD-brane chargeTodd genusBernoulli numbersAtiyah-Bott-Shapiro orientation MSpin cKUM Spin^c \to KU
2type II superstringDirac-Ramond operatorsuperstring partition function in NS-R sectorOchanine elliptic genusSO orientation of elliptic cohomology
heterotic superstringDirac-Ramond operatorsuperstring partition functionWitten genusEisenstein seriesstring orientation of tmf
self-dual stringM5-brane charge
3w4-orientation of EO(2)-theory

References

Elliptic cohomology

General

The concept of elliptic cohomology originates around:

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:

Textbook accounts:

Equivariant elliptic cohomology

On equivariant elliptic cohomology and positive energy representations of loop groups:

Relation to Kac-Weyl characters of loop group representations

The case of twisted ad-equivariant Tate K-theory:

See also:

Via derived E E_\infty-geometry

Formulation of (equivariant) elliptic cohomology in derived algebraic geometry/E-∞ geometry (derived elliptic curves):

Elliptic genera

General

The general concept of elliptic genus originates with:

Early development:

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:

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):

For the Ochanine genus:

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:

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:

On manifolds with SU(2)-action:

Twisted elliptic genera

Discussion of elliptic genera twisted by a gauge bundle, i.e. for string^c structure):

Elliptic genera as super pp-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:

Review in:

Via super vertex operator algebra

Formulation via super vertex operator algebras:

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:

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:

Via functorial QFT

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

Via conformal nets

Tentative formulation via conformal nets:

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:

Speculations on physics aspects of lifting the Witten genus to topological modular forms:

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

Discussion of the resulting elliptic genus (2d SCFT partition function) originates with:

Further discussion in:

M-string elliptic genus

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

E-string elliptic genus

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

On the elliptic genus of E-strings as M2-branes ending on M5-branes:

Last revised on January 26, 2021 at 10:35:57. See the history of this page for a list of all contributions to it.