noncommutative topology, noncommutative geometry
noncommutative stable homotopy theory
genus, orientation in generalized cohomology
Hirzebruch signature theorem?
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 $S^1$-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
In the construction from string physics this map is interpeted 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
(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
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).
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
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:
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
and write its Kac-Weyl character as
Under passing to group characters this is (Brylinski 90, p. 7(467), reviewed in KL 96, section 1.2) equivalently
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):
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).
The Witten genus a priori has coefficients the power series ring $\mathbb{Q}[ [q] ]$ over the rational numbers.
But in fact, with suitable normalization, it always 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), but see also (Zagier 86, page 6).
On manifolds with spin structure the 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).
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 ).
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 in in (Harris 12, section 4).
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 $X$:
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.
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.
partition functions in quantum field theory as indices/genera 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 | ||||
1 | spinning particle | Dirac operator | KO-theory index | A-hat genus | Bernoulli numbers |
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 | |
endpoint of type II superstring | Spin^c Dirac operator twisted by Chan-Paton gauge field | D-brane charge | Todd genus | Bernoulli numbers | |
2 | superstring | Dirac-Ramond operator | superstring partition function | elliptic genus/Witten genus | Eisenstein series |
self-dual string | M5-brane charge |
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
Clifford Taubes, $S^1$ actions and elliptic genera, Comm. Math. Phys., 122(3):455–526, 1989.
Raoul Bott, Clifford Taubes, On the rigidity theorems of Witten, J. of the Amer. Math. Soc., 2, 1989.
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, $\mathbb{H}P^2$-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
Surveys include
Further discussion of the Jacobi form-property of the Witten genus is in
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
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
and further generalized to more general vertex operator algebra representations in (DLM 02).
Review and survey of some of this is in
The Stolz conjecture on the Witten genus is due to
Reviews include
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
Michael Hopkins, Topological modular forms, the Witten Genus, and the theorem of the cube, Proceedings of the International Congress of Mathematics, Zürich 1994 (pdf)
Michael Hopkins, Algebraic topology and modular forms, Proceedings of the ICM, Beijing 2002, vol. 1, 283–309 (arXiv:math/0212397)
Matthew Ando, Michael Hopkins, Charles Rezk, Multiplicative orientations of KO-theory and the spectrum of topological modular forms, 2010 (pdf)
see also remark 1.4 of
and for more on the sigma-orientation see
Matthew Ando, The sigma orientation for analytic circle-equivariant elliptic cohomology, Geom. Topol. 7 (2003) 91-153 (arXiv:math/0201092)
Matthew Ando, Maria Basterra, The Witten genus and equivariant elliptic cohomology (arXiv:0008192)
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
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
In terms of (2,1)-dimensional Euclidean field theories and tmf:
…and in terms of factorization algebras in
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 $String^c$ Manifolds, Pacific Journal of Mathematics 266.2 (2013): 477-508. (arXiv:1206.5955)
based on formulas from
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