Witten Genus Research Articles

An index theorem for loop spaces

We formulate and prove an index theorem for loop spaces of compact manifolds in the framework of KK-theory. It is a strong candidate for the noncommutative geometrical definition (or the analytic counterpart) of the Witten genus. In order to specify an “appropriate form” of the index theorem to formulate a loop space version, we formulate and prove an equivariant index theorem for non-compact S1-manifolds with a compact fixed-point set. In order to formulate it, we use a ring of formal power series.

On a conjecture of Stolz in the toric case

In 1996 Stolz [Math. Ann. 304 (1996), pp. 785–800] conjectured that a string manifold with positive Ricci curvature has vanishing Witten genus. Here we prove this conjecture for toric string Fano manifolds and for string torus manifolds admitting invariant metrics of non-negative sectional curvature.

Supersymmetric localization, modularity and the Witten genus

Supersymmetric localization, modularity and the Witten genus

The (anti-)holomorphic sector in [formula omitted]-equivariant cohomology, and the Witten class

Atiyah's classical work on circular symmetry and stationary phase shows how the Aˆ-genus is obtained by formally applying the equivariant cohomology localization formula to the loop space of a simply connected spin manifold. The same technique, applied to a suitable “antiholomorphic sector” in the C/Λ-equivariant cohomology of the conformal double loop space Maps(C/Λ,X) of a rationally string manifold X produces the Witten genus of X. This can be seen as an equivariant localization counterpart to Berwick-Evans supersymmetric localization derivation of the Witten genus.

Witten Genus and Elliptic Genera for Proper Actions

In this paper, we construct for the first time, the Witten genus and elliptic genera on noncompact manifolds with a proper cocompact action by an almost connected Lie group and prove vanishing and rigidity results that generalise known results for compact group actions on compact manifolds. We also compute our genera for some interesting examples.

4-manifolds and topological modular forms

We build a connection between topology of smooth 4-manifolds and the theory of topological modular forms by considering topologically twisted compactification of 6d (1, 0) theories on 4-manifolds with flavor symmetry backgrounds. The effective 2d theory has (0, 1) supersymmetry and, possibly, a residual flavor symmetry. The equivariant topological Witten genus of this 2d theory then produces a new invariant of the 4-manifold equipped with a principle bundle, valued in the ring of equivariant weakly holomorphic (topological) modular forms. We describe basic properties of this map and present a few simple examples. As a byproduct, we obtain some new results on ’t Hooft anomalies of 6d (1, 0) theories and a better understanding of the relation between 2d (0, 1) theories and TMF spectra.

An analytical approach to the equivariant index and Witten genus on spin manifolds

This work is divided in two cases. In the first case, we consider a spin manifold M as the set of fixed points of an S1-action on a spin manifold X, and in the second case we consider the spin manifold M as the set of fixed points of an S1-action on the loop space of M. For each case, we build on M a vector bundle, a connection and a set of bundle endomorphisms. These objects are used to build global operators on M which define an analytical index in each case. In the first case, the analytical index is equal to the topological equivariant Atiyah–Singer index, and in the second case the analytical index is equal to a topological expression where the Witten genus appears.

Universal Formal Group for Elliptic Genus of Level N

An elliptic function of level N determines an elliptic genus of level N as a Hirzebruch genus. It is known that any elliptic function of level N is a specialization of the Krichever function that determines the Krichever genus. The Krichever function is the exponential of the universal Buchstaber formal group. In this work we give a specialization of the Buchstaber formal group such that this specialization determines formal groups corresponding to elliptic genera of level N. Namely, an elliptic function of level N is the exponential of a formal group of the form F(u, v) = (u2A(v) − v2A(u))/(uB(v) − vB(u)), where A(u), B(u) ∈ ℂ[[u]] are power series with complex coefficients such that A(0) = B(0) = 1, A″(0) = B′(0) = 0, and for m = [(N − 2)/2] and n = [(N − 1)/2] there exist parameters (a1, …, am, b1, …, bn) for which the relation \(\prod\nolimits_{j=1}^{n-1}\left(B(u)+b_{j} u\right)^{2} \cdot\left(B(u)+b_{n} u\right)^{N-2 n}=A(u)^{2} \prod\nolimits_{k=1}^{m-1}\left(A(u)+a_{k} u^{2}\right)^{2} \cdot\left(A(u)+a_{m} u^{2}\right)^{N-1-2 m}\) holds. For the universal formal group of this form, the exponential is an elliptic function of level at most N. This statement is a generalization to the case N > 2 of the well-known result that the elliptic function of level 2 determining the elliptic Ochanine–Witten genus is the exponential of a universal formal group of the form F(u, v) = (u2 − v2)/(uB(v) − vB(u)), where B(u) ∈ ℂ[[u]], B(0) = 1, and B′(0) = 0. We prove this statement for N = 3, 4, 5, 6. We also prove that the elliptic function of level 7 is the exponential of a formal group of this form. Universal formal groups that correspond to elliptic genera of levels N = 5, 6, 7 are obtained in this work for the first time.

A note on torus actions and the Witten genus

We show that the Witten genus of a string manifold $M$ vanishes, if there is an effective action of a torus $T$ on $M$ such that $\dim T>b_2(M)$. We apply this result to study group actions on $M\times G/T$, where $G$ is a compact connected Lie group and $T$ a maximal torus of $G$. Moreover, we use the methods which are needed to prove these results to the study of torus manifolds. We show that up to diffeomorphism there are only finitely many quasitoric manifolds $M$ with the same cohomology ring as $#_{i=1}^k \pm\mathbb{C} P^n$ with $k<n$.

Elliptic function of level 4

The work is dedicated to the theory of elliptic functions of level $n$. An elliptic function of level $n$ determines a Hirzebruch genus that is called elliptic genus of level $n$. Elliptic functions of level $n$ are also interesting as solutions of Hirzebruch functional equations. The elliptic function of level $2$ is the Jacobi elliptic sine. It determines the famous Ochanine--Witten genus. It is the exponential of the universal formal group of the form \[ F(u,v)=\frac{u^2 -v^2}{u B(v) - v B(u)}, \quad B(0) = 1. \] The elliptic function of level $3$ is the exponential of the universal formal group of the form \[ F(u,v)=\frac{u^2 A(v) -v^2 A(u)}{u A(v)^2 - v A(u)^2}, \qquad A(0) = 1, \quad A"(0) = 0. \] In this work we have obtained that the elliptic function of level $4$ is the exponential of the universal formal group of the form \[ F(u,v)=\frac{u^2 A(v) -v^2 A(u)}{u B(v)-v B(u)}, \text{ where } A(0) = B(0) = 1, \] and for $B'(0) = A"(0) = 0, A'(0) = A_1, B"(0) = 2 B_2$ the relation holds \[ (2 B(u) + 3 A_1 u)^2 = 4 A(u)^3 - (3 A_1^2 - 8 B_2) u^2 A(u)^2. \] To prove this result we have expressed the elliptic function of level $4$ in terms of Weierstrass elliptic functions.

Vanishing theorems of generalized Witten genus for generalized complete intersections in flag manifolds

We propose a potential function [Formula: see text] for the cohomology ring of partial flag manifolds. We prove a formula expressing integrals over partial flag manifolds by residues, which generalizes [E. Witten, The Verlinde algebra and the cohomology of the Grassmannian, in Geometry, Topology, Physics (International Press, 1995), pp. 357–422]. Using this formula, we prove a Landweber–Stong type vanishing theorem for generalized [Formula: see text] complete intersections in flag manifolds, which serves as evidence for the [Formula: see text] version of Stolz conjecture [Q. Chen, F. Han and W. Zhang, Generalized Witten genus and vanishing theorems, J. Differential Geom. 88(1) (2011) 1–39].

Elliptic genera and characteristic q-series of superconformal field theory

We analyze the characteristic series, the KO series and the series associated with the Witten genus, and their analytic forms as the q-analogs of classical special functions (in particular q-analog of the beta integral and the gamma function). q-Series admit an analytic interpretation in terms of the spectral Ruelle functions, and their relations with appropriate elliptic modular forms can be described. We show that there is a deep correspondence between the characteristic series of the Witten genus and KO characteristic series, on one side, and the denominator identities and characters of N=2 superconformal algebras, and the affine Lie (super)algebras on the other. We represent the characteristic series in the form of double series using the Hecke–Rogers modular identity.

Chiral differential operators: Formal loop group actions and associated modules

Chiral differential operators (CDOs) are closely related to string geometry and the quantum theory of 2-dimensional σ-models. This paper investigates two topics about CDOs on smooth manifolds. In the first half, we study how a Lie group action on a smooth manifold can be lifted to a “formal loop group action” on an algebra of CDOs; this turns out to be a condition on the equivariant first Pontrjagin class. The case of a principal bundle receives particular attention and gives rise to a type of vertex algebras of great interest. In the second half, we introduce a construction of modules over CDOs using the said “formal loop group actions” and semi-infinite cohomology. Intuitively, these modules should have a geometric meaning in terms of “formal loop spaces”. The first example we study leads to a new conceptual construction of an arbitrary algebra of CDOs. The other example, called the spinor module, may be useful for a geometric theory of the Witten genus.

The Cayley plane and string bordism

This paper shows that, away from 6, the kernel of the Witten genus is precisely the ideal consisting of (bordism classes of) Cayley plane bundles with connected structure group, but only after restricting the Witten genus to string bordism. It does so by showing that the divisibility properties of Cayley plane bundle characteristic numbers arising in Borel-Hirzebruch Lie-group-theoretic calculations correspond precisely to the divisibility properties arising in the Hovey-Ravenel-Wilson BP-Hopf-ring-theoretic calculation of string bordism at primes >3.

Witten genus of generalized complete intersections in products of Grassmannians

By using the Milnor ring structure of the cohomology of Grassmannian, we prove a vanishing theorem of Witten genus for generalized string complete intersections in products of Grassmannian.

Mod 3 congruence and twisted signature of 24 dimensional string manifolds

In this paper, by combining modularity of the Witten genus and the modular forms constructed by Liu and Wang, we establish mod 3 congruence properties of certain twisted signatures of 24 dimensional string manifolds.

Secondary invariants for string bordism and topological modular forms

Using spectral invariants of Dirac operators we construct a secondary version of the Witten genus, namely a bordism invariant of string manifolds in dimensions 4m−1. We prove a secondary index theorem which relates this global-analytic construction with its homotopy-theoretic analog. The latter will be calculated through its factorization over topological modular forms.

Highly connected manifolds of positive $p$-curvature

We study and in some cases classify highly connected manifolds which admit a Riemannian metric with positive $p$-curvature. The $p$-curvature was defined and studied by the second author. It turns out that positivity of $p$-curvature could be preserved under surgeries of codimension at least $p+3$. This gives a key to reduce a geometrical classification problem to a topological one, in terms of relevant bordism groups and index theory. In particular, we classify 3-connected manifolds with positive 2-curvature in terms of the spin and string bordism groups, and by means of $\alpha$-invariant and Witten genus $\phi_W$. Here we use results of Dessai, which provide appropriate generators of the rational string bordism ring in terms of geometric $\Ca P^2$-bundles, where the Cayley projective plane $\Ca P^2$ is a fiber and the structure group is $F_4$ which is the isometry group of the standard metric on $\Ca P^2$.

Quasi-topological gauged sigma models, the geometric Langlands program, and knots

We construct and study a closed, two-dimensional, quasi-topological (0, 2) gauged sigma model with target space a smooth G-manifold, where G is any compact and connected Lie group. When the target space is a flag manifold of simple G, and the gauge group is a Cartan subgroup thereof, the perturbative model describes, purely physically, the recently formulated mathematical theory of “Twisted Chiral Differential Operators”. This paves the way, via a generalized T -duality, for a natural physical interpretation of the geometric Langlands correspondence for simply-connected, simple, complex Lie groups. In particular, the Hecke eigensheaves and Hecke operators can be described in terms of the correlation functions of certain operators that underlie the infinite-dimensional chiral algebra of the flag manifold model. Nevertheless, nonperturbative worldsheet twisted-instantons can, in some situations, trivialize the chiral algebra completely. This leads to a spontaneous breaking of supersymmetry whilst implying certain delicate conditions for the existence of BeilinsonDrinfeld D-modules. Via supersymmetric gauged quantum mechanics on loop space, these conditions can be understood to be intimately related to a conjecture by Hohn-Stolz [1] regarding the vanishing of the Witten genus on string manifolds with positive Ricci curvature. An interesting connection to Chern-Simons theory is also uncovered, whence we would be able to (i) relate general knot invariants of three-manifolds and Khovanov homology to “quantum” ramified D-modules and Lagrangian intersection Floer homology; (ii) furnish physical proofs of mathematical conjectures by Seidel-Smith [2] and Gaitsgory [3, 4] which relate knots to symplectic geometry and Langlands duality, respectively. ∗E-mail: tan@ias.edu ar X iv :1 11 1. 06 91 v1 [ he pth ] 2 N ov 2 01 1

Chiral differential operators on supermanifolds

The first part of this paper provides a new description of chiral differential operators (CDOs) in terms of global geometric quantities. The main result is a recipe to define all sheaves of CDOs on a smooth cs-manifold; its ingredients consist of an affine connection ∇ and an even 3-form that trivializes p1(∇). With ∇ fixed, two suitable 3-forms define isomorphic sheaves of CDOs if and only if their difference is exact. Moreover, conformal structures are in one-to-one correspondence with even 1-forms that trivialize c1(∇). Applying our work in the first part, we then construct what may be called “chiral Dolbeault complexes” of a complex manifold M, and analyze conditions under which these differential vertex superalgebras admit compatible conformal structures or extra gradings (by fermion numbers). When M is compact, their cohomology computes (in various cases) the Witten genus, the two-variable elliptic genus and a spinc version of the Witten genus. This part contains some new results as well as provides a geometric formulation of certain known facts from the study of holomorphic CDOs and σ-models.