Contents

cohomology

# Contents

## Idea

An elliptic cohomology theory is a type of generalized (Eilenberg-Steenrod) cohomology theory associated with the datum of an elliptic curve.

Even (weakly) periodic multiplicative generalized (Eilenberg-Steenrod) cohomology theories $A$ are characterized by the formal group whose ring of functions $A(\mathbb{C}P^\infty)$ is the cohomology ring of $A$ evaluated on the complex projective space $\mathbb{C}P^\infty$ and whose group product is induced from the canonical morphism $\mathbb{C}P^\infty \times \mathbb{C}P^\infty \to \mathbb{C}P^\infty$ that describes the tensor product of complex line bundles under the identification $\mathbb{C}P^\infty \simeq \mathcal{B} U(1)$.

There are precisely three types of 1-dimensional such formal group laws:

An elliptic cohomology theory is an even periodic multiplicative generalized (Eilenberg-Steenrod) cohomology theory whose corresponding formal group is an elliptic curve, hence which is represented by an elliptic spectrum.

(e.g. Lurie, def. 1.2).

The Goerss-Hopkins-Miller-Lurie theorem shows that the assignment of generalized (Eilenberg-Steenrod) cohomology theories to elliptic curves lifts to an assignment of representing spectra in a structure-preserving way.

The homotopy limit of this assignment functor, i.e. the “gluing” of all spectra representing all elliptic cohomology theories is the spectrum that represents the cohomology theory called tmf.

## Properties

### Genera, the elliptic genus and relation to string theory

A properly developed theory of elliptic cohomology is likely to shed some light on what string theory really means. (Witten 87, very last sentence)

The following is rough material originating from notes taken live (and long ago), to be polished. See also at elliptic genus and Witten genus

Some topological invariants of manifolds that are of interest:

we restricted attention to closed connected smooth manifolds $X$

• the Euler characteristic $e(X) \in \mathbb{Z}$

• takes all values in $\mathbb{Z}$

• is the obstruction to the existence of a nowhere vanishing vector field on $X$:

$(e(X)= 0) \Leftrightarrow (\exists v \in \Gamma(T X) : \forall x \in X : v(x) \neq 0)$
• signature$sgn(X)$

this is the obstruction to $X$ being cobordant to a fiber bundle over the circle:

$X$ is bordant to a fiber bundle over $S^1$ precisely if $sgn(X) = 0$

• when $X$ has a spin structure

the index of the Dirac operator $D$:

$ind D_X \in \mathbb{Z}$
$\alpha(D) \in \left\{ \array{ \mathbb{Z} & dim X = 0 mod 4 \\ \mathbb{Z}_2 & dim X = 1, 2 mod 8 \\ 0 & otherwise } \right.$

theorem (Gromov-Lawson / Stolz) let $dim X \geq 5$ and

then $X$ admits a Riemannian metric of positive scalar curvature precisely when $\alpha(X) = 0$

These invariants share the following properties:

• they are additive under disjoint union of manifolds

• they are multiplicative under cartesian product of manifolds

• $e(X) mod 2, sgn(X), ind(D_X)$ all vanish when $X$ is a boundary, $\exists W : X = \partial W$, which means that $X$ is cobordant to the empty manifold $\emptyset$.

in other words, these invariants are genera, namely ring homomorphisms

$\Omega \to R$

form the cobordism ring $\Omega$ to some commutative ring $R$

• good genera are those which reflect geometric properties of $X$.

• now for $X$ a topological space consider the cobordism ring over $X$:

$\Omega(X) := \{(M,f)| f : M \stackrel{cont}{\to X}\}/_{bordism}$

where addition and multiplication are again given by disjoint union and cartesian product.

this assignment of rings to topological spaces is a generalized homology theory: cobordism homology theory

question given a genus $\Omega \to R$, can we find a homology theory $R(-)$ with $R = R(pt)$ its homology ring over the point and such that it all fits into a natural diagram

$\array{ \Omega &\to& R \\ \uparrow && \uparrow \\ \Omega(X) &\stackrel{\rho}{\to}& R(X) }$

This would be a parameterized extension $\rho = R(-)$ of $R$ .

Now let $X$ be a closed manifold.

consider $u_X : X \to K(\pi_1(X),1)$ (on the right an Eilenberg-MacLane space) which is the classifying map for the universal cover

$u_* \pi_1(X) \stackrel{\simeq_{canon}}{\to} \pi_1(K(\pi_1(X), 1))$

then consider

$\rho_X[X, u_X] \in R(K(\pi_1(X),1))$

theorem (Julia Weber)

take the Euler characteristic mod 2, $Eu(X)$

$\array{ \Omega^0 &\stackrel{Eu(M)\cdot t^{dim M}}{\to}& \mathbb{Z}_2[t] \\ \uparrow && \uparrow \\ \Omega^0(X) &\to& Eu(X) & \simeq H_\bullet(X; \mathbb{Z}_2[t]) }$

for $X$ smooth we have then:

$Eu_X[X, id] = Poincare\; dual\; of\; total\; Stiefel-Whitney\; class$

theorem (Minalta)

something analogous for signature genus

$\array{ \Omega_\bullet^{SO} &\to& Sig_\bullet(X) }$

$sign_X[X,u] \in sig_\bullet(K) \otimes \mathbb{Q}$

this is the Novikov higher signature

now the same for the $\alpha$-genus

$\array{ \Omega_{X}^{Spin} &\stackrel{\alpha}{\to}& KO_\bullet(pt) \\ \uparrow && \uparrow \\ \Omega_\bullet^{Spin} &\to& KO_\bullet(X) }$

now towards elliptic genera: recall the notion of string structure of a manifold $X$: a lift of the structure map $X \to \mathcal{B}O(n)$ through the 4th connected universal cover $\mathcal{B}String(n) := \mathcal{B}O(n)\langle 4\rangle \to \mathcal{B} O$:

so consider String manifolds and the bordism ring $\Omega_\bullet^{String}$ of String manifold, let $M_\bullet$ be the ring of integral modular forms, then there is a genus – the Witten genus $W$

$\array{ \Omega_\bullet^{String} &\stackrel{W}{\to}& M_\bullet \\ \uparrow && \uparrow \\ \Omega_\bullet^{String}(X) &\to& M_\bullet(X) \\ &\searrow& \\ && tmf_\bullet(X) }$

where $\Omega_\bullet^{String}(pt) \to tmf_\bullet(pt)$ is surjective

conjecture (Stolz conjecture)

If a String manifold $Y$ has a positive Ricci curvature metric, then the Witten genus vanishes.

The attempted “Proof” of this is the motivation for the Stolz-Teichner-program for geometric models for elliptic cohomology:

“Proof” If $Y$ is String, then the loop space $L Y$ is has spin structure, so if $Y$ has positive Ricci curvature the $L Y$ has positive scalar curvature which implies by the above that $ind^{S^1} D_{L Y} = 0$ which by the index formula is the Witten genus.

### Equivariant elliptic cohomology and loop group representations

The analog of the orbit method with equivariant K-theory replaced by equivariant elliptic cohomology yields (aspects of) the representation theory of loop groups. (Ganter 12)

### Chromatic filtration

chromatic homotopy theory

chromatic levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ringreal oriented cohomology theory
0ordinary cohomologyEilenberg-MacLane spectrum $H \mathbb{Z}$HZR-theory
0th Morava K-theory$K(0)$
1complex K-theorycomplex K-theory spectrum $KU$KR-theory
first Morava K-theory$K(1)$
first Morava E-theory$E(1)$
2elliptic cohomologyelliptic spectrum $Ell_E$
second Morava K-theory$K(2)$
second Morava E-theory$E(2)$
algebraic K-theory of KU$K(KU)$
3 …10K3 cohomologyK3 spectrum
$n$$n$th Morava K-theory$K(n)$
$n$th Morava E-theory$E(n)$BPR-theory
$n+1$algebraic K-theory applied to chrom. level $n$$K(E_n)$ (red-shift conjecture)
$\infty$complex cobordism cohomologyMUMR-theory

moduli spaces of line n-bundles with connection on $n$-dimensional $X$

$n$Calabi-Yau n-foldline n-bundlemoduli of line n-bundlesmoduli of flat/degree-0 n-bundlesArtin-Mazur formal group of deformation moduli of line n-bundlescomplex oriented cohomology theorymodular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory
$n = 0$unit in structure sheafmultiplicative group/group of unitsformal multiplicative groupcomplex K-theory
$n = 1$elliptic curveline bundlePicard group/Picard schemeJacobianformal Picard groupelliptic cohomology3d Chern-Simons theory/WZW model
$n = 2$K3 surfaceline 2-bundleBrauer groupintermediate Jacobianformal Brauer groupK3 cohomology
$n = 3$Calabi-Yau 3-foldline 3-bundleintermediate JacobianCY3 cohomology7d Chern-Simons theory/M5-brane
$n$intermediate Jacobian

## 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

Relation to Kac-Weyl characters of loop group representations

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

#### Via derived $E_\infty$-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)

For the Ochanine genus:

#### 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:

• 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):

### 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

and then strictly originates with:

Review in:

#### Formulations

##### 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 conformal nets

Tentative formulation via conformal nets:

#### 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):

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:

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):

• Ying-Hsuan Lin, Du Pei, Holomorphic CFTs and topological modular forms &lbrack;arXiv:2112.10724&rbrack;

• Jan Albert, Justin Kaidi, Ying-Hsuan Lin, Topological modularity of Supermoonshine $[$arXiv:2210.14923$]$

#### 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:

• 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)

Last revised on September 15, 2021 at 07:32:48. See the history of this page for a list of all contributions to it.