# nLab genus

this entry is about the notion of genus in cohomology. For classification of surfaces see genus of a surface.

### Context

#### Index theory

index theory, KK-theory

noncommutative stable homotopy theory

partition function

genus, orientation in generalized cohomology

## Definitions

operator K-theory

K-homology

## Higher genera

cohomology

### Theorems

#### Manifolds and cobordisms

manifolds and cobordisms

# Contents

## Idea

For $R$ a ring, an $R$-valued genus is a ring homomorphism

$\sigma : \Omega_n \to R$

from the bordism ring.

The cobordism ring here may be replaced by rings of cobordisms with extra structure.

### In homotopy theory / higher category theory

More generally, the notion of genus finds its natural interpretation in higher category theory, where it is refined to a morphism of symmetric monoidal ∞-groupoids

$\sigma : Bord_{(\infty,\infty)} \to R$

from the (∞,n)-category of cobordisms for $n \to \infty$ to a ring spectrum $R$. This is equivalently the Thom spectrum (see at cobordism ring for more) and so a genus may be thought of as a morphism of spectra

$\sigma : M O \to R \,.$

### In terms of quantum field theory

At least in some important cases, genera seem to be naturally understood as encoding sigma-model quantum field theories. For $G$ some structure, the Thom spectrum $M G$ is the classifying space of manifolds with G-structure, and hence may be thought of as classifying target spaces for sigma-models. The codomain spectrum $R$ itself may then be thought of as a classifying space for a certain class of QFTs, and hence the genus $\sigma : M G \to R$ can be thought of as assigning to any target space the corresponding sigma-model.

This is for instance the case at least over the point for the A-hat genus $M Spin \to K O$, which may be thought of as sending manifolds with spin structure to the corresponding (1,1)-supersymmetric EFT (“spinning particle”); and for the Witten genus $M String \to tmf$, which can be thought of as sending a manifold with string structure to the corresponding (2,1)-supersymmetric EFT (“heterotic string”).

## Examples

$d$partition function in $d$-dimensional QFTindex/genus in cohomology theory
0push-forward in ordinary cohomology: integration of differential forms
1spinning particleK-theory index
endpoint of 2d Poisson-Chern-Simons theory stringspace of quantum states of boundary phase space/Poisson manifold
endpoint of type II superstringD-brane charge
2type II superstringelliptic genus
heterotic stringWitten genus

Revised on February 28, 2014 22:35:56 by David Corfield (87.113.151.255)