Chromatic Fields and Quantum Fields

We survey some big picture ideas relating trans-chromatic phenomenon in stable homotopy theory to symmetries in quantum field theory.

Counting loops in $p$-spaces with Morava K-theory

Let $G\in G{p}^{Fin}(S)$ be a $\pi$-finite $p$-group in Anima. See pi-finite p-spaces. There’s a remarkable formula relating two invariants assigned to $BG$

$${\chi }_{K(n)}(BG)=\#\{\text{maps }{\mathbb{T}}^n\to BG\text{ (up to homotopy)}\}$$

where the LHS is defined as alternating sum of dimensions in $K(n)$ -cohomology

$${\chi }_n(BG)=\sum _{q\ge 1}(-1{)}^q{\dim }_{K(n{)}^{\bullet }(pt)}(K(n{)}^q(BG))$$

Remark: It’s a result of Ravenel-Wilson that each term here is finite.

The formula says that the $K(n)$-Euler characteristic counts $n$-fold free loops in $BG$ up to homotopy.

Detecting chromatic height

Taking $G=C_p$ the order $p$ cyclic group, the space $B{C}_p$ can be used to “detect chromatic height” in the following setting.

Let $R$ be a “cohomological (skew-) field”: that is, a $E_1$ ring spectrum $R$ such that every left $R$-module is free. The action of $R$ on $Map({\Sigma }_{+}^{\infty }B{C}_p,R)$ gives us a left $R$-module, which by the hypothesis of $R$ being a field, is free. Hence, it either

• (a) has infinite rank, in which case we say $R$ has height $\infty$
• (b) has finite rank $p^n$ for some integer $n\ge 1$, here we say $R$ has height $n$

(That these are the only finite ranks is a calculation).

Example The Morava $K$-theory $K(n)$ is a cohomological field of height $n$, hence, we have

$$di{m}_{K(n{)}^{\bullet }(pt)}(K(n{)}^{\bullet }(B{C}_p))=p^n$$

This means… height $n$ Euler characteristic of $B{C}_p$ is $p^n$?

$\pi$-finite gauge theory

A classical gauge theory is a theory of classical gauge fields. For example, we can fix a gauge group $G$, and a very simple version of gauge theory say that a gauge field living on spacetime manifold $M$ is a locally-constant $G$-torsor.

A locally-constant $G$-torsor on $M$ is classified by a map of anima

\varphi: \pi_\infty(M) \to BG $$ These live in a anima encoding the space of classical fields

M^{BG}:=Map_S(\pi_\infty M, BG)