In this post, I am exploring my thoughts on the implementation of conformal symmetry group structure and cobordism relations.

Namely, I like to know what has been done and explored in the past?

on understanding the relations of

(1). the bordism classes of $d$-manifold (in $d$-dimensions) endorsed with the conformal symmetry group structure of $G$, say classified by a bordism group: $$ \Omega_{d}^G(BG) $$


(2) Its potential classifications say, conformal field theories, through the cobordism theory.

For the conformal symmetry group structure of $G$ in $d$-dimensions, we need to consider at least:

  • In Minkowski signature $$ \text{Conf}(d-1,1)=\frac{O(d,2)}{\mathbb Z_2}. $$
  • In Euclidean signature $$ \text{Conf}(d,0)=\frac{O(d+1,1)}{\mathbb Z_2}. $$

We should can consider the global symmetries of in into the form of $$ G\equiv ({\frac{{G_{\text{spacetime} }} \quad \ltimes \quad {\mathbb{G}_{\text{internal}}\quad }}{{N_{\text{shared}}}}}), $$ where the ${G_{\text{spacetime} }}\quad$ is the spacetime symmetry, the ${\mathbb{G}_{\text{internal}} }\quad$ the internal symmetry.

Later we denote the probed background spacetime $M$ connection over the spacetime tangent bundle $TM$, e.g. as $w_j(TM)$ where $w_j$ is $j$-th Stiefel-Whitney (SW) class. We may also denote the probed background internal-symmetry/gauge connection over the principal bundle $E$, e.g. as $$w_j(E)=w_j(V_{{\mathbb{G}_{\text{internal}\quad\quad} }})\quad\quad$$ where $w_j$ is also $j$-th SW class. In some cases, we may alternatively denote the latter as $w_j'(E)=w_j'(V_{{\mathbb{G}_{\text{internal}}\quad\quad }})$.

the $\ltimes$ is a semi-direct product specifying a certain ``twisted'' operation (e.g. due to the symmetry extension from ${\mathbb{G}_{\text{internal}\quad\quad} }$ by ${G_{\text{spacetime}\quad\quad }}$) and the ${N_{\text{shared}\quad\quad}}$ is the shared common normal subgroup symmetry between the two numerator groups.

The theories and their 't Hooft anomalies that we concern are in $d$d QFTs, but the topological/cobordism invariants are defined in the $D$d = $(d+1)$d manifolds. The manifold generators for the bordism groups are actually the closed $D$d = $(d+1)$d manifolds. We should clarify that although there can be 't Hooft anomalies for $d$d QFTs so ${\mathbb{G}_{\text{internal}} }$ may not be gauge-able, the topological invariants defined in the closed $D$d = $(d+1)$d manifolds actually have ${\mathbb{G}_{\text{internal}} \quad\quad}$ always gauge-able in that $D$d = $(d+1)$d.

The new ingredient in our present question is on going after the cobordism theory of $G$ contains the conformal symmetry group. Are there some mathematical works or physical works

$$\text{"relating conformal group and conformal field theories in a framework of a cobordism theory?"}$$


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