Conformal group and cobordism

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: $$\Omega_{d}^G(BG)$$

and

(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

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