nLab p1-structure

Idea

c-structure for $c$ the first Pontryagin class $p_1$, i.e. trivialization of $p_1$.

In the presence of Spin structure then $p_1$-structures are also called $(w_1, w_2, p_1)$-structures: see Freed & Teleman 2026 §2.4. In the presence of (just) orientation (i.e. $w_1$-structure) on 3-manifolds then $p_1$-structures are also called $(w_1, p_1)$-structures (see FT26 §2.3), and are closely related to Atiyah 2-framing structures.

Properties

Relation to string-structure

A string structure induces a $(w_1, w_2, p_1)$-structure, as it includes data of a spin structure and a trivialization of $p_1/2$, the first fractional Pontryagin class. However, $\mathbb{RP}^{11}$ has a $(w_1, w_2, p_1)$-structure but is not string.

References

In the context of Chern-Simons theory/Reshetikhin-Turaev theory localized to a fully extended functorial field theory:

• Daniel S. Freed, Constantin Teleman: The role of $p_1$-structures in 3-dimensional Chern-Simons theories [arXiv:2603.11291]