Tangential Structures: Oriented, Spin, and Framed Theories

Changing the tangential structure changes the bordism category and therefore changes the classification problem. A framed fully extended TQFT is determined by a fully dualizable object; an oriented, spin, or pin theory requires a coherent homotopy fixed point for the corresponding structure group acting on that object. In two dimensions, the oriented refinement trivializes the Serre automorphism and is encoded by Calabi–Yau or symmetric Frobenius trace data in standard Morita targets. Spin theories retain information that is destroyed by forgetting the spin structure.

Required background. Bordism categories fix the geometric domain; Clifford algebras and pin/spin groups define the lifted structure groups; and characteristic classes diagnose existence of reductions and lifts.

Helpful background. Dualizability and the cobordism hypothesis provides the framed classification, while invertible phases and topological order supplies the physical comparison.

From a structure group to a bordism category

Let $\chi:G\to O(n)$ be a continuous homomorphism. A $G$-structure on an $m$-manifold, $m\le n$, is a lift of the stabilized tangent-classifying map through $BG\to BO(n)$. Equivalently, one specifies a principal $G$-bundle and an identification of its associated $\mathbb R^n$-bundle with $TM\oplus\mathbb R^{n-m}$. The resulting bordism $(\infty,n)$-category is denoted $\operatorname{Bord}_n^G$.

Important cases are distinct:

• $G=\{1\}$ gives a framing, a trivialization of the stabilized tangent bundle.
• $G=SO(n)$ gives an orientation.
• $G=\operatorname{Spin}(n)$ gives a spin structure lifting the oriented frame bundle.
• $G=\operatorname{Pin}^{\pm}(n)$ treats nonorientable manifolds with two inequivalent reflection lifts.

A change of structure is functorial only in a declared direction. Forgetting a framing can produce an orientation, and forgetting a spin lift can produce an orientation, but a theory descends along that forgetful map only when its partition functions and all lower-codimension assignments are constant on the forgotten choices. Conversely, refining an oriented manifold to spin requires a lift that may not exist and, when it exists, need not be unique.

For a symmetric monoidal target $\mathcal C$, the structured cobordism hypothesis gives

$$\operatorname{Fun}^{\otimes} \left(\operatorname{Bord}_n^G,\mathcal C\right) \simeq \left((\mathcal C^{\mathrm{fd}})^\sim\right)^{hG},$$

where the right side is the homotopy fixed-point $\infty$-groupoid for the action induced through $G\to O(n)$. Lurie states and proves this form in Lurie 2009, Theorem 2.4.26 and Examples 2.4.27–2.4.28, printed pp. 46–47. An ordinary fixed object is insufficient: the fixed point includes coherent equivalences for every group element and all higher compatibility homotopies.

Oriented refinement in two dimensions

For $n=2$, $SO(2)\simeq S^1$ acts on fully dualizable objects. The loop determined by this action is the Serre automorphism

$$S_A:A\longrightarrow A.$$

Upgrading a framed theory to an oriented one requires a coherent trivialization of this action, beginning with $S_A\simeq\mathrm{id}_A$. In the Morita $2$-category, the oriented structure is expressed by a Calabi–Yau trace

$$\operatorname{tr}:HH_0(A)\longrightarrow\mathbb C$$

whose induced pairing $A\otimes A\to\mathbb C$ is nondegenerate and cyclic. Under finite separability hypotheses this is symmetric Frobenius data. Lurie identifies $SO(2)$ homotopy fixed points with Calabi–Yau objects in Lurie 2009, Definition 4.2.6 and Remark 4.2.7, printed pp. 92–93.

The exact first application returns to topological order, invertible phases, and matter diagnostics: begin with a framed two-dimensional point object, compute its Serre automorphism, and supply the $SO(2)$ homotopy fixed-point—or equivalently the appropriate trace—needed for an oriented theory. This step adds structure; it is not automatic from full dualizability.

An independent check rotates the framing of a point through $2\pi$. The induced monodromy must agree with $S_A$. A proposed oriented trivialization must send that monodromy coherently to the identity. Merely finding an abstract isomorphism $S_A\cong\mathrm{id}$ without its higher compatibility does not complete the homotopy fixed point.

Spin dependence and the Arf test

A spin surface can carry inequivalent spin structures even when its underlying oriented surface is fixed. Invertible Arf theory assigns

$$Z_{\mathrm{Arf}}(\Sigma,\rho) =(-1)^{\operatorname{Arf}(\rho)}.$$

On a torus, three spin structures have even Arf invariant and one has odd invariant. Forgetting $\rho$ identifies all four oriented tori, but the partition function takes both signs. Therefore Arf theory does not factor through the oriented bordism category. Atiyah relates spin parity to a mod-two index in Atiyah 1971, §§3–5, pp. 55–62, and Gunningham uses spin TQFT structure in Gunningham 2016, §§1–2, pp. 1859–1878.

This supplies the adversarial failure. If a fermionic theory is defined on spin bordisms and one silently forgets the spin lift, bordisms with different spin parity become falsely equivalent. The partition-function sign exposes the error. The strongest surviving claim is a spin TQFT; no oriented descent exists unless the spin dependence is trivialized.

Exercises

Why is an orientation weaker than a framing?

Solution

An orientation reduces the structure group from $O(n)$ to $SO(n)$ but does not choose a global basis of each tangent space. A framing trivializes the stabilized tangent bundle and therefore removes the entire structure group. Many oriented manifolds are not framed.

What does an $SO(2)$ fixed point add to a fully dualizable algebra?

Solution

It coherently trivializes the circle action, whose fundamental monodromy is the Serre automorphism. In the Morita example this includes a cyclic nondegenerate trace, not merely separability of the algebra.

References

• Atiyah, Michael F. “Riemann Surfaces and Spin Structures.” Annales Scientifiques de l’École Normale Supérieure 4 (1971): 47–62. DOI; Open PDF.
• Gunningham, Sam. “Spin Hurwitz Numbers and Topological Quantum Field Theory.” Geometry & Topology 20 (2016): 1859–1907. DOI; Open PDF.
• Lurie, Jacob. “On the Classification of Topological Field Theories.” In Current Developments in Mathematics 2008, 129–280. Somerville, MA: International Press, 2009. Open PDF.