Special and general types
Manifolds and cobordisms
In the general terminology of -framing then a 2-framing of a manifold of dimension is a trivialization of .
In (Atiyah 90) the term “2-framing” is instead used for a trivialization of the double of the tangent bundle of a 3-manifold. So this is a different concept, but it turns out to be closely related to the 3-framing (in the previous sense) of surfaces.
For a compact, connected, oriented 3-dimensional manifold, write
for the fiberwise direct sum of the tangent bundle with itself. Via the diagonal embedding
this naturally induces a SO(6)-principal bundle.
Therefore the following definition makes sense
More in detail, we may also remember the groupoid of 2-framings and the smooth structure on collections of them:
The moduli stack is the homotopy pullback in
In terms of this a 2-framing on with orientation is a lift in
Relation to bounding 4-manifolds
In (Atiyah) it is shown how a framing on a compact connected oriented 3-manifold is induced by a 4-manifold with boundary . In fact, a framing is equivalently a choice of cobordism class of bounding 4-manifolds (Kerler).
Discussion of 2-framing entirely in terms of bounding 4-manifolds is for instance in (Sawin).
Relation to String-structures
By (Atiyah 2.1) an Atiyah 2-framing of a 3-manifold is equivalently a
-structure, where is the first Pontryagin class, hence is a homotopy class of a trivialization of
This perspective on Atiyah 2-framings is made explicit in (Bunke-Naumann, section 2.3). It is mentioned for instance also in (Freed, page 6, slide 5).
The notion of “2-framing” in the sense of framing of the double of the tangent bundle is due to
- Michael Atiyah, On framings of 3-manifolds , Topology, Vol. 29, No 1, pp. 1-7 (1990) (pdf)
making explicit a structure which slightly implicit in the discussion of the perturbative path integral quantization of 3d Chern-Simons theory in
- Edward Witten, Quantum field theory and the Jones Polynomial , Comm. Math. Phys. 121 (1989)
reviewed for instance in
- M. B. Young, section 2 of Chern-Simons theory, knots and moduli spaces of connections (pdf)
(see Atiyah, page 6). For more on the role of 2-framings in Chern-Simons theory see also
and for discussion in the context of the M2-brane from p. 7 on in
The relation to -structure is made explicit in
Ulrich Bunke, Niko Naumann, section 2.3 of Secondary Invariants for String Bordism and tmf, Bull. Sci. Math. 138 (2014), no. 8, 912–970 (arXiv:0912.4875)
C. Blanchet, N. Habegger, Gregor Masbaum, P.Vogel, Topological quantum field theories derived from the Kauffman bracket, Topology Vol 34, No. 4, pp. 883-927 (1995) (pdf)
More discussion in terms of bounding 4-manifolds is in
Thomas Kerler, Bridged links and tangle presentations of cobordism categories. Adv. Math., 141(2):207–281, (1999) (arXiv:math/9806114)
Stephen F. Sawin, Three-dimensional 2-framed TQFTS and surgery (2004) (pdf)
and page 9 of
- Stephen Sawin, Invariants of Spin Three-Manifolds From Chern-Simons Theory and Finite-Dimensional Hopf Algebras (arXiv:math/9910106).
and more discussion for 3-manifolds with boundary includes
- Thomas Kerler, Volodymyr Lyubashenko, section 1.6.1 of Non-semisimple topological quantum field theories for 3-manifolds with corners, Lecture notes in mathematics 2001