The Serre automorphism via homotopy actions and the Cobordism Hypothesis for oriented manifolds

Abstract

We explicitly construct an SO(2)-action on a skeletal version of the 2-dimensional framed bordism bicategory. By the 2-dimensional Cobordism Hypothesis for framed manifolds, we obtain an SO(2)-action on the core of fully-dualizable objects of the target bicategory. This action is shown to coincide with the one given by the Serre automorphism. We give an explicit description of the bicategory of homotopy fixed points of this action, and discuss its relation to the classification of oriented 2d topological quantum field theories.oriented TQFTs. It is relevant to notice that in [Lur09] the homotopy O(n)-action on the framed (∞, n)-category of cobordisms is not explicitly constructed, or even briefly sketched. For an extensive introduction to extended TQFTs and the Cobordism Hypothesis, we refer the reader to [Fre12]. Blurring the distinction between (∞, 2)-categories and bicategories, in [FHLT10] it is argued that in the case where the target is given by the bicategory Alg 2 of algebras, bimodules, and intertwiners, the fully dualizable objects are semisimple finite-dimensional algebras, and that the additional SO(2)-fixed-points structure should correspond to the structure of a symmetric Frobenius algebra. Via a direct construction, in [SP09] it is showed that the bigroupoid Frob of Frobenius algebras, Morita contexts and intertwiners indeed classifies fully extended oriented 2-dimensional TQFTs valued in Alg 2 . In [Dav11], it is observed that the SO(2)-action given by the Serre automorphism on the core of fully-dualizable objects of Alg 2 is trivializable. In a purely bicategorical setting, in [HSV17] the homotopy-fixed-point bigroupoid of the SO(2)-action on Alg 2 is computed, and it is shown that it coincides with Frob.In the present paper we provide an explicit SO(2)-action on the framed bordism bicategory, and show that the SO(2)-action induced on K (C fd ) for any symmetric monoidal bicategory C is given by the Serre automorphism, regarded as a pseudo-natural isomorphism of the identity functor. More precisely, we make use of a presentation of the framed bordism bicategory provided in [Pst14] to construct such an SO(2)-action. By the Cobordism Hypothesis for framed manifolds, which has been proven in the setting of bicategories in [Pst14], there is an equivalence of bicategories (1.1) Fun ⊗ (Cob fr 2,1,0 , C) ∼ = K (C fd ).

Truncated Affine Rozansky–Witten Models as Extended TQFTs

“…where by definition (B fd ) × is the maximal sub-2-groupoid of fully dualisable objects in B, see [HV,Prop. 2.8].…”

“…Instead of directly describing the right-hand side in terms of homotopy actions of SO(2) on B, i. e. monoidal 2-functors 2 (SO(2)) −→ Aut sm (B), we define an equivalent 2-groupoid B , following [HV,Thm. 4.3].…”

Truncated affine Rozansky--Witten models as extended TQFTs

“…Instead of directly describing the right-hand side in terms of homotopy actions of SO(2) on B, i. e. monoidal 2-functors Π 2 (SO(2)) −→ Aut sm (B), we define an equivalent 2-groupoid B , following [HV,Thm. 4.3].…”