The role of \(p_1\)-structures in
3-dimensional Chern–Simons theories
March 11, 2026
Our recent paper [1] with Claudia Scheimbauer uses the cobordism hypothesis to construct fully local Chern–Simons theories. Here we expose some physics motivations: Yang–Mills plus Chern–Simons in the bosonic case and the free Majorana–Weyl spinor field in the fermionic case. We also give expositions of tangential structures and invertible field theories, in particular the ‘gravitational Chern–Simons theory’ used by Witten to obtain topological field theories from the underlying gauge theory.
plus4.5pt minus9pt plus4.5pt plus4.5pt minus6pt truept
In the mid-late 1980s new invariants of links and 3-manifolds were introduced in rapid succession, first by Jones [2], [3] and others [4] using von Neumann algebras and more traditional methods, by Witten [5] using quantum field theory, and by Reshetikhin–Turaev [6], [7] using quantum groups. Witten’s approach inspired decades of mathematical research in topological field theory that continues unabated. (The survey [8] recounts the first 20 years of developments.) Recently, together with Claudia Scheimbauer [1], we used the cobordism hypothesis [9] to construct fully local topological field theories that encompass the invariants introduced by Witten–Reshetikhin–Turaev. In this expository paper we revisit a few field-theoretic aspects of this story.
We begin and end with nontopological quantum field theories that surround these invariants. In §[sec:sec:1] we consider 3-dimensional Yang–Mills theory with a Chern–Simons term [10], [11] and take a singular limit in which all massive modes disappear. While not precisely Witten’s approach in [5]—there he studies pure Chern–Simons theory, though with some regularization—this limit is expected to produce the same theory [12]. More precisely, the assertion that the limit is topological is only true for the underlying projective theories: a slight metric dependence remains in the limiting linear theory. Witten trades this slight metric dependence for a dependence on a tangential structure for which there are various choices: a framing [5], a “2-framing” [13], or a \(p_1\)-structure [14]. (In [1] we introduce another possibility: a complex \(p_1\)-structure.) We explain this maneuver in terms of invertible field theories. In particular, we tensor the singular limit with a classical Chern–Simons theory \(\gamma _{-c}\) that is a secondary invariant of a multiple of the first Pontrjagin class, and for that reason \(p_1\)-structures are most natural here.3 The multiple is determined by the central charge \(c\), a rational number computed 12 from the level of the Chern–Simons term. The topological field theory (which in topology is known as Chern–Simons theory) obtained after tensoring with \(\gamma _{-c}\) only “knows” the central charge modulo 24.
An important aspect of the physics is the 2-dimensional chiral conformal Wess–Zumino–Witten theory [15], [16] that lives on the boundary of topological Chern–Simons theory. We do not discuss it here, but in §[sec:sec:6] we take up another chiral 2-dimensional quantum field theory: the free Majorana–Weyl spinor field. We give three Wick-rotated variants. In the last we use an analog of the Witten maneuver to express it as the boundary theory of a topological field theory, the topological field theory whose partition function is the Adams \(e\)-invariant. In fact, the Witten maneuver here is an extension to invertible field theories of the Atiyah–Patodi–Singer expression [17] for the Adams \(e\)-invariant.
The exposition highlights aspects of Wick-rotated field theory that require attention and some further development. In both §[sec:sec:1] and §[sec:sec:6] we state carefully the domain of the Wick-rotated field theories; that is, we specify precisely the sheaf of background fields. Furthermore, it is important to evaluate theories in families of manifolds and bordisms, and we make clear the sort of families we use (e.g., holomorphic or smooth). Finally, we do not delve into unitary structures in this paper, though we point out where they enter, especially in §[sec:sec:6]. (Unitarity in the context of WZW conformal blocks remains an interesting issue; see [18], [19] for recent works.)
The middle sections are mathematical expositions of tangential structures and invertible field theory. In §[sec:sec:2] after introducing tangential structures in general, following Lashof [20], we define \(p_1\)-structures. 1 relates local changes of various tangential structures. [sec:sec:4] takes up invertible field theories, mostly focusing on the topological case in which an invertible field theory is a map out of a Madsen–Tillmann spectrum. In §[sec:sec:5] we turn to nontopological invertible field theories, which are based on differential cohomology. We sketch the construction of the theory \(\gamma _c\), which physicists call ‘gravitational Chern–Simons theory’, a variation of a prime example in [21].
We thank André Henriques, Mike Hopkins, and Greg Moore for discussions and correspondence.
Yang–Mills \(+\) Chern–Simons
We begin with the physics origin of Chern–Simons theory as a limit of a Yang–Mills theory. Namely, we define a family of nontopological theories \(F_{e,\lambda }\) and take a singular limit \(e\to \infty\). The result is (conjecturally) projectively topological, and we tensor with an invertible field theory to obtain a linear topological theory \(Z_\lambda\). We work in Wick-rotated quantum field theory as axiomatized by Segal [22], [23]; see also [24]. A theory is a linear representation of a bordism category, and to specify its flavor we use the following, specialized to dimension three.
Definition 1 ().
\(\mathop{\mathrm{Man}}_3\) is the category whose objects are smooth 3-manifolds and whose morphisms are local diffeomorphisms.
A presheaf on \(\mathop{\mathrm{Man}}_3\)* is a functor \(\mathscr{F}\colon\mathop{\mathrm{Man}}_3^\textrm{op}\to \mathop{\mathrm{Set}}\) to the category of sets.*
A sheaf* is a presheaf that satisfies the usual gluing condition for open covers [25].*
Remark 1 ().
Examples of sheaves on \(\mathop{\mathrm{Man}}_3\) that we encounter map a “test” 3-manifold \(M\) to: \[\label{eq:25} \begin{align} \mathscr{F}_{\mathop{\mathrm{Riem}}}\colon M &\longmapsto \mathop{\mathrm{Riem}}(M),\qquad &&\textrm{the set of Riemannian metrics}; \\ \mathscr{F}_{w_1}\colon M&\longmapsto \mathop{\mathrm{Orient}}(M), \qquad &&\textrm{the set of orientations}; \\ \mathscr{F}_{w_1,w_2}\colon M&\longmapsto \mathop{\mathrm{Spin}}(M), \qquad &&\textrm{the groupoid of Spin structures}.\end{align}\tag{1}\] The latter two are locally constant sheaves. The notation indicates the Stiefel–Whitney class(es) that are trivialized. We use Cartesian products of these sheaves, such as \(\mathscr{F}_{\mathop{\mathrm{Riem}},w_1}=\mathscr{F}_{\mathop{\mathrm{Riem}}}\times \mathscr{F}_{w_1}\).
To each sheaf \(\mathscr{F}\) corresponds a bordism category \(\mathop{\mathrm{Bord}}_{\langle 2,3 \rangle}(\mathscr{F})\). An object is a closed 2-manifold \(Y\) equipped with a cooriented embedding \(Y\subset X\) in a germ of a 3-manifold, together with an element of \(\mathscr{F}(X)\). That element, a section of \(\mathscr{F}\) over \(X\), is called a background field. A (Wick-rotated) 3-dimensional field theory over \(\mathscr{F}\) is a symmetric monoidal functor4 \[\label{eq:26} F\colon\bigl(\mathop{\mathrm{Bord}}_{\langle 2,3 \rangle}(\mathscr{F}),\sqcup \bigr)\longrightarrow \bigl(t\!\mathop{\mathrm{Vect}},\otimes \bigr),\tag{2}\] where \(t\!\mathop{\mathrm{Vect}}\) is the category of topological vector spaces and nuclear maps; see [22]–[24], [26] for details and expanded exposition. The theory \(F\) is topological if it factors through a locally constant sheaf \(\mathscr{F}'\) via a map \(\mathscr{F}\to \mathscr{F}'\).
Remark 2 (). Definition 2 of a field theory only evaluates the theory on a single manifold at a time. This is not sufficient. Rather, one should “sheafify” the definition over the category \(\mathop{\mathrm{Man}}\) of smooth manifolds and smooth maps. In other words, one should evaluate a field theory on families of manifolds/bordisms parametrized by a smooth manifold \(S\). (Stolz–Teichner emphasize this point in their survey [27].) For example, to a fiber bundle \(Y\to S\) with fiber closed 2-dimensional manifolds a 3-dimensional theory should assign a smooth complex vector bundle over \(S\). In a unitary theory this bundle will be equipped with a hermitian metric and compatible covariant derivative. There is also a holomorphic version of this concept. We encounter all of these variations in §[sec:sec:6].
The physics origin of Chern–Simons theory
One believes [12] that for each compact Lie group \(G\) there exists a 2-parameter family of Wick-rotated 3-dimensional field theories \[\label{eq:27} F_{e,\lambda }\colon\mathop{\mathrm{Bord}}_{\langle 2,3 \rangle}(\mathscr{F}_{\mathop{\mathrm{Riem}},w_1})\longrightarrow t\!\mathop{\mathrm{Vect}}\tag{3}\] called Yang–Mills \(+\) Chern–Simons (abbreviated YM\(+\)CS). (The Chern–Simons term was introduced into 3-dimensional gauge theory in [10], [11].) Here \(e\in {\mathbb{R}}^{>0}\) is the coupling constant and \(\lambda\) is a (cocycle for a) class in \(H^4(BG;{\mathbb{Z}})\) called the level. Fix a positive definite \(G\)-invariant inner product \(\langle -,- \rangle\) on \(\mathfrak{g}\). There is a semiclassical description as a gauge theory with Lagrangian \[\label{eq:28} L_{e,\lambda }(A) = \frac{1}{4e^2}\langle F_A\wedge *F_A\rangle\;+\;\Gamma _\lambda (A),\tag{4}\] where \(A\) is a fluctuating \(G\)-connection and \(\Gamma _\lambda\) is the Chern–Simons “form”. The Yang–Mills term uses the Riemannian metric; the Chern–Simons term uses the orientation. The Chern–Simons “form” is more properly a differential cocycle, as we indicate in §[sec:sec:5].
Assuming the level \(\lambda\) is nondegenerate,5 the theory \(F_{e,\lambda }\) is thought to be gapped. One piece of evidence: if \(G\) is abelian, then one can do an easy analysis of the free relativistic theory in Minkowski spacetime and compute the gap explicitly; see [28]. The expected gap leads to an expected singular long distance limit which is a linear theory whose projectivization is a topological theory. (See [29], [30] for more on projective field theories and anomalies; the lecture notes [26] contain a more expansive exposition of this story.) These expectations are laid out in the following, inspired by [12] and [5]; see also [31] for \(G=\mathop{\mathrm{U}}_1\).
Conjecture 1 ().
The Lagrangian 4 determines a family of Wick-rotated field theories.
The singular limit \(e\to \infty\) exists and defines a field theory \[\label{eq:29} F_{\lambda }\colon\mathop{\mathrm{Bord}}_{\langle 2,3 \rangle}(\mathscr{F}_{\mathop{\mathrm{Riem}},w_1})\longrightarrow t\!\mathop{\mathrm{Vect}}.\qquad{(1)}\]
The projectivization \(\overline{F}_\lambda\) factors through \(\mathop{\mathrm{Bord}}_{\langle 2,3 \rangle}(\mathscr{F}_{w_1})\), hence it is a projective topological* field theory.*
The projectivity (anomaly) \(\alpha ^{\vphantom{1*\prime y\vee M}}_\lambda\) of \(\overline{F}_\lambda\) extends to \(\tilde{\alpha} ^{\vphantom{1*\prime y\vee M}}_\lambda\), a 4-dimensional invertible field theory of oriented manifolds whose partition function on a closed oriented 4-manifold \(W\) is \[\label{eq:61} \tilde{\alpha} ^{\vphantom{1*\prime y\vee M}}_\lambda (W) = \exp\left( \frac{2\pi ic(\lambda )}{24}\,\langle p_1(W),[W] \rangle\right) ,\qquad{(2)}\] where \(c=c(\lambda )\in {\mathbb{Q}}\) is the central charge associated to the level \(\lambda\). (The formula for \(c(\lambda )\) is 12 below.)
Remark 3 ().
Statement (3) implies that the states spaces of \(F_{\lambda }\) are all finite dimensional.
The projectivity, or anomaly, of a 3-dimensional quantum field theory is a once-categorified invertible 3-dimensional theory [29]. In this case it is topological—it factors through a theory over \(\mathscr{F}_{w_1}\)—and as stated in 1(4) the anomaly theory \(\alpha ^{\vphantom{1*\prime y\vee M}}_\lambda\) extends to a 4-dimensional invertible theory \(\tilde{\alpha} ^{\vphantom{1*\prime y\vee M}}_\lambda\). See §[sec:sec:4] for more on invertible theories.
We discuss the central charge in §[sec:subsec:6461]. In particular, the formula for \(c(\lambda )\) in terms of \(\lambda\) is 12 . From this formula it is manifest that \(c(\lambda )\) is a rational number.
The anomaly theory \(\alpha ^{\vphantom{1*\prime y\vee M}}_\lambda\) and its extension \(\tilde{\alpha} ^{\vphantom{1*\prime y\vee M}}_\lambda\) only depend on \(c(\lambda )\pmod{24}\).
The first Pontrjagin class enters the story in ?? , and this makes the choice of \((w_1,\,p_1)\)-structures later on more natural than other tangential structures.
The topological projective theory \(\overline{F}_\lambda\) is essentially the picture of Chern–Simons theory put forward by Walker [32] long ago; see [33] for a modern account. Walker uses the quantum group construction of the topological field theory; see 4(5) below.
A version of this conjecture for \(G\) a torus is discussed in detail in [34].
Next we explain Witten’s maneuver [5] to construct a lift of the projective theory \(\overline{F}_\lambda\) to a linear topological field theory. (The singular limit \(F_\lambda\) is a linear nontopological lift of \(\overline{F}_\lambda\).) This uses another locally constant sheaf, \(\mathscr{F}_{p_1}\), equivalent to the tangential structure built from trivializing the first Pontrjagin class; we introduce it in §[sec:subsec:2463].6 The maneuver uses the commutative diagram \[\label{eq:30} \begin{gather} \xymatrix@C-1pc{&\mathscr{F}_{\mathop{\mathrm{Riem}},w_1,p_1}\ar[dl]\ar[dr] \\ \mathscr{F}_{\mathop{\mathrm{Riem}},w_1}\ar[dr] && \mathscr{F}_{w_1,p_1}\ar[dl] \\ &\mathscr{F}_{w_1}} \end{gather}\tag{5}\] of sheaves on \(\mathop{\mathrm{Man}}_3\). (The maps are projections off of Cartesian products.) The key ingredient is a family of invertible field theories \[\label{eq:31} \gamma _c\colon\mathop{\mathrm{Bord}}_{\langle 2,3 \rangle}(\mathscr{F}_{\mathop{\mathrm{Riem}},w_1,p_1})\longrightarrow \mathop{\mathrm{Line}}_{{\mathbb{C}}},\qquad c\in {\mathbb{R}},\tag{6}\] based on the intrinsic7 Chern–Simons invariant of a Riemannian 3-manifold [21]. Witten’s observation is that the theory \[\label{eq:32} Z_{\lambda }= F_{\lambda }\otimes \gamma _{-c(\lambda )} ,\tag{7}\] initially defined over \(\mathscr{F}_{\mathop{\mathrm{Riem}},w_1,p_1}\), descends to a linear topological field theory over \(\mathscr{F}_{w_1,p_1}\): the metric dependence cancels out.
Remark 4 ().
The projective theory \(\overline{F}_\lambda \bigm/ \mathscr{F}_{w_1}\) has two linear lifts in this story: (i) the long distance limit \(F_{\lambda }\bigm/\mathscr{F}_{\mathop{\mathrm{Riem}},w_1}\) of YM\(+\)CS, and (ii) the topological field theory \(Z_{\lambda }\bigm/\mathscr{F}_{w_1,p_1}\).
If \(\beta\) is an invertible 3-dimensional topological field theory of \((w_1,\,p_1)\)-manifolds, then \(Z_{\lambda }\otimes \beta\) is also a linear lift of \(\overline{F}_\lambda\). As we will see in §0.0.3, isomorphism classes of such \(\beta\) form an abelian group isomorphic to \(\raisebox{-.07em}{\rotatebox{9.9} {\tiny {\boldsymbol{/}} }}\mu\raisebox{-0.98ex}{\scalebox{2} {\color{white}\phantom{.}}}\raisebox{+0.88ex} {\color{white}\phantom{.}}_{6}\). This is one explanation for the \(\raisebox{-.07em}{\rotatebox{9.9} {\tiny {\boldsymbol{/}} }}\mu\raisebox{-0.98ex}{\scalebox{2} {\color{white}\phantom{.}}}\raisebox{+0.88ex} {\color{white}\phantom{.}}_{6}\)-extension of the Witt group that appears in [1].
The invertible field theories in (2) are not assumed to be unitary. If we impose unitarity, then the \(\raisebox{-.07em}{\rotatebox{9.9} {\tiny {\boldsymbol{/}} }}\mu\raisebox{-0.98ex}{\scalebox{2} {\color{white}\phantom{.}}}\raisebox{+0.88ex} {\color{white}\phantom{.}}_{6}\) indeterminacy of the linear lift of \(\overline{F}_\lambda\) reduces to a \(\raisebox{-.07em}{\rotatebox{9.9} {\tiny {\boldsymbol{/}} }}\mu\raisebox{-0.98ex}{\scalebox{2} {\color{white}\phantom{.}}}\raisebox{+0.88ex} {\color{white}\phantom{.}}_{3}\) indeterminacy; see 44 . This is closely related to the distinction between 3-dimensional \((w_1,\,p_1)_3\)-structures and stable \((w_1,\,p_1)_s\)-structures that we make in §[sec:subsec:2463]; see [35].
The topological theory \(Z_{\lambda }\) admits a unitary structure, as do \(F_{\lambda }\) and \(\gamma _{-c(\lambda )}\). However, the isomorphism 7 need not preserve these unitary structures.
An extension of \(Z_{\lambda }\) to a (1,2,3)-theory is what is usually called (quantum) Chern–Simons theory in topology. It was constructed rigorously by Reshetikhin–Turaev [7], [36] starting from quantum group data. More generally, one can begin with a modular tensor category and construct a (1,2,3)-Reshetikhin–Turaev Theory. In [1] we construct a fully local8 extension of Reshetikhin–Turaev theories using the cobordism hypothesis. This applies to \(Z_{\lambda }\). Of course, we can then tensor with \(\gamma _{c(\lambda )}\) to rigorously construct what should be the long distance limit of YM\(+\)CSas a fully extended theory.
The topological field theory \(Z_{\lambda }\bigm/\mathscr{F}_{w_1,p_1}\) only sees \(c\pmod{24}\). There are further statements along these lines in [1]. On the other hand, \(\gamma _c\) requires a specification of \(c\in {\mathbb{R}}\), hence by 7 so too does the limit \(F_{\lambda }\bigm/\mathscr{F}_{\mathop{\mathrm{Riem}},w_1}\) of YM\(+\)CS.
For all \(N\in {\mathbb{Z}}\), the theory \(\gamma ^{\vphantom{1*\prime y\vee M}}_{24N}\) does not depend on a \(p_1\)-structure; it factors through \(\mathscr{F}_{\mathop{\mathrm{Riem}},w_1}\). The tensor product \(F_{e,\lambda }\otimes \gamma ^{\vphantom{1*\prime y\vee M}}_{24N}\) has the same underlying projective theory as \(F_{e,\lambda }\), hence so too the same singular limit \(e\to \infty\) as a projective theory. This is another reason why we only expect the topological field theories constructed from YM\(+\)CSto have a central charge in \({\mathbb{Q}}/24{\mathbb{Z}}\) rather than in \({\mathbb{Q}}\).
We execute the Witten maneuver for a free spinor field in §[sec:subsec:6464].
The Spin variation
There are Chern–Simons theories whose level \(\lambda\) lies not in ordinary cohomology, but rather in a slightly exotic cohomology theory. Even on the classical level these theories require a Spin structure, not just an orientation. Aspects of Spin Chern–Simons theory are studied in [37]–[41]. In these theories the analog of the modular tensor category is enriched over the category of super vector spaces and there are other new features. In [1] we consider Spin theories as well as non-Spin theories. Here, in §[sec:sec:6], we discuss another Spin theory: the free 2-dimensional spinor field and its anomaly, in various guises.
The key change over §[sec:subsec:1461] is that 5 is replaced with \[\label{eq:33} \begin{gather} \xymatrix@C-1pc{&\mathscr{F}_{\mathop{\mathrm{Riem}},w_1,w_2,p_1}\ar[dl]\ar[dr] \\ \mathscr{F}_{\mathop{\mathrm{Riem}},w_1,w_2}\ar[dr] && \mathscr{F}_{w_1,w_2,p_1}\ar[dl] \\ &\mathscr{F}_{w_1,w_2}} \end{gather}\tag{8}\] Each sheaf includes a Spin structure: a trivialization of both \(w_1\) and \(w_2\). Now YM\(+\)CSis defined over Riemannian Spin manifolds, and the projective long distance limit \(\overline{F}_\lambda\) is defined over Spin manifolds. The tangential structure derived from \(\mathscr{F}_{w_1,w_2,p_1}\) is defined in §[sec:subsec:2464].
Remark 5 ().
The anomaly theory, as a theory of \(\mathop{\mathrm{Spin}}_3\)-manifolds, only depends on \(c(\lambda )\pmod6\). (Compare 3(4).)
The indeterminacy of 4(2) due to the cyclic group \(\raisebox{-.07em}{\rotatebox{9.9} {\tiny {\boldsymbol{/}} }}\mu\raisebox{-0.98ex}{\scalebox{2} {\color{white}\phantom{.}}}\raisebox{+0.88ex} {\color{white}\phantom{.}}_{6}\) of invertible \((w_1,\,p_1)_3\)-theories is now a \(\raisebox{-.07em}{\rotatebox{9.9} {\tiny {\boldsymbol{/}} }}\mu\raisebox{-0.98ex}{\scalebox{2} {\color{white}\phantom{.}}}\raisebox{+0.88ex} {\color{white}\phantom{.}}_{48}\times \raisebox{-.07em}{\rotatebox{9.9} {\tiny {\boldsymbol{/}} }}\mu\raisebox{-0.98ex}{\scalebox{2} {\color{white}\phantom{.}}}\raisebox{+0.88ex} {\color{white}\phantom{.}}_{2}\) indeterminacy from invertible \((w_1,\,w_2, \,p_1)_3\)-theories; see §0.0.4.
The theory \(\gamma _c\) is the same as in §[sec:subsec:1461]: there is no Spin structure dependence introduced.
A formula for the central charge
The choice of \(c(\lambda )\in {\mathbb{Q}}\)—the central charge—in 7 is given as follows, assuming that \(\lambda\) is positive definite (as defined shortly).
Let \(G\) be a compact Lie group and let \(\lambda\) be a cocycle for a class in \(H^4(BG;{\mathbb{Z}})\). The real image \([\lambda _{{\mathbb{R}}}]\in H^4(BG;{\mathbb{R}})\) is equivalently a \(G\)-invariant symmetric bilinear form \[\label{eq:56} \langle -,- \rangle_\lambda \colon\mathfrak{g}\times \mathfrak{g}\longrightarrow {\mathbb{R}}.\tag{9}\] We say that \(\lambda\) is positive definite if this form is positive definite, which we now assume. The negative of one-half the Killing form is a nondegenerate \(G\)-invariant symmetric bilinear form \[\label{eq:57} \langle -,- \rangle_h\colon\mathfrak{g}\times \mathfrak{g}\longrightarrow {\mathbb{R}}.\tag{10}\] Here \([2h]\in H^4(BG;{\mathbb{Z}})\) is the first Pontrjagin class of the adjoint representation. Define the symmetric endomorphism \(S_\lambda \in \mathop{\mathrm{End}}(\mathfrak{g})\) by \[\label{eq:58} \langle \xi _1,\xi _2 \rangle_\lambda = \langle \xi _1,S_\lambda (\xi _2) \rangle_{\lambda +h},\qquad \xi _1,\xi _2\in \mathfrak{g}.\tag{11}\] Then the central charge is the trace \[\label{eq:59} c(\lambda ) = \mathop{\mathrm{tr}}(S_\lambda ).\tag{12}\] For example, if \(G=\mathop{\mathrm{SU}}_2\) and the level \(\lambda\) is \(k\) times the positive generator of \(H^4(BG;{\mathbb{Z}})\), then 12 reduces to the usual formula \[\label{eq:60} c(\lambda )= \frac{3k}{k+2}.\tag{13}\] Equation 12 can be derived from the Segal–Sugawara construction of the boundary WZW theory.
Tangential structures
Let \(B\!\relax_n\) be a choice of classifying space of the orthogonal group. The inclusions \(\relax_1\hookrightarrow \relax_2\hookrightarrow \cdots\) induce maps \(B\!\relax_1\longrightarrow B\!\relax_2\longrightarrow \cdots\); the colimit of this sequence of topological spaces is denoted \(B\!\relax\). The following definition was introduced by Lashof [20].
Definition 2 ().
For \(n\in {\mathbb{Z}}^{>0}\) an \(n\)-dimensional tangential structure* is a continuous map \(\pi_n\colon\mathscr{X}_n\to B\!\relax_n\).*
A stable tangential structure* is a continuous map \(\pi \colon\mathscr{X}\to B\!\relax\).*
Equivalently, an \(n\)-dimensional tangential structure is a pair \((\mathscr{X}_n,\mathscr{E}_n)\), where \(\mathscr{X}_n\) is a topological space and \(\mathscr{E}_n\to \mathscr{X}_n\) is a rank \(n\) real vector bundle; there is a similar formulation in the stable case. A stable tangential structure induces an \(n\)-dimensional tangential structure by pullback: \[\label{eq:1} \begin{gather} \xymatrix{\mathscr{X}_n\ar@{-->}[r]^{} \ar@{-->}[d]_{\pi_n} & \mathscr{X}\ar[d]^{\pi} \\ B\!\relax_n\ar[r]^{} & B\!\relax} \end{gather}\tag{14}\] Similarly, an \(n\)-dimensional tangential structure induces an \(m\)-dimensional tangential structure for all \(m<n\).
Definition 3 (). Suppose an \(n\)-dimensional tangential structure \(\pi_n\colon\mathscr{X}_n\to B\!\relax_n\) is given.
A \(\pi_n\)-structure* on a smooth \(n\)-manifold \(M\) is a lift \(s_n\) of a classifying map9 of its tangent bundle: \[\label{eq:2} \begin{gather} \xymatrix@C+1pc{&\mathscr{X}_n\ar[d]^{\pi_n} \\ M\ar[ur]^{s_n}\ar[r]^{\tau _M} & B\!\relax_n} \end{gather}\tag{15}\] Two \(\pi _n\)-structures are isomorphic if the lifts are homotopic.*
Suppose \(\pi_n\) is a principal fibration with fiber \(F\). A change of \(\pi_n\)-structure* on \(M\) is a map \(M\to F\). A change of isomorphism class of \(\pi_n\)-structure is a homotopy class of maps \(M\to F\).*
There is a space of \(\pi _n\)-structures. If \(p\colon\mathscr{X}\to B\!\relax\) is a stable tangent structure, there are analogous definitions of a \(\pi\)-structure and a change of \(\pi\)-structure on a smooth manifold.
Lemma 1 (). Let \(\pi_n\colon\mathscr{X}_n\to B\!\relax_n\) be induced from \(\pi\colon\mathscr{X}\to B\!\relax\), as in 14 , and suppose \(M\) is an \(n\)-manifold. Then there is a homeomorphism between the space of \(\pi_n\)-structures on \(M\) and the space of \(\pi\)-structures on \(M\).
Proof. In the diagram \[\label{eq:3} \begin{gather} \xymatrix@C+=3pc{& \mathscr{X}_n\ar[r]^{} \ar[d]_{\pi_n} & \mathscr{X}\ar[d]^{\pi} \\ M\ar@{-->}[ur]^{s_n} \ar@{-->}[urr]^<<<<<<<<<<<<<<<<<<<<<<<<s \ar[r]^<<<<<<<<<{\tau_M}&B\!\relax_n\ar[r]^{} & B\!\relax} \end{gather}\tag{16}\] the right hand square is a pullback. ◻
Remark 6 ().
For some tangential structures more rigid models are possible, such as for framings, orientations, and Spin structures. Rigid models—reductions of the principal bundle of frames of a smooth \(n\)-manifold—exist for \(n\)-dimensional tangential structures \(BG_n\to B\!\mathop{\mathrm{GL}}_n\!{\mathbb{R}}\) that are induced from a homomorphism \(G_n\to \mathop{\mathrm{GL}}_n\!{\mathbb{R}}\) of Lie groups. (In the preceding we have replaced \(\mathop{\mathrm{GL}}_n\!{\mathbb{R}}\) by the homotopy equivalent maximally compact subgroup \(\relax_n\subset \mathop{\mathrm{GL}}_n\!{\mathbb{R}}\).)
In 3(2) the entire \(\pi_n\)-structure is changed; all that is left fixed is the underlying smooth manifold. We will later consider changes of structure in which an underlying orientation or Spin structure is left fixed.
For the purposes of this paper, we set \(n=3\) and introduce the relevant tangential structures.
Framings
Let \(*\) denote the singleton topological space. A framing or 3-framing is the tangential structure10 \[\label{eq:4} \mathop{\mathrm{fr}}_3\colon*\longrightarrow B\!\relax_3\tag{17}\] A stable framing is the tangential structure \[\label{eq:5} \mathop{\mathrm{fr}}\colon*\longrightarrow B\!\relax\tag{18}\] Observe that 17 is truly unstable: while it is induced from a 4-dimensional tangential structure, it is not induced from a 5-dimensional tangential structure.11 A 3-framing on a 3-manifold \(M\) is also called a parallelism. A change of parallelism (3(2)) is effected by a map \(M\to \relax_3\); if it preserves the induced orientation then it is a map \(M\to \mathop{\mathrm{SO}}_3\). This can be seen from the rigid model of \(s\) as a global basis of the tangent bundle, or from the diagram \[\label{eq:6} \begin{gather} \xymatrix@C+2pc{&\relax_3\ar[d] \\ & \ast\ar[d]^{\mathop{\mathrm{fr}}_3} \\ M\ar@{-->}[uur] \ar[ur]^<<<<<<<<<<<<<s \ar[r]^<<<<<<<<<<<{\tau _M} & B\!\relax_3} \end{gather}\tag{19}\] A similar comment applies to stable framings: orientation-preserving changes are maps \(M\to \mathop{\mathrm{SO}}\).
Remark 7 (). A framing induces a stable framing, as in 16 , but not every stable framing comes from a framing. For example, \(S^2\) admits stable framings (unique up to homotopy), but the hairy ball theorem obstructs the existence of framings. On the other hand, \(S^3\) admits 3-framings.
Lemma 2 (). The map \[\label{eq:7} \pi _3\mathop{\mathrm{SO}}_3\longrightarrow \pi _3\mathop{\mathrm{SO}}\qquad{(3)}\] from orientation-preserving changes of framings of \(S^3\) to orientation-preserving changes of stable framings of \(S^3\) is multiplication by 2.
Each of \(\pi _3\mathop{\mathrm{SO}}_3\), \(\pi _3\mathop{\mathrm{SO}}\) is infinite cyclic; the lemma asserts that ?? maps a generator of \(\pi _3\mathop{\mathrm{SO}}_3\) to twice a generator of \(\pi _3\mathop{\mathrm{SO}}\).
Remark 8 (). Homotopy groups are defined using pointed maps \(S^3\to \mathop{\mathrm{SO}}_3\) and \(S^3\to \mathop{\mathrm{SO}}\), whereas changes of (stable) framings are unpointed. A straightforward argument proves that \(\pi _3\mathop{\mathrm{SO}}_N\to [S^3,\mathop{\mathrm{SO}}_N]\) is bijective for all \(N\in {\mathbb{Z}}^{\ge3}\).
Proof. Compute ?? by passing to the double cover \[\label{eq:11} \pi _3\mathop{\mathrm{Spin}}_3\longrightarrow \pi _3\mathop{\mathrm{Spin}}\tag{20}\] Under the special isomorphisms of low dimensional Spin groups, the inclusions \[\label{eq:12} \xymatrix@1{\mathop{\mathrm{Spin}}_3\;\ar@{^{(}->}[r] & \;\mathop{\mathrm{Spin}}_4\;\ar@{^{(}->}[r]& \;\mathop{\mathrm{Spin}}_6}\tag{21}\] become the inclusions \[\label{eq:13} \xymatrix@1{\mathop{\mathrm{SU}}_2\;\ar@{^{(}->}[r]^<<<<<\Delta & \;\mathop{\mathrm{SU}}_2\times \mathop{\mathrm{SU}}_2\;\ar@{^{(}->}[r]& \;\mathop{\mathrm{SU}}_4}\tag{22}\] where \(\Delta\) is the diagonal inclusion and the second map is the block diagonal inclusion. This composition represents twice a generator of \(\pi _3\mathop{\mathrm{SU}}_4\), and we are well into the stable range. ◻
Orientations and Spin structures
Each is a stable tangential structure, as witnessed by the pullback diagrams \[\label{eq:8} \begin{gather} \xymatrix{B\!\mathop{\mathrm{SO}}_3\ar[r]^{} \ar[d]_{\pi_3} & B\!\mathop{\mathrm{SO}}\ar[d]^{\pi} \\ B\!\relax_3\ar[r]^{} & B\!\relax}\qquad \qquad \xymatrix{B\!\mathop{\mathrm{Spin}}_3\ar[r]^{} \ar[d]_{\pi_3} & B\!\mathop{\mathrm{Spin}}\ar[d]^{\pi} \\ B\!\relax_3\ar[r]^{} & B\!\relax} \end{gather}\tag{23}\] The fibers of the vertical maps are12 \(\raisebox{-.07em}{\rotatebox{9.9} {\tiny {\boldsymbol{/}} }}\mu\raisebox{-0.98ex}{\scalebox{2} {\color{white}\phantom{.}}}\raisebox{+0.88ex} {\color{white}\phantom{.}}_{2}=\{\pm1\}\) and \(\raisebox{-.07em}{\rotatebox{9.9} {\tiny {\boldsymbol{/}} }}\mu\raisebox{-0.98ex}{\scalebox{2} {\color{white}\phantom{.}}}\raisebox{+0.88ex} {\color{white}\phantom{.}}_{2}\times B\!\raisebox{-.07em}{\rotatebox{9.9} {\tiny {\boldsymbol{/}} }}\mu\raisebox{-0.98ex}{\scalebox{2} {\color{white}\phantom{.}}}\raisebox{+0.88ex} {\color{white}\phantom{.}}_{2}\;\sim\; \raisebox{-.07em}{\rotatebox{9.9} {\tiny {\boldsymbol{/}} }}\mu\raisebox{-0.98ex}{\scalebox{2} {\color{white}\phantom{.}}}\raisebox{+0.88ex} {\color{white}\phantom{.}}_{2}\times K(\raisebox{-.07em}{\rotatebox{9.9} {\tiny {\boldsymbol{/}} }}\mu\raisebox{-0.98ex}{\scalebox{2} {\color{white}\phantom{.}}}\raisebox{+0.88ex} {\color{white}\phantom{.}}_{2},1)\), respectively. Of course, there are rigid geometric models. (The paper [43] is a useful reference for Spin structures, especially in low dimensions.)
An orientation of a smooth manifold \(M\) is a trivialization of the orientation double cover \(\widehat{M}\to M\), and \(\widehat{M}\to M\) represents the Stiefel–Whitney class \(w_1\) of the tangent bundle \(TM\to M\). Hence we sometimes use ‘\(w_1\)-structure’ in place of ‘orientation’. In a similar vein, we use ‘\((w_1,w_2)\)-structure’ synonymously with ‘Spin structure’.
Remark 9 (). Stabilization induces an isomorphism of the set of orientations and the set of stable orientations, as in 1. Let \(M\) be a 3-manifold. The corresponding equivalence of groupoids of (stable) Spin structures follows from the diagram \[\label{eq:9} \begin{gather} \xymatrix{P\ar@{-->}[rr]^{} \ar@{-->}[d]^{\raisebox{-.07em}{\rotatebox{9.9} {\tiny {\boldsymbol{/}} }}\mu\raisebox{-0.98ex}{\scalebox{2} {\color{white}\phantom{.}}}\raisebox{+0.88ex} {\color{white}\phantom{.}}_{2}} && P'\ar[d]^{\raisebox{-.07em}{\rotatebox{9.9} {\tiny {\boldsymbol{/}} }}\mu\raisebox{-0.98ex}{\scalebox{2} {\color{white}\phantom{.}}}\raisebox{+0.88ex} {\color{white}\phantom{.}}_{2}} \\ \mathop{\mathrm{SO}}(M)\ar@{^{(}->}[rr]^{} \ar[dr]^{\mathop{\mathrm{SO}}_3} && SO'(M)\ar[dl]_<<<<<<<<{\mathop{\mathrm{SO}}_N} \\ &M} \end{gather}\tag{24}\] in which a stable Spin structure \(P'\xrightarrow{\;\;\mathop{\mathrm{Spin}}_N\;\;}M\), \(N\gg 3\), pulls back to a Spin structure \(P\xrightarrow{\;\;\mathop{\mathrm{Spin}}_3\;\;}M\) over the 3-manifold \(M\). (This is the rigid model analog of 16 .) By contrast, recall (7) that framings and stable framings differ.
\((w_1,\,p_1)\)-structures
The remaining tangential structures that we introduce do not admit rigid models in the sense of 6. A \(p_1\)-structure is a trivialization of the first Pontrjagin class. Restrict to oriented manifolds. Fix models \(\Sigma ^{k}{\mathbb{Z}}\), \(k\in {\mathbb{Z}}^{\ge 0}\), for Eilenberg–MacLane spaces with nonzero homotopy group \(\pi _k\cong {\mathbb{Z}}\). Also fix maps that represent the first Pontrjagin class \(p_1\), and define \(\mathscr{X}_3,\mathscr{X}\) as the homotopy fibers of \(p_1\) in the diagram \[\label{eq:10} \begin{gather} \xymatrix@C-1pc{\mathscr{X}_3\ar@{-->}[rr]^{} \ar@{-->}[d]^{} && \mathscr{X}\ar@{-->}[d]^{} \\ B\!\mathop{\mathrm{SO}}_3\ar[rr]^{} \ar[dr]^<<<<<<<{p_1} && B\!\mathop{\mathrm{SO}}\ar[dl]_<<<<<<<<{p_1} \\ &\Sigma ^{4}{\mathbb{Z}}} \end{gather}\tag{25}\] After composition with \(B\!\mathop{\mathrm{SO}}_3\to B\!\relax_3\) and \(B\!\mathop{\mathrm{SO}}\to B\!\relax\) we obtain the pullback diagram \[\label{eq:14} \begin{gather} \xymatrix{\mathscr{X}_3\ar[r]^{} \ar[d]_{(w_1,\,p_1)^{\vphantom{1*\prime y\vee M}}_3} & \mathscr{X}\ar[d]^{(w_1,\,p_1)_s} \\ B\!\relax_3\ar[r]^{} & B\!\relax} \end{gather}\tag{26}\] in which the vertical maps define the stable tangential structure \((w_1,\,p_1)_s\) and its pullback to a 3-dimensional tangential structure \((w_1,\,p_1)^{\vphantom{1*\prime y\vee M}}_3\). We also call these the ‘stable oriented \(p_1\)-structure’ (or ‘\(\mathop{\mathrm{SO}}^{p_1}\)-structure’) and the ‘3-dimensional oriented \(p_1\)-structure’ (or ‘\(\mathop{\mathrm{SO}}^{p_1}(3)\)-structure’). By 1 there is an equivalence between stable and unstable structures.
Let \(M\) be an oriented 3-manifold. A change of oriented \(p_1\)-structure on \(M\) that fixes the underlying orientation is a map \[\label{eq:15} \begin{gather} \xymatrix@C+=2.5pc{ &\Sigma ^{3}{\mathbb{Z}}\ar@{=}[rr]\ar[d] && \Sigma ^{3}{\mathbb{Z}}\ar[d]\\ &\mathscr{X}_3\ar[rr]^{} \ar[d]^{} && \mathscr{X}\ar@{-->}[d]^{} \\ M\ar@{-->}[uur] \ar[ur]^<<<<<<<<<s \ar[r]^<<<<<<<{\tau _M} & B\!\mathop{\mathrm{SO}}_3\ar[rr]^{} \ar[dr]^{p_1} && B\!\mathop{\mathrm{SO}}\ar[dl]_<<<<<<<<{p_1} \\ &&\Sigma ^{4}{\mathbb{Z}}} \end{gather}\tag{27}\] (The dotted map in the diagram shows a change of 3-dimensional structure; a map to the other \(\Sigma ^{3}{\mathbb{Z}}\) is a change of stable structure.) Here \(\Sigma ^{3}{\mathbb{Z}}\) is the homotopy fiber of \(\mathscr{X}_3\to B\!\mathop{\mathrm{SO}}_3\) and of \(\mathscr{X}\to B\!\mathop{\mathrm{SO}}\); since the lower triangle homotopy commutes, the induced map \(\Sigma ^{3}{\mathbb{Z}}\to \Sigma ^{3}{\mathbb{Z}}\) is a homotopy equivalence. The homotopy class of a change of (stable) oriented \(p_1\)-structure on \(M\) is given by a class in \(H^3(M;{\mathbb{Z}})\).
Lemma 3 (). Let \(M\) be an oriented 3-manifold. The map \[\label{eq:16} H^3(M;{\mathbb{Z}})\longrightarrow H^3(M;{\mathbb{Z}})\qquad{(4)}\] from changes of 3-dimensional oriented \(p_1\)-structures to change of stable oriented \(p_1\)-structures is an isomorphism.
Proof. Diagram 27 . ◻
Remark 10 (). Fix an oriented 3-manifold \(M\). Isomorphism classes of compatible 3-dimensional \(p_1\)-structures on \(M\) form a torsor \(\mathcal{P}(M)\) over \(H^3(M;{\mathbb{Z}})\). If \(M\) is closed and connected, then \(\mathcal{P}(M)\) is a \({\mathbb{Z}}\)-torsor. It is useful to contemplate this and related torsors in some of what follows.
\((w_1,\,w_2, \,p_1)\)-structures
These are also called ‘3-dimensional Spin \(p_1\)-structures’ and ‘stable Spin \(p_1\)-structures’. Replace \(\mathop{\mathrm{SO}}_3\) with \(\mathop{\mathrm{Spin}}_3\) and \(\mathop{\mathrm{SO}}\) with \(\mathop{\mathrm{Spin}}\) in §[sec:subsec:2463] to make the constructions. For the analog of 26 we use the diagram \[\label{eq:17} \begin{gather} \xymatrix{\mathscr{Y}_3\ar[r]^{} \ar[d]_{(w_1,\,w_2, \,p_1)^{\vphantom{1*\prime y\vee M}}_3} & \mathscr{Y}\ar[d]^{(w_1,\,w_2, \,p_1)_s} \\ B\!\relax_3\ar[r]^{} & B\!\relax} \end{gather}\tag{28}\] Here \(\mathscr{Y}_3\) is the homotopy fiber of \(p_1\colon B\!\mathop{\mathrm{Spin}}_3\to \Sigma ^{4}{\mathbb{Z}}\) and \(\mathscr{Y}\) is the homotopy fiber of \(p_1\colon B\!\mathop{\mathrm{Spin}}\to \Sigma ^{4}{\mathbb{Z}}\). The Spin version of 3 holds.
Remark 11 (). The first Pontrjagin class \(p_1\) is divisible by 2 on \(B\!\mathop{\mathrm{Spin}}\) and its pullback to \(B\!\mathop{\mathrm{Spin}}_3\) is divisible by 4. This leads to cousins of the \((w_1,\,w_2, \,p_1)\)-structures. With evident notation, a \((w_1,\,w_2, \,p_1/4)^{\vphantom{1*\prime y\vee M}} _3\)-structure is equivalent to a 3-framing, and a \((w_1,\,w_2, \,p_1/2)^{\vphantom{1*\prime y\vee M}} _3\)-structure is equivalent to a stable framing on manifolds of dimension \(\le3\).
Remark 12 (). Fix a Spin 3-manifold \(M\). Then isomorphism classes of compatible 3-dimensional \(p_1/4\)-structures, \(p_1/2\)-structures, and \(p_1\)-structures form \(H^3(M;{\mathbb{Z}})\)-torsors; there are torsor morphisms \[\label{eq:34} \mathcal{P}_{1/4}(M)\longrightarrow \mathcal{P}_{1/2}(M)\longrightarrow \mathcal{P}(M)\tag{29}\] compatible with the group homomorphisms \[\label{eq:35} H^3(M;{\mathbb{Z}}) \xrightarrow{\;\;2\;\;}H^3(M;{\mathbb{Z}}) \xrightarrow{\;\;2\;\;}H^3(M;{\mathbb{Z}}) .\tag{30}\] Therefore, the morphisms 29 are injective and each image has index 2.
Maps of tangential structures
The relevant maps are encoded in the diagram \[\begin{gather} \label{eq:18} \vcenter{ \xymatrix@C+3.5pc{\ast \ar[r]\ar[ddr]|{\mathop{\mathrm{fr}}_3} \ar[drrr] & \mathscr{Y}_3 \ar[r]\ar[dd]|{(w_1,\,w_2, \,p_1)^{\vphantom{1*\prime y\vee M}}_3}\ar[drrr] & \mathscr{X}_3 \ar[ddl]|{(w_1,\,p_1) ^{\vphantom{1*\prime y\vee M}}_3} \ar[drrr]\\ &&& \ast \ar[r]\ar[dr]|{\mathop{\mathrm{fr}}_s} & \mathscr{Y} \ar[r]\ar[d]|{(w_1,\,w_2, \,p_1)_s} & \mathscr{X}\ar[dl]|{(w_1,\,p_1)_s} \\ &B\!\relax_3\ar[rrr] &&&B\!\relax}} \end{gather}\tag{31}\] As for local changes of structures, which are changes on \(M=S^3\), we have the following. Let \(\Delta (p)\) denote the infinite cyclic group of local changes of structure13 \(\pi\).
Proposition 1 (). Local changes of structure are related by the commutative diagram \[\label{eq:19} \begin{gather} \xymatrix@R+.5pc@C+.5pc{\Delta (\mathop{\mathrm{fr}})_3\ar[r]^<<<<<{4} \ar[d]^{2} & \Delta (w_1,\,w_2, \,p_1)^{\vphantom{1*\prime y\vee M}}_3\ar[d]^{1} \ar[r]^<<<<<1& \Delta (w_1,\,p_1)^{\vphantom{1*\prime y\vee M}}_3 \ar[d]^1\\ \Delta (\mathop{\mathrm{fr}})_s\ar[r]^<<<<<{2} & \Delta (w_1,\,w_2, \,p_1)_s\ar[r]^<<<<<1 & \Delta (w_1,\,p_1)_s} \end{gather}\qquad{(5)}\] of homomorphisms of infinite cyclic groups.
Proof. The vertical arrows are computed in 2 and 3, together with the Spin version of the latter. We compute the composite of the top horizontal arrows from the diagram \[\label{eq:20} \begin{gather} \xymatrix@C-1pc{\mathop{\mathrm{SO}}_3\ar[rr]^{f} \ar[d]^{} && \Sigma ^{3}{\mathbb{Z}}\ar[d]^{} \\ \ast\ar[rr]^{f} \ar[dr]^{} && \mathscr{X}_3\ar[dl] \\ &B\!\mathop{\mathrm{SO}}_3} \end{gather}\tag{32}\] This induces a pullback map of Leray spectral sequences for cohomology with integer coefficients:
(0,0) (0,2) (2,2) (0,3) (3,0) (4,0) [̣"d_3" xshift=1.4pc, yshift=1.1pc]3(0,2) [̣"d_2" xshift=1.4pc, yshift=1.1pc]2(0,3)
(0,0) (0,3) (3,0) (4,0) [̣"d_4" yshift=2pc]4(0,3)
Here \(2a=2b=0\). The class \(2\kappa\) survives to the \(E_4\)-page, and \(d_4(2\kappa )=p_1\). Hence \[\label{eq:21} d_4f^*(\iota ) = f^*(d_4\iota ) = f^*(p_1) = p_1,\tag{33}\] from which \(f^*(\iota) =2\kappa\). This implies that \(f_*\colon H_3(\mathop{\mathrm{SO}}_3)\to H_3(\Sigma ^{3}{\mathbb{Z}})\) is multiplication by 2. Since the Hurewicz homomorphism \(\pi _3\mathop{\mathrm{SO}}_3\to H_3(\mathop{\mathrm{SO}}_3)\) is also multiplication by 2, the composite of the top horizontal arrows is multiplication by 4. (Recall from 2 that local changes of oriented framings are computed in \(\pi _3\mathop{\mathrm{SO}}_3\).)
A similar argument shows that the composition of the bottom horizontal arrows is multiplication by 2; the Hurewicz homomorphism \(\pi _3\mathop{\mathrm{SO}}\to H_3(\mathop{\mathrm{SO}})\) is an isomorphism. (Logically, we do not need this step in the proof.)
We claim that the rightmost horizontal arrows in ?? are isomorphisms. For the arrow in the top row we use the diagram \[\label{eq:23} \begin{gather} \xymatrix@C-1pc{\Sigma ^{3}{\mathbb{Z}}\ar[rr]^{h} \ar[d]^{} && \Sigma ^{3}{\mathbb{Z}}\ar[d]^{} \\ \mathscr{Y}_3\ar[rr]^{h} \ar[d]^{} && \mathscr{X}_3\ar[d] \\ B\!\mathop{\mathrm{Spin}}_3\ar[rr]^h&&B\!\mathop{\mathrm{SO}}_3} \end{gather}\tag{34}\] The induced map of spectral sequences is now
(0,0) (0,3) (4,0) [̣"d_4" yshift=2pc]4(0,3)
where \(4\mu =h^*(p_1)\). (In the first spectral sequence the class \(\mu\) generates the infinite cyclic group \(H^4(B\!\mathop{\mathrm{SO}}_3;{\mathbb{Z}})\).) Then \[\label{eq:24} d_4h^*(\iota ) = h^*(d_4\iota ) = h^*(p_1) = 4\mu = d_4(\iota '),\tag{35}\] from which \(h^*(\iota )=\iota '\). The argument for the bottom row (stable analog) is the same except that \(p_1\) is only divisible by 2 in \(H^4(B\!\mathop{\mathrm{Spin}};{\mathbb{Z}})\). ◻
Remark 13 (). The two leftmost horizontal arrows in ?? can also be computed using 11 and 12 for \(M=S^3\).
Invertible field theories
The basic notion is straightforward, stated here in the 3-dimensional case. We use notation from 2 .
Definition 4 (). A 3-dimensional field theory \[\label{eq:36} A\colon\mathop{\mathrm{Bord}}_{\langle 2,3 \rangle}(\mathscr{F})\longrightarrow t\!\mathop{\mathrm{Vect}},\qquad{(6)}\] is invertible* if it factors through the Picard groupoid of complex lines: \[\label{eq:37} \begin{gather} \xymatrix@R-1pc@C+1pc{&\mathop{\mathrm{Line}}\ar[dd] \\ \mathop{\mathrm{Bord}}_{\langle 2,3 \rangle}(\mathscr{F})\ar@{-->}[ur]^<<<<<<<<{\widehat\alpha } \ar[dr]^<<<<<<{A} \\ & t\!\mathop{\mathrm{Vect}}} \end{gather}\tag{36}\] *
The factorization is unique if it exists. As is true for general field theories, an invertible field theory may or may not be topological. In both the topological and nontopological cases there is a “homological” interpretation. And in both cases we consider fully local field theories, defined on a bordism 3-category \(\mathop{\mathrm{Bord}}_{\langle 0,1,2,3 \rangle}(\mathscr{F})\). In this section we consider topological theories. In §[sec:sec:5] we describe the family 6 of nontopological invertible theories.
First, we examine the arrow \(\widehat\alpha\) in 36 more closely. It is a symmetric monoidal functor whose codomain is a Picard groupoid. Therefore, \(\widehat\alpha\) factors through the Picard groupoid quotient of the domain (also known as the group completion): \[\label{eq:38} \begin{gather} \xymatrix@R-1pc@C+1pc{\mathop{\mathrm{Bord}}_{\langle 2,3 \rangle}(\mathscr{F})\ar[dd]\ar[rd]^{\widehat\alpha } \\ &\mathop{\mathrm{Line}}\\|\mathop{\mathrm{Bord}}_{\langle 2,3 \rangle}(\mathscr{F})|\ar@{-->}[ur]^<<<<<<<<{\alpha } } \end{gather}\tag{37}\] The geometric realization of a Picard groupoid is an infinite loop space, or equivalently a connective spectrum. For example, the geometric realization of the Picard groupoid \(\mathop{\mathrm{Line}}\) is the Eilenberg–MacLane spectrum \(\Sigma H{\mathbb{Z}}/2{\mathbb{Z}}\). The geometric realization of a topological bordism category was determined in [44]; it is a Madsen–Tillmann spectrum [45]. So too is the geometric realization of a fully local higher topological bordism category [46]. In the fully local case the codomain is also replaced by a richer spectrum. There is a universal choice \(\Sigma ^3I{\mathbb{C}}^\times\), which is dual to the sphere spectrum. Bottom line: a 3-dimensional invertible topological field theory is a map from a Madsen–Tillmann spectrum to \(\Sigma ^3I{\mathbb{C}}^\times\). See [35] and the references therein for a more generous account.
Remark 14 ().
Other choices of codomain are possible, but \(\Sigma ^3I{\mathbb{C}}^\times\) enjoys a universal property: the abelian group of isomorphism classes of 3-dimensional invertible topological field theories with codomain \(\Sigma ^3I{\mathbb{C}}^\times\) is isomorphic to the Pontrjagin dual group to \(\pi _3\) of the appropriate Madsen–Tillmann spectrum. More colloquially, with codomain \(\Sigma ^3I{\mathbb{C}}^\times\) the partition function determines the invertible theory.
Unitarity is not included in the definition 2 of a Wick-rotated quantum field theory. In the invertible case, if we impose unitarity then the Madsen–Tillmann spectrum is replaced by a stabilized version [35].
Some invertible topological field theories
We rely on the bordism computations in [1]. In the cases we consider, the corresponding groups of invertible topological field theories are sums of finite cyclic groups. Here we tell the partition function of generating theories.
The Madsen–Tillmann spectrum built from 3-framed 3-manifolds is the sphere spectrum \(\mathbb{S}\). The relevant homotopy group is the 3-stem \[\label{eq:39} \pi _3\mathbb{S}\cong {\mathbb{Z}}/24{\mathbb{Z}}.\tag{38}\] The Lie group \(\mathop{\mathrm{SU}}_2\) with left-invariant framing represents a generator.
By the general remarks above, the group of 3-framed invertible 3-dimensional topological field theories is isomorphic to the Pontrjagin dual \(\raisebox{-.07em}{\rotatebox{9.9} {\tiny {\boldsymbol{/}} }}\mu\raisebox{-0.98ex}{\scalebox{2} {\color{white}\phantom{.}}}\raisebox{+0.88ex} {\color{white}\phantom{.}}_{24}\) to \({\mathbb{Z}}/24{\mathbb{Z}}\). A generating theory14 \(\psi\) can be constructed using the Adams \(e\)-invariant [17], [47] as follows. Let \(X\) be a closed 3-framed 3-manifold. Choose15 \(W\) a compact Spin 4-manifold with \(\partial W=X\). The 3-framing on \(X\) determines a relative first Pontrjagin class \(p_1(W,X)\in H^4(W,X;{\mathbb{Z}})\). The partition function is \[\label{eq:40} \psi (X) = \exp\left( \frac{2\pi i}{48}\bigl\langle p_1(W,X),[W,X] \bigr\rangle \right) ,\tag{39}\] where \([W,X]\in H_4(W,X)\) is the relative fundamental class of the orientation. (Observe that \(p_1\in H^4(B\!\mathop{\mathrm{Spin}};{\mathbb{Z}})\) is divisible by two.)
Whereas every closed Spin 3-manifold bounds a compact Spin 4-manifold, if we impose a 3-dimensional reduction of structure on the tangent bundle of the 4-manifold this is no longer true: \[\label{eq:41} \pi _3\Sigma ^3MT\!\mathop{\mathrm{Spin}}_3\cong {\mathbb{Z}}/2{\mathbb{Z}}.\tag{40}\] (See [48] for exposition of this kind of bordism and Madsen–Tillmann spectra.) The 3-sphere with its unique Spin structure represents the generator. The nontrivial invertible 3-dimensional Spin theory \(\nu\) has the following partition function. Let \(X\) be a closed Spin 3-manifold. Write \(X=\partial W\) for a compact Spin 4-manifold \(W\). Then \[\label{eq:42} \nu (X) = \exp\left( \frac{2\pi i}{2}\mathop{\mathrm{Euler}}(W)\right) ,\tag{41}\] where \(\mathop{\mathrm{Euler}}(W)\) is the Euler number of \(W\). If \(X\) is 3-framed, then \(\nu (X)=\psi (X)^{12}\).
Remark 15 (). For \(X=S^3\) with its unique Spin structure, we can choose \(W=D^4\) to be the 4-disk, and so \(\nu (S^3)=-1\). This implies that \(\nu\) does not admit a reflection positive structure, since \(S^3\) is a double and the partition function of a double is positive in a reflection positive theory [35].
For rank 3 structures the relevant bordism group is16 \[\label{eq:43} \pi _3\Sigma ^3MT\!\mathop{\mathrm{SO}}_3^{p_1}\cong {\mathbb{Z}}/6{\mathbb{Z}}.\tag{42}\] The left parallelism on \(\mathop{\mathrm{SU}}_2\) induces a \((w_1,\,p_1)_3\)-structure, which represents a generator of 42 . In other words, the map \(\pi _3\mathbb{S}\to \pi _3\Sigma ^3MT\!\mathop{\mathrm{SO}}_3^{p_1}\) is surjective. A generator \(\beta\) of the group \(\raisebox{-.07em}{\rotatebox{9.9} {\tiny {\boldsymbol{/}} }}\mu\raisebox{-0.98ex}{\scalebox{2} {\color{white}\phantom{.}}}\raisebox{+0.88ex} {\color{white}\phantom{.}}_{6}\) of invertible 3-dimensional \((w_1,\,p_1)_3\)-theories has the following partition function. Let \(X\) be a closed \((w_1,\,p_1)\) 3-manifold, and write \(X=\partial W\) for a compact oriented 4-manifold \(W\). Then \[\label{eq:44} \beta (X) = \exp\left( \frac{2\pi i}{6}\Bigl\{ \bigl\langle p_1(W,X),[W,X] \bigr\rangle + 3\mathop{\mathrm{Euler}}(W) \Bigr\} \right) .\tag{43}\] If \(X\) is 3-framed, then \(\beta (X)=\psi (X)^{-4}\).
Turning now to stable \((w_1,\,p_1)_s\)-structures, the relevant bordism group is \[\label{eq:45} \pi _3MT\!\mathop{\mathrm{SO}}^{p_1}\cong {\mathbb{Z}}/3{\mathbb{Z}};\tag{44}\] the order 3 theory \(\beta ^2\) factors to a \((w_1,\,p_1)_s\)-theory that generates the group \(\raisebox{-.07em}{\rotatebox{9.9} {\tiny {\boldsymbol{/}} }}\mu\raisebox{-0.98ex}{\scalebox{2} {\color{white}\phantom{.}}}\raisebox{+0.88ex} {\color{white}\phantom{.}}_{3}\) of invertible 3-dimensional \((w_1,\,p_1)_s\)-theories.
Here we meet noncyclic bordism groups. Namely, the relevant bordism group of rank 3 \((w_1,\,w_2, \,p_1)_3\)-structures is \[\label{eq:46} \pi _3\Sigma ^3MT\!\mathop{\mathrm{Spin}}_3^{p_1}\cong {\mathbb{Z}}/48{\mathbb{Z}}\;\oplus\; {\mathbb{Z}}/2{\mathbb{Z}},\tag{45}\] whereas the relevant bordism group of stable \((w_1,\,w_2, \,p_1)_s\)-structures is \[\label{eq:47} \pi _3MT\!\mathop{\mathrm{Spin}}^{p_1} \cong {\mathbb{Z}}/48{\mathbb{Z}}.\tag{46}\] Fix an orientation of \(S^3\) and its (unique up to isomorphism) refinement to a Spin structure. Let \(\mathscr{F}_s(S^3)\) denote the \({\mathbb{Z}}\)-torsor of oriented stable framings of \(S^3\), and recall from 12 the \({\mathbb{Z}}\)-torsor \(\mathcal{P}(S^3)\) of \(p_1\)-structures compatible with the Spin structure. Then by 1 the map \[\label{eq:48} \mathscr{F}_s(S^3)\longrightarrow \mathcal{P}(S^3)\tag{47}\] has image of index \(2\). Let \(\mathfrak{p}\in \mathcal{P}(S^3)\) be the image of the Lie group framing on \(S^3\cong \mathop{\mathrm{SU}}_2\). Then \(\mathfrak{p}-1\in \mathcal{P}(S^3)\) generates \(\pi _3MT\!\mathop{\mathrm{Spin}}^{p_1}\). Furthermore, it represents an element \(x\) of order 48 in \(\pi _3\Sigma ^3MT\!\mathop{\mathrm{Spin}}_3^{p_1}\). The Spin 3-sphere \(S^3\) with a bounding \(p_1\)-structure represents an element \(y\) of order 2 in \(\pi _3\Sigma ^3MT\!\mathop{\mathrm{Spin}}_3^{p_1}\) that does not equal \(24x\). The elements \(x,y\) together generate 45 .
Remark 16 (). The preceding follows from [1], in which the natural map \[\label{eq:49} \pi _3\Sigma ^3MT\!\mathop{\mathrm{Spin}}_3^{p_1} \longrightarrow \pi _3\Sigma ^3MT\!\mathop{\mathrm{Spin}}^{p_1}\;\oplus \;\pi _3\Sigma ^3MT\!\mathop{\mathrm{Spin}}_3\tag{48}\] is shown to be the isomorphism 45 .
Turning to the corresponding invertible field theories, whose isomorphism classes form a group isomorphic to \(\raisebox{-.07em}{\rotatebox{9.9} {\tiny {\boldsymbol{/}} }}\mu\raisebox{-0.98ex}{\scalebox{2} {\color{white}\phantom{.}}}\raisebox{+0.88ex} {\color{white}\phantom{.}}_{48}\times \raisebox{-.07em}{\rotatebox{9.9} {\tiny {\boldsymbol{/}} }}\mu\raisebox{-0.98ex}{\scalebox{2} {\color{white}\phantom{.}}}\raisebox{+0.88ex} {\color{white}\phantom{.}}_{2}\), the pullback of \(\nu\) to \(\Sigma ^3MT\!\mathop{\mathrm{Spin}}_3^{p_1}\) generates the \(\raisebox{-.07em}{\rotatebox{9.9} {\tiny {\boldsymbol{/}} }}\mu\raisebox{-0.98ex}{\scalebox{2} {\color{white}\phantom{.}}}\raisebox{+0.88ex} {\color{white}\phantom{.}}_{2}\) factor. A generator \(\lambda\) of the \(\raisebox{-.07em}{\rotatebox{9.9} {\tiny {\boldsymbol{/}} }}\mu\raisebox{-0.98ex}{\scalebox{2} {\color{white}\phantom{.}}}\raisebox{+0.88ex} {\color{white}\phantom{.}}_{48}\) factor has the following partition function. Let \(X\) be a closed Spin 3-manifold equipped with a \(p_1\)-structure. Choose a compact Spin 4-manifold \(W\) with \(\partial W=X\). Then \[\label{eq:50} \lambda (X) = \exp\left( \frac{2\pi i}{48}\bigl\langle p_1(W,X),[W,X] \bigr\rangle \right) .\tag{49}\] Comparing with 39 we see that \(\lambda\) is the extension of \(\psi\) to Spin \(p_1\)-manifolds.
Tangential Chern–Simons theory
In this section we construct the invertible 3-dimensional field theory \(\gamma _c\), \(c\in {\mathbb{R}}\), that was introduced in 6 . This theory is nontopological, so it is not described by a map out of a Madsen–Tillmann spectrum. Rather, it is constructed using the theory of differential cohomology. We begin with a lightning review of ordinary differential cohomology. Then we construct the theory \(\gamma _c\) of 6 , which recall is an invertible 3-dimensional theory over the sheaf \(\mathscr{F}_{\mathop{\mathrm{Riem}},w_1,p_1}\to \mathop{\mathrm{Man}}_3\).
Differential cocycles
A comprehensive treatment together with extensive history and referencing may be found in [49]; see also the survey [50] and the references therein. A foundational paper is [51].
As in §[sec:sec:1] we work with sheaves on \(\mathop{\mathrm{Man}}_3\), though in this subsection all sheaves extend to \(\mathop{\mathrm{Man}}\), the category of smooth manifolds and smooth maps between them. (See [25] for background.) There are sheaves \(\Omega ^q\), \(q\in {\mathbb{Z}}\), whose sections over a smooth manifold \(M\) form the vector space \(\Omega ^q(M)\) of differential \(q\)-forms on \(M\). There are also sheaves \(\Omega ^q_{\textrm{cl}}\) of closed differential forms of degree \(q\). As well, there are sheaves \(Z^q\) of smooth singular cocycles with integer coefficients. The sheaf \(\widehat{Z}\) of differential cocycles is constructed as a homotopy pullback in the references at the beginning of this subsection. A differential cocycle has a curvature, which is a closed differential form, and also an underlying smooth singular cocycle; this is expressed in the diagram of sheaves \[\label{eq:51} \begin{gather} \xymatrix{\widehat{Z}^q\ar[r]^{\omega } \ar[d]^{\mu } & \Omega ^q_{\textrm{cl}}\\ Z^q} \end{gather}\tag{50}\] Suppose \(M\in \mathop{\mathrm{Man}}\), \(\hat{c}\in \widehat{Z}^q(M)\), and \(\tau \in C^{q-1}(M)\) is a trivialization of \(\mu (\hat{c})\) in the sense that \(\delta \tau =\mu (\hat{c})\). Then \(\hat{c}\) lifts17 uniquely to a differential \((q-1)\)-form on \(M\). In terms of sheaves there is a fibration \[\label{eq:52} \begin{gather} \xymatrix{\Omega ^{q-1}\ar[d]\\ \widehat{Z}^q\ar[d]^{\mu } \\ Z^q } \end{gather}\tag{51}\]
The preceding is based on integral Eilenberg–MacLane cohomology \(H{\mathbb{Z}}\). There are differential refinements of generalized cohomology theories as well. Particular cases are used to define (classical) Spin Chern–Simons invariants; see [39], [52], for example.
The theory \(\gamma _c\)
We first recast the classical Chern–Simons form [21] as a differential cocycle. Let \(G\) be a Lie group with finitely many components, and fix a cocycle \(\lambda\) for a class (the level) in \(H^4(BG;{\mathbb{Z}})\). Let \(\mathscr{F}\!_{B^{\vphantom{1*\prime y\vee M}}_{\nabla }G}\) denote the simplicial sheaf of \(G\)-connections. The Chern–Simons differential cocycle is a map \(\Gamma\) that fits into the diagram \[\label{eq:53} \begin{gather} \xymatrix@C+1pc{\mathscr{F}\!_{B^{\vphantom{1*\prime y\vee M}}_{\nabla }G} \ar[r]^{\Gamma } & \widehat{Z}^4\ar[r]^\omega \ar[d]^{\mu } & \Omega ^4_{\textrm{cl}} \\ & Z^4 } \end{gather}\tag{52}\] where \(\omega \circ \Gamma\) is the Chern–Weil 4-form of the level \(\lambda\), and we can assume that \(\mu \circ \Gamma =\lambda\).
Fix \(G=\mathop{\mathrm{SO}}_3\) and let the level be the first Pontrjagin class \(p_1\in H^4(B\!\mathop{\mathrm{SO}}_3;{\mathbb{Z}})\). In the following diagram \(\Theta ^{\textrm{LC}}\) is the Levi–Civita connection: \[\label{eq:54} \begin{gather} \xymatrix@C+1pc{\mathscr{F}_{\mathop{\mathrm{Riem}},p_1} \ar@{-->}[rr]^\phi \ar[d] &&\Omega ^3\ar[r]^d \ar[d] & \Omega ^4_{\textrm{cl}}\ar@{=}[d] \\ \mathscr{F}_{\mathop{\mathrm{Riem}}} \ar[r]^{\Theta ^{\textrm{LC}}}\ar[d]_{p_1} & \mathscr{F}\!_{B^{\vphantom{1*\prime y\vee M}}_{\nabla }\mathop{\mathrm{SO}}_3} \ar[r]^\Gamma & \widehat{Z}^4\ar[r]^\omega \ar[d]^\mu & \Omega ^4_{\textrm{cl}}\\ Z^4\ar@{=}[rr] && Z^4 } \end{gather}\tag{53}\] The construction of \(\phi\) uses 51 : a Riemannian manifold with a \(p_1\)-structure produces a 3-form whose differential is the Chern–Weil 4-form that represents \(p_1\).
The theory \(\gamma _c\), \(c\in {\mathbb{R}}\), is constructed by integrating \(c\phi /24\) and exponentiating. So if \(X\) is a closed oriented Riemannian 3-manifold with \(p_1\)-structure, then the partition function is \[\label{eq:55} \gamma _c(X) = \exp\left( \frac{2\pi ic}{24}\int_{X}\phi \right) .\tag{54}\]
Remark 17 (). As stated in 2, one should evaluate field theories in families of bordisms. Let us consider the theory \(\gamma _c\) in families. First, let \(\pi \colon X\to S\) be a fiber bundle of smooth manifolds whose fibers are closed 3-manifolds. Additionally, the relative tangent bundle \(T(X/S)\to X\) carries an orientation and a \(p_1\)-structure, and there is a relative Riemannian structure18 on \(\pi\): that is, an inner product on \(T(X/S)\to X\) together with a horizontal distribution on \(\pi\). Then \[\label{eq:66} \gamma _c(X/S) = \exp\left( \frac{2\pi ic}{24}\int_{X/S}\phi \right)\tag{55}\] is a smooth function \(\gamma _c(X/S)\colon S\to \mathbb{T}\subset {\mathbb{C}}^{\times }\). For \(\pi \colon Y\to S\) a smooth fiber bundle with fibers closed 2-manifolds—equipped with the same geometric structure as above—the imaginary 1-form \[\label{eq:67} \frac{2\pi ic}{24}\int_{Y/S}\phi\tag{56}\] is a connection form on the trivial line bundle over \(S\).
Remark 18 (). The composition \(\mu \circ \Gamma \circ \Theta ^{\textrm{LC}}\) in 53 leads to a 4-dimensional invertible topological field theory with “integer values” [53], [35]; in the framework of §[sec:sec:4] it is a spectrum map \[\label{eq:68} \zeta \colon\Sigma ^4MT\!\mathop{\mathrm{SO}}_4\longrightarrow \Sigma ^4H{\mathbb{Z}}.\tag{57}\] On a fiber bundle \(X\to S\) of closed oriented Riemannian 3-manifolds, the Chern–Simons theory built from \(\Gamma \circ \Theta ^{\textrm{LC}}\) returns the tangential Chern–Simons invariant \(S\to {\mathbb{R}}/{\mathbb{Z}}\); the topological theory \(\zeta\) in 57 only tracks the homotopy class of this map (or equivalently the associated \({\mathbb{Z}}\)-torsor over \(S\)). On a fiber bundle \(Y\to S\) of closed oriented Riemannian 2-manifolds, the Chern–Simons invariant is a principal \({\mathbb{R}}/{\mathbb{Z}}\)-bundle with connection over \(S\); the topological theory \(\zeta\) returns the underlying principal \({\mathbb{R}}/{\mathbb{Z}}\)-bundle without connection over \(S\) (or equivalently the associated \({\mathbb{Z}}\)-gerbe over \(S\)).
The pullback of 57 to oriented manifolds with \(p_1\)-structure is trivialized, as follows from 53 . Let \(\tau\) be the trivialization: \[\label{eq:69} \begin{gather} \begin{tikzcd} \Sigma^4\text{MTSO}_4^{p_1} \arrow[r] \arrow[rr, bend right=30, "\mathbb{1}"' pos=0.522, ""{name=B, above, pos=0.522}] & |[alias=T]| \Sigma^4\text{MTSO}_4 \arrow[r, "\zeta"] & \Sigma^4H\mathbb{Z} \arrow[Rightarrow, from=B, to=T, "\tau" right, shorten >=-2pt] \end{tikzcd} \end{gather}\tag{58}\] On a closed Riemannian \((w_1,\,p_1)\)-manifold the trivialization \(\tau\) of the \({\mathbb{Z}}\)-torsor \(\zeta (X)\) lifts the Chern–Simons invariant from \({\mathbb{R}}/{\mathbb{Z}}\) to \({\mathbb{R}}\), evident from its expression as the integral \(\int_{X}\phi\). On a fiber bundle \(Y\to S\) of closed \((w_1,\,p_1)\) Riemannian 2-manifolds, the trivialization \(\tau\) is a (not-necessarily-flat) section of the principal \({\mathbb{R}}/{\mathbb{Z}}\)-bundle with connection over \(S\) that is the Chern–Simons invariant.
There is a similar trivialization of the “integer-valued” theory \([\gamma _c]\) underlying \(\gamma _c\) for any \(c\in {\mathbb{R}}\).
The free spinor field
We treat a chiral 2-dimensional spinor field (Majorana–Weyl) in three Wick-rotated guises. Before Wick rotation this free theory is defined as a relativistic quantum mechanical system on 2-dimensional Minkowski spacetime. The translation group acting on this affine space decomposes under the Lorentz Spin group \(\mathop{\mathrm{Spin}}_{1,1}\) into a sum of left- and right-moving translations. The representation of \(\mathop{\mathrm{Spin}}_{1,1}\) that defines the chiral spinor field is the double cover of the left-moving translations. The resulting free theory leads to an irreducible 1-particle representation of the Poincaré group.
We present the Wick-rotated theory, which is an anomalous theory, in three incarnations.
Remark 19 (). As pointed out in 2, it is important to evaluate the free spinor field on families of manifolds/bordisms. We use both holomorphic and smooth families. As will be apparent, theories differ depending on the choice of families, as well as the nature of the output.
Remark 20 (). We comment on unitary structures but leave their detailed development to the future.
Holomorphic presentation
Perhaps the most familiar presentation of the chiral spinor field is as an anomalous theory over the sheaf \(\mathscr{F}_{\mathop{\mathrm{Conf}},w_1,w_2}\to \mathop{\mathrm{Man}}_2\), i.e., a theory of 2-dimensional Spin conformal manifolds. An orientation plus conformal structure on a 2-manifold \(Y\) is equivalent to a complex structure on \(Y\), and a compatible Spin structure is then equivalent to a choice of square root \(K^{1/2}_Y\to Y\) of the canonical bundle \(K^{\vphantom{1*\prime y\vee M}}_Y\to Y\). The Dirac operator is the \({\overline{\partial}}\)-operator coupled to \(K^{1/2}_Y\to Y\): \[\label{eq:62} D_Y={\overline{\partial}}_Y(K^{1/2}_Y)\colon\Omega ^{1/2,\,0}_Y\longrightarrow \Omega ^{1/2,\,1}_{Y},\tag{59}\] where \(\Omega ^{1/2,\,q}_Y=\Omega ^{0,\,q}_Y(K^{1/2}_Y)\). This first-order differential operator is complex skew-adjoint if \(Y\) is closed. Then the partition function of the free spinor field is the pfaffian of the Dirac operator, which is an element of the Pfaffian line: \[\label{eq:63} \mathop{\mathrm{pfaff}}D_Y\in \mathop{\mathrm{Pfaff}}D_Y.\tag{60}\] This is the top level of an anomalous 2-dimensional field theory \(F\) over \(\mathscr{F}_{\mathop{\mathrm{Conf}},w_1,w_2}\to \mathop{\mathrm{Man}}_2\).
In this version of the theory it is natural to consider holomorphic families. Thus to a holomorphic fiber bundle \(Y\to S\) of closed 1-dimensional complex manifolds is associated a holomorphic line bundle \[\label{eq:70} \mathop{\mathrm{Pfaff}}D_{Y/S}\longrightarrow S\tag{61}\] together with a holomorphic section \(\mathop{\mathrm{pfaff}}D_{Y/S}\). There is no hermitian metric or connection on the line bundle 61 in this variant. In other words, there is no unitary structure on the theory \(F\).
Remark 21 (). The anomaly theory \(\alpha\) comes to us as a once-shifted (or once-categorified) invertible 2-dimensional theory over the sheaf \(\mathscr{F}_{\mathop{\mathrm{Conf}},w_1,w_2}\to \mathop{\mathrm{Man}}_2\); the value of \(\alpha\) on a holomorphic family \(Y\to S\) is the holomorphic Pfaffian line bundle 61 . Apparently \(\alpha\) does not extend to an invertible conformal 3-dimensional theory, though we have not worked out a proof of this assertion. When we pull back from 2-dimensional conformal structures to 2-dimensional Riemannian structures, and we replace holomorphic families by smooth families, then there is an extension, as we explain in §[sec:subsec:6463].
Remark 22 (). The rational Chern class of the line bundle 61 is nonzero for some families of surfaces. Introduce a \(p_1\)-structure: pull back to the sheaf \(\mathscr{F}_{\mathop{\mathrm{Conf}},w_1,w_2,p_1}\to \mathop{\mathrm{Man}}_2\). Then in all holomorphic families of closed surfaces, now with \(p_1\)-structure, the rational Chern class does vanish. Even more is true: now there is a flat structure on the Pfaffian line bundle. This line bundle with flat structure does not depend on conformal structures. In fact, this flat Pfaffian line bundle is part of a 3-dimensional topological field theory over \(\mathscr{F}_{w_1,w_2,p_1}\to \mathop{\mathrm{Man}}_3\): the theory \(\lambda\) in 49 . In the remainder of this section we introduce Riemannian metrics to derive this flat structure. Not only does it exist for families of closed surfaces, but we will see \(\lambda\) emerge as a fully local invertible 3-dimensional topological field theory.
Riemannian presentation
As in §[sec:subsec:1461], let \(\mathscr{F}_{\mathop{\mathrm{Riem}},w_1,w_2}\to \mathop{\mathrm{Man}}_2\) be the sheaf of Riemannian metrics plus Spin structures. Consider the pullback of the anomalous theory \(F\) with anomaly \(\alpha\) along the fiber bundle \(\mathscr{F}_{\mathop{\mathrm{Riem}},w_1,w_2}\to \mathscr{F}_{\mathop{\mathrm{Conf}},w_1,w_2}\) of sheaves over \(\mathop{\mathrm{Man}}_2\). On a Riemannian Spin 2-manifold \(Y\), the Pfaffian line \(\mathop{\mathrm{Pfaff}}D_Y\) carries a Quillen metric. The pullback of the anomaly theory \(\alpha\) extends to a unitary invertible 3-dimensional theory \(\widehat{\alpha }\) over the sheaf \(\mathscr{F}_{\mathop{\mathrm{Riem}},w_1,w_2}\to \mathop{\mathrm{Man}}_3\): the partition function of \(\widehat{\alpha }\) on a closed Riemannian Spin 3-manifold \(X\) is \(\exp(2\pi i\xi _X/2)\), where \(\xi _X\) is the Atiyah–Patodi–Singer \(\xi\)-invariant.19 The pullback \(\widehat{F}\) of the free spinor field theory \(F\) is a boundary theory of \(\widehat{\alpha }\) (as theories over \(\mathscr{F}_{\mathop{\mathrm{Riem}},w_1,w_2}\to \mathop{\mathrm{Man}}_3\)).
In this Riemannian variant of the theory, it is natural to evaluate on smooth families. Thus suppose \(\pi \colon Y\to S\) is a smooth fiber bundle with a Spin structure on the relative tangent bundle \(T(Y/S)\to Y\) and a relative Riemannian structure on \(\pi\), as in 17. Then \(\widehat{\alpha }\) evaluates to the smooth Pfaffian line bundle \[\label{eq:71} \mathop{\mathrm{Pfaff}}D_{Y/S}\longrightarrow S\tag{62}\] equipped with its Quillen metric and compatible covariant derivative [55]. The Quillen metric and covariant derivative are part of a unitary structure on \(\widehat{\alpha }\). The free spinor field theory \(\widehat{F}\) returns the section \(\mathop{\mathrm{pfaff}} D_{Y/S}\) as in 60 .
Remark 23 (). The anomaly theory \(\widehat{\alpha }\) can be constructed using differential \(KO\)-theory. At least at the top levels, the equivalence between the construction with Dirac operators and the differential \(KO\) construction can be proved using geometric variants of Atiyah–Singer index theory.
The Witten maneuver and the Adams \(e\)-invariant
Now we execute the move in 7 . This maneuver is the extension to invertible field theory of the Atiyah–Patodi–Singer expression [17] for the Adams \(e\)-invariant. (The Adams \(e\)-invariant for framed manifolds appears in §0.0.1.) For this we pull back along the fibration \(\mathscr{F}_{\mathop{\mathrm{Riem}},w_1,w_2,p_1}\to \mathscr{F}_{\mathop{\mathrm{Riem}},w_1,w_2}\) of sheaves over \(\mathop{\mathrm{Man}}_3\) . Then the (exponentiated) Adams \(e\)-invariant of \(X\), now a closed 3-dimensional Riemannian Spin manifold with a \(p_1\)-structure, is \[\label{eq:64} \exp(2\pi i\xi _X/2)\cdot \gamma _{-1/2}(X) = \exp\left( 2\pi i\left[ \frac{\xi _X}{2} - \frac{1}{48}\int_{X}\phi \right] \right) ,\tag{63}\] where \(\gamma _{-1/2}\) is defined in 54 . As explained in 18, the invertible theory \(\gamma _{-1/2}\) carries a nonflat trivialization \(\tau\). Recall that a boundary theory of a field theory is a domain wall to the trivial theory. Hence a (nonflat) trivialization \(\tau\) of an invertible field theory is equivalent to a (nonflat) invertible boundary theory \(\widehat{\tau }\). Then as theories over \(\mathscr{F}_{\mathop{\mathrm{Riem}},w_1,w_2,p_1}\to \mathop{\mathrm{Man}}_3\), we can say that \(\widehat{F}\otimes \widehat{\tau }\) is a boundary theory of \(\widehat{\alpha }\otimes \gamma _{-1/2}\). Furthermore, \(\widehat{\alpha }\otimes \gamma _{-1/2}\) is a topological field theory: it factors to a theory over \(\mathscr{F}_{w_1,w_2,p_1}\to \mathop{\mathrm{Man}}_3\). This is precisely the theory \(\lambda\) in 49 .
Remark 24 (). If we drop the unitary structure, then the 2-dimensional boundary theory \(\widehat{F}\otimes \widehat{\tau }\) depends only on a conformal structure, not a Riemannian structure, and we identify it with the holomorphic theory \(F\) in §[sec:subsec:6462], now lifted along \(\mathscr{F}_{\mathop{\mathrm{Conf}},w_1,w_2,p_1}\to \mathscr{F}_{\mathop{\mathrm{Conf}},w_1,w_2}\). Explicitly, \[\label{eq:65} F \bigm/ \mathscr{F}_{\mathop{\mathrm{Conf}},w_1,w_2,p_1}\;\textrm{ is a boundary theory of }\; \lambda \bigm/\mathscr{F}_{w_1,w_2,p_1}.\tag{64}\] This achieves the goal set out in 22, and it is a standard picture of the free chiral spinor field in two dimensions: a chiral 2-dimensional conformal field theory as the boundary theory of a 3-dimensional topological field theory.
These theories are evaluated on smooth families of manifolds/bordisms. Since \(\lambda\) is a topological theory, its values are locally constant. Thus, for example, to a smooth fiber bundle \(Y\to S\) of closed 2-manifolds with relative \((w_1,\,w_2, \,p_1)\)-structure, the theory \(\lambda\) produces a line bundle over \(S\) with a flat covariant derivative. The boundary theory \(F\) returns the pfaffian section \(\mathop{\mathrm{pfaff}}D_{Y/S}\).
Remark 25 (). 20The unitary structures in §[sec:subsec:6463] do not descend to unitary structures on \(F\) and \(\lambda =\widehat{\alpha }\otimes \gamma _{-1/2}\). In fact, \(\lambda\) carries a (rather trivial) unitary structure, being a topological theory of finite order, but it does not lift to the unitary structure on \(\widehat{\alpha } \otimes \gamma _{-1/2}\) given by the Quillen metric and the hermitian structure on the Chern–Simons line. (Compare 4(4).) In this vein, we remark that the conformal anomaly obstructs the descent of the unitary structure on \(\widehat{\alpha }\) along the fibration \(\mathscr{F}_{\mathop{\mathrm{Riem}},w_1,w_2}\to \mathscr{F}_{\mathop{\mathrm{Conf}},w_1,w_2}\).
DSF is supported by the Simons Foundation Award 888988 as part of the Simons Collaboration on Global Categorical Symmetries. This work was performed in part at Aspen Center for Physics, which is supported by National Science Foundation grant PHY-2210452.↩︎
CT is supported by the Simons Foundation Award 824143 as part of the Simons Collaboration on Global Categorical Symmetries.↩︎
In the context [1] of the cobordism hypothesis, framings are most natural, at least at first.↩︎
The symmetric monoidal structures are disjoint union \(\sqcup\) and tensor product \(\otimes\).↩︎
The real image \(\lambda _{{\mathbb{R}}}\in H^4(BG;{\mathbb{R}})\) is equivalent to a symmetric bilinear form on the Lie algebra \(\mathfrak{g}\), and it is the nondegeneracy of this form to which we refer.↩︎
In this context \(p_1\)-structures first appeared in [14]. Atiyah [13] used “2-framings” instead of \(p_1\)-structures. We will also consider framings and stable framings in §[sec:sec:2]. But, as already stated, \(p_1\)-structures are most natural here.↩︎
Physicists call this the “gravitational Chern–Simons invariant”. It is a secondary invariant associated to the first Pontrjagin class of the tangent bundle. We sketch the construction in §[sec:sec:5].↩︎
a (0,1,2,3)-theory↩︎
The classifying map is a contractible choice: the category of smooth manifolds and smooth maps is equivalent to a category whose objects are pairs of a smooth manifold and a choice of classifying map for its tangent bundle.↩︎
One can replace \(*\) with a contractible space \(E\!\relax_3\) on which \(\relax_3\) acts freely.↩︎
If it were, then in the 5-dimensional tangential structure \(\mathscr{X}_5\to B\!\relax_5\) the space \(\mathscr{X}_5\) would be a delooping of the Stiefel manifold \(O_5/O_3\), and this can be proved not to exist using [42].↩︎
\(\raisebox{-.07em}{\rotatebox{9.9} {\tiny {\boldsymbol{/}} }}\mu\raisebox{-0.98ex}{\scalebox{2} {\color{white}\phantom{.}}}\raisebox{+0.88ex} {\color{white}\phantom{.}}_{n}\) is the cyclic group of \(n^{\textrm{th}}\) roots of unity in \({\mathbb{C}}\). \(K(\pi ,q)\) is an Eilenberg–MacLane space with \(\pi _qK(\pi ,q)=\pi\).↩︎
For oriented and Spin \(p_1\)-structures these are changes of the \(p_1\)-structure that fix the underlying orientation or Spin structure.↩︎
In [1] this is called the “topological free fermion theory”.↩︎
Every closed Spin 3-manifold bounds a compact Spin 4-manifold. Similarly, every closed oriented 3-manifold bounds a compact oriented 4-manifold. A 3-framing on a 3-manifold fixes a Spin structure.↩︎
The notation reflects ‘\((w_1,\,p_1)\)-structure’ = ‘oriented \(p_1\)-structure’. The topological group \(\mathop{\mathrm{SO}}^{p_1}_3\) is defined as the loop space of the homotopy fiber of the map \(p_1\colon B\!\mathop{\mathrm{SO}}_3\to K({\mathbb{Z}},4)\). We use notations ‘\((w_1,\,p_1)_3\)’ and ‘\((w_1,\,p_1)_s\)’ to distinguish the rank 3 structure from the stable structure, as in §[sec:subsec:2463].↩︎
This construction is spelled out in a particular cochain model in [52]. In the notation of that reference, if \(\hat{c}=(c,h,\omega )\) is a cocycle in \(\widehat{C}(q)^q(M)\), then \(\mu (\hat{c})=(c,h,\omega )\) is its image in \(\widehat{C}(q-1)^q(M)\). The trivializing cochain of \(\mu (\hat{c})\) has the form \(\tau =(b,k,\phi )\in \widehat{C}(q-1)^{q-1}(M)\), and \(\phi\) is the desired \((q-1)\)-form.↩︎
This structure gives a Levi–Civita covariant derivative on the relative tangent bundle \(T(X/S)\to X\).↩︎
Recall that \(\xi _X= \frac{1}{2}(\eta _X+\dim\ker D_X)\), where \(\eta _X\) is the Atiyah–Patodi–Singer \(\eta\)-invariant. The \((2,3)\)-truncation of \(\widehat{\alpha }\) is constructed in [54].↩︎
We thank Greg Moore for a discussion about unitary structures.↩︎