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Bordism categories and orientations of
gauge theory moduli spaces
Dominic Joyce and Markus Upmeier

This is the second paper of a series that develops a bordism-theoretic point of view on orientations in enumerative geometry. This paper focuses on those applications to gauge theory that can be established purely using formal arguments and calculations from algebraic topology.

We prove that the orientability of moduli spaces of connections in gauge theory for all principal $G$-bundles $P\ra X$ over compact spin $n$-manifolds at once is equivalent to the vanishing of a certain morphism $\Om_n^{\Spin}(\cL BG)\ra\Z_2$ on the $n$-dimensional spin bordism group of the free loop space of the classifying space of $G,$ and we give a complete list of all compact, connected Lie groups $G$ for which this holds. Moreover, we apply bordism techniques to prove that mod-$8$ Floer gradings exist for moduli spaces of $G_2$-instantons for all principal $\SU(2)$-bundles.

We also prove that there are canonical orientations for all principal $\U(m)$-bundles $P\ra X$ over compact spin $8$-manifolds satisfying $c_2(P)-c_1(P)^2=0.$ The proof is based on an interesting relationship to principal $E_8$-bundles. These canonical orientations play an important role in many conjectures about Donaldson–Thomas type invariants on Calabi–Yau $4$-folds, and resolve an apparent paradox in these conjectures.

§ INTRODUCTION

Coherent orientations of moduli spaces play an important role in gauge theory and for enumerative geometry [30, 31, 32, 41, 72]. Despite their central role, the absence of a general framework means that orientations often remain poorly understood. Indeed, the motivating question for this work was whether one can  construct canonical orientations for Donaldson–Thomas type invariants on Calabi–Yau $4$-folds [32]; we will find in [45] that the answer is always affirmative.

This is the second in a series of papers [71], [45] which establishes a general framework for studying orientations based on bordism categories: bordism theory enters because the first paper [71] by the second author establishes a fundamental new technique that formalizes the idea that orientations can be `propagated' along bordisms; we recall this result as Theorem <ref> below.

In this paper, we develop the general framework, explore these new ideas, and give those applications to gauge theory that can be deduced purely by means of algebraic topology. Thus, Theorem <ref> expresses the orientability of moduli spaces $\B_P$ (see <ref> for notation) for  all compact, simply-connected spin Riemannian $n$-manifolds $(X,g)$ and all principal $G$-bundles $P\ra X$ at once in terms of a $\Z_2$-valued morphism on the bordism group of $\cL BG,$ the free loop space of the classifying space of the Lie group $G.$ By combining this result with Theorem <ref> concerning certain Lie group morphisms of `complex type' and previous results by the authors <cit.> and by Cao–Gross–Joyce <cit.>, we establish in Theorems <ref> and <ref> below a complete classification for which compact, connected Lie groups $G$ the moduli space $\B_P$ is orientable for all principal $G$-bundles over simply-connected spin Riemannian 7- and 8-manifolds.

Another application of the general framework is to establish the existence of mod-$8$ (or mod-$6$) Floer gradings on the moduli spaces $\B_P$ for principal $\SU(2)$ (or $\SU(3)$) bundles $P\ra X$ over compact spin Riemannian $7$-manifolds, see Theorem <ref> and <ref> for the terminology. Floer gradings refer to the spectral geometry of the differential operators that govern the deformation theory; the role of our bordism-theoretic calculations is to reduce the question to a short list of basic examples. Such Floer gradings would be important if one wanted to define instanton Floer homology groups of $G_2$-manifolds using $G_2$-instantons and $\Spin(7)$-instantons, by analogy with instanton Floer homology groups of 3-manifolds, as in Donaldson [29].

It is perhaps not too surprising that orientability results can be deduced using algebraic topology alone; after all, the orientability of a moduli space can be expressed as the vanishing of a certain families index in real $K$-theory, see <cit.> (these indices, however, were not computable up to this point), and by the Atiyah–Singer Index Theorem [4] such indices are always topological.

It is more surprising that in Theorem <ref> we can in some cases construct canonical orientations using only `formal techniques' from algebraic topology and the results of [71]. These orientations will be induced from the exceptional Lie group $G=E_8$ as the gauge group and they depend on hardly any data. As discussed in <ref>, canonical orientations of this kind are a crucial prerequisite for numerous conjectures about DT4 invariants of Calabi–Yau $4$-folds.

The sequel [45] will deal with canonical orientations in the more difficult case when they depend on additional structure. While we study orientations in gauge theory here, the next paper dually studies orientations in calibrated geometry. This, combined with the bordism-theoretic point of view, naturally leads to a general concept of flag structures, originally invented by the first author in the special case of associative $3$-folds in $G_2$-manifolds [41]. We will show that flag structures can be used for constructing canonical orientations in gauge theory, in the general case, and that indeed orientations in gauge theory and calibrated geometry are essentially equivalent.

This research was partly funded by a Simons Collaboration Grant on `Special Holonomy in Geometry, Analysis and Physics'.

§ BACKGROUND MATERIAL

§.§ Connection moduli spaces 𝒜ₚ,ℬₚ and orientations

The following definitions are taken from Joyce, Tanaka and Upmeier <cit.>.

Suppose we are given the following data:

(a) A compact, connected manifold $X$ of dimension $n>0$.

(b) A Lie group $G$, with $\dim G>0$, and centre $Z(G)\subseteq G$, and Lie algebra $\g$.

(c) A principal $G$-bundle $\pi:P\ra X$. We write $\Ad(P)\ra X$ for the vector bundle with fibre $\g$ defined by $\Ad(P)=(P\t\g)/G$, where $G$ acts on $P$ by the principal bundle action, and on $\g$ by the adjoint action.

Write $\A_P$ for the set of connections $\nabla_P$ on the principal bundle $P\ra X$. This is a real affine space modelled on the infinite-dimensional vector space $\Ga^\iy(\Ad(P)\ot T^*X)$, and we make $\A_P$ into a topological space using the $C^\iy$ topology on $\Ga^\iy(\Ad(P)\ot T^*X)$. Here if $E\ra X$ is a vector bundle then $\Ga^\iy(E)$ denotes the vector space of smooth sections of $E$. Note that $\A_P$ is contractible.

Write $\G_P=\Aut(P)$ for the infinite-dimensional Lie group of $G$-equivariant diffeomorphisms $\ga:P\ra P$ with $\pi\ci\ga=\pi$. Then $\G_P$ acts on $\A_P$ by gauge transformations, and the action is continuous for the topology on $\A_P$.

There is an inclusion $Z(G)\hookra\G_P$ mapping $z\in Z(G)$ to the principal bundle action of $z$ on $P$. This maps $Z(G)$ into the centre $Z(\G_P)$ of $\G_P$, so we may take the quotient group $\G_P/Z(G)$. The action of $Z(G)\subset\G_P$ on $\A_P$ is trivial, so the $\G_P$-action on $\A_P$ descends to a $\G_P/Z(G)$-action.

Each $\nabla_P\in\A_P$ has a (finite-dimensional) stabilizer group $\Stab_{\G_P}(\nabla_P)\subset\G_P$ under the $\G_P$-action on $\A_P$, with $Z(G)\subseteq\Stab_{\G_P}(\nabla_P)$. As $X$ is connected, $\Stab_{\G_P}(\nabla_P)$ is isomorphic to a closed Lie subgroup $H$ of $G$ with $Z(G)\subseteq H$. As in <cit.> we call $\nabla_P$ irreducible if $\Stab_{\G_P}(\nabla_P)=Z(G)$, and reducible otherwise. Write $\A_P^\irr,\A_P^\red$ for the subsets of irreducible and reducible connections in $\A_P$. Then $\A_P^\irr$ is open and dense in $\A_P$, and $\A_P^\red$ is closed and of infinite codimension in the infinite-dimensional affine space $\A_P$.

We write $\B_P=[\A_P/\G_P]$ for the moduli space of gauge equivalence classes of connections on $P$, considered as a topological stack in the sense of Metzler [54] and Noohi [56, 57]. Write $\B_P^\irr=[\A_P^\irr/\G_P]$ for the substack $\B_P^\irr\subseteq\B_P$ of irreducible connections.

Define variations $\ovB_P=[\A_P/(\G_P/Z(G))]$, $\ovB_P^\irr=[\A_P^\irr/(\G_P/Z(G))]$ of $\B_P,\ab\B_P^\irr$. Then $\ovB_P$ is a topological stack, but as $\G_P/Z(G)$ acts freely on $\A_P^\irr$, we may consider $\ovB_P^\irr$ as a topological space (which is an example of a topological stack). There are natural morphisms $\Pi_P:\B_P\ra\ovB_P$, $\Pi_P^\irr:\B^\irr_P\ra\ovB^\irr_P$.

We define orientation bundles $O^{E_\bu}_P,\bar O^{E_\bu}_P$ on the moduli spaces $\B_P,\ovB_P$:

Work in the situation of Definition <ref>, with the same notation. Suppose we are given real vector bundles $E_0,E_1\ra X$, of the same rank $r$, and a linear elliptic partial differential operator $D:\Ga^\iy(E_0)\ra\Ga^\iy(E_1)$, of degree $d$. As a shorthand we write $E_\bu=(E_0,E_1,D)$. With respect to connections $\nabla_{E_0}$ on $E_0\ot\bigot^iT^*X$ for $0\le i<d$, when $e\in\Ga^\iy(E_0)$ we may write
D(e)=∑_i=0^d a_i·∇_E_0^ie,

where $a_i\in \Ga^\iy(E_0^*\ot E_1\ot S^iTX)$ for $i=0,\ldots,d$. The condition that $D$ is elliptic is that $a_d\vert_x\cdot\ot^d\xi:E_0\vert_x\ra E_1\vert_x$ is an isomorphism for all $x\in X$ and $0\ne\xi\in T_x^*X$, and the symbol $\si(D)$ of $D$ is defined using $a_d$.

Let $\nabla_P\in\A_P$. Then $\nabla_P$ induces a connection $\nabla_{\Ad(P)}$ on the vector bundle $\Ad(P)\ra X$. Thus we may form the twisted elliptic operator

D^∇_(P)   :^((P)E_0)^((P)E_1),

D^∇_(P)   :e⟼∑_i=0^d (𝕀_(P)a_i)·∇_(P)E_0^ie,

where $\nabla_{\Ad(P)\ot E_0}^i$ are the connections on $\Ad(P)\ot E_0\ot\bigot^iT^*X$ for $0\le i\le d$ induced by $\nabla_{\Ad(P)}$ and $\nabla_{E_0}$.

Since $D^{\nabla_{\Ad(P)}}$ is a linear elliptic operator on a compact manifold $X$, it has finite-dimensional kernel $\Ker(D^{\nabla_{\Ad(P)}})$ and cokernel $\Coker(D^{\nabla_{\Ad(P)}})$. The determinant $\det(D^{\nabla_{\Ad(P)}})$ is the 1-dimensional real vector space

where if $V$ is a finite-dimensional real vector space then $\det V=\La^{\dim V}V$. Recall that the index is $\ind_P^{E_\bu}=\dim\Ker(D^{\nabla_{\Ad(P)}})-\dim\Coker(D^{\nabla_{\Ad(P)}})\in\Z.$

These operators $D^{\nabla_{\Ad(P)}}$ vary continuously with $\nabla_P\in\A_P$, so they form a family of elliptic operators over the base topological space $\A_P$. Thus as in Atiyah and Singer [4], there is a natural real line bundle $\hat L{}^{E_\bu}_P\ra\A_P$ with fibre $\hat L{}^{E_\bu}_P\vert_{\nabla_P}=\det(D^{\nabla_{\Ad(P)}})$ at each $\nabla_P\in\A_P$. It is equivariant under the actions of $\G_P$ and $\G_P/Z(G)$ on $\A_P$, and so pushes down to real line bundles $L^{E_\bu}_P\ra\B_P$, $\bar L^{E_\bu}_P\ra\ovB_P$ on the topological stacks $\B_P,\ovB_P$, with $L^{E_\bu}_P\cong\Pi_P^*(\bar L_P^{E_\bu})$. We call $L^{E_\bu}_P,\bar L^{E_\bu}_P$ the determinant line bundles of $\B_P,\ovB_P$. The restriction $\bar L^{E_\bu}_P\vert_{\ovB_P^\irr}$ is a topological real line bundle in the usual sense on the topological space $\ovB_P^\irr$.

For a real line bundle $L\ra\A$ we write $O(L)=(L\setminus 0_\A)/(0,\iy)$ for the principal $\Z_2$-bundles of (fibrewise) orientations on $L.$ That is, we take the complement of the zero section of $L$ and quotient by $(0,\iy)$ acting on the fibres by scalar multiplication.

Define orientation bundles $\hat O^{E_\bu}_P=O(\hat L^{E_\bu}_P)\ra\A_P,$ $O^{E_\bu}_P=O(L^{E_\bu}_P)\ra\B_P,$ and $\bar O^{E_\bu}_P=O(\bar L^{E_\bu}_P)\ra\ovB_P$. There is then a canonical isomorphism $O^{E_\bu}_P\cong\Pi_P^*(\bar O_P^{E_\bu})$. The fibres of $O^{E_\bu}_P\ra\B_P$, $\bar O^{E_\bu}_P\ra\ovB_P$ are orientations on the real line fibres of $L^{E_\bu}_P\ra\B_P$, $\bar L^{E_\bu}_P\ra\ovB_P$. The restriction $\bar O^{E_\bu}_P\vert_{\ovB^\irr_P}$ is a principal $\Z_2$-bundle on the topological space $\ovB^\irr_P$, in the usual sense.

We say that $\B_P$ is orientable if $O^{E_\bu}_P$ is isomorphic to the trivial principal $\Z_2$-bundle $\B_P\t\Z_2\ra\B_P$. An orientation $\om$ on $\B_P$ is an isomorphism $\om:O^{E_\bu}_P\,{\buildrel\cong\over\longra}\,\B_P\t\Z_2$ of principal $\Z_2$-bundles. We make the same definitions for $\ovB_P$ and $\bar O^{E_\bu}_P$. Since $\Pi_P:\B_P\ra\ovB_P$ is a fibration with fibre $[*/Z(G)]$, which is connected and simply-connected, and $O^{E_\bu}_P\cong\Pi_P^*(\bar O_P^{E_\bu})$, we see that $\B_P$ is orientable if and only if $\ovB_P$ is, and orientations of $\B_P$ and $\ovB_P$ correspond. As $\B_P$ is connected, if $\B_P$ is orientable it has exactly two orientations.

Here is a variation on Definition <ref>, for skew-adjoint elliptic operators $E_\bu$.

Continue in the situation of Definition <ref>, and suppose that $E_0=E_1$ and $E_\bu$ is
formally skew-adjoint, $D^*=-D,$ with respect to some choice of metrics on $E_0=E_1$ and $X$. Then $D^{\nabla_{\Ad(P)}}$ is also skew-adjoint, so $\Ker(D^{\nabla_{\Ad(P)}})\cong\Coker(D^{\nabla_{\Ad(P)}})$, and we see from bc2eq3 that $\det(D^{\nabla_{\Ad(P)}})$ and $\hat O^{E_\bu}_P,O^{E_\bu}_P,\bar O^{E_\bu}_P$ are canonically trivial, so the orientation problem is boring.

However, as in Freed <cit.>, we can define the Pfaffian line bundle $\hat\Pf{}^{E_\bu}_P\ra\A_P$ to be the $\Z_2$-graded real line bundle with fibres

\begin{equation*}
\hat\Pf{}^{E_\bu}_P\big\vert_{\na_P}=\det\Ker\bigl(D^{\na_{\Ad(P)}}\bigr)
\end{equation*}

placed in degree of the skew index

\begin{equation*}
\skewind_P^{E_\bu}=\dim\Ker\bigl(D^{\na_{\Ad(P)}}\bigr)\pmod{2}.
\end{equation*}

The Pfaffian line bundle is a kind of square root of $\det(D^{\nabla_{\Ad(P)}})$, defined for skew-adjoint elliptic operators, and need not be trivial. It is equivariant under the actions of $\G_P$ and $\G_P/Z(G)$ on $\A_P$, and so pushes down to real line bundles $\Pf{}^{E_\bu}_P\ra\B_P$, $\bar\Pf{}^{E_\bu}_P\ra\ovB_P$ on the topological stacks $\B_P,\ovB_P$, with $\Pf{}^{E_\bu}_P\cong\Pi_P^*(\bar\Pf{}_P^{E_\bu})$. We call $\Pf{}^{E_\bu}_P,\bar\Pf{}^{E_\bu}_P$ the Pfaffian line bundles of $\B_P,\ovB_P$.

Define Pfaffian orientation bundles $\hat O^{E_\bu}_{\Pf,P}=O(\hat\Pf{}^{E_\bu}_P)\ra\A_P,$ $O^{E_\bu}_{\Pf,P}=O(\Pf{}^{E_\bu}_P)\ra\B_P,$ and $\bar O^{E_\bu}_{\Pf,P}=O(\bar\Pf{}^{E_\bu}_P)\ra\ovB_P$. We say that $\B_P$ is Pfaffian orientable if $O^{E_\bu}_{\Pf,P}$ is isomorphic to the trivial principal $\Z_2$-bundle $\B_P\t\Z_2\ra\B_P$. A Pfaffian orientation $\om$ on $\B_P$ is an isomorphism $\om:O^{E_\bu}_{\Pf,P}\,{\buildrel\cong\over\longra}\,\B_P\t\Z_2$ of principal $\Z_2$-bundles. As $\B_P$ is connected, if $\B_P$ is Pfaffian orientable it has exactly two Pfaffian orientations.

A more general, bordism-theoretic point of view on orientation problems will be developed in Sections <ref> and <ref> below.

(i) Up to continuous isotopy, and hence up to isomorphism, $L^{E_\bu}_P,O^{E_\bu}_P$ in Definition <ref> depend on the elliptic operator $D:\Ga^\iy(E_0)\ra\Ga^\iy(E_1)$ up to continuous deformation amongst elliptic operators, and thus only on the symbol $\si(D)$ of $D$ (essentially, the highest order coefficients $a_d$ in bc2eq1), up to deformation.

(ii) For orienting moduli spaces of `instantons' in gauge theory, as in <ref>–<ref>, we usually start not with an elliptic operator on $X$, but with an elliptic complex
@C=28pt 0 [r]     ^(E_0) [r]^D_0     ^(E_1) [r]^(0.55)D_1     ⋯[r]^(0.4)D_k-1     ^(E_k) [r]     0.

If $k>1$ and $\nabla_P$ is an arbitrary connection on a principal $G$-bundle $P\ra X$ then twisting bc2eq4 by $(\Ad(P),\nabla_{\Ad(P)})$ as in bc2eq2 may not yield a complex (that is, we may have $D^{\nabla_{\Ad(P)}}_{i+1}\ci D^{\nabla_{\Ad(P)}}_i\ne 0$), so the definition of $\det(D_\bu^{\nabla_{\Ad(P)}})$ does not work, though it does work if $\nabla_P$ satisfies the appropriate instanton-type curvature condition. To get round this, we choose metrics on $X$ and the $E_i$, so that we can take adjoints $D_i^*$, and replace bc2eq4 by the elliptic operator
@C=90pt ^(_0≤i≤k/2E_2i) [r]^(0.48)∑_i(D_2i+D_2i-1^*)     ^(_0≤i< k/2E_2i+1),

and then Definitions <ref>–<ref> work with bc2eq5 in place of $E_\bu$.

Let $\io:G\ra H$ be a morphism of Lie groups, with induced Lie algebra morphism $\io_*:\g\ra\h$. We say that $\io:G\ra H$ is of complex type if $\io_*:\g\ra\h$ is injective, and the quotient $G$-representation $\m=\h/\io_*(\g)$ is of complex type, that is, the real vector space $\m$ may be made into a complex vector space such that the action of $G$ on $\m$ is complex linear.

The next theorem collects results from Joyce–Tanaka–Upmeier <cit.>, except (d), which is easily deduced from <cit.>.

Let $X$ be a compact $n$-manifold and $E_\bu=(E_0,E_1,D)$ a linear elliptic operator on $X,$ and choose an orientation for $\det D$. In the following we consider principal $G$-bundles $P\ra X$ for Lie groups $G,$ including the trivial principal $G$-bundle $P=X\t G,$ the associated connection moduli space $\B_P,$ and its orientation bundle $\pi:O^{E_\bu}_P\ra\B_P$.

(a) If $G$ is abelian e.g. $G=\U(1)^k$ then $\B_P$ is orientable for any principal $G$-bundle $P\ra X,$ and has a canonical orientation, which depends on the orientation for $\det D$.
(b) If $G_1,G_2$ are Lie groups then any principal $G_1\t G_2$-bundle $P\ra X$ is canonically isomorphic to a fibre product $P_1\t_XP_2,$ where $P_i\ra X$ is a principal $G_i$-bundle for $i=1,2$ given by $P_i=P/G_{3-i}$. There is a canonical isomorphism $\B_P\cong\B_{P_1}\t\B_{P_2}$ which identifies $O^{E_\bu}_P\cong O^{E_\bu}_{P_1}\bt O^{E_\bu}_{P_2}$. Thus, $\B_P$ is orientable if and only if $\B_{P_1},\B_{P_2}$ are orientable, and orientations on $\B_{P_1},\B_{P_2}$ induce an orientation on $\B_P$.

(c) Let $\io:G\ra H$ be a morphism of Lie groups, and $P\ra X$ be a principal $G$-bundle. Then $Q=(P\t H)/G$ is a principal $H$-bundle, where $G$ acts on $P$ by the principal $G$-bundle action, and on $H$ by $g:h\mapsto h\cdot\io(g)^{-1}$. Any connection $\nabla_P$ on $P$ induces a connection $\nabla_Q$ on $Q,$ and mapping $[\nabla_P]\mapsto[\nabla_Q]$ induces a morphism of topological stacks $\Up_P^Q:\B_P\ra\B_Q$.

Now suppose that $\io:G\ra H$ is of complex type, as in Definition <ref>. Then there is a canonical isomorphism $\up_P^Q:O_P^{E_\bu}\ra(\Up_P^Q)^*(O_Q^{E_\bu})$. Thus, if $\B_Q$ is orientable then $\B_P$ is orientable, and an orientation on $\B_Q$ induces one on $\B_P$.

(d) Suppose $\io:G\t\U(1)^k\ra H$ is a morphism of connected Lie groups of complex type for $k\ge 0,$ and write $\jmath:=\io\vert_{G\t\{1\}}:G=G\t\{1\}\ra H$. Let $P\ra X$ be a principal $G$-bundle, so that $P\t\U(1)^k\ra X$ is a principal $G\t\U(1)^k$-bundle, and set $Q=(P\t H)/G=(P\t\U(1)^k\t H)/(G\t\U(1)^k),$ so that $Q$ is the principal $H$-bundle induced from both $P,\jmath:G\ra H$ and $P\t\U(1)^k,\io$. Then we have a commutative diagram

\begin{equation*}
\xymatrix@C=50pt@R=11pt{
\B_P \ar[rr]_{\Up_P^Q} \ar[dr]_(0.4){(\id_{\B_P},\nabla_0)} && \B_Q, \\
& \B_P\t\B_{X\t\U(1)^k}\cong\B_{P\t\U(1)^k} \ar[ur]_(0.6){\Up_{P\t\U(1)^k}^Q} }
\end{equation*}

and combining (a)–(c) gives an isomorphism $\ul{\up}_P^Q:O_P^{E_\bu}\ra(\Up_P^Q)^*(O_Q^{E_\bu})$.

Recall that a continuous map $f:S\ra T$ between connected topological spaces $S,T$ is called $p$-connected if the induced maps of homotopy groups $\pi_i(f):\pi_i(S)\ra\pi_i(T)$ are isomorphisms for $i<p$ and surjective for $i=p$. Suppose $\jmath:G\ra H$ is $p$-connected for some $p>n$. Then $\Up_P^Q:\B_P\ra\B_Q$ induces an isomorphism $\pi_1(\Up_P^Q):\pi_1(\B_P)\ra\pi_1(\B_Q)$. Since orientability of $\B_P$ depends on morphisms $\pi_1(\B_P)\ra\Z_2,$ it follows that $\B_P$ is orientable if and only if $\B_Q$ is orientable, and choices of orientations for $\B_P$ and $\B_Q$ are equivalent.

If $E_\bu$ is skew-adjoint, the analogues hold with Pfaffian orientations.

To apply Theorem <ref> it will be helpful to have a list of Lie group morphisms $\io:G\ra H$ of complex type, and to know when the conditions of Theorem <ref>(d) are satisfied. The next theorem will be proved in <ref>.

(a) Here is a list of Lie group morphisms $\io:G\ra H$ of complex type, as in Definition <ref>, for all $m\ge 1$:
E_7(1)   E_8,     E_6(1)^2   E_8,     (14)(1)   E_8,

(8)(1)   E_8,       (3)(1)   F_4,     (7)(1)   F_4,

G_2   (8),     (m)   (m+1),     (m)   (m).

(b) Here is a list of Lie group morphisms $\io:G\t\U(1)^k\ra H$ of complex type, where the $\U(1)^k$ factor is written as the final factor in the domain, such that $\jmath:=\io\vert_{G\t\{1\}}:G=G\t\{1\}\ra H$ is $p$-connected for the specified $p$:

(m)(1)   (m+1)     (p   =2m),

(m)(1)   (m+1)     (p   =4m+2),

(m)(1)   (m+2)     (p   =m-1),

(m)(1)   (m+2)     (p   =m-1).

Here we do not specify the actual morphisms $\io$, although these are implicit in the proof, as we will not need them later. In (b), when we say $\jmath$ is $p$-connected, this may not be the maximum such $p$. To prove Theorem <ref>, we will show that:

(i) Suppose a Lie group $H$ has a torus subgroup $T\subseteq H$, and write $G=Z(T)$ for the centralizer of $T$. Then $\inc:G\hookra H$ is of complex type.

(ii) Let $\io:G\ra H$ be a morphism of connected Lie groups which is a covering map, e.g. $\Spin(n)\,{\buildrel 2:1\over\longra}\,\SO(n)$. Then $\io$ is of complex type.

(iii) Compositions of complex type morphisms are of complex type.

(iv) Suppose $\inc:G\hookra H$ is an inclusion of a Lie subgroup, and $Y=H/G$ has $\pi_i(Y)=0$ for $i\le p$. Then $\inc$ is $p$-connected.

Using these we can easily construct many examples of complex type morphisms.

§.§ G₂-instantons on G₂-manifolds

Part (a) of the next theorem follows from Walpuski <cit.> and <cit.>, and part (b) is proved by Joyce–Upmeier <cit.>.

Let $X$ be a compact, oriented, spin Riemannian $7$-manifold, and $E_\bu$ be the Dirac operator $\slashed{D}:\Ga^\iy(S)\ra\Ga^\iy(S)$ on $X$ in Definition <ref>. Suppose $P\ra X$ is a principal $G$-bundle for $G=\U(m)$ or $\SU(m)$. Then:

(a) $\B_P$ and $\ovB_P$ are orientable, that is, $O_P^{E_\bu}\ra\B_P$ and $\bar O^{E_\bu}_P\ra\ovB_P$ are trivializable principal $\Z_2$-bundles.

(b) An orientation on $\det\slashed{D}$ and a flag structure on $X,$ as in Joyce <cit.>, determine canonical trivializations of $O_P^{E_\bu},\bar O^{E_\bu}_P$ for all $P$. Here flag structures are an algebro-topological structure on $7$-manifolds $X$, related to `linking numbers' of disjoint homologous $3$-submanifolds $Y_1,Y_2\subset X$.

Theorem <ref> is related to a 7-dimensional gauge theory discussed by Donaldson and Thomas [32] and Donaldson and Segal [31]. Suppose $X$ is a compact 7-manifold and $(\vp,g)$ a $G_2$-structure on $X$ in the sense of <cit.> which is coclosed (i.e. $\d(*\vp)=0$). Let $G$ be a Lie group, and $P\ra X$ a principal $G$-bundle. A $G_2$-instanton on $P$ is a connection $\nabla_P$ on $P$ with $F^{\nabla_P}\w*\vp=0$ in $\Ga^\iy(\Ad(P)\ot\La^6T^*X)$.

Write $\M_P^{G_2}$ for the moduli space of irreducible $G_2$-instantons on $P$, regarded as a subspace of $\ovB_P^\irr\subset\ovB_P$. As $\d(*\vp)=0$, the deformation theory of $\M_P^{G_2}$ is controlled by an elliptic complex. Then $\M_P^{G_2}$ is a derived manifold of virtual dimension 0. If $\vp$ is generic in its cohomology class, $\M_P^{G_2}$ is an ordinary 0-manifold. Examples and constructions of $G_2$-instantons are given in [53, 60, 61, 73, 74, 75]. As in <cit.>, the orientation bundle of $\M_P^{G_2}$ is the restriction to $\M_P^{G_2}$ of $\bar O^{E_\bu}_P\ra\ovB_P$, for $E_\bu$ the Dirac operator of the spin structure on $X$ induced by $(\vp,g)$, so we may orient $\M_P^{G_2}$ by restricting orientations on $\bar O^{E_\bu}_P\ra\ovB_P$. Thus Theorem <ref> implies <cit.>:

Let $X$ be a compact $7$-manifold and $(\vp,g)$ a coclosed $G_2$-structure on $X,$ and fix an orientation on $\det\slashed{D}$ and a flag structure on $X$. Then for any principal $G$-bundle $P\ra X$ for $G=\U(m)$ or $\SU(m),$ we can construct a canonical orientation on $\M_P^{G_2}$.

Donaldson and Segal [31] propose defining enumerative invariants of $(X,\vp,g)$ by counting $\M_P^{G_2}$, with signs, and adding correction terms from associative 3-folds in $X$. To determine the signs we need an orientation of $\M_P^{G_2}$. Thus, Corollary <ref> contributes to the Donaldson–Segal programme.

§.§ Spin(7)-instantons on Spin(7)-manifolds

Here is Cao–Gross–Joyce <cit.>, an analogue of Theorem <ref>(a).

Let $X$ be a compact, oriented, spin Riemannian $8$-manifold, and $E_\bu$ be the positive Dirac operator $\slashed{D}_+:\Ga^\iy(S_+)\ra\Ga^\iy(S_-)$ on $X$ in Definition <ref>. Suppose $P\ra X$ is a principal $G$-bundle for $G=\U(m)$ or $\SU(m)$. Then $\B_P$ and $\ovB_P$ are orientable, that is, $O_P^{E_\bu}\ra\B_P$ and $\bar O^{E_\bu}_P\ra\ovB_P$ are trivializable principal $\Z_2$-bundles.

This begs the question of whether there is an 8-dimensional analogue of Theorem <ref>(b), which we will answer in this paper and the sequel [45].

Again, Theorem <ref> is related to an 8-dimensional gauge theory discussed by Donaldson and Thomas [32]. Let $X$ be a compact 8-manifold and $(\Om,g)$ a $\Spin(7)$-structure on $X$ in the sense of <cit.>, which need not have $\d\Om=0$. Then there is a natural splitting $\La^2T^*X=\La^2_7T^*X\op\La^2_{21}T^*X$ into vector subbundles of ranks 7 and 21. Suppose $G$ is a Lie group and $P\ra X$ a principal $G$-bundle. A $\Spin(7)$-instanton on $P$ is a connection $\nabla_P$ on $P$ with $\pi^2_7(F^{\nabla_P})=0$ in $\Ga^\iy(\Ad(P)\ot\La^2_7T^*X)$. Write $\M_P^{\Spin(7)}$ for the moduli space of irreducible $\Spin(7)$-instantons on $P$, regarded as a subspace of $\ovB_P^\irr\subset\ovB_P$. Then $\M_P^{\Spin(7)}$ is a derived manifold in the sense of [38, 39, 40, 42], and an ordinary manifold if $\Om$ is generic (amongst non-closed 4-forms). Examples of $\Spin(7)$-instantons were given by Lewis [48], Tanaka [67], and Walpuski [76].

As in <cit.>, the orientation bundle of $\M_P^{\Spin(7)}$ is the restriction to $\M_P^{\Spin(7)}$ of $\bar O^{E_\bu}_P\ra\ovB_P$, for $E_\bu$ the positive Dirac operator $\slashed{D}_+$ of the spin structure on $X$ induced by $(\Om,g)$, so we may orient $\M_P^{\Spin(7)}$ by restricting orientations on $\bar O^{E_\bu}_P\ra\ovB_P$. Thus Theorem <ref> implies <cit.>:

Let $X$ be a compact $8$-manifold with $\Spin(7)$-structure $(\Om,g)$. Then $\M_P^{\Spin(7)}$ is orientable for any principal $\U(m)$- or $\SU(m)$-bundle $P\ra X$.

§.§ DT4 invariants of Calabi-Yau 4-folds

Suppose $X$ is a Calabi–Yau 4-fold, and write $\M$ and $\bcM$ for the classical and derived moduli stacks of objects in $\coh(X)$, with inclusion $i:\M\hookra\bcM$. Then $\bcM$ has a $-2$-shifted symplectic structure in the sense of Pantev–Toën–Vaquié–Vezzosi [59]. Also $\bL_i:i^*(\bL_{\bcM})\ra\bL_\M$ is a 4-Calabi–Yau obstruction theory on $\M$, a classical truncation of the $-2$-shifted symplectic structure on $\bcM$.

Borisov–Joyce [10] defined virtual classes for proper $-2$-shifted symplectic derived $\C$-schemes, using Derived Differential Geometry [38, 39, 40, 42]. More recently, Oh–Thomas [58] gave a new, algebro-geometric definition of 4-Calabi–Yau virtual classes, equivalent to [10], in the style of Behrend–Fantechi [5].

Oh–Thomas [58] define their virtual class $[\M]_\virt$ only when $\M$ is a projective moduli scheme of Gieseker stable sheaves on a Calabi–Yau 4-fold $X$. However, Kiem–Park <cit.> provide an alternative definition which works for $\M$ a proper Deligne–Mumford stack with a 4-Calabi–Yau obstruction theory satisfying an `isotropic cone' condition. Invariants defined by integrating universal cohomology classes over 4-Calabi–Yau virtual classes of moduli spaces of semistable coherent sheaves or complexes on $X$ are known as DT4 invariants.

To define a 4-Calabi–Yau virtual class we need a choice of orientation on $\M$, defined in Borisov–Joyce <cit.>.

Let $\M$ be an Artin or higher $\C$-stack with a 4-Calabi–Yau obstruction theory $\phi:\cF^\bu\ra\bL_\M$, $\th:\cF^\bu\,{\buildrel\sim\over\longra}\,(\cF^\bu)^\vee[2]$. Then we have a determinant line bundle $\det(\cF^\bu)\ra \M$, and $\th$ induces an isomorphism $\det\th:\det\cF^\bu\ra(\det\cF^\bu)^*$. An orientation for $(\M,\phi,\th)$ is a choice of isomorphism $\la:\det\cF^\bu\ra\O_\M$ with $\la^*\ci\la=\det\th$.

Here $\la$ is basically a square root of $\det\th$. Locally on $\M$ in the étale topology there are two choices for $\la$, and there is a principal $\Z_2$-bundle $O_{\cF^\bu}\ra \M$ parametrizing choices of $\la$. We say that $(\M,\phi,\th)$ is orientable if $O_{\cF^\bu}$ is trivializable, and an orientation is a trivialization $O_{\cF^\bu}\cong \M\t\Z_2$.

The next theorem summarizes parts of Cao–Gross–Joyce <cit.>, plus background material from Joyce–Tanaka–Upmeier <cit.>.

Let $X$ be a projective Calabi–Yau $4$-fold.

(a) Write $\M$ for the moduli stack of objects $G^\bu$ in $D^b\coh(X),$ a higher stack. It has a decomposition $\M=\coprod_{\al\in K^0_\top(X)}\M_\al,$ where $\M_\al$ is the substack of complexes $G^\bu$ with class $\lb G^\bu\rb=\al$ in the topological K-theory of the underlying $8$-manifold of $X$. There is a natural $4$-Calabi–Yau obstruction theory $\phi:\cF^\bu\ra\bL_\M$, $\th:\cF^\bu\,{\buildrel\sim\over\longra}\,(\cF^\bu)^\vee[2]$ on $\M,$ and hence a principal $\Z_2$-bundle $O^{\cF^\bu}\ra\M$ of orientations on $\M$ as in Definition <ref>, restricting to $O^{\cF^\bu}_\al\ra\M_\al$.

Write $\M^\top$ for the topological realization of $\M,$ a topological space natural up to homotopy equivalence, as in Simpson [63], Blanc <cit.>, and <cit.>. Then $O_{\cF^\bu}$ lifts to a principal $\Z_2$-bundle $O^{\cF^\bu,\top}\ra\M^\top,$ restricting to $O^{\cF^\bu,\top}_\al\ra\M^\top_\al,$ such that trivializations of $O^{\cF^\bu}_\al$ and $O^{\cF^\bu,\top}_\al$ are naturally in 1-1 correspondence.

(b) Write $\cC=\Map_{C^0}(X, B\U\t\Z),$ where $B\U=\varinjlim_{n\ra\iy}B\U(n)$ is the unitary classifying space. It has a natural decomposition $\cC=\coprod_{\al\in K^0_\top(X)}\cC_\al,$ where $\cC_\al$ is connected. Taking the elliptic operator $E_\bu\ra X$ to be the positive Dirac operator $\slashed{D}_+$ of the spin structure on $X$ induced by the Calabi–Yau $4$-fold structure, which for a Calabi–Yau $4$-fold $X$ may be written

\begin{equation*}
\slashed{D}_+=\db+\db^*:\Ga^\iy(\La^{0,{\rm even}}T^*X)\longra \Ga^\iy(\La^{0,{\rm odd}}T^*X),
\end{equation*}

in <cit.> we construct a principal $\Z_2$-bundle $O_\cC\ra\cC,$ restricting to $O_{\cC_\al}\ra\cC_\al$. It is thought of as a bundle of orientations on $\cC,$ and is obtained from the bundles $O_P^{E_\bu}\ra\B_P$ in <ref> for $\U(m)$-bundles $P\ra X$ in a limiting process as $m\ra\iy$.

From the definition of $O_{\cC_\al},$ if $k\in\N$ then and $\Xi_{\al,k}:\cC_\al\ra\cC_{\al+k\lb\cO_X\rb}$ is the homotopy equivalence induced by direct sum with the trivial vector bundle $\bigop^{k}\cO_X\ra X,$ then there is a canonical isomorphism $O_{\cC_\al}\cong \Xi_{\al,k}^*(O_{\cC_{\al+k\lb\cO_X\rb}})$.

Actually, for a general spin $8$-manifold, $O_{\cC_\al}$ and $\Xi_{\al,k}^*(O_{\cC_{\al+k\lb\cO_X\rb}})$ differ by the $\Z_2$-torsor $\Or(\det\slashed{D}_+)^{\ot^k},$ so in general we should restrict to $k$ even. But as $X$ is a Calabi–Yau $4$-fold there is a canonical isomorphism $\Or(\det\slashed{D}_+)\cong\Z_2$.

(c) We relate (a),(b) as follows: using the classifying morphism of the universal complex $\cU^\bu\ra X\t\M,$ as in <cit.> we can define a continuous map $\Phi:\M^\top\ra\cC,$ natural up to homotopy, restricting to $\Phi_\al:\M_\al^\top\ra\cC_\al$ for $\al\in K^0_\top(X)$. Then there are natural isomorphisms $O^{\cF^\bu,\top}_\al\cong \Phi^*(O_{\cC_\al})$ of principal $\Z_2$-bundles on $\M^\top_\al$. Hence, a trivialization of $O_{\cC_\al}$ induces trivializations of $O^{\cF^\bu,\top}_\al$ and $O^{\cF^\bu}_\al$.

(d) Let $P\ra X$ be a principal $\U(m)$-bundle, and $O_P^{E_\bu}\ra\B_P$ be as in <ref> and <ref> for $E_\bu\ra X$ the positive Dirac operator $\slashed{D}_+$ of the spin structure on $X$ induced by the Calabi–Yau $4$-fold structure. Write $\be=\lb P\rb\in K^0_\top(X)$.

Write $\B_P^\top$ for the topological realization of the topological stack $\B_P,$ a topological space natural up to homotopy equivalence. Then $O_P^{E_\bu}$ lifts to a principal $\Z_2$-bundle $O_P^{E_\bu,\top}\ra\B_P^\top,$ such that trivializations of $O_P^{E_\bu}$ and $O_P^{E_\bu,\top}$ are naturally in 1-1 correspondence.

(e) We relate (b),(d) as follows: using the universal principal $\U(m)$-bundle $U_P\ra X\t\B_P$ we can define a continuous map $\Psi_\be:\B_P^\top\ra\cC_\be,$ natural up to homotopy. Then the construction of $O_{\cC_\be}$ implies that there is a natural isomorphism $O_P^{E_\bu,\top}\cong \Psi_\be^*(O_{\cC_\be})$ of principal $\Z_2$-bundles on $\B_P^\top$. Hence, a trivialization of $O_{\cC_\be}$ induces trivializations of $O_P^{E_\bu}$ and $O_P^{E_\bu,\top}$.

(f) In (d),(e), suppose $m\ge 5$. Then $\Psi_\be:\B_P^\top\ra\cC_\be$ induces isomorphisms $\pi_i(\B_P^\top)\ra\pi_i(\cC_\be)$ for $i=0,1$. Therefore (e) induces a 1-1 correspondence between trivializations of $O_{\cC_\al},$ $O_P^{E_\bu},$ and $O_P^{E_\bu,\top},$ so in particular, a trivialization of $O_P^{E_\bu}$ induces a trivialization of $O_{\cC_\be}$.

(g) Let $\al\in K^0_\top(X)$  and set $k=\max(5-\rank\al,0),$ $m=\min(5,\rank\al),$ and $\be=\al+k\lb\cO_X\rb$. Then there exists a principal $\U(m)$-bundle $P\ra X,$ unique up to isomorphism, with $\lb P\rb=\be$ in $K^0_\top(X)$. By (a)–(f), we now see that a trivialization of $O_P^{E_\bu}$ induces trivializations of $O_P^{E_\bu,\top},O_{\cC_\be},O_{\cC_\al},O^{\cF^\bu,\top}_\al,$ and $O^{\cF^\bu}_\al$. That is, an orientation on $\B_P$ induces an orientation on $\M_\al$.

We offer some explanation of Theorem <ref>. For simplicity, let us start with moduli spaces $\M_\al^{\vect,\ss}(\tau)$ of Gieseker stable vector bundles $E\ra X$ in class $\al\in K^0_\top(X)$ with $c_1(\al)=0$, where $\rank\al=r\ge 4$.

By the Hitchin–Kobayashi correspondence, every such $E\ra X$ admits a natural Hermitian–Einstein connection $\nabla_E$, and then $(E,\nabla_E)$ is a $\Spin(7)$-instanton. Every $\Spin(7)$-instanton connection on the complex vector bundle $E\ra X$ comes from an algebraic vector bundle structure on $E$ in this way. As $r\ge 4$, every complex vector bundle $E'\ra X$ with $\lb E'\rb=\al$ has $E'\cong E$.

This induces an isomorphism from $\M_\al^{\vect,\ss}(\tau)$ to the moduli space $\M_P^{\Spin(7)}$ of irreducible $\Spin(7)$-instantons on the principal $\U(r)$-bundle $P\ra X$ associated to $E$, and hence an inclusion $\M_\al^{\vect,\ss}(\tau)\hookra\B_P$. Since DT4 orientations on $\M_\al^{\vect,\ss}(\tau)$ are basically orientations of $\Spin(7)$ instanton moduli spaces, as in <ref>, an orientation on $\B_P$ pulls back to a DT4 orientation of $\M_\al^{\vect,\ss}(\tau)$.

Now $\M_\al^{\vect,\ss}(\tau)$ is a finite-dimensional $\C$-scheme, whereas $\B_P$ is an infinite-dimensional topological stack. One might think that $\M_\al^{\vect,\ss}(\tau)$ is a simpler object, but in fact orientations on $\B_P$ are much easier to understand. In examples it is difficult to describe $\M_\al^{\vect,\ss}(\tau)$ explicitly. It could have $N\gg 0$ connected components, so that $\M_\al^{\vect,\ss}(\tau)$ would have $2^N$ orientations, but $\B_P$ is connected and so has only 2 orientations. Thus pulling back orientations from $\B_P$ to $\M_\al^{\vect,\ss}(\tau)$ gives orientations with fewer arbitrary choices.

Theorem <ref> gives orientations not just on moduli spaces of vector bundles $\Vect(X)$, but also of coherent sheaves $\coh(X)$, and complexes in $D^b\coh(X)$. The rough analogue in Differential Geometry of passing from $\Vect(X)$ to $D^b\coh(X)$ is taking the limit $r\ra\iy$, for $r=\rank E$. More precisely, the analogue in Topology is passing from $\coprod_{r\ge 0}\Map_{C^0}(X,B\U(r))$ to $\Map_{C^0}(X,B\U\t\Z)$, where $B\U=\varinjlim_{n\ra\iy}B\U(n)$, and the $\Z$ factor keeps track of the rank $r$.

Combining Theorems <ref> and <ref> yields <cit.>:

Let $X$ be a projective Calabi–Yau $4$-fold. Then the orientation bundle $O^{\cF^\bu}\ra\M$ from Theorem <ref>(a) is trivializable, i.e. $\M$ is orientable.

Corollary <ref> has important applications in the programme of defining and studying `DT4 invariants' of Calabi–Yau 4-folds proposed by Borisov and Joyce [10] and Cao and Leung [19], as orientations are necessary to define DT4 invariants. Theorem <ref> and Corollary <ref> are extended to noncompact Calabi–Yau 4-folds by Bojko [7].

The higher $\C$-stack $\M$ in Theorem <ref> and Corollary <ref> contains as open Artin $\C$-substacks the moduli stacks $\M^{\rm coh},\M^{\rm coh,ss},\M^{\rm vect}$ of coherent sheaves, and semistable coherent sheaves, and algebraic vector bundles on $X$, respectively. The principal $\Z_2$-bundle $O^{\cF^\bu}\ra\M$, and orientations on $\M$, may be restricted to $\M^{\rm coh},\ldots,\M^{\rm vect}$. Thus, Theorem <ref> and Corollary <ref> are still interesting if we only care about $\M^{\rm coh},\ldots,\M^{\rm vect}$ rather than $\M$.

§.§ Results and conjectures on DT4 invariants

There is a growing literature on DT4 invariants of Calabi–Yau 4-folds $X$. One frequent theme, which appears in the papers [8, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27] and we summarize in Conjecture <ref> below, are conjectured relations of the schematic form
Conventional invariants of $X$≃DT4 invariants of $X$,

where by `conventional invariants' of $X$ we mean things like the Euler characteristic and Gromov–Witten invariants, and the relation `$\simeq$' may involve change of variables in a generating function, etc.

For this paper, the thing that interests us about bc2eq8 is that the left hand side is orientation-independent, but the right hand side involves a choice of orientation on the relevant DT4 moduli spaces $\M_\al^\ss(\tau)\subseteq\M_\al$. Thus, for bc2eq8 to make sense, it seems that there should exist canonical orientations on $\M_\al$ for all those $\al$ involved in bc2eq8. Corollary <ref> tells us only that $\M_\al$ is orientable, not that it has a canonical orientation. One of our goals is to construct canonical orientations for all moduli spaces $\M_\al$ with $c_2(\al)-c_1(\al)^2=0$, which will be sufficient for the relations bc2eq8 in [8, 13, 14, 16, 17, 18, 19, 22, 23, 20, 21, 24, 25, 26, 27].

Let $X$ be a projective Calabi–Yau $4$-fold. Then:

(a) Cao–Kool <cit.> propose an explicit generating function for invariants $\int_{\Hilb^n(X)}c_n(L^{[n]})$ for $L\ra X$ a line bundle. See also [8, 24].

(b) Bojko [8] proposes formulae for integrals of Segre classes, Verlinde classes and Nekrasov genera over $\Hilb^n(X)$.

(c) Cao–Maulik–Toda <cit.> relate genus $0$ Gromov–Witten invariants of $X$ and $1$-dimensional DT4 invariants. Cao–Toda <cit.> make a related conjecture. See also [13, 14].

(d) Cao–Maulik–Toda <cit.> relate genus $0,1$ Gromov–Witten invariants of $X$ and Pandharipande–Thomas style DT4 invariants. Cao–Toda <cit.> make a related conjecture. See also [18, 25].

(e) Cao–Kool <cit.> relate genus $0,1$ Gromov–Witten invariants of $X$ and rank $1$ DT4 invariants. See also [19].

(f) For holomorphic symplectic $4$-folds $X,$ Cao–Oberdieck–Toda <cit.> relate reduced genus $0,1,2$ Gromov–Witten invariants of $X$ and reduced DT4 invariants counting $1$-dimensional sheaves, and also <cit.> to reduced Pandharipande–Thomas style DT4 invariants.

Although we state this as a conjecture, we emphasize that the cited papers also contain many theorems. All parts of Conjecture <ref> involve only moduli spaces $\M_\al$ on $X$ with $c_1(\al)=c_2(\al)=0$.

§.§ Background on bordism theory

§.§.§ Tangential structures

To define bordism groups with different flavours, we first define tangential structures. Our treatment is based on Lashof [47] and Stong [65].

Let $B\O=\colim_{n\ra\iy}B\O(n)$ be the classifying space of the stable orthogonal group, the direct limit of the classifying spaces $B\O(n)$ of the orthogonal groups $\O(n)$ under the inclusions $\O(n)\hookra\O(n+1)$. There are natural continuous maps $\io_{B\O(n)}:B\O(n)\ra B\O$ coming from the direct limit.

The inclusions $\O(m)\t\O(n)\ra\O(m+n)$ induce a binary operation $\mu_{B\O}:B\O\t B\O\ra B\O$ which is associative, unital, and commutative up to homotopy. Hence $B\O$ is a commutative H-space.

If $X$ is a smooth $n$-manifold (possibly with boundary or corners) then choosing a Riemannian metric $g$ on $X$ gives $TX$ an $\O(n)$-structure, so we have a classifying map $\phi_{TX}:X\ra B\O(n)$, which we compose with $\io_{B\O(n)}:B\O(n)\ra B\O$ to get a map $\phi^\rst_{TX}:X\ra B\O$ classifying the stable tangent bundle of $X$. Up to contractible choice this is unique and independent of the Riemannian metric $g$ on $X,$ which permits us to fix the choice of $\phi^\rst_{TX}$ below. We have a homotopy commutative diagram

\begin{equation*}
\xymatrix@C=60pt@R=15pt{ X \ar[d]^{\phi_{TX}} \ar[dr]^(0.7){\phi_{TX\op\ul{\R}}} \ar@/^.8pc/[drr]^(0.7){\phi^\rst_{TX}} \\
B\O(n) \ar[r] & B\O(n+1) \ar[r]^{\io_{B\O(n+1)}} & B\O, }
\end{equation*}

where $\ul{\R}\ra X$ is the trivial line bundle. The vector bundle isomorphism $\id_{TX}\op -\id_{\ul\R}:TX\op\ul{\R}\ra TX\op\ul{\R}$ induces a homotopy $-1_{\phi_{TX\op\ul{\R}}}:\phi_{TX\op\ul{\R}}\Ra\phi_{TX\op\ul{\R}}$ whose square is homotopic to the constant homotopy $\Id_{\phi_{TX\op\ul{\R}}}$. We define $-1_{\phi^\rst_{TX}}:\phi^\rst_{TX}\Ra\phi^\rst_{TX}$ to be the horizontal composition of this with $\Id_{\io_{B\O(n)}}$.

If $X$ has boundary or corners then $TX\vert_{\pd X}\cong T(\pd X)\op\ul{\R}$, where $\ul{\R}\ra X$ is the trivial line bundle. Thus we have a homotopy commutative diagram

\begin{equation*}
\xymatrix@C=120pt@R=15pt{
*+[r]{\pd X} \drtwocell_{}\omit^{}\omit{^{}} \ar[r]_(0.35){i_{\pd X}} \ar[d]^{\phi_{T\pd X}}  & *+[l]{X} \ar[d]_{\phi_{TX}} \\
*+[r]{B\O(n-1)} \ar[r] & *+[l]{B\O(n).\!} }
\end{equation*}

Composing with $B\O(n)\ra B\O$ shows that

A tangential structure $\bs B=(B,\be)$ is a topological space $B$ and a continuous map $\be:B\ra B\O$. We say that $\bs B$ has products if we are given a continuous map $\mu_{\bs B}:B\t B\ra B$, which is homotopy commutative and associative, in a homotopy commutative diagram

\begin{equation*}
\xymatrix@C=120pt@R=15pt{
*+[r]{B\t B} \ar[r]_(0.35){\mu_{\bs B}} \ar[d]^{\be\t\be} \drtwocell_{}\omit^{}\omit{^{\eta_{\bs B}\,\,\,\,}} & *+[l]{B} \ar[d]_{\be} \\
*+[r]{B\O\t B\O} \ar[r]^(0.65){\mu_{B\O}} & *+[l]{B\O.\!} }
\end{equation*}

A $\bs B$-structure $\bs\ga_X=(\ga_X,\eta_X)$ on a smooth manifold $X$ (possibly with boundary or corners) is a homotopy commutative diagram of continuous maps

_^_    η_X     B [dr]^

X [ur]^_X [rr]^(0.35)ϕ_X^        BØ.

An isomorphism of tangential structures $\bs\ga_X=(\ga_X,\eta_X)$ and $\bs\ga_X'=(\ga_X',\eta_X')$ is represented by a homotopy $\eta:\ga_X\Ra\ga_X'$ such that the diagram

\begin{equation*}
\begin{tikzcd}
\be\circ\ga_X\arrow[rd,Rightarrow,"\eta_X"]\arrow[rr,Rightarrow,"\Id_\be\circ\eta"] && \be\circ\ga_X'\arrow[ld,Rightarrow,"\eta_X'"]\\
\end{tikzcd}
\end{equation*}

commutes up to homotopy (of homotopies). Here we only care about $\eta$ up to homotopy and often we will only care about isomorphism classes of $\bs B$-structures.

The opposite $\bs B$-structure $-\bs\ga_X$ is obtained by composing homotopies across the diagram

\begin{equation*}
\xymatrix@C=70pt@R=13pt{
\drrtwocell_{}\omit^{}\omit{_{\,\,\,\,\eta_X}} & B \ar[dr]^\be  \\
X \drrtwocell_{}\omit^{}\omit{_{\qquad -1_{\phi^\rst_{TX}}}} \ar@/_2pc/[rr]_(0.15){\phi_X^\rst} \ar[ur]^{\ga_X} \ar[rr]_(0.3){\phi_X^\rst} && B\O. \\
&& }
\end{equation*}

Often we just write a manifold with $\bs B$-structure $(X,\bs\ga_X)$ as $X$, omitting $\bs\ga_X$ from the notation. In this case we write $-X$ as a shorthand for $(X,-\bs\ga_X)$, that is, $X$ with the opposite $\bs B$-structure.

From bc2eq9 we see that if $X$ has boundary or corners then composing bc2eq10 with $i_X:\pd X\ra X$ gives a restriction $\bs\ga_X\vert_{\pd X}$ which is a $\bs B$-structure on $\pd X$.

If $\bs B=(B,\be)$, $\bs B'=(B',\be')$ are tangential structures, we say that $\bs B$ factors through $\bs B'$ if there is a homotopy commutative diagram

_^_     B' [dr]^'

B [ur] [rr]^(0.35)        BØ.

Composing with this diagram, a $\bs B$-structure on $X$ induces a $\bs B'$-structure.

Here are some examples, including well known geometric structures such as orientations and spin structures.

(a) The orthogonal tangential structure is $\bs\BO=(B\O,\id_{B\O})$. Every manifold $X$ has a $\bs\BO$-structure unique up to homotopy.

(b) The special orthogonal tangential structure is $\bs\BSO=(B\SO,\be_\SO)$, where $B\SO=\colim_{n\ra\iy}B\SO(n)$ and $\be_\SO:B\SO\ra B\O$ is induced by the inclusions $\SO(n)\hookra\O(n)$. A $\bs\BSO$-structure on $X$ is equivalent to an orientation on $X$. The opposite $\bs\BSO$-structure is equivalent to the opposite orientation.

(c) The spin tangential structure is $\bs\BSpin\!=\!(B\Spin,\be_\Spin)$, where $B\Spin\!=\!\colim_{n\ra\iy}\ab B\Spin(n)$ and $\be_\Spin:B\Spin\ra B\O$ is induced by $\Spin(n)\ra\O(n).$ A $\bs\BSpin$-structure on $X$ is equivalent to an orientation and a spin structure.

(d) The spin$^c$ tangential structure is $\bs\BSpinc=(B\Spinc,\be_{\Spinc})$, for $B\Spinc=\colim_{n\ra\iy}\ab B\Spinc(n)$ and $\be_{\Spinc}:B\Spinc\ra B\O$ induced by $\Spinc(n)\ra\O(n)$. A $\bs\BSpinc$-structure on $X$ amounts to an orientation and a spin$^{\rm c}$ structure.

(e) The unitary tangential structure is $\bs\BU=(B\U,\be_\U)$, where $B\U=\colim_{m\ra\iy}\ab B\U(m)$ and $\be_\U:B\U\ra B\O$ is induced by the commutative diagram

\begin{equation*}
\xymatrix@C=20pt@R=15pt{
\cdots \ar[r] & \U(m) \ar[r] \ar[d] & \U(m) \ar[r] \ar[d] & \U(m+1) \ar[r] \ar[d] & \U(m+1) \ar[r] \ar[d] & \cdots \\
\cdots \ar[r] & \O(2m) \ar[r]  & \O(2m+1) \ar[r] & \O(2m+2) \ar[r] & \O(2m+3) \ar[r] & \cdots.\! }
\end{equation*}

A $\bs\BU$-structure on $X$ is equivalent to a stable almost complex structure on $X.$

(f) The special unitary tangential structure is $\bs\BSU=(B\SU,\be_\SU)$, where $B\SU=\colim_{m\ra\iy}\ab B\SU(m)$ and $\be_\SU:B\SU\ra B\O$ is defined as in (e).

(g) The quaternionic tangential structure is $\bs\BSp=(B\Sp,\be_\Sp)$, where $B\Sp=\colim_{m\ra\iy}\ab B\Sp(m)$ and $\be_\Sp:B\Sp\ra B\O$ is defined in a similar way to (e).

All of the tangential structures in (a)–(g) have products. Also (b)–(g) factor through $\bs\BSO$, but (a) does not.

§.§.§ Bordism (generalized) homology theory

Let $\bs B$ be a stable tangential structure. Then $\bs B$-bordism $\Om_*^{\bs B}(-)$ is a generalized homology theory of topological spaces $T$, in which the `$n$-chains' are continuous maps $f:X\ra T$ for $X$ a compact $n$-manifold with a $\bs B$-structure. The subject began with the work of Thom [69]. Bordism was introduced by Atiyah [2], and good references are Conner <cit.> and Stong [65].

Let $\bs B$ be a tangential structure, $T$ be a topological space, and $n\in\N$. Consider triples $(X,\bs\ga_X,f)$, where $X$ is a compact manifold with $\dim X=n$, $\bs\ga_X$ is a $\bs B$-structure on $X$, and $f:X\ra T$ is a continuous map. Given two such triples, a bordism from $(X_0,\bs\ga_{X_0},f_0)$ to $(X_1,\bs\ga_{X_1},f_1)$ is a triple $(W,\bs\ga_W,e)$, where:

(i) $W$ is a compact $(n+1)$-manifold with boundary, with a given identification $\pd W\cong X_0\amalg X_1$.

(ii) $\bs\ga_W$ is a $\bs B$-structure on $W$, with a given isomorphism of $\bs B$-structures $\bs\ga_W\vert_{\pd W}\ab\cong -\bs\ga_{X_0}\amalg\bs\ga_{X_1}$ on $\pd W\cong X_0\amalg X_1.$

(iii) $e:W\ra T$ is a continuous map such that $e\vert_{\pd W}\cong f_0\amalg f_1$ under the identification $\pd W\cong X_0\amalg X_1.$

Write $(X_0,\bs\ga_{X_0},f_0)\sim(X_1,\bs\ga_{X_1},f_1)$ if there exists such a bordism $(W,\bs\ga_W,e)$. Then `$\sim$' is an equivalence relation, called $\bs B$-bordism, and the equivalence class $[X,\bs\ga_X,f]$ is called a $\bs B$-bordism class. The $n^{\it th}$ $\bs B$-bordism group $\Om^{\bs B}_n(T)$ is the set of $\bs B$-bordism classes $[X,\bs\ga_X,f]$ with $\dim X=n,$ where the group structure has zero element $0_T=[\es,\es,\es],$ addition $[X,\bs\ga_X,f]+[X',\bs\ga_{X'},f']=[X\amalg X',\bs\ga_X\amalg\bs\ga_{X'},f\amalg f'],$ and inverse $-[X,\bs\ga_X,f]=[X,-\bs\ga_X,f].$

When $T$ is a point we may omit the necessarily constant map $f$ from the notation, and write elements of $\Om^{\bs B}_n(*)$ as $[X,\bs\ga_X]$.

If $T$ is a smooth manifold then, as smooth maps are dense in continuous maps, we can take $f:X\ra T$ and $e:W\ra F$ above to be smooth.

Now suppose that $\bs B'$ is another tangential structure and that $\bs B$ factors through $\bs B'$ as in bc2eq11 of Definition <ref>. Then a $\bs B$-structure $\bs\ga_X$ on a manifold $X$ induces a $\bs B'$-structure $\Pi_{\bs B}^{\bs B'}(\bs\ga_X)$ on $X$. This defines a group morphism

\begin{equation*}
\Om_n^{\bs B}(T)\overset{\Pi_{\bs B}^{\bs B'}}{\longra}\Om_n^{\bs B'}(T),\qquad[X,\bs\ga_X,f]\longmapsto\bigl[X,\Pi_{\bs B}^{\bs B'}(\bs\ga_X),f\bigr].
\end{equation*}

If $\io: U\hookrightarrow T$ is a subspace we can define relative bordism groups $\Om_n^{\bs B}(T,U)$, whose elements $[X,\bs\ga_X,f]$ are bordism classes of triples $(X,\bs\ga_X,f)$ with $X$ a compact $n$-manifold with boundary and $\bs B$-structure $\bs\ga_X$, and $f:X\ra T$ a continuous map with $f(\pd X)\subseteq U\subseteq T$. These fit into a long exact sequence

\begin{equation*}
\xymatrix@C=21pt{ \cdots \ar[r] & \Om_n^{\bs B}(U) \ar[r]^{\io_*} & \Om_n^{\bs B}(T) \ar[r]^(0.45){\pi_*} & \Om_n^{\bs B}(T,U) \ar[r]^(0.53)\pd & \Om_{n-1}^{\bs B}(U) \ar[r] & \cdots. }
\end{equation*}

If $T$ is path-connected we define the reduced bordism groups to be $\ti\Om_n^{\bs B}(T)=\Om_n^{\bs B}(T,\{t_0\})$, for $t_0\in T$ any base point. As the inclusion of $U=\{t_0\}$ has a left inverse, the long exact sequence reduces to short exact sequences
@C=30pt 0 [r]     _n^B(*) [r]^_*     _n^B(T) [r]^π_*     _n^B(T) [r]     0.

$n\!=\!0$     $n\!=\!1$     $n\!=\!2$     $n\!=\!3$     $n\!=\!4$     $n\!=\!5$     $n\!=\!6$     $n\!=\!7$     $n\!=\!8$     $n\!=\!9$

$\Om_n^{\bs\SO}(*)$     $\Z$     0     0     0     $\Z$     $\Z_2$     0     0     $\Z^2$

$\Om_n^{\bs\O}(*)$     $\Z_2$     0     $\Z_2$     0     $\Z_2^2$     $\Z_2$     $\Z_2^3$     $\Z_2$     $\Z_2^5$

$\Om_n^{\bs\Spin}(*)$     $\Z$     $\Z_2$     $\Z_2$     0     $\Z$     0     0     0     $\Z^2$     $\Z_2^2$

$\Om_n^{\bs\Spinc}(*)$     $\Z$     0     $\Z$     0     $\Z^2$     0     $\Z^2$     0     $\Z^4$     0

$\Om_n^{\bs\U}(*)$     $\Z$     0     $\Z$     0     $\Z^2$     0     $\Z^3$     0     $\Z^5$

$\Om_n^{\bs\SU}(*)$     $\Z$     $\Z_2$     $\Z_2$     0     $\Z$     0     $\Z$     0     $\Z^2$

$\bs B$-bordism groups of the point

As $\bs B$-bordism is a generalized homology theory, there is a spectral sequence $H_p(T,\Om^{\bs B}_q(*))\Ra\Om^{\bs B}_{p+q}(T)$, where $*$ is the point. If $T$ is path-connected then as the splitting $\Om^{\bs B}_*(T)=\Om^{\bs B}_*(*)\op\ti\Om^{\bs B}_*(T)$ is functorial, this induces a spectral sequence $\ti H_p(T,\Om^{\bs B}_q(*))\Ra\ti\Om^{\bs B}_{p+q}(T)$. Thus, a lot of important behaviour of bordism depends on the bordism groups $\Om_n^{\bs B}(*)$ of the point, so much effort has gone into calculating these. Table <ref> gives values of $\Om_n^{\bs B}(*)$ for $\bs B=\bs\BSO,\bs\BO,\bs\BSpin,\bs\BSpinc,\bs\BU,\bs\BSU$ and $n\le 8$, which are taken from Stong [65], Anderson, Brown and Peterson [1], and Gilkey <cit.>. Note that we omit the letter $\bs{\rm B}$ from the notation for the bordism group for the classical tangential structures defined in Example <ref>.

$n$     $0$     $1$     $2$     $4$

[top][4ex][c]1.3cm$\Om^\Spin_n(*)$     $\Z\an{1}$     $\Z_2\an{\al_1}$     $\Z_2\an{\al_1^2}$     $\Z\an{\al_2}$

$\;\al_1=[\cS^1,\bs\ga'_{\cS^1}],\; \al_2=[K3,\bs\ga_{K3}]$
Explicit presentation of $\Om^\Spin_n(*)$, $n\le 4$

A presentation of $\Om^\Spin_n(*)$, $n\le 4$ is given in Table <ref>, where
$\bs\ga'_{\cS^1}$ is the spin structure on $\cS^1$ which is not the restriction of a spin structure on the closed unit disc $D^2\subset\R^2$, and $\bs\ga_{K3}$ is the unique spin structure on the $K3$ surface.

§.§.§ Free loop spaces, based loop spaces, and their bordism

Let $T$ be a topological space, which we suppose is path-connected, with a basepoint $t_0\in T$. Write $\cS^1=\R/\Z$, with basepoint $\ul 0=0+\Z$. The free loop space of $T$ is $\cL T=\Map_{C_0}(\cS^1,T)$, with the compact-open topology. Points of $\cL T$ are continuous maps $\ga:\cS^1\ra T$, that is, loops in $T$. The based loop space of $T$ is $\Om T=\Map_{C_0}((\cS^1,\ul{0}),(T,t_0))$. Points of $\Om T$ are continuous maps $\ga:\cS^1\ra T$ with $\ga(\ul{0})=t_0$, that is, based loops in $T$.

Mapping $\ga\mapsto\ga(\ul{0})$ defines an evaluation map $\ev_{\ul{0}}:\cL T\ra T$, with $\ev_{\ul{0}}^{-1}(0)=\Om T$. It is a homotopy fibration, with fibre $\Om T$. Let $\bs B$ be a tangential structure. As $\bs B$-bordism $\Om_*^{\bs B}(-)$ is a generalized homology theory, the homotopy fibration gives an Atiyah–Hirzebruch spectral sequence

The fibration has a section $s_T:T\ra\cL T$ mapping $t\in T$ to the constant loop $\cS^1\ra\{t\}\subset T$, with $\ev_{\ul{0}}\ci s_T=\id_T$. The morphisms $(\ev_{\ul{0}})_*:\Om_n^{\bs B}(\cL T)\ra \Om_n^{\bs B}(T)$ and $(s_T)_*:\Om_n^{\bs B}(T)\ra \Om_n^{\bs B}(\cL T)$ induce a splitting

_n^B(T)=_n^B(T)_n^B(T;T),    where

Write $\Pi_n^{\bs B}(T):\Om_n^{\bs B}(\cL T)\ra\Om_n^{\bs B}(\cL T;T)$ for the projection in this direct sum. We regard $\Om_n^{\bs B}(\cL T;T)$ as a relative $\bs B$-bordism group.

The decomposition bc2eq14 of $\Om_n^{\bs B}(\cL T)$ corresponds in the spectral sequence bc2eq13 to the decomposition $\Om_q^{\bs B}(\Om T)=\Om_q^{\bs B}(*)\op\ti\Om_q^{\bs B}(\Om T)$. Therefore bc2eq13 splits as the direct sum of two spectral sequences, with the first $H_p\bigl(T,\Om_q^{\bs B}(*)\bigr)\Ra \Om_{p+q}^{\bs B}(T)$ the usual spectral sequence for computing $\Om_*^{\bs B}(T)$, and the second being

Let $f:S\ra T$ be a continuous map of connected topological spaces, and write $\cL f:\cL S\ra\cL T$ for the induced map of free loop spaces. Then $\cL f_*:\Om_n^{\bs B}(\cL S)\ra\Om_n^{\bs B}(\cL T)$ is compatible with the splittings bc2eq14. Write

for the restriction of $\cL f_*$ to relative $\bs B$-bordism.

Now a continuous map $\phi:X\ra\cL T$ is equivalent to a continuous map $\phi':X\t\cS^1\ra T$, by the tautological definition $\phi'(x,y)=\phi(x)(y)$ for $x\in X$ and $y\in\cS^1$. Define a morphism $\xi_n^{\bs B}(T):\Om_n^{\bs B}(\cL T)\ra\Om_{n+1}^{\bs B}(T)$ by

where $\phi'$ is as above and $\bs\ga_{\cS^1}$ is the $\bs B$-structure on $\cS^1$ induced from the standard $\bs B$-structure on the closed unit disc $D^2\subset\R^2$ by identifying $S^1=\pd D^2$.

Equation bc2eq17 is compatible with equivalences $(X_0,\bs\ga_{X_0},\phi_0)\sim\ab(X_1,\ab\bs\ga_{X_1},\ab\phi_1)$, and so is well defined. Note that $[\cS^1,\bs\ga_{\cS^1}]=0$ in $\Om_1^{\bs B}(*)$. When $\bs B=\Spin$ there is also a second $\Spin$-structure $\bs\ga'_{\cS^1}$ on $\cS^1$ with $[\cS^1,\bs\ga'_{\cS^1}]\ne 0$ in $\Om_1^\Spin(*)$; it is important that we use $\bs\ga_{\cS^1}$ rather than $\bs\ga'_{\cS^1}$ in bc2eq17.

Consider the diagram

0 [r]     _n^B(T) @<-1ex>[dr]_0 [r]^(s_T)_*     _n^B(T) [r]^Π_n^B(T) [d]^ξ_n^B(T)     _n^B(T;T) @<1ex>@..>[dl]^ξ_n^B(T) [r]     0

The top row is exact. If $[X,\bs\ga_X,\psi]\in \Om_n^{\bs B}(T)$ then

\begin{align*}
&\xi_n^{\bs B}(T)\ci (s_T)_*\bigl([X,\bs\ga_X,\psi]\bigr)=\xi_n^{\bs B}(T)\bigl([X,\bs\ga_X,s_T\ci\psi]\bigr)
\\
&\;\> =[X\t\cS^1,\bs\ga_X\t\bs\ga_{\cS^1},\psi\ci\pi_X]=[X,\bs\ga_X,\psi]*[\cS^1,\bs\ga_{\cS^1}]=[X,\bs\ga_X,\psi]*0=0,
\end{align*}

where $*:\Om_n^{\bs B}(T)\t\Om_q^{\bs B}(*)\ra\Om_{n+1}^{\bs B}(T)$ is the natural product. Thus the left hand triangle of bc2eq18 commutes, so there exists a unique morphism $\ti\xi_n^{\bs B}(T):\Om_n^{\bs B}(\cL T;T)\ra\Om_{n+1}^{\bs B}(T)$ making the right hand triangle commute.

§.§.§ Spin bordism of some classifying spaces

The next theorem, proved in <ref>, does some bordism calculations we will need to prove Theorem <ref>, which will be key to proving our applications in <ref>–<ref>. Part (d) is what we actually need for Theorem <ref>. Parts (a)–(c) will be used to prove (d), via the spectral sequences bc2eq15 for $T=B\SU$ and $T=K(\Z,4)$, noting that there are homotopy equivalences $\Om B\SU\simeq\SU$ and $\Om K(\Z,4)\simeq K(\Z,3)$, so (a)–(c) help us understand the terms $\ti\Om_q^{\bs B}(\Om T)$ in bc2eq15.

(a) Write $\SU=\varinjlim_{n\ra\iy}\SU(n)$. The reduced spin bordism groups $\ti\Om^\Spin_n(\SU)$ for $n\le 8$ are given by

$n$     $0,1,2,4,6$     $3$     $5$     $7$     $8$

[top][4ex][c]1.6cm$\ti\Om^\Spin_n(\SU)$     $0$     $\Z$     $\Z$     $\Z^2$     $\Z$

Writing the integral cohomology ring as $H^*(\SU,\Z)=\La_\Z[b_2,b_3,\ldots]$ with $\deg b_i\ab =2i-1,$ the isomorphisms in bc2eq19 are given explicitly by

^_3(),        [X,ϕ]⟼∫_Xϕ^*(b_2),

^_5(),        [X,ϕ]⟼∫_Xϕ^*(b_3),

^_7()^2,        [X,ϕ]⟼(∫_Xϕ^*(b_4),

^_8(),        [X,ϕ]⟼∫_Xϕ^*(b_2∪b_3).

(b) Write $K(\Z,k)$ for the Eilenberg–MacLane space with $\pi_k(K(\Z,k))\cong\Z$. The reduced spin bordism groups $\ti\Om^\Spin_n(K(\Z,3))$ for $n\le 8$ are given by

$n$     $0,1,2,4,5,6$     $3$     $7$     $8$

[top][4ex][c]2.6cm$\ti\Om^\Spin_n(K(\Z,3))$     $0$     $\Z$     $\Z$     $\Z_2$

The isomorphisms in bc2eq21 are given explicitly by

^_3(K(,3)),        [X,ϕ]⟼∫_Xϕ^*(d_3),

^_7(K(,3)),        [X,ϕ]⟼1/8∫_Xp_1(X)∪ϕ^*(d_3),

^_8(K(,3))_2,        [X,ϕ]⟼∫_Xϕ^*(d̅_3∪^2(d̅_3)).

Here $d_3\in H^3(K(\Z,3),\Z)$ is the universal cohomology class, as $K(\Z,3)$ is the classifying space for $H^3(-,\Z),$ and $\bar d_3\in H^3(K(\Z,3),\Z_2)$ its mod $2$ reduction, and $\Sq^2(\bar d_3)\in H^5(K(\Z,3),\Z_2)$ the Steenrod square of $\bar d_3$.

(c) Write $\la:\SU\ra K(\Z,3)$ for the classifying map of $b_2\in H^3(\SU,\Z),$ so that $\la^*(d_3)=b_2$. Then under the identifications bc2eq20, bc2eq22, the morphisms $\la_*:\ti\Om^\Spin_n(\SU)\ra\ti\Om^\Spin_n(K(\Z,3))$ for $n=3,7,8$ are given by

_*:^_3()=^_3(K(,3))=,        n⟼n,

_*:^_7()=^2^_7(K(,3))=,        (m,n)⟼3n,

_*:^_8()=^_8(K(,3))=_2,        n⟼n+2.

(d) Write $B\SU=\varinjlim_{n\ra\iy}B\SU(n)$. Then $H^*(B\SU,\Z)=\Z[c_2,c_3,\ldots],$ where $c_i$ is the $i^{\rm th}$ Chern class, with $\deg c_i=2i$. The reduced spin bordism groups $\ti\Om^\Spin_n(\cL B\SU;B\SU)$ for $n\le 7$ are given by

$n$     $0,1,2,4,6$     $3$     $5$     $7$

[top][4ex][c]3cm$\ti\Om^\Spin_n(\cL B\SU;B\SU)$     $0$     $\Z$     $\Z$     $\Z^3$

where the isomorphisms are given explicitly by

^_3(B;B),         [X,ϕ]⟼∫_X^1c_2(P),

^_5(B;B),        [X,ϕ]⟼∫_X^1c_3(P),

^_7(B;B)^3,        [X,ϕ]⟼

1/6∫_X^1 c_4(P) - 1/12∫_X^1 c_2(P)^2,
1/48∫_X^1 p_1(X)∪c_2(P),
1/2∫_X^1 c_2(P)^2
where $P\ra X\t\cS^1$ is the pullback of the universal bundle over $B\SU$ along the adjoint $X\t\cS^1\ra B\SU$ of the map $\phi\colon X\ra\cL B\SU$ the proof will show that the above integrals are always integers.

Write $\mu:B\SU\ra K(\Z,4)$ for the classifying map of $c_2\in H^4(B\SU,\Z)$. Consider the morphisms

μ_^𝕀__2   :_n^(B;B)__2

where $\mu_\rel^\Spin$ is as in bc2eq16. Then bc2eq26 is surjective for $n=7,8$.

§.§.§ The exceptional Lie group E₈ and its classifying space

Let $G$ be a Lie group. Then, as in May <cit.> or Milnor–Stasheff [55], $G$ has a classifying space $BG$, a connected topological space natural up to homotopy equivalence, with a principal $G$-bundle $\pi:EG\ra BG$ with $EG$ contractible, such that if $X$ is a (nice) topological space then isomorphism classes of principal $G$-bundles $\pi:P\ra X$ are in natural correspondence with homotopy classes of continuous maps $f_P:X\ra BG$, with $P\cong f^*(EG)$. The classifying space has a free loop space $\cL BG=\Map_{C^0}(\cS^1,BG)$, the topological space of continuous maps $\ga:\cS^1\ra BG$, where $\cS^1=\R/\Z$.

The exceptional Lie group $E_8$ is a compact, simply-connected, simple Lie group of dimension $248$ and rank $8$. It will be important later for two reasons: firstly, as the only nonzero homotopy group $\pi_d(E_8)$ for $d\le 14$ is $\pi_3(E_8)=\Z$, homotopy-theoretic calculations for $E_8$ and $BE_8$ are not that difficult. And secondly, because of Theorem <ref>, once we have proved orientability results for $E_8$ we can deduce orientability results for many other Lie groups. The next theorem will be proved in <ref>, deduced from Theorem <ref>(d). It will be essential for the applications in <ref>–<ref>.

Write $\io:\SU(8)\ra E_8$ for the Lie group morphism  defined as the composition $\SU(8)\,{\buildrel(\id,1)\over\longra}\,\SU(8)\t\U(1)\,{\buildrel\eq{bc2eq6}\over\longra}\, E_8,$ and $B\io:B\SU(8)\ra BE_8$ be the induced morphism of classifying spaces. Consider the morphisms

B_^𝕀__2   :^_n(B(8);B(8))__2

where $B\io_\rel^\Spin$ is as in bc2eq16. Then bc2eq27 is surjective for $n=7,8$.

§.§ Categorical groups and Picard groupoids

Categorical groups may be viewed as a categorification of the concept of a group. Similarly, Picard groupoids categorify abelian groups. These will appear as tools in this paper, so we briefly review them as background here and state a classification result due to Sinh [64].

For more background on symmetric monoidal categories, we refer to Joyal–Street [36] and MacLane <cit.>.

A monoidal category $(\cC,\ot,\bf 1,\al)$ is a category $\cC$ with a tensor product functor $\ot:\cC\t\cC\ra\cC,$ a unit object $\bf 1\in\cC,$ a natural associativity isomorphism $\al,$ and unit isomorphisms. Usually, we will not make these explicit, which is justified by MacLane's coherence theorem. To simplify our exposition, we will usually assume that all unit isomorphisms are identities. The set $\pi_0(\cC)$ of isomorphism classes of objects of a monoidal category is a (possibly non-commutative) monoid. Moreover, the operation induced by the tensor product and the ordinary composition agree in the automorphism group $\pi_1(\cC)=\Aut_\cC(\bf 1),$ which implies that $\pi_1(\cC)$ is an abelian group (by the Eckmann–Hilton argument). We write $\pi_0(\cC)$ multiplicatively and $\pi_1(\cC)$ additively. A categorical group is a monoidal category $(\cG,\ot,\bf 1,\al)$ in which all morphisms are invertible and for which the monoid $\pi_0(\cG)$ is a group.

This means that every object $x$ has a dual, an object $x^*$ for which there exist isomorphisms $\ep_x: x^*\ot x\cong{\bf 1}$ and $\eta_x:{\bf 1}\cong x\ot x^*$ (one usually requires some axioms, which play no role here). In a categorical group, all of the automorphism groups can be identified with each other via

\begin{equation*}
\pi_1(\cG)\longra\Aut_\cG(x),\enskip \left({\bf 1}\xrightarrow{\varphi}{\bf 1}\right)\longmapsto \left(x\cong {\bf 1}\ot x\xrightarrow{\varphi\ot x}{\bf 1}\ot x\cong x\right).
\end{equation*}

Given a group $\pi_0$ and an abelian group $\pi_1,$ let $\cG=\pi_0\quotstack\pi_1$ denote the category of $\pi_0$-graded $\pi_1$-torsors. In other words, the objects of $\cG$ are all pairs $(S,x),$ where $x\in\pi_0$ and $S$ is a set with a free, transitive left action of the group $\pi_1.$ If $x=y,$ then $\Hom_\cG\bigl((S,x),(T,y)\bigr)$ is the set of all $\pi_1$-equivariant maps $\varphi: S\ra T,$ otherwise the morphism set is defined to be empty. Define the tensor product of objects by $(S_0,x_0)\ot(S_1,x_1)=(S_0\ot_{\pi_1} S_1,x_0x_1),$ where $S_0\ot_{\pi_1} S_1=(S_0\t S_1)/\pi_1$ is the quotient by the anti-diagonal $\pi_1$-action.

As any two $\pi_1$-torsors are isomorphic and every isomorphism is multiplication by a group element, $\cG$ is a categorical group with $\pi_0(\cG)=\pi_0,$ $\pi_1(\cG)=\pi_1,$ and a trivial conjugation action of $\pi_0(\cG)$ on $\pi_1(\cG).$

In case $\pi_1=0$ the construction of the category $\pi_0\quotstack\pi_1$ boils down to the abelian group $\pi_0$ viewed as a discrete monoidal category in the usual way.

A monoidal structure on a functor $F:\cC\ra\cD$ of monoidal categories $(\cC,\ot_\cC,\bf 1_\cC)$ and $(\cD,\ot_\cD,\bf 1_\cD)$ is a collection of isomorphisms
F(x)_F(y) F(x_y),  1_ F(1_),

for all objects $x, y$ of $\cC,$ compatible with the associativity and unit isomorphisms in $\cC$ and $\cD,$ see <cit.>. A monoidal transformation of such functors is a natural transformation $F\Rightarrow G$ that maps the isomorphisms bc2eq28 for $F$ and $G$ onto each other, see <cit.>. A monoidal equivalence is a pair of monoidal functors whose composites either way admit monoidal natural isomorphisms to the identity functors of $\cC$ and $\cD.$

A symmetry $\si$ on a monoidal category $(\cC,\ot,\bf 1,\al)$ is a natural isomorphism $\si_{x,y}: x\ot y\ra y\ot x$ such that $\si_{y,x}\circ\si_{x,y}=1_{x\ot y}$ and such that the unit and the hexagon coherence diagrams of <cit.> commute. A Picard groupoid is a categorical group $\cG$ equipped with a symmetry $\si.$

In particular, $\pi_0(\cG)$ is then a commutative monoid. From now on, we write the abelian groups $\pi_0, \pi_1$ additively.

Recall that for a symmetric monoidal functor of symmetric monoidal categories the isomorphisms bc2eq28 are required to commute with the symmetry, see <cit.>. A symmetric monoidal functor between Picard groupoids is also called a morphism of Picard groupoids. There are no further conditions for a monoidal transformation between symmetric monoidal functors.

Picard groupoids are classified by a linear, quadratic invariant defined from the symmetry. We first recall some terminology.

Let $\pi_0$ and $\pi_1$ be abelian groups.

(i) A map $q:\pi_0\ra\pi_1$ is quadratic if $b_q(x,y)=q(x+y)-q(x)-q(y)$ defines a bilinear map. This implies $q(\la x)=\la^2q(x)$ for $\la\in\Z.$ Let $\Quad(\pi_0,\pi_1)$ be the abelian group of all quadratic maps.

A bilinear map $\al:\pi_0\t\pi_0\ra\pi_1$ is alternating if $\al(x,x)=0$ for all $x\in\pi_0$ and skew-symmetric if $\al(x,y)+\al(y,x)=0$ for all $x,y\in\pi_0.$ (This is also a good definition when $\al$ is not bilinear.) Let $\Alt(\pi_0,\pi_1)$ be the abelian group of all alternating bilinear maps and let $\Skew(\pi_0,\pi_1)$ be the abelian group of all skew-symmetric bilinear maps.

By expanding $\al(x+y,x+y)=0,$ one finds $\Alt(\pi_0,\pi_1)\subset\Skew(\pi_0,\pi_1).$ If a quadratic map $q:\pi_0\ra\pi_1$ is also a linear map, then $q(2x)=4q(x)$ and $q(2x)=q(x+x)=q(x)+q(x),$ so $2q(x)=0.$ Therefore, $q$ factors through a linear map $\pi_0/2\pi_0\ra\pi_1.$ Conversely, every linear map $\pi_0/2\pi_0\ra\pi_1$ determines a linear quadratic map by precomposing with the canonical projection. Hence $\Hom(\pi_0/2\pi_0,\pi_1)\subset\Quad(\pi_0,\pi_1)$ is the subset of linear quadratic maps.

There is a short exact sequence

\begin{equation*}
\begin{tikzcd}[column sep=3ex]
0\rar&\Alt(\pi_0,\pi_1)\rar&\Skew(\pi_0,\pi_1)\rar{\De^*} & \Hom(\pi_0/2\pi_0,\pi_1)\rar & 0,
\end{tikzcd}
\end{equation*}

where $\De^*$ maps $\al\in\Skew(\pi_0,\pi_1)$ to the quadratic map $q(x)=\al(x,x).$

A self-equivalence of a Picard groupoid changes $\si$ by a $2$-cocycle, which explains why only the linear quadratic form $q(x)=\si(x,x)$ is an invariant of $\cG,$ called the symmetry invariant. The triple $(\pi_0,\pi_1,q)$ is a complete invariant of Picard groupoids. We have the following classification result from Sinh [64].

(a) Let $\pi_0$ and $\pi_1$ be abelian groups. Up to equivalence, Picard groupoids $\cG$ with $\pi_0(\cG)=\pi_0$ and $\pi_1(\cG)=\pi_1$ are classified by their symmetry invariant, which is a linear quadratic form $q:\pi_0(\cG)\ra\pi_1(\cG).$ Conversely, every triple $(\pi_0,\pi_1,q)$ occurs as the invariants of some Picard groupoid.

Let $\cG$ and $\cG'$ be Picard groupoids with symmetry invariants $q$ and $q'.$ Let $f_0:\pi_0(\cG)\ra\pi_0(\cG')$ and $f_1:\pi_1(\cG)\ra\pi_1(\cG')$ be group morphisms. There exists a symmetric monoidal functor $F:\cG\ra\cG'$ with $\pi_0(F)=f_0$ and $\pi_1(F)=f_1$ if and only if $q'\circ f_0=f_1\circ q.$

It follows that every Picard groupoid is equivalent to a category of $\pi_0$-graded $\pi_1$-torsors $\cG=\pi_0\quotstack\pi_1$ (see Example <ref>) with symmetry isomorphism determined by $\si\in\Skew(\pi_0,\pi_1)$ as

\begin{align*}
s_0\ot_\si s_1&\longmapsto\si(x_0,x_1)(s_1\ot_\si s_0).
\end{align*}

These Picard groupoids are equivalent if the `diagonal' quadratic forms $\si(x,x)$ coincide. We may therefore view a symmetry isomorphism as a sign convention when commuting objects past each other. From this point of view, Theorem <ref> classifies all possible sign conventions on $\pi_0\quotstack\pi_1$ up to equivalence. Sign conventions are very important in the construction of the Quillen determinant line bundle (see [70]) and they are equally important here.

§ BORDISM CATEGORIES AND GAUGE THEORY

For each dimension $n\ge 0$, tangential structure $\bs B$, and Lie group $G$, we will define categories $\Bord_n^{\bs B}(BG),\Bord_n^{\bs B}(\cL BG)$. If $X$ is a compact $n$-manifold we also define a category $\Bord_X(BG)$. All three will be called bordism categories. Parts of <ref>, <ref>, <ref> and <ref> were discussed in the previous paper [71], in less detail.

§.§ Bordism categories Bordₙᴮ(BG)

Fix a dimension $n\ge 0$, a tangential structure $\bs B$ in the sense of <ref>, and a Lie group $G$. We will define a symmetric monoidal category $\Bord_n^{\bs B}(BG)$ that we call a bordism category.

(a) Objects of $\Bord_n^{\bs B}(BG)$ are pairs $(X,P)$, where $X$ is a compact $n$-manifold without boundary with a $\bs B$-structure $\bs\ga_X$, which we generally omit from the notation, and $P\ra X$ is a principal $G$-bundle.

(b) Morphisms $[W,Q]:(X_0,P_0)\ra(X_1,P_1)$ in $\Bord_n^{\bs B}(BG)$ are equivalence classes of pairs $(W,Q),$ see (c), where $W$ is a compact $(n+1)$-manifold with $\bs B$-structure $\bs\ga_W$, there is a chosen isomorphism $\pd W\cong -X_0\amalg X_1$ of the boundary preserving $\bs B$-structures (where $-X_0$ indicates that $X_0$ has the opposite $\bs B$-structure $-\bs\ga_{X_0}$), and $Q\ra W$ is a principal $G$-bundle with a chosen isomorphism $Q\vert_{\pd W}\cong P_0\amalg P_1$. We suppress the isomorphisms from the notation.

(c) In the situation of (b), let $(W_0,Q_0)$ and $(W_1,Q_1)$ be two choices for $(W,Q)$. We say that $(W_0,Q_0)\sim(W_1,Q_1)$ if there exists a pair $(V,R)$, where $V$ is a compact $(n+2)$-manifold with corners and $\bs B$-structure $\bs\ga_V$, with a chosen isomorphism of boundaries identifying $\bs B$-structures
V≅(-X_0[0,1])⨿(X_1[0,1]) ⨿-W_0⨿W_1

such that along $\pd^2V$ we identify $\pd W_i$ with $(-X_0\amalg X_1)\t\{i\}$ for $i=0,1$ in the obvious way, and $R\ra V$ is a principal $G$-bundle such that under bc3eq1 we have

with the obvious compatibility with the chosen isomorphisms $Q_i\vert_{\pd W_i}\cong P_0\amalg P_1$ over $X_0\t\{i\}\amalg X_1\t\{i\}$. It is easy to see that `$\sim$' is an equivalence relation, so the equivalence classes $[W,Q]$ are well defined.

(d) If $[W,Q]:(X_0,P_0)\ra(X_1,P_1)$ and $[W',Q']:(X_1,P_1)\ra(X_2,P_2)$ are morphisms, the composition is

That is, we glue $W,W'$ along their common boundary component $X_1$ to make a manifold $W'\amalg_{X_1}W$ with $\bs B$-structure and boundary $\pd(W'\amalg_{X_1}W)=-X_0\amalg X_2$. To define the smooth structure on $W'\amalg_{X_1}W$ we should choose `collars' $X_1\t(-\ep,0]\subset W$, $X_1\t[0,\ep)\subset W'$ of $X_1$ in $W,W'$, and similarly for $Q,Q'$, but the choices do not change the equivalence class $[W'\amalg_{X_1}W,Q'\amalg_{P_1}Q]$. Composition is associative.

(e) If $(X,P)$ is an object in $\Bord_n^{\bs B}(BG),$ the identity morphism is

(f) If $[W,Q]:(X_0,P_0)\ra(X_1,P_1)$ is a morphism, we can prove that it has an inverse morphism

[W,Q]^-1   =[-W⨿(W⨿_X_0⨿X_1-W),Q⨿(Q⨿_P_0⨿P_1-Q)]:

noting that $\pd(-W)=-(-X_0\amalg X_1)=-X_1\amalg X_0$. Thus the category $\Bord_n^{\bs B}(BG)$ is a groupoid, that is, all morphisms are isomorphisms.

(g) Define a monoidal structure $\ot$ on $\Bord_n^{\bs B}(BG)$ by, on objects

and if $[W,Q]:(X_0,P_0)\ra(X_1,P_1)$, $[W',Q']:(X_0',P_0')\ra(X_1',P_1')$ are morphisms, then

This is compatible with `$\sim$', and with compositions and identities.

(h) The identity in $\Bord_n^{\bs B}(BG)$ is $\boo=(\es,\es)$.

(i) If $(X,P)\in\Bord_n^{\bs B}(BG)$ we write $-(X,P)=(-X,P)$, that is, we give $X$ the opposite $\bs B$-structure $-\bs\ga_X$. Observe that we have an isomorphism

Thus $-(X,P)$ is an inverse for $(X,P)$ under `$\ot$'.

(j) The symmetry isomorphism $\si_{(X,P),(X',P')}=[W,Q]\colon(X,P)\ot(X',P')\to(X',P')\ot(X,P)$
has $(W,Q)=((X\amalg X')\t[0,1],(P\amalg P')\t[0,1])$ with the obvious identification of $\6 W$ with the disjoint union of $-(X\amalg X')$ and $X'\amalg X.$

Hence $\Bord_n^{\bs B}(BG)$ is a Picard groupoid, as in <ref>.

In the case $G=\{1\}$ we will write $\Bord_n^{\bs B}(*)$ instead of $\Bord_n^{\bs B}(B\{1\})$. By definition, objects of $\Bord_n^{\bs B}(*)$ are pairs $(X,P)$, where $P\ra X$ is a principal $\{1\}$-bundle. But as principal $\{1\}$-bundles are trivial (we may take $P\ra X$ to be $\id_X:X\ra X$) we may omit $P$, and write objects of $\Bord_n^{\bs B}(*)$ as $X$, morphisms as $[W]:X_0\ra X_1$, and so on. This is equivalent to $\Bord_n^{\bs B}(*)$ in Definition <ref>.

If $\ga:G_1\ra G_2$ is a morphism of Lie groups, there is an obvious functor

mapping $P\mapsto (P\t G_2)/G_1$ on objects and $[W,\psi]\mapsto[W,(Q\t G_2)/G_1]$ on morphisms, where $G_1$ acts on $P\t G_2$ by the principal bundle action on $P$, and by $g_1:g_2\mapsto g_2\cdot\ga(g_1)^{-1}$ on $G_2$. In particular, the morphisms $\{1\}\hookra G$, $G\twoheadrightarrow\{1\}$ induce functors $\Bord_n^{\bs B}(*)\ra\Bord_n^{\bs B}(BG)$ and $\Bord_n^{\bs B}(BG)\ra\Bord_n^{\bs B}(*)$.

Similarly, a morphism of tangential structures induces a functor.

The next proposition, proved in <ref>, motivates the name bordism category, and the choice of notation `$BG$' in $\Bord_n^{\bs B}(BG)$. It shows the $\Bord_n^{\bs B}(BG)$ can be understood explicitly using homotopy-theoretic methods. The groups $\Om_n^{\bs B}(BG)$ are often explicitly computable, as in <ref>.

Work in the situation of the Definition <ref>. Then:

(i) As in Definition <ref>, write $\pi_0(\Bord_n^{\bs B}(BG))$ for the set of isomorphism classes $[X,P]$ of objects $(X,P).$ Make $\pi_0(\Bord_n^{\bs B}(BG))$ into an abelian group with product, identity, and inverses, induced by $\ot,\boo,-$ in Definition <ref>(g)–(i). Then there is a canonical isomorphism

where $BG$ is the topological classifying space of the Lie group $G$ and $\Om_n^{\bs B}(BG)$ is the bordism group with tangential $\bs B$-structures.

(ii) In Definition <ref>, there is a canonical group isomorphism

(iii) The isomorphisms bc3eq10 and bc3eq11 are compatible with change of group functors, in particular with $\{1\}\ra G.$

(iv) As in every Picard groupoid, a morphism $\la:(X_0,P_0)\ab\to(X_1,P_1)$ in $\Bord_n^{\bs B}(BG)$ determines a bijection

given by composition in the diagram of bijections

*+[r]_n+1^B(BG) [d]^bc3eq11 [r]_(0.22)≅     *+[l]__n^B(BG)((X_0,P_0),(X_1,P_1)) @=[d]
+[r]__n^B(BG)(,) [r]^(0.34)      *+[l]__n^B(BG)((X_0,P_0),(X_1,P_1)).

Theorem <ref>(b) now shows that $\Bord_n^{\bs B}(BG)$ is classified up to equivalence as a Picard groupoid by the abelian groups $\Om_n^{\bs B}(BG),\Om_{n+1}^{\bs B}(BG)$ and linear quadratic map $q:\Om_n^{\bs B}(BG)\ra\Om_{n+1}^{\bs B}(BG)$.

§.§ Bordism categories Bordₙᴮ(ℒBG)

Fix a dimension $n\ge -1$, a tangential structure $\bs B$ in the sense of <ref>, and a Lie group $G$. We will define another symmetric monoidal category $\Bord_n^{\bs B}(\cL BG)$ that we call a bordism category. It is a simple modification of Definition <ref>: we replace the principal $G$-bundles $P\ra X$, $Q\ra W$, $R\ra V$ by principal $G$-bundles $P\ra X\t\cS^1$, $Q\ra W\t\cS^1$, $R\ra V\t\cS^1$.

(a) Objects of $\Bord_n^{\bs B}(\cL BG)$ are pairs $(X,P)$, where $X$ is a compact $n$-manifold without boundary with a $\bs B$-structure $\bs\ga_X$, which we generally omit from the notation, and $P\ra X\t\cS^1$ is a principal $G$-bundle.

(b) Morphisms $[W,Q]:(X_0,P_0)\ra(X_1,P_1)$ in $\Bord_n^{\bs B}(\cL BG)$ are equivalence classes of pairs $(W,Q),$ see (c), where $W$ is a compact $(n+1)$-manifold with $\bs B$-structure $\bs\ga_W$, there is a chosen isomorphism $\pd W\cong -X_0\amalg X_1$ of the boundary preserving $\bs B$-structures (where $-X_0$ indicates that $X_0$ has the opposite $\bs B$-structure $-\bs\ga_{X_0}$), and $Q\ra W\t\cS^1$ is a principal $G$-bundle with a chosen isomorphism $Q\vert_{\pd W}\cong P_0\amalg P_1$. We suppress the isomorphisms from the notation.

(c) In the situation of (b), let $(W_0,Q_0)$ and $(W_1,Q_1)$ be two choices for $(W,Q)$. We say that $(W_0,Q_0)\sim(W_1,Q_1)$ if there exists a pair $(V,R)$, where $V$ is a compact $(n+2)$-manifold with corners and $\bs B$-structure $\bs\ga_V$, with a chosen isomorphism bc3eq1 of boundaries identifying $\bs B$-structures, such that along $\pd^2V$ we identify $\pd W_i$ with $(-X_0\amalg X_1)\t\{i\}$ for $i=0,1$ in the obvious way, and $R\ra V\t\cS^1$ is a principal $G$-bundle such that as for bc3eq2, under bc3eq1 we have

\begin{equation*}
R\vert_{\pd V\t\cS^1}\cong (P_0\t[0,1])\amalg (P_1\t[0,1])\amalg Q_0\amalg Q_1,
\end{equation*}

with the obvious compatibility with the chosen isomorphisms $Q_i\vert_{\pd W_i}\cong P_0\amalg P_1$ over $X_0\t\{i\}\amalg X_1\t\{i\}$. It is easy to see that `$\sim$' is an equivalence relation, so the equivalence classes $[W,Q]$ are well defined.

(d) If $[W,Q]:(X_0,P_0)\ra(X_1,P_1)$ and $[W',Q']:(X_1,P_1)\ra(X_2,P_2)$ are morphisms, the composition $[W',Q']\ci [W,Q]$ is defined as in bc3eq3. Composition is associative.

(e) If $(X,\phi)$ is an object in $\Bord_n^{\bs B}(\cL BG),$ the identity morphism $\id_{(X,P)}$ is defined as in bc3eq4.

(f) If $[W,Q]:(X_0,P_0)\ra(X_1,P_1)$ is a morphism, it has an inverse morphism $[W,Q]^{-1}$ defined as in bc3eq5. Thus the category $\Bord_n^{\bs B}(\cL BG)$ is a groupoid, that is, all morphisms are isomorphisms.

(g) Define a monoidal structure $\ot$ on $\Bord_n^{\bs B}(\cL BG)$ as in bc3eq6–bc3eq7.

(h) The identity in $\Bord_n^{\bs B}(\cL BG)$ is $\boo=(\es,\es)$.

(i) If $(X,\phi)\in\Bord_n^{\bs B}(\cL BG)$ we write $-(X,\phi)=(-X,\phi)$, that is, we give $X$ the opposite $\bs B$-structure $-\bs\ga_X$. As in bc3eq8, $-(X,\phi)$ is an inverse for $(X,\phi)$ under `$\ot$'.

(j) The symmetry isomorphism is as in Definition <ref>.

Hence $\Bord_n^{\bs B}(\cL BG)$ is a Picard groupoid, as in <ref>.

Here if $n=-1$, by definition the only manifold $X$ with $\dim X=-1$ is $X=\es$, so the only object in $\Bord_{-1}^{\bs B}(\cL BG)$ is $(\es,\es)$, but morphisms $[W,\psi]:(\es,\es)\ra(\es,\es)$ can still be nontrivial, with $W$ a 0-manifold.

In the case $G=\{1\}$ we will write $\Bord_n^{\bs B}(*)$ instead of $\Bord_n^{\bs B}(\cL B\{1\})$. Then the data $P\ra X\t\cS^1$, $Q\ra W\t\cS^1$ is trivial, so we may write objects of $\Bord_n^{\bs B}(*)$ as $X$, morphisms as $[W]:X_0\ra X_1$, and so on.

If $\ga:G_1\ra G_2$ is a morphism of Lie groups, as in bc3eq9 there is a functor

mapping $P\mapsto (P\t G_2)/G_1$ on objects and $[W,\psi]\mapsto[W,(Q\t G_2)/G_1]$ on morphisms. In particular, the morphisms $\{1\}\hookra G$, $G\twoheadrightarrow\{1\}$ induce functors $\Bord_n^{\bs B}(*)\ra\Bord_n^{\bs B}(\cL BG)$ and $\Bord_n^{\bs B}(\cL BG)\ra\Bord_n^{\bs B}(*)$.

Similarly, a morphism of tangential structures induces a functor.

We relate the categories of Definitions <ref> and <ref>.

Let $n\ge 0$ and $\bs B,G$ be as above. Define a functor

to act on objects by $I_n^{\bs B}(G):(X,P)\mapsto (X\t\cS^1,P)$, and on morphisms by $I_n^{\bs B}(G):[W,\psi]\mapsto [W\t\cS^1,Q]$. Here given the $\bs B$-structures on $X,W$, to define the $\bs B$-structures on $X\t\cS^1,W\t\cS^1$ we use the standard $\bs B$-structure on $\cS^1=\R/\Z$, which is invariant under the action of $\R/\Z\cong\U(1)$. So, for example, when $\bs B=\Spin$, we use the $\Spin$-structure on $\cS^1$ whose principal $\Spin(1)$-bundle is the trivial bundle $(\R/\Z)\t\Spin(1)\ra\R/\Z$. It is easy to check that $I_n^{\bs B}(G)$ is a well-defined symmetric monoidal functor. Also $I_n^{\bs B}(G_1),I_n^{\bs B}(G_2)$ commute with the change-of-group functors $F_\ga$ in bc3eq14 and bc3eq9 in the obvious way.

Here is the analogue of Proposition <ref>, proved in <ref>. It motivates the choice of notation `$\cL BG$' in $\Bord_n^{\bs B}(\cL BG)$.

Work in the situation of Definition <ref>. Then:

(i) As in Definition <ref>, write $\pi_0(\Bord_n^{\bs B}(\cL BG))$ for the set of isomorphism classes $[X,P]$ of objects $(X,P)$ in $\Bord_n^{\bs B}(\cL BG)$. Make it into an abelian group with product, identity, and inverses, induced by $\ot,\boo,-$ in Definition <ref>(g)–(i). Then there is a canonical isomorphism

where $\Om_n^{\bs B}(\cL BG)$ is the bordism group of the free loop space $\cL BG=\Map_{C^0}(\cS^1,BG)$ of the topological classifying space $BG$ of $G,$ with tangential $\bs B$-structures from <ref>.

(ii) In Definition <ref>, there is a canonical group isomorphism

(iii) The isomorphisms bc3eq16 and bc3eq17 are compatible with change of group functors, in particular with $\{1\}\ra G.$

(iv) As in every Picard groupoid, a morphism $\la:(X_0,P_0)\ab\to(X_1,P_1)$ in $\Bord_n^{\bs B}(\cL BG)$ determines a bijection

\begin{equation*}
\Om_{n+1}^{\bs B}(\cL BG)\longra\Hom_{\Bord_n^{\bs B}(\cL BG)}\bigl((X_0,P_0),(X_1,P_1)\bigr)
\end{equation*}

given by composition in the diagram of bijections

\begin{equation*}
\xymatrix@C=150pt@R=15pt{
*+[r]{\Om_{n+1}^{\bs B}(\cL BG)} \ar[d]^{\eq{bc3eq17}} \ar[r]_(0.22)\cong &  *+[l]{\Hom_{\Bord_n^{\bs B}(\cL BG)}\bigl((X_0,P_0),(X_1,P_1)\bigr)} \ar@{=}[d] \\
*+[r]{\Hom_{\Bord_n^{\bs B}(\cL BG)}(\boo,\boo)} \ar[r]^(0.34){\ot\la}  & *+[l]{\Hom_{\Bord_n^{\bs B}(\cL BG)}\bigl(\boo\ot(X_0,P_0),\boo\ot(X_1,P_1)\bigr).}
\end{equation*}

(v) There is a commutative diagram

*+[r]_n^B(BG) [d]^bc3eq17_≅[r]_ξ_n^B(BG)      *+[l]_n+1^B(BG) [d]^bc3eq11_≅
+[r]__n-1^B(BG)() [r]^I_n^B(G)_bc3eq15      *+[l]__n^B(BG)(),

where $\xi_n^{\bs B}(BG)$ is as in Definition <ref>.

Theorem <ref>(a) now shows that $\Bord_n^{\bs B}(\cL BG)$ is classified up to equivalence as a Picard groupoid by the abelian groups $\Om_n^{\bs B}(\cL BG),\Om_{n+1}^{\bs B}(\cL BG)$ and a linear quadratic form $q:\Om_n^{\bs B}(\cL BG)\ra\Om_{n+1}^{\bs B}(\cL BG)$.

§.§ Bordism categories Bordₓ(BG)

The next definition is a variation of Definition <ref>, in which we fix the $n$-manifold $X$, and take $W=X\t[0,1]$ and $V=X\t[0,1]^2$.

Let $X$ be a compact $n$-manifold and $G$ a Lie group. Define $\Bord_X(BG)$ to be the category with objects $P$ for $P\ra X$ a principal $G$-bundle, and morphisms $[Q]:P_0\ra P_1$ be $\sim$-equivalence classes $[Q]$ of principal $G$-bundles $Q\ra X\t[0,1]$ with chosen isomorphisms $Q\vert_{X\t\{i\}}\cong P_i$ for $i=0,1$. If $Q,Q'$ are alternative choices for $Q$, we write $Q\sim Q'$ if there exists a principal $G$-bundle $R\ra X\t[0,1]^2$ with chosen isomorphisms

\begin{align*}
R\vert_{X\t\{0\}\t[0,1]}&\cong P_0\t[0,1],  & R\vert_{X\t\{1\}\t[0,1]}&\cong P_1\t[0,1], \\ R\vert_{X\t[0,1]\t\{0\}}&\cong Q, & R\vert_{X\t[0,1]\t\{1\}}&\cong Q',
\end{align*}

which are compatible over $X\t\{0,1\}^2$ with the given isomorphisms $Q\vert_{X\t\{i\}}\cong P_i\cong Q'\vert_{X\t\{i\}}$. To define composition of morphisms $[Q]:P_0\ra P_1$ and $[Q']:P_1\ra P_2$ we set $[Q']\ci[Q]=[Q'']$, where $Q''\ra X\t[0,1]$ is given by $Q''\vert_{X\t\{t\}}=Q\vert_{X\t\{2t\}}$ for $t\in[0,\ha]$, and $Q''\vert_{X\t\{t\}}=Q'\vert_{X\t\{2t-1\}}$ for $t\in[\ha,1]$, and when $t=\ha$ we identify $Q''\vert_{X\t\{\frac{1}{2}\}}=Q\vert_{X\t\{1\}}=Q'\vert_{X\t\{0\}}$ via the given isomorphisms $Q\vert_{X\t\{1\}}\cong P_1\cong Q'\vert_{X\t\{0\}}$. To define the smooth structure on $Q''$ near $X\t\{\ha\}$ we use collars as in Definition <ref>(d).

It is then easy to show that composition is associative, so that $\Bord_X(BG)$ is a category, where identity morphisms are $\id_P=[P\t[0,1]]:P\ra P$. Every morphism in $\Bord_X(BG)$ is invertible, where the inverse of $[Q]:P_0\ra P_1$ is $[Q]^{-1}=[Q']:P_1\ra P_0$, with $Q'\vert_{X\t\{t\}}=Q\vert_{X\t\{1-t\}}$ for $t\in[0,1]$.

Now suppose that $\bs B$ is a tangential structure, and $X$ has a $\bs B$-structure $\bs\ga_X$. Since the stable tangent bundles of $X\t[0,1]$ and $X\t[0,1]^2$ are the pullbacks of the stable tangent bundle of $X$, pullback of $\bs\ga_X$ along the projections $X\t[0,1]\ra X$, $X\t[0,1]^2\ra X$ induces $\bs B$-structures on $X\t[0,1]$ and $X\t[0,1]^2$. Define a functor

to map $P\mapsto(X,P)$ on objects and $[Q]\mapsto\bigl[X\t[0,1],Q\bigr]$ on morphisms, using the $\bs B$-structures on $X,X\t[0,1]$. This is well defined as writing $W=X\t[0,1]$ and $V=X\t[0,1]^2$, the definitions above of the equivalence $\sim$ on $Q$ and $(X\t[0,1],Q)$, and of compositions of morphisms, and so on, map to those in Definition <ref>.

If $P\ra X$ is a principal $G$-bundle, we write $\Bord_X(BG)_P\subset\Bord_X(BG)$ to be the full subcategory with one object $P$ in $\Bord_X(BG)$. Write $\Pi_{X,P}^{\bs B}$ for the restriction of $\Pi_X^{\bs B}$ to $\Bord_X(BG)_P\subset\Bord_X(BG)$.

If $\ga:G_1\ra G_2$ is a morphism of Lie groups as for bc3eq9 there is a functor

mapping $(X,P)\mapsto (X,(P\t G_2)/G_1)$ on objects and $[Q]\mapsto[(Q\t G_2)/G_1]$ on morphisms.

Next suppose $X,X'$ are compact $n$-manifolds with $\bs B$-structures $\bs\ga_X,\bs\ga_{X'}$, and set $X''=X\amalg X'$ with $\bs B$-structure $\bs\ga_{X''}=\bs\ga_X\amalg\bs\ga_{X'}$. There is a diagram of functors and natural transformations:

\begin{equation*}
\xymatrix@C=180pt@R=15pt{
*+[r]{\Bord_X(BG)\t\Bord_{X'}(BG)} \ar[r]_(0.6){\amalg} \ar[d]^{\Pi_X^{\bs B}\t\Pi_{X'}^{\bs B}} \drtwocell_{}\omit^{}\omit{^{\al_{X,X'}^{\bs B}\,\,\,\,\,\,\,\,\,\,\,\,\,}} & *+[l]{\Bord_{X''}(BG)} \ar[d]_{\Pi_{X''}^{\bs B}}  \\
*+[r]{\Bord_n^{\bs B}(BG)\t\Bord_n^{\bs B}(BG)} \ar[r]^(0.6)\ot &*+[l]{\Bord_n^{\bs B}(BG).}
\end{equation*}

Here the functor $\amalg$ on the top row acts by $(P,P')\mapsto P\amalg P'$ on objects and $([Q],[Q'])\mapsto[Q\amalg Q']$ on morphisms, and the functor $\ot$ on the bottom row is the monoidal structure on $\Bord_n^{\bs B}(BG)$ from Definition <ref>. The natural isomorphism $\al_{X,X'}^{\bs B}$ is just the identity, mapping $(P,P')\mapsto\id_{(X\amalg X',\bs\ga_X\amalg\bs\ga_{X'},P\amalg P')}$.

In a similar way to Propositions <ref> and <ref>, we can use homotopy theory to give a partial description of the categories $\Bord_X(BG)$ and functors $\Pi_X^{\bs B}$. The next proposition is proved in <ref>.

Suppose $\bs B$ is a tangential structure, $X$ a compact $n$-manifold with $\bs B$-structure $\bs\ga_X,$ $G$ a Lie group, and $P\ra X$ a principal $G$-bundle. Then $P$ is an object in $\Bord_X(BG),$ and $(X,\bs\ga_X,P)$ an object in $\Bord_n^{\bs B}(BG),$ and $\Pi_X^{\bs B}:P\mapsto(X,\bs\ga_X,P)$. We have a commutative diagram

*+[r]__X(BG)(P) [r]_(0.45)Π_X^B [d]^χ_P^B     *+[l]__n^B(BG)(X,_X,P)
+[r]_n^B(BG) [r]^(0.45)ξ_n^B(BG)     *+[l]_n+1^B(BG), [u]^bc3eq12_≅

where $\xi_n^{\bs B}(BG)$ is in Definition <ref>, the right hand column is the bijection  bc3eq12, and $\chi_P^{\bs B}$ is defined as follows: let $\phi_P:X\ra BG$ be a classifying map for $P$. Then for $[Q]:P\ra P$ in $\Aut_{\Bord_X(BG)}(P),$ as $Q\ra X\t[0,1]$ is a principal $G$-bundle with chosen isomorphisms $Q\vert_{X\t\{0\}}\cong P\cong Q\vert_{X\t\{1\}},$ we can choose a classifying map $\phi_Q:X\t[0,1]\ra BG$ for $Q$ such that $\phi_Q\vert_{X\t\{0\}}=\phi_Q\vert_{X\t\{1\}}=\phi_P$. Writing $\cS^1=\R/\Z=[0,1]/(0\sim 1)$ with projection $\pi:[0,1]\ra\cS^1,$ define $\bar\phi_Q:X\t\cS^1\ra BG$ by $\bar\phi_Q\ci(\id_X\t\pi)=\phi_Q$. Let $\ti\phi_Q:X\ra \cL BG=\Map_{C^0}(\cS^1,BG)$ be the induced map. Then define

§.§ Orientation functors on categories Bordₓ(BG)

We now come to the bordism-theoretic point of view on the orientations from <ref>. For this, we first rephrase the $\G_P$-equivariance of the orientation bundles $\hat O^{E_\bu}_P\ra\A_P$ for the gauge group action in terms of the functoriality of constructions $\sO_X^{E_\bu,G}:\Bord_X(BG)\ra\sZtor$. While this is straightforward, it sets the stage for the generalization stated in the next section.

We define the categories $\Ztor, \sZtor.$ Recall that a $\Z_2$-torsor is just a two point set $T=\{a,b\}$, with the obvious free transitive $\Z_2$-action. Morphisms of $\Z_2$-torsors are bijections $\io:\{a,b\}\ra\{c,d\}$. Write $\Ztor$ for the category of $\Z_2$-torsors, which is a Picard groupoid in the sense of <ref>, with unit object $\Z_2$ and monoidal structure $T\ot T'=(T\t T')/\Z_2,$ where we take the quotient by the anti-diagonal $\Z_2$-action on $T\t T'.$ The symmetric structure identifies $T\ot T'\cong T'\ot T$ by $(t,t')\Z_2\cong(t',t)\Z_2$ in the obvious way.

A super $\Z_2$-torsor (also called a $\Z_2$-graded $\Z_2$-torsor) is a pair $(T,(-1)^\ep)$ of a $\Z_2$-torsor $T$ and $\ep\in\Z_2=\{\ul 0,\ul 1\}.$ Morphisms $(T,(-1)^\ep)\ra (T',(-1)^{\ep'})$ are only defined if $\ep=\ep'$, and then are morphisms of $\Z_2$-torsors $\io:T\ra T'$. Write $\sZtor$ for the category of super $\Z_2$-torsors, a Picard groupoid with identity $(\Z_2,(-1)^{\ul{0}})$ and monoidal structure

\begin{equation*}
\bigl(T,(-1)^\ep\bigr)\ot \bigl(T',(-1)^{\ep'}\bigr)=\bigl(T\ot_{\Z_2}T',(-1)^{\ep+\ep'}\bigr).
\end{equation*}

The nontrivial part of the definition is in the symmetric structure, which identifies $\bigl(T,(-1)^\ep\bigr)\ot \bigl(T',(-1)^{\ep'}\bigr)\ra \bigl(T',(-1)^{\ep'}\bigr)\ot \bigl(T,(-1)^\ep\bigr)$ by

\begin{equation*}
\bigl((t,t')\Z_2,(-1)^{\ep+\ep'}\bigr)\cong\bigl((-1)^{\ep\ep'}(t',t)\Z_2,(-1)^{\ep+\ep'}\bigr),
\end{equation*}

that is, the natural isomorphism $T\ot T'\cong T'\ot T$ is twisted by $(-1)^{\ep\ep'}$.

There is an inclusion functor $I_\sZtor:\Ztor\ra\sZtor$ mapping $T\mapsto (T,(-1)^{\ul 0})$ on objects which preserves symmetric monoidal structures. There is also a forgetful functor $\Pi_\Ztor:\sZtor\ra\Ztor$ mapping $(T,(-1)^\ep)\mapsto T$ on objects which preserves monoidal structures, but not symmetric structures.

We will use (super) $\Z_2$-torsors to study orientations on gauge-theoretic moduli spaces as in <ref>. In the situation of Definition <ref> the set of orientations on $\det(D^{\nabla_{\Ad(P)}})\cong\R$ is a nonempty $\Z_2$-torsor $\Or(\det(D^{\nabla_{\Ad(P)}})),$ which we make into a super $\Z_2$-torsor by placing it in degree $(-1)^{\ind D^{\nabla_{\Ad(P)}}}$ (of course, we can even define a $\Z$-graded $\Z_2$-torsor in degree $\ind D^{\nabla_{\Ad(P)}}\in\Z,$ but only the induced $\Z_2$-grading is important for the symmetric structure).

If we work only on one fixed manifold $X$, then the `super' part is not important and we could work with ordinary $\Z_2$-torsors. But in <ref> we will want to compare gauge theory orientations over $X_1,X_2,$ and $X_1\amalg X_2,$ and to do this we will need the twisted symmetric structure on $\sZtor$.

As in Definition <ref> let $X$ be a compact manifold, $E_\bu$ an elliptic differential operator on $X,$ and $G$ a Lie group. We now define a functor
that encodes the orientations on $\A_P,\B_P$ from <ref>.

Recall the principal $\Z_2$-bundle of orientations $\hat O^{E_\bu}_P\ra\A_P$ from Definition <ref>. Since $\A_P$ is contractible, the space of global continuous sections $\Ga_{C^0}(\hat O^{E_\bu}_P)$ is a nonempty $\Z_2$-torsor. Moreover, the index $\ind D^{\nabla_{\Ad(P)}}$ of $E_\bu$ twisted by $\nabla_P\in\A_P$ is independent of $\nabla_P$. We define $\sO_X^{E_\bu,G}$ on objects $P\ra X$ in $\Bord_X(BG)$ by

\begin{align*}
\sO_X^{E_\bu,G}(P)&=\bigl(\Ga_{C^0}(\hat O^{E_\bu}_P),(-1)^{\ind D^{\nabla_{\Ad(P)}}}\bigr).
\end{align*}

If $[Q]:P_0\ra P_1$ is a morphism in $\Bord_X(BG),$ then $Q_t=Q\vert_{X\t\{t\}}$ is a principal $G$-bundle over $X$ for all $t\!\in\![0,1]$, and so defines a $\Z_2$-torsor $\Ga_{C^0}(\hat O^{E_\bu}_{Q_t})$. This depends continuously on $t\in[0,1]$ and agrees with $\Ga_{C^0}(\hat O^{E_\bu}_{P_t})$ at $t=0,1$ via the given isomorphisms $Q_t\vert_{X\t\{t\}}\cong P_t.$ Parallel transport in $[0,1]$ along the continuous family of $\Z_2$-torsors determines an isomorphism of $\Z_2$-torsors

\begin{equation*}
\sO_X^{E_\bu,G}(Q):\Ga_{C^0}(\hat O^{E_\bu}_{P_0})\longra\Ga_{C^0}(\hat O^{E_\bu}_{P_1}).
\end{equation*}

Since $P_0,P_1$ are isomorphic, $\ind D^{\nabla_{\Ad(P_0)}}=\ind D^{\nabla_{\Ad(P_1)}}$ and thus $\sO_X^{E_\bu,G}(Q):\sO_X^{E_\bu,G}(P_0)\ra\sO_X^{E_\bu,G}(P_1)$ is a morphism in $\sZtor$. This defines $\sO_X^{E_\bu,G}$ on morphisms, and it is easy to check that is defines a functor bc3eq23.

For a fixed principal $G$-bundle $P\ra X$ we also write $\sO_{X,P}^{E_\bu,G}$ for the restriction of $\sO_X^{E_\bu,G}$ to the subcategory $\Bord_X(BG)_P\subseteq\Bord_X(BG)$ from Definition <ref>.

In Definition <ref> we explained that if $E_\bu$ is skew-adjoint then the orientation bundle $\hat O^{E_\bu}_P\ra\A_P$ is canonically trivial, but we can instead define the Pfaffian orientation bundle $\hat O^{E_\bu}_{\Pf,P}\ra\A_P$. This gives an analogue of Definition <ref>.

As in Definition <ref> let $X$ be a compact manifold, $E_\bu$ a skew-adjoint elliptic differential operator on $X,$ and $G$ a Lie group. We define a functor
that encodes the Pfaffian orientations on $\A_P,\B_P$ from <ref>. The definition is exactly as in Definition <ref>, but replacing $\hat O^{E_\bu}_P\ra\A_P$ by $\hat O^{E_\bu}_{\Pf,P}\ra\A_P$ throughout. For a fixed principal $G$-bundle $P\ra X$ we also write $\sO_{\Pf,X,P}^{E_\bu,G}$ for the restriction of $\sO_{\Pf,X}^{E_\bu,G}$ to the subcategory $\Bord_X(BG)_P\subseteq\Bord_X(BG)$ from Definition <ref>.

The next proposition expresses the orientations on $\B_P$ from <ref> in terms of natural isomorphisms of the functor $\sO_{X,P}^{E_\bu,G}$. The proof is straightforward: the fact the functor $\boo:\Bord_X(BG)_P\ra\Ztor$ that takes every object to $\Z_2$ and every morphism to $\id_{\Z_2},$ maps all morphisms $P\ra P$ in $\Bord_X(BG)_P$ to $\id_{\Z_2}$ means that the orientations on $\A_P$ chosen by the natural isomorphism $\be_P$ are invariant under the action of $\G_P$ on $\A_P$, and therefore descend to $\B_P$.

Work in the situation of Definition <ref>. Let $P\ra X$ be a principal $G$-bundle, and consider the existence of a natural isomorphism $\be_P$ in the diagram of functors

*+[r]_X(BG)_P [rrrr]^(0.55)_X,P^E_,G @/_1pc/[drrrr]_(0.4)    _^_     _P           *+[l] [d]_Π_

(a) Such $\be_P$ exists if and only if $\B_P$ is orientable in the sense of <ref>.

(b) A choice of $\be_P$ is equivalent to an orientation on $\B_P$.

If $E_\bu$ is skew-adjoint, the analogue holds for $\sO_{\Pf,X,P}^{E_\bu,G}$ in Definition <ref> and Pfaffian orientations in Definition <ref>.

Let $X,X'$ be compact $n$-manifolds, $E_\bu,E'_\bu$ linear elliptic partial differential operators on $X,X'$ of the same order, and $G$ a Lie group. Set $X''=X\amalg X'$, and let $E''_\bu=E_\bu\amalg E'_\bu$, which is an elliptic operator on $X''$. We will define a diagram of functors and natural transformations:

\begin{equation*}
\xymatrix@C=180pt@R=15pt{
*+[r]{\Bord_X(BG)\t\Bord_{X'}(BG)} \ar[r]_(0.6){\amalg} \ar[d]^{\sO_X^{E_\bu,G}\t\sO_{X'}^{E'_\bu,G}} \drtwocell_{}\omit^{}\omit{^{\ga_{X,X'}^{E_\bu,E'_\bu,G}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}} & *+[l]{\Bord_{X''}(BG)} \ar[d]_{\sO_{X''}^{E''_\bu,G}}  \\
*+[r]{\sZtor\t\sZtor} \ar[r]^(0.6)\ot &*+[l]{\sZtor.}
\end{equation*}

Here the columns are from Definition <ref>, the functor $\amalg $ on the top row acts by $(P,P')\mapsto P\amalg P'$ on objects and $([Q],[Q'])\mapsto[Q\amalg Q']$ on morphisms, and the functor $\ot$ on the bottom row is the monoidal structure on $\sZtor$ from Definition <ref>. To define the natural isomorphism $\ga_{X,X'}^{E_\bu,E'_\bu,G}$ for all principal $G$-bundles $P\ra X$, $P'\ra X'$ we have to define an isomorphism of $\Z_2$-torsors

\begin{equation*}
\ga_{X,X'}^{E_\bu,E'_\bu,G}(P,P'):\Ga_{C^0}(\hat O^{E_\bu}_P)\ot_{\Z_2}\Ga_{C^0}(\hat O^{E'_\bu}_{P'})\longra\Ga_{C^0}(\hat O^{E''_\bu}_{P\amalg P'})
\end{equation*}

satisfying certain commutation conditions. After choosing $\nabla_P\in\A_P$, $\nabla_{P'}\in\A_{P'}$, this is equivalent to defining an isomorphism

\begin{equation*}
\Or\bigl(\det(D^{\nabla_{\Ad(P)}})\bigr)\ot_{\Z_2}\Or\bigl(\det(D^{\nabla_{\Ad(P')}})\bigr)\longra\Or\bigl(\det(D^{\nabla_{\Ad(P\amalg P')}})\bigr).
\end{equation*}

There is a natural choice, but it requires an orientation convention, and it depends on the order of $X,X'$: exchanging $X,X'$ changes the sign of the isomorphism by a factor $(-1)^{\ind D^{\nabla_{\Ad(P)}}\cdot\ind D^{\nabla_{\Ad(P')}}}$. It is then easy to check that $\ga_{X,X'}^{E_\bu,E'_\bu,G}$ is a natural isomorphism. If $E_\bu,E'_\bu$ are skew-adjoint, the analogue holds for $\sO_{\Pf,X}^{E_\bu,G},\sO_{\Pf,X'}^{E_\bu',G},\sO_{\Pf,X''}^{E_\bu'',G}$ in the obvious way.

We applied the projection $\Pi_\Ztor$ in bc3eq25 as if we had just considered functors to $\sZtor$, we would have had to replace the functors $\boo$ by functors mapping $P$ to either $(\Z_2,1)$ or $(\Z_2,-1)$, depending on whether $\ind D^{\nabla_{\Ad(P)}}$ is even or odd. This is an indication that we should expect difficulties in choosing canonical orientations in problems when $\ind D^{\nabla_{\Ad(P)}}$ can be odd, as these might not respect the symmetric structures, and the sign changes mentioned in Definition <ref>.

§.§ Factorizing orientation functors via Bordₙᴮ(BG)

So far, most of <ref>–<ref> is just notation, and Proposition <ref> is just an alternative point of view on orientations on $\B_P$. We now introduce a powerful new technique, the bordism invariance of orientations: for certain elliptic operators $E_\bu$ and tangential structures $\bs B$, the functor $\sO_X^{E_\bu,G}\colon\Bord_X(BG)\ra\sZtor$ from bc3eq23 factorizes via a functor $\sO_n^{\bs B}\colon\Bord_n^{\bs B}(BG)\ra\sZtor$.

When this works it is a useful tool for analyzing orientation problems, as we can reduce questions about orientability and choice of orientations to questions about morphisms $\Om_n^{\bs B}(\cL BG;BG)\ra\Z_2$, which can then hopefully be answered by explicit computation. Effectively, we study gauge theory orientations on all $n$-manifolds $X$ at once by factoring the gauge group action through a more fundamental action of a `categorical group' of bordisms between possibly different manifolds as in <ref>.

We restrict ourselves here to the important case of real Dirac operators. This handles all the applications we have in mind, which are to orienting moduli spaces of $G_2$-instantons on compact $G_2$-manifolds, to $\Spin(7)$-instantons on compact $\Spin(7)$-manifolds, and to coherent sheaves on Calabi–Yau 4-folds for DT4 invariants, see <ref>–<ref>. There are probably other interesting classes of elliptic operators for which this approach also works.

The next theorem is proved in the second author <cit.>. The restriction to dimensions $n\equiv 1,7,8\pmod 8$ is because in dimensions $n\equiv 2,3,4,5,6\pmod 8$ the real Dirac operator $\slashed{D}$ is $\C$- or $\H$-linear, so moduli spaces $_P$ with orientation bundles $O^E__P$ have canonical orientations for essentially trivial reasons.

\begin{thm}
\label{bc3thm1}
Let\/ $\bs B$ be a tangential structure factoring via\/ $\Spin,$ and\/ $G$ be a Lie group. For all\/ $n\ge 0$ with\/ $n\equiv 1,7,$ or\/ $8\pmod 8$ there exists a functor
\e
\sO_n^{\bs B}\colon\Bord_n^{\bs B}(BG)\longra\sZtor,
\label{bc3eq26}
\e
which maps to\/ $\Ztor\subset\sZtor$ if\/ $n\equiv 7\pmod 8,$ such that:
\begin{itemize}
\setlength{\itemsep}{0pt}
\setlength{\parsep}{0pt}
\item[{\bf(a)}] $\sO_n^{\bs B}$ is a symmetric monoidal functor.
\item[{\bf(b)}] Let\/ $X$ be a compact\/ $n$-manifold with a\/ $\bs B$-structure and define an elliptic operator\/ $E_\bu$ on\/ $X$ by
\e
\text{skew Dirac operator $\slashed{D}_X^\skew$}& \text{if\/ $n\equiv 1\pmod{8},$}\\
\text{Dirac operator $\slashed{D}_X$} & \text{if\/ $n\equiv 7\pmod{8},$}\\
\text{positive Dirac operator $\slashed{D}_X^+$}& \text{if\/ $n\equiv 8\pmod{8}.$}
\end{cases}
\label{bc3eq27}
\e
Then if\/ $n\equiv 7$ or\/ $8\pmod 8$ there exists a natural isomorphism\/ $\de_X^{E_\bu,G}$ making the following diagram commute:
\e
\begin{gathered}
\xymatrix@!0@C=32pt@R=30pt{
*+[r]{\Bord_X(BG)} \ar[rrrr]^(0.48){\Pi_X^{\bs B}} \ar@/_1pc/[drrrr]_(0.35){\sO_X^{E_\bu,G}} & \drrtwocell_{}\omit^{}\omit{_{\,\,\,\,\,\,\,\,\,\,\,\,\de_X^{E_\bu,G}}} &&& *+[l]{\Bord_n^{\bs B}(BG)} \ar[d]_{\sO_n^{\bs B}}  \\
&&&& *+[l]{\sZtor.}
\end{gathered}
\label{bc3eq28}
\e
If\/ $n\equiv 1\pmod 8$ then\/ $E_\bu$ is skew-adjoint, and the analogue of \eq{bc3eq28} holds with\/ $\sO_{\Pf,X}^{E_\bu,G}$ from \eq{bc3eq24} in place of\/ $\sO_X^{E_\bu,G}$. In other words,\/ $\sO_n^{\bs B}$ in \eq{bc3eq26} encodes gauge theory orientations \textup(or Pfaffian orientations\textup) as in Proposition\/~\textup{\ref{bc3prop4}} for the Dirac operators \eq{bc3eq27}.

\item[{\bf(c)}] Let\/ $\io:G\ra H$ be a morphism of Lie groups of complex type in the sense of Definition\/ {\rm\ref{bc2def4}}. Then for\/ $F_\io$ as in \eq{bc3eq9} there exists a canonical natural isomorphism\/ $\ep_{n,G}^{\bs B,H}$ making the following diagram commute:
\e
\begin{gathered}
\xymatrix@!0@C=32pt@R=30pt{
*+[r]{\Bord_n^{\bs B}(BG)} \ar[rrrr]^(0.48){F_\io} \ar@/_1pc/[drrrr]_(0.35){\sO_n^{\bs B}} & \drrtwocell_{}\omit^{}\omit{_{\,\,\,\,\,\,\,\,\,\,\,\,\ep_{n,G}^{\bs B,H}}} &&& *+[l]{\Bord_n^{\bs B}(BH)} \ar[d]_{\sO_n^{\bs B}}  \\
&&&& *+[l]{\sZtor.}
\end{gathered}
\label{bc3eq29}
\e
\item[{\bf(d)}] Let\/ $G_1,G_2$ be Lie groups. Then there exists a canonical natural isomorphism\/ $\ze_{n,G_1,G_2}^{\bs B}$ making the following diagram commute:
\end{itemize}
\ea
\begin{gathered}
\xymatrix@C=175pt@R=15pt{
*+[r]{\Bord_n^{\bs B}(B(G_1\t G_2))} \ar[r]_(0.7){\sO_n^{\bs B,G_1\t G_2}} \ar[d]^{(F_{\Pi_{G_1}},F_{\Pi_{G_2}})} \drtwocell_{}\omit^{}\omit{^{\ze_{n,G_1,G_2}^{\bs B}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}} & *+[l]{\sZtor}  \\
*+[r]{\Bord_n^{\bs B}(BG_1)\t\Bord_n^{\bs B}(BG_1)} \ar[r]^(0.62){\sO_n^{\bs B,G_1}\t\sO_n^{\bs B,G_2}} & *+[l]{\sZtor\!\t\!\sZtor.} \ar[u]^\ot }
\end{gathered}
\label{bc3eq30}
\ea
\end{thm}

\subsection{Applications to orientability of moduli spaces}
\label{bc36}

Here is a criterion, proved in \S\ref{bc54}, for when the moduli spaces $_P$ in \S\ref{bc21} for a principal $G$-bundle $PX$ are orientable for all possible~$X,P$.

\begin{thm}
\label{bc3thm2}
Work in the situation of Theorem\/ {\rm\ref{bc3thm1},} with\/ $n\equiv 1,7,8\pmod{8}$ and\/ $G$ a fixed Lie group. Consider the commutative diagram
\e
\begin{gathered}
\xymatrix@!0@C=90pt@R=35pt{
& \Aut_{\Bord_{n-1}^{\bs B}(\cL BG)}(\boo)  \ar[rr]^(0.37){I_n^{\bs B}(G)}_(0.37){\eq{bc3eq15}} && *+[l]{\Aut_{\Bord_n^{\bs B}(BG)}(\boo)} \ar[dd]^(0.4){\sO_n^{\bs B}}_(0.4){\eq{bc3eq26}} \\
*+[r]{\Om_n^{\bs B}(\cL BG)} \ar[ur]^{\eq{bc3eq17}}_\cong \ar[d]^{\Pi_n^{\bs B}(BG)} \ar[rr]^(0.4){\xi_n^{\bs B}(BG)}_(0.4){\eq{bc2eq17}}  && \Om_{n+1}^{\bs B}(BG)  \ar[ur]^(0.4){\eq{bc3eq11}}_(0.4)\cong \\
*+[r]{\Om_n^{\bs B}(\cL BG;BG)} \ar@{..>}[rrr]^(0.43){\Xi_n^{\bs B,G}} \ar[urr]_(0.8){\ti\xi_n^{\bs B}(BG)} &&& *+[l]{\Z_2=\Aut_{\sZtor}(\Z_2,\ul{0}),\!} }\!\!\!
\end{gathered}
\label{bc3eq31}
\e
where\/ $\xi_n^{\bs B}(BG),$ $\ti\xi_n^{\bs B}(BG),$ $\Pi_n^{\bs B}(BG)$ are as in Definition\/ {\rm\ref{bc2def8}}. The top parallelogram commutes by \eq{bc3eq18}. The bottom left triangle commutes by \eq{bc2eq18}. Define\/ $\Xi_n^{\bs B,G}$ to be the unique morphism making the bottom right quadrilateral commute.

Then\/ $\B_P$ is orientable \textup(if\/ $n\equiv 7,8\pmod{8}$\textup) or Pfaffian orientable \textup(if\/ $n\equiv 1\pmod{8}$\textup) for every compact Riemannian\/ $n$-manifold\/ $(X,g)$ with\/ $\bs B$-structure\/ $\bs\ga_X$ and every principal\/ $G$-bundle\/ $P\ra X$ if and only if\/ $\Xi_n^{\bs B,G}\equiv\ul{0},$ where the \textup(Pfaffian\textup) orientation bundles\/ $O^{E_\bu}_P,O^{E_\bu}_{\Pf,P}\ra\B_P$ are defined using\/ $E_\bu$ as in~\eq{bc3eq27}.
\end{thm}

The next proposition follows easily from Theorems \ref{bc3thm1}(c),(d) and \ref{bc3thm2}.

\begin{prop}
\label{bc3prop5}
The morphisms\/ $\Xi_n^{\bs B,G}$ in Theorem\/ {\rm\ref{bc3thm2}} satisfy:
\begin{itemize}
\setlength{\itemsep}{0pt}
\setlength{\parsep}{0pt}
\item[{\bf(a)}] Let\/ $\io:G\ra H$ be a morphism of Lie groups of complex type, in the sense of Definition\/ {\rm\ref{bc2def4}}. Then the following diagram commutes:
\begin{equation*}
\xymatrix@C=140pt@R=15pt{
*+[r]{\Om_n^{\bs B}(\cL BG;BG)} \ar[r]_{B\io_\rel^{\bs B}}<EMAIL_ADDRESS>B,G}} & *+[l]{\Om_n^{\bs B}(\cL BH;BH)} \ar[d]_{\Xi_n^{\bs B,H}} \\
& *+[l]{\Z_2.\!} }
\end{equation*}
\item[{\bf(b)}] Let\/ $G_1,G_2$ be Lie groups. Then the following diagram commutes:
\begin{equation*}
\xymatrix@C=186pt@R=15pt{
*+[r]{\Om_n^{\bs B}(\cL B(G_1\t G_2);B(G_1\t G_2))} \ar[r]_(0.75){\Xi_n^{\bs B,G_1\t G_2}} \ar[d]^{((B\Pi_{G_1})^{\bs B}_\rel,(B\Pi_{G_2})^{\bs B}_\rel)} & *+[l]{\Z_2}  \\
*+[r]{\Om_n^{\bs B}(\cL BG_1;BG_1)\t\Om_n^{\bs B}(\cL BG_2;BG_2)} \ar[r]^(0.75){\Xi_n^{\bs B,G_1}\t\Xi_n^{\bs B,G_2}} & *+[l]{\Z_2\t\Z_2.\!} \ar[u]^+ }
\end{equation*}
\end{itemize}
\end{prop}

We now restrict to $B=$ and $n=7,8$ for the rest of \S\ref{bc3}. The next theorem will be proved in~\S\ref{bc55}.

\begin{thm}
\label{bc3thm3}
{\bf(a)} In Theorem\/ {\rm\ref{bc3thm2}} let\/ $n=7$ or\/ $8,$ $\bs B=\Spin,$ and\/ $G$ be one of the following compact, connected Lie groups:
\e
\label{bc3eq32}
E_8,\; E_7,\; E_6,\; G_2,\; \Spin(3),\; \SU(m),\; \U(m),\; \Spin(2m),
\quad
\text{where $m\ge 1.$}
\e
\textup(Here,\/ $E_6,$ $E_7$ are the simply-connected versions.\textup)

Then\/ $\Xi_n^{\Spin,G}=0$ in \eq{bc3eq31}, so\/ $\B_P$ is orientable for every compact spin Riemannian\/ $n$-manifold\/ $(X,g)$ and every principal\/ $G$-bundle\/~$P\ra X$.
\smallskip

\noindent{\bf(b)} Part {\bf(a)} also holds if\/ $G$ is any finite product of the groups in \eq{bc3eq32}.
\smallskip

\noindent{\bf(c)} Let\/ $G=(G_1\t\cdots\t G_k)/K$ be the quotient of any finite product of groups\/ $G_1\t\cdots\t G_k$ in \eq{bc3eq32} by a finite normal subgroup\/ $K,$ e.g.\ $G=\SO(2m)=\Spin(2m)/\Z_2$ or\/ $G=\mathop{\rm PSU}(m)=\SU(m)/\Z_m$. Then\/ $\B_P$ is orientable for every compact, \begin{bfseries}simply-connected\end{bfseries} spin Riemannian\/ $n$-manifold\/ $(X,g)$ and every principal\/ $G$-bundle\/~$P\ra X$.
\end{thm}

Here is an outline of the proof:
\begin{itemize}
\setlength{\itemsep}{0pt}
\setlength{\parsep}{0pt}
\item[(i)] Theorems \ref{bc2thm3}, \ref{bc2thm4} and \ref{bc3thm2} imply that $\Xi_n^{\Spin,\SU(m)}=0$ for $n=7,8$.
\item[(ii)] Using Theorem \ref{bc2thm7} we deduce that $\Xi_n^{\Spin,E_8}=0$.
\item[(iii)] Theorem \ref{bc3thm3}(a),(b) then follow from Proposition \ref{bc3prop5} and Theorem \ref{bc2thm2}.
\item[(iv)] We deduce (c) from the fact that $(G_1\t\cdots\t G_k)/K$-bundles can be lifted to $G_1\t\cdots\t G_k$-bundles over simply-connected manifolds.
\end{itemize}

The next theorem will be proved in \S\ref{bc56}. The proof uses examples in \cite[\S 2.4]{JoUp1} and \cite[Ex.~1.14]{CGJ} of principal $(2)$-bundles $X(2)X$ with $_X(2)$ non-orientable for $X=7,8$, and Proposition \ref{bc3prop5}.

\begin{thm}
\label{bc3thm4}
{\bf(a)} In Theorem\/ {\rm\ref{bc3thm2},} let\/ $n=7$ or\/ $8,$ $\bs B=\Spin,$ and\/ $G$ be one of the following compact, connected Lie groups:
\e
\begin{aligned}
& F_4, &
&\Sp(m+1), &
\quad
\text{where $m\ge 1.$}
\end{aligned}
\label{bc3eq33}
\e

Then\/ $\Xi_n^{\Spin,G}\ne 0$. Hence there exists a compact spin Riemannian\/ $n$-manifold\/ $(X,g)$ and a principal\/ $G$-bundle $P\ra X$ for which\/ $\B_P$ is not orientable. This is the case for\/ $X=\Sp(2)\t_{\Sp(1)\t\Sp(1)}\Sp(1)$ when\/ $n=7$ and\/ $X=(\Sp(2)\t_{\Sp(1)\t\Sp(1)}\Sp(1))\t\cS^1$ when\/ $n=8,$ and\/~$P=X\t G$.
\smallskip

\noindent{\bf(b)} Part {\bf(a)} also holds for any Lie group\/ $G=(G_1\t G_2)/K,$ where\/ $G_1$ is on the list\/ {\rm\eq{bc3eq33}},\/ $G_2$ is any Lie group, and\/ $K$ is a discrete normal subgroup of\/ $G_1\t G_2$. For example, we can take\/ $G=\SO(2m+3)=(\Spin(2m+3)\t\{1\})/\Z_2$.
\end{thm}

Now \eq{bc3eq32}--\eq{bc3eq33} include $(1)$ and all compact, simply-connected, simple Lie groups. But by the classification of Lie groups, every compact, connected Lie group $G$ is of the form $G=((1)^kG_1⋯G_l)/K$, where $G_1,…,G_l$ are compact, simply-connected, simple Lie groups and $K⊂(1)^kG_1⋯G_l$ is a finite normal subgroup. Thus we deduce:

\begin{cor}
\label{bc3cor1}
Every compact, connected Lie group\/ $G$ satisfies the conditions of either Theorem\/ {\rm\ref{bc3thm3}(c),} or Theorem\/ {\rm\ref{bc3thm4}(b),} but not both. Thus, for every compact, connected Lie group\/ $G,$ Theorems\/ {\rm\ref{bc3thm3}--\ref{bc3thm4}} provide a complete answer to when\/ $\B_P$ is orientable for all compact, simply-connected spin Riemannian\/ $n$-manifolds\/ $(X,g)$ with\/ $n=7,8$ and principal\/ $G$-bundles\/ $P\ra X$.
\end{cor}

\subsection{Applications to Floer gradings}
\label{bc37}

For $n≡3,78,$ the second author actually proves a stronger version of Theorem \ref{bc3thm1} in \cite[\S 3.2]{Upme2}: Since the Dirac operator $E_$ from \eqref{bc3eq27} is self-adjoint in these dimensions, we can improve Definition~\ref{bc3def6} and actually define a functor $_,X^E_,G_X(BG),$ which is closely related to the idea of Floer gradings, and [71] shows that this functor factors via $_n^B(BG).$ Hence bordism techniques are available for the study of Floer gradings. Based on Theorem \ref{bc2thm6} we show that for principal $(2)$ (or $(3)$) bundles, mod-$8$ (or mod-$6$) Floer gradings exist on the moduli spaces of $G_2$-instantons.

\begin{dfn}
\label{bc3def9}
There is a principal $\Z$-bundle $\hat\Sp{}_P^{E_\bu}\ra\cA_P,$ the \emph{spectral bundle}, whose fiber over $\na_P$ is the set of enumerations $\cdots\le\la_{-1}\le\la_0\le\la_1\le\cdots$ (with eigenvalues repeated according to multiplicity) of the spectrum of the self-adjoint twisted Dirac operator $\slashed{D}^{\na_{\Ad(P)}}$; for the definition of the topology on the bundle $\hat\Sp{}_P^{E_\bu}\ra\cA_P$ we refer to the second author's paper~\cite[\S 3.4]{Upme1}. The bundle is equivariant under the actions of $\G_P$ and $\G_P/Z(G)$ on $\A_P$ and therefore factors through principal $\Z$-bundles $\Sp{}^{E_\bu}_P\ra\B_P,$ $\bar\Sp{}^{E_\bu}_P\ra\ovB_P$ on the topological stacks $\B_P,\ovB_P$ with $\Sp{}^{E_\bu}_P\cong\Pi_P^*(\bar\Sp{}_P^{E_\bu}).$ We call $\Sp{}^{E_\bu}_P,$ $\bar\Sp{}^{E_\bu}_P$ the {\it spectral bundles\/} of~$\B_P,$ $\ovB_P,$ and we say that $\B_P$ admits {\it mod-$k$ Floer gradings\/} if $\Sp{}^{E_\bu}_P\!/k\Z$ is isomorphic to the trivial principal $\Z_k$-bundle $\B_P\t\Z_k\ra\B_P$. Then a {\it mod-$k$ Floer grading\/} on $\B_P$ is an isomorphism $\om\colon\Sp{}^{E_\bu}_P\!/k\Z\,{\buildrel\cong\over\longra}\,\B_P\t\Z_k$ of principal $\Z_k$-bundles. As $\B_P$ is connected, there are either zero or $k$ different mod-$k$ Floer gradings.
\end{dfn}

\begin{rem}
\label{bc3rem4}
{\bf(a)} Essentially, a mod-$k$ Floer grading is a continuous enumeration $\cdots\le\la_{-1}(\na_P)\ab\le\ab\la_0(\na_P)\le\la_1(\na_P)\le\cdots$ of the spectra of the associated twisted Dirac operators $D^{\na_{\Ad(P)}}$ for all $\na_P\in\A_P$ for which elements of the gauge group $\G_P$ only shift the spectrum of the Dirac operator by multiples of $k,$ so the parallel transport of $\{\,\la_i(\ga^*(\na_P));i\in\Z\,\}\in\hat\Sp{}_P^{E_\bu}|_{\ga^*(\na_P)}$ from $\ga^*(\na_P)$ to $\na_P$ along any path is $\{\,\la_{i+k\ell}(\na_P);i\in\Z\,\}\in\hat\Sp{}_P^{E_\bu}|_{\na_P}$ for some $\ell\in\Z$.
\smallskip

\noindent
{\bf(b)} The definition of Floer grading here is new and should be compared to the `relative' version of Floer gradings $\de(\na_P,\na_P')\in\Z$ as defined by Donaldson~\cite[\S 3.2.2]{Dona} for acyclic flat connections $\na_P,$ $\na_P'.$ From our point of view $\de(\na_P,\na_P')$ is the parallel transport isomorphism in $\hat\Sp{}^{E_\bu}_P\ra\A_P$ along a path from $\na_P$ to $\na_P'.$ The invertibility of the operators (as the connections are acyclic) at the endpoints implies $\hat\Sp{}^{E_\bu}_P|_{\na_P}\cong\Z,$ $\hat\Sp{}^{E_\bu}_P|_{\na_P'}\cong\Z$ and is used to identify the parallel transport isomorphism $\hat\Sp{}^{E_\bu}_P|_{\na_P}\ra\hat\Sp{}^{E_\bu}_P|_{\na_P'}$ with an integer $\de(\na_P,\na_P').$
\end{rem}

\begin{dfn}
\label{bc3def10}
Let $X$ be a compact Riemannian spin $n$-manifold with $n\equiv 3,7\pmod{8},$ $E_\bu$ a self-adjoint elliptic differential operator on $X,$ and $G$ be a Lie group. The functor
\e
\label{bc3eq34}
\sO_{\Sp,X}^{E_\bu,G}\colon\Bord_X(BG)\ra\ZZtor
\e
is defined on objects by $(X,P)\longmapsto\Ga_{C^0}(\hat \Sp{}_P^{E_\bu})$ and on morphisms by using parallel transport in the same way as in Definition~\ref{bc3def6}.
\end{dfn}

The functor \eqref{bc3eq34} refines $_X^E_,G$ from \eqref{bc3eq23} in the sense that the diagram
\begin{equation*}
\xymatrix@C=120pt@R=15pt{
*+[r]{\Bord_X(BG)} \drtwocell_{}\omit^{}\omit{^{}} \ar[r]_(0.65){\sO_{\Sp,X}^{E_\bu,G}} \ar[d]^{\sO_X^{E_\bu,G}} & *+[l]{\ZZtor} \ar[d]_{\Pi_\Ztor} \\
*+[r]{\sZtor} \ar[r]^(0.65){\Pi_\Ztor}  & *+[l]{\Ztor} }
\end{equation*}
commutes up to natural isomorphism. Hence $_,X^E_,G(X,P)≅_X^E_,G(X,P)/2$ for all objects $(X,P)$ and, in particular, mod-2 Floer gradings can be identified with orientations as in Definition~\ref{bc2def2}. The second author \cite[Cor.~3.8]{Upme2} implies that \eqref{bc3eq34} can be factored as in Theorem~\ref{bc3thm1} through a symmetric monoidal functor $_,n^B_n^B(BG).$ The analogues of parts (a), (b), and (d) of Theorem~\ref{bc3thm1} continue to hold but there is no analogue of part (c). This implies the following version of Theorem~\ref{bc3thm2}, whose proof is exactly the same as the one given in \S\ref{bc54} and is left to the reader.

\begin{thm}
\label{bc3thm5}
Work in the situation of Theorem\/ {\rm\ref{bc3thm1},} with\/ $n\equiv 3,7\pmod{8}$ and\/ $G$ a fixed Lie group. Consider the commutative diagram
\begin{equation*}
\xymatrix@!0@C=90pt@R=35pt{
& \Aut_{\Bord_{n-1}^{\bs B}(\cL BG)}(\boo)  \ar[rr]^(0.37){I_n^{\bs B}(G)}_(0.37){\eq{bc3eq15}} && *+[l]{\Aut_{\Bord_n^{\bs B}(BG)}(\boo)} \ar[dd]_{\sO_{\Sp,n}^{\bs B}} \\
*+[r]{\Om_n^{\bs B}(\cL BG)} \ar[ur]^{\eq{bc3eq17}}_\cong \ar[d]^{\Pi_n^{\bs B}(BG)} \ar[rr]^(0.4){\xi_n^{\bs B}(BG)}_(0.4){\eq{bc2eq17}}  && \Om_{n+1}^{\bs B}(BG)  \ar[ur]^(0.4){\eq{bc3eq11}}_(0.4)\cong \\
*+[r]{\Om_n^{\bs B}(\cL BG;BG)} \ar@{..>}[rrr]^(0.43){\Xi_{\Sp,n}^{\bs B,G}} \ar[urr]_(0.8){\ti\xi_n^{\bs B}(BG)} &&& *+[l]{\Z=\Aut_{\ZZtor}(\Z).\!} }\!\!\!
\end{equation*}
Define\/ $\Xi_{\Sp,n}^{\bs B,G}$ to be the unique morphism making the bottom right quadrilateral commute. Then\/ $\B_P$ admits mod-$k$ Floer gradings for every compact Riemannian\/ $n$-manifold\/ $(X,g)$ with\/ $\bs B$-structure and principal\/ $G$-bundle\/ $P\ra X$ if and only if\/ $\Xi_{\Sp,n}^{\bs B,G}\bmod{k}\equiv\ul{0}\in\Z_k$.
\end{thm}

In \S\ref{bc57} we will use Theorems \ref{bc2thm6}(d) and \ref{bc3thm5} to prove:

\begin{thm}
\label{bc3thm6}
Let\/ $(X,g)$ be a compact Riemannian spin\/ $7$-manifold, and\/ $P\ra X$ be a principal\/ $\SU(r)$-bundle. Then
\begin{itemize}
\item[\bf (i)] If\/ $r=2,$ then mod-$8$ Floer gradings exist on\/ $\B_P.$
\item[\bf (ii)] If\/ $r=3,$ then mod-$6$ Floer gradings exist on\/ $\B_P.$
\item[\bf (iii)] If\/ $r\ge 4$ then mod-$2$ Floer gradings exist on\/ $\B_P$ (equivalently, $\B_P$ is orientable, as in Theorem\/ {\rm\ref{bc2thm3}(a)}), but for general\/ $X,P$ mod-$k$ Floer gradings do not exist on\/ $\B_P$ for any $k>2$. For example, this is true for $X=\cS^4\t\cS^3$ and\/ $P\ra X$ an $\SU(r)$-bundle with\/ $c_2(P)=\mathop{\rm Pd}(\{\rm{pt}\}\t\cS^3)$.
\end{itemize}
In particular, such Floer gradings exist on the moduli spaces of\/ $G_2$-instantons.
\end{thm}

\begin{rem}
\label{bc3rem5}
{\bf(a)} The Floer gradings of Theorem \ref{bc3thm6} would be important if (in the spirit of Donaldson--Thomas [32] and Donaldson--Segal [31]) one wanted to define instanton Floer homology groups of compact $G_2$-manifolds using $G_2$-instantons and $\Spin(7)$-instantons, by analogy with instanton Floer homology groups of compact 3-manifolds, as in Donaldson [29].
\smallskip

\noindent{\bf(b)} In contrast to orientations, Floer gradings are sensitive to the passage $\SU(r)\subset\SU(r+1)$; for orientations, `stability' was based on \cite[\S2.2.9]{JTU} and \cite[p.~18]{JTU}, which are not available here. Instead, the existence of mod-$k$ Floer gradings is preserved under~$\SU(r)\subset\SU(r+k).$
\smallskip

\noindent{\bf(c)} Because of {\bf(b)}, there is an alternative way to define Floer gradings: if $r<r'$, given an $\SU(r)$-bundle $P\ra X$ we can extend it to an $\SU(r')$-bundle $P'=(P\t\SU(r'))/\SU(r)$ by adding a trivial $\C^{r'-r}$ factor, and compute Floer gradings on $\B_P$ using $P'$ rather than $P$. (Note that for monodromy calculations round loops in $\B_P$ we do {\it not\/} consider general $\SU(r')$-bundles $Q'\ra X\t\cS^1$, but only $Q'=(Q\t\SU(r'))/\SU(r)$, so that $c_m(Q')=0$ for $r<m\le r'$.) For $r=2,3$, computing Floer gradings using $\SU(r')$-bundles in this way can yield different answers to Theorem \ref{bc3thm6}(i),(ii). For example, it follows from \eq{bc5eq12} below that if $r=2,3$ and $r'\equiv 6\pmod{12}$ then $\B_P$ admits mod-$24$ Floer gradings.
\smallskip

\noindent{\bf(d)} Divisibility properties of indices can be difficult to determine. A common approach in the literature is to make a fortuitous choice of twisting for which the integrality property of the twisted index implies the divisibility result for the index in question. This works for orientations, $k=2,$ as in \cite[p.~149]{Walp1} or for mod-$8$ Floer gradings in Chern--Simons theory on $3$-manifolds as in \cite[\S 3.3.2]{Dona}, but is challenging here. The role of bordism theory here is to reduce the divisibility question for the index to a simple index calculation in a few basic examples.
\end{rem}

\subsection{Applications to canonical orientation of moduli spaces}
\label{bc38}

The next theorem will be proved in \S\ref{bc58}.

\begin{thm}
\label{bc3thm7}
Let\/ $(X,g)$ be a compact, oriented, spin, Riemannian\/ $8$-mani\-fold, and fix an orientation for\/ $\det(\slashed{D}_+),$ where\/ $\slashed{D}_+$ is the positive Dirac operator on\/ $(X,g)$. Suppose\/ $P\ra X$ is a principal\/ $\SU(m)$-bundle for\/ $m\ge 1$ with\/ $c_2(P)=0$ in\/ $H^4(X,\Z)$. Then the moduli space\/ $\B_P$ has a \begin{bfseries}canonical\end{bfseries} orientation.

Similarly, if\/ $Q\ra X$ is a principal\/ $\U(m)$-bundle for\/ $m\ge 1$ with\/ $c_2(Q)-c_1(Q)^2=0$ in\/ $H^4(X,\Z)$ then\/ $\B_Q$ has a canonical orientation.
\end{thm}

The main idea of the proof is that if $PX$ is a principal $(8)$-bundle with $c_2(P)=0$ then the associated principal $E_8$-bundle $R=(PE_8)/(8)X$ is trivializable. We use Theorem \ref{bc3thm3}(a) to show that $_R$ has a canonical orientation, which we pull back to a canonical orientation on $_P$ using Theorem \ref{bc2thm1}. We extend from $(8)$ to $(m)$ for any $m≥1$ by stabilization.

Combining Theorems \ref{bc2thm5} and \ref{bc3thm7} we deduce

\begin{cor}
\label{bc3cor3}
Let\/ $X$ be a projective Calabi--Yau\/ $4$-fold, and\/ $\al\in K^0_\top(X)$ with\/ $c_2(\al)-c_1(\al)^2=0$ in\/ $H^4(X,\Z)$. As in Theorem\/ {\rm\ref{bc2thm5}(a)} we have a  moduli stack\/ $\M_\al$ of objects\/ $G^\bu$ in\/ $D^b\coh(X)$ with class\/ $\lb G^\bu\rb=\al$ in\/ $K^0_\top(X)$. It has a\/ $4$-Calabi--Yau obstruction theory and a principal\/ $\Z_2$-bundle\/ $O^{\cF^\bu}_\al\ra\M_\al$ of orientations as in Definition\/ {\rm\ref{bc2def5}}. Then\/ $\M_\al$ has a \begin{bfseries}canonical\end{bfseries} orientation.
\end{cor}

\begin{proof}
Theorem \ref{bc2thm5}(g) constructs a principal $\U(m)$-bundle $P\ra X$, unique up to isomorphism, such that orientations on $\B_P$ induce orientations on $\M_\al$. We have $c_2(P)-c_1(P)^2=c_2(\al)-c_1(\al)^2=0$. As in Theorem \ref{bc2thm5}(b), there is a canonical isomorphism $\Or(\det\slashed{D}_+)\cong\Z_2$. Thus Theorem \ref{bc3thm7} gives a canonical orientation on $\B_P$, which induces a canonical orientation on $\M_\al$. This is independent of the choice of $P$, as the canonical orientations in Theorem \ref{bc3thm7} are preserved by isomorphisms~$P\cong P'$.
\end{proof}

\begin{rem}
\label{bc3rem6}
What Theorem \ref{bc3thm7} and Corollary \ref{bc3cor3} really mean is that we have an {\it algorithm\/} for constructing orientations on $\B_P,\B_Q,\M_\al$, which depends only on $(X,g)$, the orientation for $\det(\slashed{D}_+)$, and~$P,Q,\al$.

In contrast, Theorem \ref{bc2thm3}(b) says that to construct orientations on $\B_R$ for $R\ra Y$ a principal $\U(m)$- or $\SU(m)$-bundle over a compact spin 7-manifold $Y$, we need to choose the additional algebro-topological data of a {\it flag structure\/} on $Y$. Theorem \ref{bc3thm7} and Corollary \ref{bc3cor3} show that no analogue of a flag structure is needed in the 8-dimensional case provided~$c_2(\al)-c_1(\al)^2=0$.

Note that other algorithms are possible, which would yield orientations on $\B_P,\B_Q,\M_\al$ differing from those in Theorem \ref{bc3thm7} and Corollary \ref{bc3cor3} by a sign depending on natural invariants in the problem such as $m$, $\rank\al$, $\chi(X)$, $\int_Xc_1(Q)^4$ and $\int_Xc_1(\al)c_3(X)$. We have no way to say which of these algorithms is `best', if this even makes sense.
\end{rem}

Corollary \ref{bc3cor3} has applications to the theory of DT4 invariants of Calabi--Yau 4-folds discussed in \S\ref{bc24}--\S\ref{bc25}. In particular, since all parts of Conjecture \ref{bc2conj1} involve only moduli spaces $_$ on $X$ with $c_1()=c_2()=0$, we deduce:

\begin{cor}
\label{bc3cor4}
As in Conjecture\/ {\rm\ref{bc2conj1},} for a Calabi--Yau\/ $4$-fold\/ $X$ there are conjectures in {\rm[8, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]} of the form \eq{bc2eq8} relating conventional invariants of\/ $X$ \textup(which require no choice of orientation\textup) and DT4 invariants of\/ $X$ \textup(which do require a choice of orientation\textup), an apparent paradox.

Corollary\/ {\rm\ref{bc3cor3}} provides canonical orientations for all the moduli spaces\/ $\M_\al$ occurring in Conjecture\/ {\rm\ref{bc2conj1},} resolving this paradox.
\end{cor}

\section{Proofs of theorems in \S\ref{bc2}}
\label{bc4}

\subsection{Proof of Theorem \ref{bc2thm2}}
\label{bc41}

We will first show that:
\begin{itemize}
\setlength{\itemsep}{0pt}
\setlength{\parsep}{0pt}
\item[(i)] Suppose a Lie group $H$ has a torus subgroup $T\subseteq H$, and write $G=Z(T)$ for the centralizer of $T$. Then $\inc:G\hookra H$ is of complex type.
\item[(ii)] Let $\io:G\ra H$ be a morphism of connected Lie groups which is a covering map. Then $\io$ is of complex type.
\item[(iii)] Compositions of complex type morphisms are of complex type.
\end{itemize}

Parts (ii),(iii) are obvious. For (i), write $,$ for the Lie algebras of $G,H$. Under the adjoint representation of $T$ on $$ we have a splitting $h=$, where $$ is a trivial $T$-representation and $$ contains only nontrivial $T$-representations. Let $(1)⊆T$ be a sufficiently general $(1)$-subgroup. Then $$ contains only nontrivial $(1)$ representations, so we may split $=_k>0V_k_^2[k]$ as $(1)$-representations, where $V_k$ is a real vector space and $^2[k]$ is the irreducible real $(1)$-representation with action $e^iþ↦(coskþ    sinkþ
-sinkþ   coskþ)$.

We make $$ into a complex vector space by identifying $^2[k]≅$ with $i∈$ acting by $(0    1
-1     0)$. As the $G$- and $(1)$-actions on $$ commute and the complex structure on $$ is determined by the $(1)$-action, it is preserved by $G$. Hence $:GH$ is of complex type.

Suppose now that $H$ is a compact, connected, simply-connected, simple Lie group corresponding to a Dynkin diagram $$, e.g. $H=E_8$. Then $H$ has a maximal torus $(1)^_0$ with $(1)$ factors corresponding to the set of vertices $_0$ of $$. Choose $k$ vertices $v_1,…,v_k$ in $$, corresponding to a subgroup $(1)^k⊂H$. Then $:Z((1)^k)H$ is of complex type by (i).

By Lie theory, it is easy to show the Lie algebra of $Z((1)^k)$ is $𝔷((1)^k)=(̆1)^^k$, where $$ is the semisimple Lie algebra whose Dynkin diagram $'$ is the result of deleting vertices $v_1,…,v_k$ and any edges meeting them from $$. Write $G$ for the compact, connected, simply-connected, semisimple Lie group with Dynkin diagram $'$. It is then nearly true that $Z((1)^k)=(1)^kG$.

In fact $Z((1)^k)$ could have finitely many connected components, and its identity component $Z((1)^k)_1$ is of the form $Z((1)^k)_1=((1)^kG)/K$ for $K⊂(1)^kG$ a finite normal subgroup. But $Z((1)^k)H$ of complex type implies that $Z((1)^k)_1H$ is of complex type, which implies that $(1)^kGH$ is of complex type by (ii),(iii).

In \eq{bc2eq6}, the morphisms $E_7(1)E_8$, $E_6(1)^2E_8$, $(14)(1)E_8$, $(8)(1)E_8$, $(3)(1)F_4$, and $(7)(1)F_4$, all arise this way by deleting 1 or 2 vertices from the Dynkin diagrams $E_8,F_4$.

For $G_2(8)$, we have inclusions $G_2(7)(8)$, where in Lie algebras $𝔰𝔭𝔦𝔫(7)/_2$ and $𝔰𝔭𝔦𝔫(8)/𝔰𝔭𝔦𝔫(7)$ are both the irreducible 7-dimensional $G_2$-representation $_7$. Hence $𝔰𝔭𝔦𝔫(7)/_2≅_7_7≅_7_$, so $G_2(8)$ is of complex type. Also $(m)(m)$ is by~(ii).

Next consider the three embeddings of Lie groups:
\begin{align*}
&{\rm(A)} & \U(1)&\longra\SU(m+1), 	& e^{i\th}&\longmapsto \mathop{\rm diag}\bigl(e^{i\th},\ldots,  e^{i\th},e^{-im\th}\bigr), \\
&{\rm(B)} & \U(1)&\longra\Sp(m+1), 	& e^{i\th}&\longmapsto \mathop{\rm diag}\bigl(1,\ldots,1, e^{i\th}\bigr), \\
&{\rm(C)} & \U(1)&\longra\SO(m+2), 	& e^{i\th}&\longmapsto \begin{pmatrix}
1 & 0 & \cdots  & 0 & 0 & 0 \\
0 & 1 & 0 & \cdots   & 0 & 0\\
\vdots & 0 & \ddots & \ddots & \vdots & \vdots \\
0 & \vdots & \ddots & 1 & 0 & 0 \\
0 & 0 & \cdots & 0 & \cos\th & \sin\th \\
0 & 0 & \cdots & 0 & -\sin\th & \cos\th
\end{pmatrix}.
\end{align*}

For (A), $Z((1))≅(m)⊂(m+1)$, where the embedding $(m)(m+1)$ maps $A↦( A     0
0     A^-1 )$. Hence $(m)(m+1)$ is of complex type by (i), completing Theorem \ref{bc2thm2}(a). Also $(m)(1)(m)$ is a covering map, so $(m)(1)(m+1)$ is of complex type by~(ii),(iii).

For (B), $Z((1))≅(m)(1)⊂(m+1)$, so $(m)(1)(m+1)$ is of complex type by (i). For (C), $Z((1))≅(m)(2)⊂(m+2)$ with $(2)≅(1)$, so $(m)(1)(m+2)$ is of complex type by (i). We show $(m)(1)(m+2)$ is of complex type by lifting to Spin groups. We have now constructed the four complex type morphisms in \eq{bc2eq7}.

To prove the claims on $p$-connectedness, we first show:
\begin{itemize}
\setlength{\itemsep}{0pt}
\setlength{\parsep}{0pt}
\item[(iv)] Suppose $\inc:G\hookra H$ is an inclusion of a Lie subgroup, and $Y=H/G$ has $\pi_i(Y)=0$ for $i\le p$. Then $\inc$ is $p$-connected.
\end{itemize}
This holds because $HH/G$ is a fibration with fibre $G$, so we have a long exact sequence of homotopy groups
\begin{equation*}
\xymatrix@C=20pt{ \cdots \ar[r] & \pi_{i+1}(H/G) \ar[r] & \pi_i(G) \ar[r] & \pi_i(H) \ar[r] & \pi_i(H/G) \ar[r] & \cdots. }
\end{equation*}
We now see that
\begin{itemize}
\setlength{\itemsep}{0pt}
\setlength{\parsep}{0pt}
\item[(v)] $\SU(m+1)/\SU(m)\cong\cS^{2m+1}$, with $\pi_i(\cS^{2m+1})=0$ for $i\le 2m$, so $\SU(m)\hookra\SU(m+1)$ is $2m$-connected.
\item[(vi)] $\Sp(m+1)/\Sp(m)\cong\cS^{4m+3}$, with $\pi_i(\cS^{4m+3})=0$ for $i\le 4m+2$, so $\SU(m)\hookra\SU(m+1)$ is $(4m+2)$-connected.
\item[(vii)] $\SO(m+1)/\SO(m)\cong\cS^m$, with $\pi_i(\cS^m)=0$ for $i\le m-1$, so $\SO(m)\hookra\SO(m+1)$ is $m-1$-connected. Similarly $\SO(m+1)\hookra\SO(m+2)$ is $m$-connected, so the composition $\SO(m)\hookra\SO(m+2)$ is $(m-1)$-connected.
\item[(viii)] Lifting (vii) to Spin groups, $\Spin(m)\hookra\Spin(m+2)$ is $(m-1)$-connected.
\end{itemize}
This completes the proof of Theorem \ref{bc2thm2}.

\subsection{Proof of Theorem \ref{bc2thm6}}
\label{bc42}

\subsubsection{The (co)homology of $\SU$}
\label{bc421}