group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
noncommutative topology, noncommutative geometry
noncommutative stable homotopy theory
genus, orientation in generalized cohomology
Hirzebruch signature theorem?
integration over supermanifolds, Berezin integral, fermionic path integral
Kontsevich integral, Selberg integral, elliptic Selberg integral
Generally, for $E$ an E-∞ ring spectrum, and $P \to X$ a sphere spectrum-bundle, an $E$-orientation of $P$ is a trivialization of the associated $E$-bundle.
Specifically, for $P = Th(V)$ the Thom space of a vector bundle $V \to X$, an $E$-orientation of $V$ is an $E$-orientation of $P$.
More generally, for $A$ an $E$-algebra spectrum, an $E$-bundle is $A$-orientable if the associated $A$-bundle is trivializable. For more on this see (∞,1)-vector bundle.
Let $E$ be a E-∞ ring spectrum. Write $\mathbb{S}$ for the sphere spectrum.
Write $R^\times$ or $GL_1(R)$ for the general linear group of the $E_\infty$-ring $R$: it is the subspace of the degree-0 space $\Omega^\infty R$ on those points that map to multiplicatively invertible elements in the ordinary ring $\pi_0(R)$.
Since $R$ is $E_\infty$, the space $GL_1(R)$ is itself an infinite loop space. Its one-fold delooping $B GL_1(R)$ is the classifying space for $GL_1(R)$-principal ∞-bundles (in Top): for $X \in Top$ and $\zeta : X \to B GL_1(R)$ a map, its homotopy fiber
is the $GL_1(R)$-principal $\infty$-bundle $P \to X$ classified by that map.
For $R = \mathbb{S}$ the sphere spectrum, we have that $B GL_1(\mathbb{S})$ is the classifying space for spherical fibrations.
There is a canonical morphism
from the classifying space of the orthogonal group to that of the infinity-group of units of the sphere spectrum, called the J-homomorphism. Postcomposition with this sends real vector bundles $V \to X$ to sphere bundles. This is what is modeled by the Thom space construction
which sends each fiber to its one-point compactification.
For $P \to X$ a $GL_1(R)$-principal ∞-bundle there is canonically the corresponding associated ∞-bundle with fiber $R$. Precisely, in the stable (∞,1)-category $Stab(Top)$ of spectra, regarded as the stabilization of the (∞,1)-topos Top
the associated bundle is the smash product over $\Sigma^\infty GL_1(R)$
This is the generalized Thom spectrum. For $R = K O$ the real K-theory spectrum this is given by the ordinary Thom space construction on a vector bundle $V \to X$.
An $E$-orientation of a vector bundle $V \to X$ is a trivialization of the $E$-module bundle $E \wedge S^V$, where fiberwise form the smash product of $E$ with the Thom space of $V$.
For $f : R \to S$ a morphism of $E_\infty$-rings, and $\zeta : X \to B GL_1(R)$ the classifying map for an $R$-bundle, the corresponding associated $S$-bundle classified by the composite
is given by the smash product
This appears as (Hopkins, bottom of p. 6).
For $X \stackrel{\zeta}{\to} B GL_1(\mathbb{S})$ a sphere bundle, an $E$-orientation on $X^\zeta$ is a trivialization of the associated $R$-bundle $X^\zeta \wedge R$, hence a trivialization (null-homotopy) of the classifying morphism
where the second map comes from the unit of $E_\infty$-rings $\mathbb{S} \to R$ (the sphere spectrum is the initial object in $E_\infty$-rings).
Specifically, for $V : X \to B O$ a vector bundle, an $E$-orientation on it is a trivialization of the $R$-bundle associated to the associated Thom space sphere bundle, hence a trivialization of the morphism
This appears as (Hopkins, p.7).
A natural $R$-orientation of all vector bundles is therefore a trivialization of the morphism
Similarly, an $R$-orientation of all spinor bundles is a trivialization of
and an $R$-orientation of all string group-bundles a trivialization of
and so forth, through the Whitehead tower of $B O$.
Now, the Thom spectrum MO is the spherical fibration over $B O$ associated to the $O$-universal principal bundle. In generalization of the way that a trivialization of an ordinary $G$-principal bundle $P$ is given by a $G$-equivariant map $P \to G$, one finds that trivializations of the morphism
correspond to $E_\infty$-maps
from the Thom spectrum to $R$. Similarly trivialization of
corresponds to morphisms
and trivializations of
to morphisms
and so forth.
This is the way orientations in generalized cohomology often appear in the literature.
The construction of the string orientation of tmf, hence a morphism
is discussed in (Hopkins, last pages).
For $H$ the multiplicative cohomology theory corresponding to $E$, and $V \to X$ a vector bundle of rank $n$, an $H$-orientation of $V$ is an element $u \in H^n(Th(V))$ in the cohomology of the Thom space of $V$ – a Thom class – with the property that its restriction $i^* u$ along $i : S^n \to Th(V)$ to any fiber of $Th(V)$ is
where
$\epsilon \in H^0(S^0)$ is a multiplicatively invertible element;
$\gamma_n \in H^n(S^n)$ is the image of the multiplicative unit under the suspension isomorphism $H^0(S^0) \stackrel{\simeq}{\to}H^n(S^n)$.
Multiplication with $u$ induces hence an isomorphism
This is called the Thom isomorphism.
The existence of an $H$-orientation is necessary in order to have a notion of fiber integration in $H$-cohomology.
The relation (equivalence) between choices of Thom classes and trivializations of (∞,1)-line bundles is discussed e.g. in Ando-Hopkins-Rezk 10, section 3.3
Let $G$ be a topological group equipped with a homomorphism to the stable orthogonal group, and write $B G \to B O$ for the corresponding map of classifying spaces. Write $J \colon B O \longrightarrow B GL_1(\mathbb{S})$ for the J-homomorphism.
For $E$ an E-∞ ring, there is a canonical homomorphism $B GL_1(\mathbb{S}) \to B GL_1(E)$ between the deloopings of the ∞-groups of units. A trivialization of the total composite
is a universal $E$-orientation of G-structures. Under (∞,1)-colimit in $E Mod$ this induces a homomorphism of $E$-∞-modules
from the universal $G$-Thom spectrum to $E$.
If here $G \to GL_1(\mathbb{S})$ is the $\Omega^\infty$-component of a map of spectra then this $\sigma$ is a homomorphism of E-∞ rings and in this case there is a bijection between universal orientations and such $E_\infty$-ring homomorphisms (Ando-Hopkins-Rezk 10, prop. 2.11).
The latter, on passing to homotopy groups, are genera on manifolds with G-structure.
For $V \to X$ a vector bundle of rank $k$, an ordinary orientation is a trivialization of the line bundle $\wedge^k V$. This is indeed equivalently a trivialization of $V \wedge H(\mathbb{R})$ of smashing with the Eilenberg-MacLane spectrum.
For spin orientation of vector bundle the construction is given by forming Clifford algebra bundles.
For string orientation of vector bundle the construction is supposed to be given by forming free fermion local net-bundle. See Andre Henriques’ website.
An $E_\infty$ complex oriented cohomology theory $E$ is indeed equipped with a universal complex orientation given by an $E_\infty$-ring homomorphism $MU \to E$, see here.
A comprehensive account of the general abstract persepctive (with predecessors in Ando-Hopkins-Rezk 10) is in
Lecture notes include
which are motivated towards constructing the string orientation of tmf, based on
Orientation of vector bundles in $E$-cohomology is discussed for instance in