# Equivariant K-theory and Resolution
II: Non-Abelian actions

Panagiotis Dimakis and Richard Melrose  Department of Mathematics, Stanford
University  Department of Mathematics, Massachusetts Institute of Technology
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

The smooth action of a compact Lie group on a compact manifold can be resolved
to an iterated space, as made explicit by Pierre Albin and the second author.
On the resolution the lifted action has fixed isotropy type corresponding to
the open stratum and also in an iterated sense, with connecting equivariant
fibrations over the boundary hypersurfaces covering the resolutions of the
other strata. This structure descends to a resolution of the quotient as a
stratified space. For an Abelian group action the equivariant K-theory can
then be described in terms of bundles over the bases ‘dressed’ by the
representations of the isotropy types with morphisms covering the connecting
maps. A similar model is given here covering the non-Abelian case. Now the
reduced objects are torsion-twisted bundles over finite covers of the bases,
corresponding to the projective action of the normalizers on the
representations of the isotropy groups, again with morphisms over all the
boundaries. This leads to a closely related iterated deRham model for
equivariant cohomology and, now with values in forms twisted by flat bundles
of representation rings over the bases, for delocalized equivariant
cohomology. We show, as envisioned by Baum, Brylinksi and MacPherson, that the
usual equivariant Chern character, mapping to equivariant cohomology, factors
through a natural Chern character from equivariant K-theory to delocalized
equivariant cohomology with the latter giving an Atiyah-Hirzebruch
isomorphism.

###### Contents

  1. 1 Resolution and Lifting
  2. 2 Reduction
  3. 3 Single Isotropy type
  4. 4 Equivariant fibrations
  5. 5 Reduced models
  6. 6 Atiyah-Hirzebruch isomorphism

## Introduction

In this continuation of [6] the description, in terms of the normal
resolution, of the equivariant K-theory of a compact manifold with an action
by a compact Lie group is extended to the general, non-Abelian, case. Doing so
suggests a deRham realization of _delocalized_ equivariant cohomology, in the
sense of Baum, Brylinksi and MacPherson [2]. The natural Chern character

(1) $\operatorname{Ch}:K^{0}_{G}(M)\longrightarrow
H^{\operatorname{even}}_{G,\mathrm{dl}}(M)$

yields an Atiyah-Hirzebruch isomorphism over $\mathbb{R}.$

The simplest cases of smooth $G$ action can be analyzed without resolution. If
$G$ acts freely then the compact manifold is the total space of a principal
bundle $\kappa:M\longrightarrow Y,$ over the smooth orbit space $Y=G\backslash
M.$ The action on an equivariant bundle, $V,$ over $M$ gives descent data
defining a bundle $W$ over $Y.$ Conversely, every equivariant bundle is
equivariantly isomorphic to the pull-back, $V=\kappa^{*}W,$ of a bundle over
$Y$ with $G$ action $gw_{x}=w_{gx}$ and equivariant bundle isomorphisms over
$M$ descend uniquely to bundle isomorphisms over $Y.$ Passing to the
Grothendieck groups this induces a natural identification as rings

(2) $K_{G}(M)=K(Y).$

Thus the equivariant K-theory of $M,$ as a ring, does _not_ depend on the
bundle structure but only on the quotient. Note however that the
representation ring of $G$ acts on $K_{G}(X)$ by taking the tensor product
with a representation viewed as an equivariant bundle; the composite is a
‘transfer map’

(3) $\tau:\mathrm{\mathcal{R}}(G)\longrightarrow K(Y)$

which does involve the geometry of the principal bundle.

The second elementary case is that of a trivial action. Since this corresponds
to the isotropy group below we denote the compact Lie group acting as $H.$ Let
$H^{\prime}$ be a complete set of irreducible unitary representations, with
$M(\pi)$ the Hermitian vector space on which $\pi\in H^{\prime}$ acts

(4)
$H^{\prime}\ni\pi\Longrightarrow\pi:H\longrightarrow\operatorname{U}(M(\pi)).$

Each fiber of an $H$-equivariant bundle over $M$ may then be decomposed under
the action to a sum

(5) $V_{x}=\bigoplus_{\pi\in H^{\prime}}V_{x}(\pi),\ V(\pi)=W(\pi)\otimes
M(\pi)$

where the $W(\pi)$ form an unrestricted bundle with compact support on
$M\times H^{\prime}.$

An $H$-equivariant isomorphism of $V$ induces a bundle isomorphism of each of
the $W(\pi)$ and the decomposition of representations under tensor product
then results in the identification as rings

(6) $K_{H}(M)=K(M)\otimes_{\mathbb{Z}}\mathcal{R}(H).$

The more general case in which the $G$-action has a single stratum is
intermediate between these two cases, where now all the isotropy groups are
conjugate. Borel showed that such a smooth action of a compact Lie group with
fixed isotropy type can be reduced to a principal action by the quotient
$Q=N(H)/H$ of the normalizer in $G$ for a choice of isotropy group

(7)
---
$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X^{H}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y=X/G}$

where $X^{H}$ is the submanifold where the isotropy group is $H.$ As rings

(8) $K_{G}(X)=K_{N}(X^{H}).$

In this setting the equivariant K-theory was identified by Wassermann [10] in
terms of twisted K-theory. Our description of K-theory in the general case
follows from a related decomposition of equivariant vector bundles in terms of
twisted bundles, or gerbe modules, rather than the $C^{*}$-theoretic approach
in [10].

Thus, suppose that $H\subset N$ is a closed normal subgroup (the notation
indicating that $N$ is perhaps the normalizer of $H$ in a larger group)
acting, with fixed isotropy group $H,$ on a compact manifold (allowed to have
corners) $Z;$ so $Z$ is a principal $Q=N/H$ bundle over a compact manifold
with corners $Y.$ The action of $N$ on $H$ by conjugation induces a locally
trivial action of the quotient group $Q=N/H$ on the irreducible
representations, $H^{\prime},$ so with finite orbits. This action lifts to an
action on $Z\times H^{\prime}$ with the same isotropy group, $H.$ Our
‘reduced’ description of $N$-equivariant bundles over $Z$ utilizes the finite
covers

(9) $Y(\pi)=(Z\times N\pi)/N\longrightarrow Y$

given by replacing $H^{\prime}$ by the orbit $N\pi\subset H^{\prime}.$

Schur’s Lemma shows that the projective action of $N$ on the representation
space of $\pi$ lifts to a projective action of $N$ on the sum over the orbit
of the representations and so to a central extension

(10)
$\textstyle{\operatorname{U}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\check{N}(\pi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N(\pi)}$

which corresponds to Mackey’s obstruction. Since $H$ lifts naturally into
$\check{N}(\pi)$ this defines a central extension $\check{Q}(\pi)$ of $Q$ for
each orbit. Making a choice of representatives $\pi\in H^{\prime}$
parameterizing the orbit space $H^{\prime}_{N}=H^{\prime}/N$ the space

(11) $\textstyle{Z(\pi)=(Z\times
N\pi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Q}$$\textstyle{Y(\pi)}$

is a principal bundle for the action of $Q.$

The decomposition (5) may be captured by considering the $H$-equivariant
homomorphisms

(12) $W_{x}(\pi)=\\{l:M(\pi)\longrightarrow V_{x}(\pi);h\circ l=l\circ\pi(h)\
\forall\ h\in H\\}.$

This has dimension, $n,$ the multiplicity of $\pi$ in $V_{x}.$ The rigidity of
representations of compact groups means that $n$ is constant on components of
$X.$ Let $W(\pi)$ denote the collective bundle over $Z(\pi)$ with fiber
$W_{x}(\pi^{\prime})$ at $(x,\pi^{\prime})\in(X,N\pi).$

The action of $N$ on $V$ induces a $\widehat{Q}(\pi)$-equivariant action on
$W(\pi)$ by the reverse extension to $\check{Q}(\pi).$ This makes the $W(\pi)$
into twisted bundles over $Y(\pi)$ or equivalently into gerbe modules for the
lifting bundle gerbe defined by the central extension $\widehat{Q}(\pi)$ and
the principal bundle $Z(\pi)\longrightarrow Y(\pi).$ Functorially this is
associated with the (torsion) Dixmier-Douady class $\alpha(\pi)\in
H^{3}(Y(\pi);\mathbb{Z})$ which is the transgression of the Mackey class.
Denoting the category of twisted bundles
$\operatorname{\mathfrak{B}un}(Y(\pi);\alpha(\pi))$ (in which the morphisms
depend on the trivialization of $\alpha(\pi))$ this gives an equivalence of
categories

(13) $\operatorname{\mathfrak{B}un}_{N}(Z)\longrightarrow\bigoplus_{\pi\in
H^{\prime}_{N}}\operatorname{\mathfrak{B}un}(Y(\pi);\alpha(\pi))$

where the sums on the right are finite. Wassermann’s ([10]) description of the
equivariant K-theory for an action with fixed isotropy type

(14) $K_{G}(M)\simeq\bigoplus_{\pi\in H^{\prime}_{N}}K(Y(\pi);\alpha(\pi))$

is a direct consequence. Note that this is a significantly functorial
statement because the twisted K-groups on the right depend on the
representations of the Dixmier-Douady class with different representations
leading to isomorphisms corresponding to tensoring with line bundles.

The localized representation ring, $L(H),$ consisting of the $H$-invariant
polynomials on the Lie algebra of $H$ (so $L(H)$ is the real representation
ring if $H$ is connected), is acted upon by the normalizer, $N,$ of $H$ and
hence by the quotient $Q.$ This action is locally trivial so the quotient

(15) $Z\times L(H)/N=L_{H}(Y)$

is a flat bundle over $Y$ modeled on $L(H).$ By an argument similar to that
for free actions (see for example [7], [8]), the Cartan model for the
equivariant cohomology of $X$ can be reduced to the deRham cohomology of the
complex of forms with values in $L_{H}$

(16) ${\mathcal{C}}^{\infty}(YZ;\Lambda^{*}\otimes L_{H}).$

Replacing $L(H)$ by the real representation ring generated by $H^{\prime},$
$\mathrm{\mathcal{R}_{\mathbb{R}}}(H)=\oplus(H^{\prime}\otimes\mathbb{R}),$
the quotient of $Z\times\mathrm{\mathcal{R}_{\mathbb{R}}}(H)$ by the action of
$Q$ is again the total space of a flat coefficient bundle which we denote
$\mathrm{\mathcal{R}_{\mathbb{R}}}(Y).$ Then delocalized equivariant
cohomology, in the sense of Baum, Brylinski and MacPherson, corresponds to the
deRham cohomology of

(17)
${\mathcal{C}}^{\infty}(Y;\Lambda^{*}\otimes\mathrm{\mathcal{R}_{\mathbb{R}}}(Y)).$

As a consequence of the properties of torsion-twisted K-theory, in particular
that the Chern character is a deRham class in the usual sense, the Chern
character applied to the twisted bundles $W(\pi)$ over $Y(\pi)$ gives an
element of (17) and defines the delocalized Chern character (1). The usual
equivariant Chern character to equivariant deRham cohomology corresponds to
applying the twisted Chern character to the (twisted) coefficient bundles
defined by the elements of $H^{\prime}$ and so factors through (1) by a
surjective localization map.

A general smooth action by a compact Lie group, $G,$ on a compact manifold,
$M,$ is reduced to an iterated version of the case of a single isotropy type
by resolution. The ‘normal resolution’ of the action (which may not be minimal
as a resolution), see [1], is obtained by the radial blow up of the isotropy
types, the strata, of the action in any order compatible with the (reversed)
partial inclusion order on the isotropy classes. The result is a compact
manifold with corners with iterated blow-down map

(18) $\beta:X\longrightarrow M$

with interior the preimage of the open isotropy type. The resolutions of the
closures, $M_{a},$ of the components of the non-open isotropy types are in 1-1
correspondence with the boundary hypersurfaces $H_{a}$ of $X$ as the images of
equivariant fibrations

(19) $\phi_{a}:H_{a}\longrightarrow X_{a},\ \beta_{a}:X_{a}\longrightarrow
M_{a}$

with the action on each $X_{a}$ having the same conjugacy class of closed
subgroups as the action on the interior of $M_{a}.$ These fibrations give $X$
an iterated structure, in particular the fibrations are jointly compatible at
each boundary face of $X,$ forming a chain under fiber inclusion with respect
to the isotropy partial order.

As shown in [6] smooth equivariant bundles over $M$ lift to smooth equivariant
bundles over each of the $X_{a},$ related by pull-back under the $\phi_{a}.$
This functor to iterated equivariant bundles over $X$ projects to an
equivalence on the Grothendieck groups, i.e. the iterated equivariant bundles
on $X$ lead to the equivariant K-theory of $M.$

On each of the compact manifolds with corners, $X_{a},$ including $X=X_{0},$
the description of equivariant bundles proceeds through the Borel reduction to
corresponding manifolds $Z_{a}$ where the action has a fixed isotropy group.
The quotient manifolds $Y_{a}$ with quotient fibrations $\phi_{a}$ form an
iterated manifold giving a resolution of the stratified space $M/G.$ The main
step in defining the iterated twisted bundles representing equivariant
K-theory on the resolution of the quotient is to describe the pull-back maps
corresponding to the equivariant fibrations $\phi_{a}.$ It is significant here
that the existence of an equivariant fibration for a space,
$H_{a}\longrightarrow X_{a}$ with $G$-actions on both $H_{a}$ and $X_{a}$
having single isotropy type, imposes a partial triviality condition on the
action on $H_{a}.$

Changing notation, let $\phi:X_{0}\longrightarrow X_{1}$ be a fibration of
smooth manifolds with corners, equivariant for $G$-actions with fixed isotropy
type, and with quotient fibration $\tilde{\phi}:Y_{0}\longrightarrow Y_{1}.$
We may assume that $Y_{0}$ is connected. First pass to the Borel reduction
$Z_{1}=X^{H_{1}}_{1}$ of $X_{1}$ for a choice of isotropy group $H_{1},$ with
normalizer $N_{1}.$ The preimage $Z_{0}=\phi^{-1}(Z_{1})$ is a ‘reduction’ of
$X_{0}$ generalizing Borel’s reduction and made precise below. Choose a
component of $Z_{0}$ and an isotropy group $H_{0}$ at some point in it. The
isotropy groups at all points of this component are necessarily $H_{1}$
conjugate to $H_{0},$ not just conjugate under the $N_{1}$-action. Let
$Z_{0}^{\prime}\subset Z_{0}$ be the union of the components where all
isotropy groups are $H_{1}$ conjugate to $H_{0},$ this is a reduction of
$Z_{0},$ and hence $X_{0},$ with group action by $N_{1}^{\prime}\subset N_{1}$
the subgroup under which $H_{0}$ is mapped to an $H_{1}$ conjugate. The image,
$Z_{1}^{\prime}=\phi(Z_{0}^{\prime})$ is a reduction of $Z_{1}$ with the same
group, $N_{1}^{\prime}$ acting. Passing to the Borel reduction
$Z_{0}^{\prime\prime}\subset Z_{0}^{\prime}$ with its $N^{\prime}_{1}$-action
gives spaces with isotropy groups $H_{0}\subset H_{1}$ and actions by groups
$N_{0}$ and $N_{1}^{\prime}$ connected by an equivariant fibration. As for
Borel reduction under these successive steps the $G$-equivariant bundles over
$X_{0}$ and $X_{1},$ respectively, are identified with with $N_{0}$ and
$N_{1}^{\prime}$-equivariant bundles over $Z^{\prime\prime}_{0}$ and
$Z_{1}^{\prime}.$

This ‘double’ reduction of the $G$-action on the boundary hypersurfaces
$H_{a}\subset X$ corresponds to a refinement of the decomposition of
$G$-equivariant bundles into twisted bundles and allows pull-back under
$\phi_{1}$ for equivariant bundles to be identified with an ‘augmented’ pull-
back, $\phi_{1}^{\\#},$ on the corresponding twisted bundles. Applied
iteratively this gives the compatibility conditions on twisted bundles and a
functorial reduction

(20)
$\begin{gathered}\operatorname{\mathfrak{B}un}_{G}(X)\equiv\operatorname{\mathfrak{B}un}_{\operatorname{red}}(Y_{*})\subset\bigoplus_{\pi\in
H^{\prime}_{N}}\operatorname{\mathfrak{B}un}(Y(\pi);\alpha(\pi)),\\\
\oplus_{\alpha}W_{\alpha}\in\operatorname{\mathfrak{B}un}_{\operatorname{red}}(Y_{*})\Longleftrightarrow
W_{\alpha}\big{|}_{H_{\beta}}=\phi^{\\#}_{\alpha,\beta}W_{\beta}\ \forall\
\beta>\alpha\end{gathered}$

identifying $G$-equivariant bundles over the original $G$-space $M$ with
iterated twisted bundles over the resolution of the quotient. Notice however
that the pull-back maps here involve additional information not found in the
quotient itself, and the same is true of the twistings.

There is a corresponding description of equivariant cohomology in terms of
iterated forms on the resolved quotient with coefficients in the bundles
formed from the localized representation rings of the isotropy groups. These
are linked by augmented pull-back maps. Then delocalized equivariant
cohomology is defined by analogy where the coefficient rings are the full real
representation rings. This results in an Atiyah-Hirzebruch isomorphism (1), as
envisioned by Baum, Brylinski and MacPherson. This is shown using the six term
exact excision sequences corresponding to pruning the isotropy tree,
essentially as in [6].

The authors thank Victor Guillemin, Eckhard Meinrenken and David Vogan for
helpful discussions.

##  1\. Resolution and Lifting

We refer to the corresponding sections of [6] for a description of the normal
resolution of a compact manifold with a smooth action by a compact Lie group
and the categorical lifting of $G$-bundles. If the initial manifold has
corners then we demand that the action be ‘boundary free’ in the sense that if
one connected boundary hypersurface is mapped to another by an element of the
group then they do not intersect. If this is not the case then it can be
arranged by replacing the manifold by its total boundary blowup. This
boundary-free property is preserved under normal resolution and so applies
throughout.

##  2\. Reduction

Borel, see [4], showed how to reduce the geometry of the action of a compact
Lie group with a fixed isotropy type to the special case of a fixed, i.e.
normal, isotropy group. The geometry is thereby reduced to that of a principal
bundle for the quotient of the normalizer by the chosen isotropy group. We
need a more general version of this construction which we term ‘reduction’
below.

Let $X$ be a compact manifold (meaning possibly with corners always) with a
smooth action by a compact Lie group, $G,$ with unique isotropy type. This is
a fibre bundle $X\longrightarrow Y=X/G$ with fibre modeled on $G/H$ for a
choice of isotropy group, $H.$ By a reduction of $X$ we mean a smooth
submanifold $Z\subset X$ which meets every orbit, so defines a subfibration
over $Y,$ is stabilized by a closed subgroup $K$ in the strong sense that

(2.1) $g\in G,\ x\in Z\text{ then }gx\in Z\text{ iff }g\in K$

and on which $K$ acts with fixed isotropy type. In particular Borel’s
reduction is the case where $Z=X^{H}$ is the set of points where the isotropy
group is a given element of the conjugacy class. Equivariant K-theory behaves
in the same way in this more general setting.

######  Proposition 1.

If $X$ is a compact manifold with a smooth action, with unique isotropy type,
by a compact Lie group, $G,$ and $Z\subset X$ is a reduction of it, with
stabilizer $K,$ as defined above, then restriction gives an equivalence of
categories

(2.2)
$\operatorname{\mathfrak{B}un}_{G}(X)\to\operatorname{\mathfrak{B}un}_{K}(Z).$

###### Proof.

If $V$ is a $G$-equivariant bundle over $X$ then its restriction to $Z$ is
$K$-equivariant. This defines a map (2.2) which is clearly functorial with
respect to equivariant morphisms. Conversely if $U$ is a $K$-equivariant
bundle over $Z$ then it lifts to $G\times Z$ under projection to the second
factor. The left $G$-action induces a trivial $G$-action on the lifted bundle.
The diagonal $K$-action in which $k$ maps $(g,z)$ to $(gk^{-1},kz)$ is free,
commutes with the $G$-action and is covered by the action of $K$ on $U.$ By
(2.1) the quotient space given by the $K$-action is $X$ and the $K$-action on
the bundle gives descent data defining a $G$-equivariant bundle. That these
bundles and operations are smooth follows from the local product
decompositions implied by the uniqueness of the isotropy types.

Clearly restriction is a left equivalence for this extension. To see it is a
right equivalence set $U=V\big{|}_{Z}$ for a $G$-equivariant bundle $V$ on
$X.$ Then the action on $V$ gives a bundle map of fibre isomorphisms

(2.3) $\\{g\\}\times V_{z}\longrightarrow V_{gz}$

which is equivariant for the $G$ actions and commutes with the $K$ action
$\\{g\\}\times V{z}\longrightarrow\\{gk^{-1}\\}\times V_{kz}$ and so extends
to an equivariant isomorphism of $V$ and the extension of $V\big{|}_{Z}.$ ∎

Borel reduction corresponds to a choice among the isotropy groups $H\subset
G.$ The normalizer $N(H)=N_{G}(H)$ acts on $X^{H}$ with the subgroup $H$
acting trivially and the quotient group $Q=N_{G}(H)/H$ acts freely. Thus
$X^{H}$ is a principal $Q$-bundle over the orbit space $Y=Q\backslash X^{H}$

(2.4)
$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X^{H}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y}$

which is independent of the choice of $H$ up to equivalence.

However, the choice of a particular isotropy group is not natural, in general,
and at points below we need to allow changes of this choice. The closed
subgroups conjugate to $H$ in $G$ may be identified with the homogeneous space
$G/N(H)$ over which the space $X$ fibres with the fibre above $H_{g}=gHg^{-1}$
being a principal $Q(H_{g})=gQg^{-1}$-bundle. In this more equivariant picture

(2.5)
---
$\textstyle{Q(H_{g})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{G/N(H_{g})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y}$

the group $G$ still acts as a fibre-preserving map.

It is convenient to consider another setting closely related to reduction.
Thus, suppose that a compact Lie group $N$ acts on a compact manifold with
corners $Z$ with a fixed, hence normal, isotropy group $H.$ A smooth
submanifold $Z^{\prime}\subset Z$ will be called a _restriction_ of the action
if it is a subbundle of $Z$ as a principal $N/H$ bundle with the subgroup
$N^{\prime}\subset N$ acting fibre-transversally on $Z^{\prime}$ with isotropy
group $H^{\prime}=H\cap N^{\prime}.$ Clearly there is then a restriction
functor

(2.6)
$\operatorname{\mathfrak{B}un}_{N}(Z)\longrightarrow\operatorname{\mathfrak{B}un}_{N^{\prime}}(Z^{\prime}).$

This functor arises in the connecting maps in the reduced picture of
equivariant K-theory below.

Borel’s model for the equivariant cohomology of a $G$-space $X,$ for a compact
Lie group $G,$ is to take a classifying bundle for $G,$ so a contractible
space $EG$ on which $G$ acts freely. Then the product action on $X\times EG$
is free. Writing the quotient space as $X\times_{G}EG$

(2.7) $H_{G}(X)=H(X\times_{G}EG)$

where we are interested in the case of real coefficients.

######  Proposition 2.

Under a reduction of a smooth action of a compact Lie group, as defined above

(2.8) $H_{N}(Z)=H_{G}(X)$

###### Proof.

Consider the product $G$ action on $G\times Z\times EG$ which commutes with
the $N$ action recovering $X$ from $G\times Z.$ This has $G$ action has the
transversal $\\{e\\}\times Z\times EG$ so the quotient may be identified with
$Z\times EG.$ Projected to this transversal the action of $N$ on $G\times
Z\times EG$ is identified with the product action on $Z\times EG$ with
quotient $Z\times_{N}EG.$ On the other hand taking the quotient by the $N$
action first this is also identified with the quotient $X\times_{G}EG.$ Thus
it follows that there is a natural identification (2.8) for any reduction of a
$G$ action. ∎

Note that for a reduction the image $gZ$ of $Z$ under any element $g\in G$ is
a reduction with group action by $gNg^{-1}$ and (2.8) extends to provide
natural identifications between the various $H_{gNg^{-1}}(gZ).$

From the naturality of these constructions the $G$-equivariant Chern character
of a $G$ bundle over $X$ can be realized as the $N$-equivariant Chern
character for the restriction of the bundle to a reduction $Z$ and this is
natural with respect to restriction in the Cartan model for equivariant
cohomology.

##  3\. Single Isotropy type

The resolution of the manifold with group action reduces it to a tree of
compact manifolds with corners on each of which the group acts with all
isotropy groups conjugate; the resolved pieces are linked by equivariant
fibrations over the boundary hypersurfaces. Borel’s reduction replaces each
component manifold $X$ by the submanifold $Z=X^{H}$ where the isotropy group,
$H,$ is fixed with the group replaced by the normalizer, i.e. $H\subset N$ is
a normal subgroup. Then $Z$ is the total space of a principal $Q=N/H$-bundle;
by treating the components separately we can assume that the base, $Y,$ is
connected. We proceed to show how equivariant bundles are reduced to twisted
bundles over finite covers of the base.

Let $H^{\prime}$ be the category of irreducible unitary representations of $H$
with $M$ the tautological ‘bundle’ over $H^{\prime}$ with fibre the
representation space. The conjugation action of $N$ on $H$ induces an action
of $N$ on $H^{\prime}$ with finite orbits, $o(\pi)=N\pi.$ Then over $Z\times
o(\pi)$ consider the trivial bundle obtained by pulling back the tautoligical
bundle from $H^{\prime},$

(3.1) $M^{\prime}\longrightarrow Z(\pi)=Z\times o(\pi),$

so with fibre at $\\{z\\}\times\\{\pi^{\prime}\\}$ the representation space
$M(\pi^{\prime})$ of $\pi^{\prime}\in o(\pi).$ Each $n\in N$ conjugates the
representation $\pi^{\prime}$ to $n\pi^{\prime}$ and by Schur’s Lemma the
unitary conjugation is determined up to a multiple of the identity. This
determines a central extension

(3.2) $\operatorname{U}(1)\longrightarrow\check{N}(\pi)\longrightarrow N.$

Thus $\check{N}(\pi)$ acts equivariantly on $M^{\prime}$ covering its action
through $N$ on $Z(\pi).$ We write $\check{N}(\pi)$ since it is the ‘opposite’
extension, $\widehat{N}(\pi),$ given by the dual of the circle bundle (3.2),
which appears in the decomposition of equivariant bundles.

The action by $N$ on $Z(\pi)$ has isotropy group $H$ which naturally includes
in $\check{N}(\pi)$ since it acts consistently on each $M(\pi).$ Thus $Z(\pi)$
is the total space of a principal bundle with base $Y(\pi)$ and structure
group $Q=N/H$ whereas $M^{\prime}$ is a bundle over $Z(\pi)$ with an
equivariant action of $\check{N}(\pi).$

We now turn to the decomposition of an $N$-equivariant bundle, $V,$ over $Z$
on which $N$ acts with fixed isotropy group $H.$ The pointwise action of $H$
induces a finite decomposition into subbundles

(3.3) $V=\bigoplus_{\pi\in H^{\prime}}V(\pi)$

where the action of $H$ on $V(\pi)$ is conjugate to the action on
$\mathbb{C}^{n}\otimes M(\pi)$ for some $n.$ This can be formalized by
considering the bundle over $Z\times H^{\prime}$ with fibre at
$(z,\pi^{\prime})$ the space of $H$-equivariant homomorphisms

(3.4) $W(z,\pi^{\prime})=\\{l:M(\pi^{\prime})\longrightarrow
V_{z};l\circ\pi^{\prime}(h)=h\circ l\\}=\hom_{H}(M(\pi^{\prime}),V_{z},).$

The rigidity of representations implies that these are smooth bundles,
although the dimension of $W(z,\pi^{\prime})$ may vary between components of
$Z.$

######  Proposition 3.

The fibres (3.4) define a smooth bundle bundle $W$ over $Z\times H^{\prime}$
which carries an equivariant action of $\widehat{Q}(\pi)=\widehat{N}(\pi)/H$
on the restriction $W(\pi)$ to each orbit $Z(\pi)$ and is such that summing
over the fibres

(3.5) $V=(\pi_{Z})_{*}(W\otimes M^{\prime})$

as $N$-equivariant bundles over $Z.$ This defines an equivalence of categories

(3.6) $\operatorname{\mathfrak{B}un}_{N}(Z)\longrightarrow\bigoplus_{\pi\in
H^{\prime}/N}\operatorname{\mathfrak{B}un}_{\widehat{Q}(\pi)}(Z(\pi)).$

###### Proof.

The given equivariant bundle $V$ decomposes as in (3.3) and so lifts to a
bundle $V^{\prime}$ over $Z\times H^{\prime}$ where the fibre at $(z,\pi)$ is
the image of $W(z,\pi).$ Then

(3.7)
$W=\hom_{H}(M^{\prime},V^{\prime})=V^{\prime}\otimes_{H}(M^{\prime})^{*}$

is the bundle of $H$-equivariant homomorphisms over $Z\times H^{\prime}.$
Since the $\check{N}(\pi)$ action on $M^{\prime}$ induces an
$\widehat{N}(\pi)$ action on the dual, $W$ has (over each orbit $Z(\pi))$ an
action of $\widehat{N}(\pi)$ on the second factor and of $N$ on the first.
Combined this gives an equivariant action of $\widehat{N}(\pi)$ in which $H$
acts trivially. Thus $W$ carries an equivariant action of $\widehat{Q}(\pi).$
Then $V^{\prime}$ is recovered as the bundle $W\otimes M^{\prime}$ over
$Z\times H^{\prime}$ in which the extensions in the $\widehat{N}(\pi)$ and
$\check{N}(\pi)$ actions cancel to give an $N$-equivariant bundle which pushes
forward, i.e. sums over the finite number of representations present, to give
$V.$

This relationship is an equivalence of categories. ∎

If $W$ was actually $Q$-equivariant its restriction to $Z(\pi)$ would descend
to a bundle over the base, $Y(\pi),$ for the action of $Q.$ Since the action
is only projective it becomes a twisted bundle over $Y(\pi).$ We treat these
in the context of _gerbe modules_ which we now recall. The central extension
of $Q(\pi)$ induces a lifting bundle gerbe, in the sense of Murray [9], over
$Y(\pi):$

(3.8) |
---|---
$\textstyle{C(\pi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{Q}(\pi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Q(\pi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
Z^{[2]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Q(\pi)}$$\textstyle{Y(\pi)}$

where $C(\pi)$ is the pull-back of the circle giving the central extension.

This gerbe is classified by its (torsion) Dixmier-Douady class $\alpha(\pi)\in
H^{3}(Y(\pi);\mathbb{Z})$ which in this case is the transgression of the
Mackey class of the extension. A vector bundle $W(\pi)$ over $Z(\pi)$ with a
$\widehat{Q}(\pi)$-action covering the $Q(\pi)$-action is a gerbe module in
the sense of [5] since the induced isomorphisms over each point $(p,q)\in
Z^{[2]}$

(3.9) $\gamma_{p,q}:W_{p}\otimes C(\pi)_{p,q}\longrightarrow W_{q}$

are consistent with the gerbe structure. Thus:

######  Proposition 4.

A $G$-equivariant vector bundle on $X,$ for an action with unique isotropy
type, induces a gerbe module over $Y(\pi)$ for each of the lifting bundle
gerbes (3.8), trivial outside a finite subset of $H^{\prime}/N$ and
conversely; $G$-equivariant bundle isomorphisms correspond uniquely to
$\widehat{Q}(\pi)$-equivariant bundle isomorphisms, and hence gerbe module
isomorphisms resulting in an equivalence of categories

(3.10) $\operatorname{\mathfrak{B}un}_{G}(X)=\bigoplus_{\pi\in
H^{\prime}/N}\operatorname{\mathfrak{B}un}(Y(\pi);\alpha(\pi)).$

Thus, each $G$-equivariant bundle over $X$ induces gerbe modules over a finite
collection of the lifting gerbes (3.8) with $\pi$ lying in different orbits
for the action of $N$ on $H^{\prime}.$ Different choices of base point in the
orbits induce gerbe module isomorphisms.

######  Theorem 5 (A. Wassermann).

For the action of a compact group $G$ on a compact manifold $X$ with unique
isotropy type, the equivariant K-theory can be identified with the collective
twisted K-theory with coefficients in the representation ring of the isotropy
group

(3.11) $K_{G}(X)=K(Y(\pi);\alpha(\pi))\otimes_{H^{\prime}/N}\mathcal{R}(H)$

###### Proof.

This is a direct consequence of Proposition 4, by passing to the corresponding
Grothendieck groups, and the characterization of twisted K-theory in terms of
gerbe modules by Bouwknegt, Carey, Varghese, Murray and Stevenson in [5]. ∎

Although (3.10) gives a complete description of equivariant bundles in this
case, the twisted bundles can be further reduced, in the sense of §2, and this
is relevant to the discussion of pull-back maps below.

In the action of $N$ on $H^{\prime}$ for a normal subgroup $H,$ consider the
stabilizer $N(\pi)$ and its normalizer $\Gamma(\pi).$ The submanifold

(3.12) $Z^{\prime}(\pi)=Z\times\Gamma(\pi)\pi\subset Z(\pi)=Z\times N\pi$

is a reduction of $Z(\pi)$ for the action of $\widehat{Q}(\pi)$ which is
reduced to the action of $\widehat{Q}^{\prime}(\pi)=\widehat{\Gamma}(\pi)/H$
where $\widehat{\Gamma}(\pi)$ is the central extension of $\Gamma(\pi)$
induced by $\widehat{N}(\pi).$

As noted in the Introduction, and proved already by Cartan, the equivariant
cohomology for a free action is the cohomology of the base. This can be seen
directly in terms of the Borel model and there is a corresponding argument for
the Cartan model using a connection, see the book of Guillemin and Sternberg,
[7]. Namely if $\alpha$ is a connection on $X$ as a principal $G$-bundle over
$Y$ then the trace functional on $\mathfrak{g}\otimes\mathfrak{g}^{*}$ defines
a linear map

(3.13) ${\mathcal{C}}^{\infty}(X;\Lambda^{*}\otimes S(\mathfrak{g}^{*}))\ni
u\longrightarrow Au=\operatorname{Tr}_{\mathfrak{g}}(\alpha\wedge
u)\in{\mathcal{C}}^{\infty}(X;\Lambda^{*}\otimes S(\mathfrak{g^{*}})).$

Here $S(\mathfrak{g}^{*})$ is realized as the totally symmetric polynomials
with the trace acting in the first variable. This lowers the polynomial order
by one. Acting on $G$ -invariant forms which are homogeneous of degree $p$ as
polynomials on $\mathfrak{g}$

(3.14) $d_{G}(Au)-Ad_{G}+pu=pB_{p}u$

where $B_{p}$ is given in terms of contraction with the curvature 2-form of
$\alpha$ and is of degree $p-1$ as a polynomial. Repeated application of
$B_{p}$ to the terms of highest positive degree, eventually stabilizes on each
form, replacing an equivariant form by a basic one as discussed by Meinrenken
[8].

The same approach can be used to study the Cartan cohomology in the case of an
action with single isotropy type. It follows from the relationship between the
Borel and Cartan models, see [7, Chap 4], that under Borel reduction the
restriction map

(3.15) ${\mathcal{C}}^{\infty}(X;\Lambda^{*}\otimes
S(\mathfrak{g}^{*}))^{G}\longrightarrow{\mathcal{C}}^{\infty}(X^{H};\Lambda^{*}\otimes
S(\mathfrak{n}^{*}))^{N}$

induces an homotopy equivalence of complexes realizing the isomorphism of
$H^{*}_{G}(X)$ and $H^{*}_{N}(X^{H})$ where $N=N(H)$ is the normalizer.

To see directly that the restriction, both on the manifold and the Lie
algebra, gives an isomorphism the contraction operator $A$ in (3.13) given in
terms of a connection form, can be replaced by contraction with a ‘partial
connection’. For convenience, give the group an invariant metric. By such a
partial connection form we mean a $G$-equivariant 1-form on $X,$ with values
in $\mathfrak{g},$ which restricted to the tangent space to the orbit at each
point restricts to the identity on the orthocomplement of the Lie algebra of
the normalizer of the isotropy group at that point. For the homogeneous space
$G/H$ such a form can be constructed by orthogonal projection of the Maurier-
Cartan form and in general by using a $G$-invariant metric on $X$ to project
the orthogonal subspace to the Lie algebra of the normalizer in the tangent
space to the orbits. Application of the analogue of (3.13) lowers the
polynomial order pointwise in this subspace and gives a homotopy equivalence
for corresponding to (3.15).

In the reduced case written as the action by $N$ on $Z$ with fixed isotropy
group $H$ the conjugation action of $N$ on the Lie algebra $\mathfrak{h}$
induces a locally trivial equivariant action on
$Z\times(S^{*}(\mathfrak{h^{*}})^{H})$ defining a flat bundle $S_{H}$ over the
base.

######  Proposition 6.

If $N$ acts on a compact manifold $Z$ with unique isotropy group $H$ and base
$Y,$ a choice of invariant inner product on $N$ and a connection on $Z$ as the
total space of a principal $Q=N/H$ bundle defines a homotopy equivalence
intertwining the equivariant deRham differential on $Z$ and the deRham
differential on $Y$ with coefficients

(3.16) $B^{\infty}:\left({\mathcal{C}}^{\infty}(Z;\Lambda^{*}\otimes
S(\mathfrak{n}^{*})\right)^{N}\longrightarrow{\mathcal{C}}^{\infty}(Y;\Lambda^{*}\otimes
S_{H}).$

###### Proof.

The given connection form
$\alpha\in{\mathcal{C}}^{\infty}(Z;\Lambda^{1}\otimes\mathfrak{q})$ for $Z$ as
a principal $Q$ bundle can be interpreted as a form with values in
$\mathfrak{n}$ using the embedding of $\mathfrak{q}\subset\mathfrak{n}$ as the
space orthogonal to $\mathfrak{h}.$ Then the identity (3.14) allow a
successive lowering of the polynomial order in $\mathfrak{q}$ of Cartan forms
on $Z$ so identifying the equivariant cohomology with the deRham cohomology of
basic $Q$-invariant sections of
${\mathcal{C}}^{\infty}(Z;\Lambda^{*}\otimes(S^{*}(\mathfrak{h^{*}})^{H})).$
The action of $Q$ is locally trivial and defines the descent to the cohomology
of $Y$ with coefficients in $S_{H}.$ ∎

For the extension to a general action below it is important that (3.16)
defines an explicit homotopy equivalence.

In this reduced case of an action with single isotropy group, an equivariant
bundle $V$ decomposes as a $\widehat{Q}(\pi)$-equivariant bundle $W(\pi)$ over
each $Z(\pi).$ the Chern character can be see directly in terms of Proposition
6. Namely, given an equivariant connection, the $\widehat{Q}(\pi)$-equivariant
Chern character of $W(\pi)$ is a $Q(\pi)$-equivariant form on $Z(\pi).$ The
character map given by the trace defines a section of the trivial polynomial
bundle

(3.17) $\operatorname{tr}:H^{\prime}\longrightarrow(S(\mathfrak{h}^{*}))^{H}$

over $Z(\pi)$ which is invariant under the action of $Q$ projected from the
action of $\check{Q}(\pi)$ on $M^{\prime}.$ The form and section combine to
give a form over the quotient, $Y(\pi)$ with values in the flat descended
bundle and this further pushes forward, summing over the fibres of each
$Y(\pi),$ and over $H^{\prime}/N,$to give the closed form

(3.18)
$\operatorname{Ch}(V)\in{\mathcal{C}}^{\infty}(Y;\Lambda^{\operatorname{even}}\otimes
S_{H}).$

This descends to the equivariant Chern character in $H_{N}(Z)=H_{G}(X).$

The _delocalized_ equivariant cohomology in this case of a single isotropy
group is obtained by replacing the character ring by the real representation
ring $\mathcal{R}(H).$ As a vector space this is freely generated by
$H^{\prime}.$ Thus there is a linear subspace $o(\pi)\otimes\mathbb{R}$
associated with each orbit of the action of $Q$ on $H^{\prime}$ which can be
viewed as the push forward under the map from $o(\pi)$ to a point. The action
of $Q$ on $Z\times o(\pi)$ therefore generates a flat real line bundle over
$Y(\pi)$ which pushes forward to $Y$ to define a flat bundle. The formal sum
over $H^{\prime}/N$ of these spaces is the flat bundle $\mathcal{R}_{H}(Y).$

######  Definition 1.

The _delocalized equivariant cohomology,_ $H^{*}_{G,\mathrm{dl}}(X),$ of a
manifold with smooth action, with unique isotropy type, by a compact Lie group
is the deRham cohomology of the base with coefficients in
$\mathcal{R}_{H}(Y).$

Note that the forms appearing here, in
${\mathcal{C}}^{\infty}(X/G;\Lambda^{*}\otimes\mathcal{R}_{H}(Y)),$ are each
required to have coefficients in a fixed finite span of the subspaces
corresponding to the orbits of the normalizer of an isotropy group on its
representation ring. This space of forms is functorial under change of
isotropy group as in (2.5).

######  Theorem 7 (Following [2]).

For the action of a compact Lie group $G$ on a compact manifold $X$ with
unique isotropy type, the equivariant Chern character factors through a
natural ‘delocalized Chern character’ as in (1)

(3.19)
---
$\textstyle{K^{0}_{G}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Ch}_{\mathrm{dl}}}$$\scriptstyle{\operatorname{Ch}}$$\textstyle{H^{\operatorname{even}}_{G,\mathrm{dl}}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{loc}}$$\textstyle{H^{\operatorname{even}}_{G}(M)}$

with the top map an isomorphism after tensoring with $\mathbb{R}.$

So this is an Atiyah-Hirzebruch isomorphism.

###### Proof.

As in the discussion of the standard equivariant Chern character above, an
equivariant K-class is represented by a formal difference
$V\ominus\mathbb{C}^{N}$ where the trivial bundle contributes a constant form.
The decomposition (5) yields the bundle $W(\pi)$ over each of the principal
$Q$ bundles $Z(\pi).$ A $\widehat{Q}$-invariant connection on $W(\pi)$ yields
a $Q$-invariant form on $Z(\pi).$ Tensored with the canonical section on the
trivial $\mathbb{R}$-bundle over $H^{\prime}$ with its $Q$ action this
descends to a form with values in the flat line bundle over the quotient
$Y(\pi)$ and then pushes forward to a closed element of
${\mathcal{C}}^{\infty}(Y;\Lambda^{\operatorname{even}}\otimes\mathcal{R}_{H}.)$
The deRham class of this form is well-defined, giving the map
$\operatorname{Ch}_{\mathrm{dl}}$ in (3.19). The trace map on representations
induces the localization map completing the commutative diagram.

For torsion-twisted K-theory there is an Atiyah-Hirzebruch isomorphism

(3.20) $\operatorname{Ch}:K(Y;\alpha)\otimes\mathbb{R}\longrightarrow
H^{\operatorname{even}}(Y;\mathbb{R}).$

The crucial injectivity of this map follows from the multiplicative property
of twisted K-theory

(3.21) $K(Y;\alpha)\times K(Y;\beta)\longrightarrow K(Y;\alpha+\beta)$

(given consistent trivializations of the torsion classes). At the level of
bundles (3.20) corresponds to the fact that if $n\alpha=0$ then the $n$-fold
tensor product of a twisted bundle is untwisted and division by $n$ is an
isomorphism over $\mathbb{R}.$

Applying (3.20), at the level of bundles over the $Y(pi),$ to (3.10) shows the
injectivity of the delocalized Chern character over $\mathbb{R}.$ Since the
surjectivity of the usual equivariant Chern character is known the delocalized
Atiyah-Hirzebruch isomorphism follows. ∎

##  4\. Equivariant fibrations

Consider two compact manifolds $X_{i},$ $i=0,1,$ with smooth $G$ actions with
fixed isotropy type where $X_{0}$ is the total space of a $G$-equivariant
fibre bundle over $X_{1}$

(4.1) $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\hbox{$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
30.93124pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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72.80066pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{$\textstyle{Y_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
30.93124pt\raise-38.63776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
72.80066pt\raise-38.63776pt\hbox{\hbox{\kern 0.0pt\raise
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0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{Y_{1}}$}}}}}}}\ignorespaces}}}}\ignorespaces\text{with
}Y_{0}\text{ connected.}$

In the absence of the connectedness condition on $Y_{0}$ (which becomes
significant in the stability of subgroups) we proceed component by component.

First choose an isotropy group $H_{1}$ for the lower action and pass to the
Borel reduction $Z_{1}=X_{1}^{H_{1}}\subset X_{1}$ with its $N_{1}=N(H_{1})$
action. Next consider the preimage

(4.2) $Z_{0}=\phi^{-1}(X_{1}^{H_{1}})\subset X_{0}.$

As the preimage of a smooth manifold under a fibration this is smooth and
meets each orbit of the $G$-action on $X_{0}.$ Furthermore the pointwise
stabilizer is exactly $N_{1}$ since if $x\in\phi^{-1}(X_{1}^{H_{1}})$ and
$gx\in\phi^{-1}(X_{1}^{H_{1}})$ then $\phi(x)\in X_{1}^{H_{1}}$ and
$g\phi(x)=\phi(gx)\in X_{1}^{H_{1}}$ so $g\in N_{1}$ and conversely. The
isotropy groups at points of $Z_{0}$ remain unchanged since they are
necessarily subgroups of $H_{1}\subset N_{1}.$ Thus $Z_{0}$ is a reduction of
the action on $X_{0}.$

We have thereby reduced the equivariant fibration to the special case where
the lower action has unique isotropy group $H_{1}$

(4.3)
$\textstyle{N_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Id}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
Z_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{Y_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\phi}}$$\textstyle{N_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{1}}$$\textstyle{\kern-10.0pt\text{
with normal isotropy group }H_{1}.}$

Now, consider the components of $Z_{0}.$ Since $Y_{0}$ is connected each of
these connected submanifolds meets every orbit in $Z_{0}$ and hence projects
onto $Y_{0}.$ Choose a point in $Z_{0}$ at which the isotropy group is
$H_{0}.$ The rigidity of subgroups, already utilized above, shows that within
the component containing this point all isotropy groups fall in the
$H_{1}$-conjugacy class of $H_{0}$ as a subgroup of $H_{1}.$ Indeed, near the
chosen point, the isotropy groups are contained in $H_{1}$ and are conjugate
to $H_{0}$ by an element of $N_{1}$ near the identity and hence by the product
$n_{0}h_{1}$ with $n_{0}$ in the normalizer of $H_{0}$ and $h_{1}\in H_{1}.$
Thus they are $H_{1}$-conjugate to $H_{0},$ in fact by an element in the
component of the identity in $H_{1}.$ This therefore remains true over the
component of $Z_{0}.$ Let $Z_{0}^{\prime}$ be the union of the images of this
component under the action of $H_{1};$ all the isotropy groups on
$Z_{0}^{\prime}$ are $H_{1}$-conjugate to $H_{0}$ and all such groups occur in
each orbit. The stabilizer of $Z^{\prime}_{0}$ is the subgroup
$N_{1}^{\prime}\subset N_{1}$ (of finite index) fixing the $H_{1}$-isotropy
class of $H_{0}.$ This is a reduction of the group action on $Z_{0}$ and hence
of that on $X_{0}.$

The image $\phi(Z_{0}^{\prime})=Z_{1}^{\prime}$ is a reduction of the action
on $Z_{1}$ to the subgroup $N_{1}^{\prime};$ each of the components of
$Z^{\prime}_{0}$ projects to a smooth submanifold and for two components the
images are either equal or disjoint. Thus $Z_{1}^{\prime}\subset Z_{1}$ is
fixed by the condition that at every point in the preimage in $Z_{0}$ the
isotropy group is $H_{1}$-conjugate to $H_{0}.$ Thus we arrive at the reduced
equivariant fibration

(4.4)
$\textstyle{N^{\prime}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Id}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
Z^{\prime}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{Y_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\phi}}$$\textstyle{\kern-25.0pt\text{
with isotropy groups }H_{1}\text{-conjugate to
}H_{0}}$$\textstyle{N^{\prime}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z^{\prime}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{1}}$$\textstyle{\kern-60.0pt\text{
with normal isotropy group }H_{1}.}$

From this we deduce

######  Proposition 8.

For an equivariant fibration (4.1) with $Y_{0}$ connected there are choices of
isotropy groups $H_{0}\subset H_{1}$ for the two actions and a (‘ _double_ ’)
reduction, $Z^{\prime\prime}_{0},$ of $X_{0}$ with fixed isotropy group
$H_{0}$ and group action of $N_{0}=N_{G}(H_{0})\cap N_{G}(H_{1})$ and
$Z_{1}^{\prime}$ of $X_{1}$ with isotropy groups $H_{1}$ and group action by
$N_{1}^{\prime}=N_{0}H_{1}$ giving an equivariant fibration

(4.5)
$\textstyle{N_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
Z^{\prime\prime}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{Y_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\phi}}$$\textstyle{\kern-25.0pt\text{with
normal isotropy group
}H_{0}}$$\textstyle{N_{1}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z^{\prime}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{1}}$$\textstyle{\kern-25.0pt\text{
with normal isotropy group }H_{1}.}$

###### Proof.

Having arrived at (4.4) we make the Borel reduction of the upper action
corresponding to the isotropy group $H_{0}$ and with action by $N_{0},$ the
normalizer of $H_{0}$ in $N_{1}^{\prime}.$ The fibres of $\phi$ above points
of $Z_{1}^{\prime}$ have actions by $H_{1}$ and isotropy groups, for the
$N_{1}^{\prime}$ action on $Z_{1}^{\prime},$ $H_{1}$ conjugate to $H_{0}.$
Thus, each fibre meets $Z_{0}^{\prime\prime}$ which fibres over
$Z_{1}^{\prime}.$ The fact that the $N_{1}^{\prime}$-conjugates of $H_{0}$ are
also $H_{1}$-conjugates means that for each $n_{1}\in N_{1}^{\prime}$ there
exists $h_{1}\in H_{1}$ such that

(4.6) $n_{1}H_{0}n_{1}^{-1}=h_{1}H_{0}h_{1}^{-1}\Longrightarrow
h_{1}^{-1}n_{1}H_{0}(h_{1}^{-1}n_{1})^{-1}=H_{0}\Longrightarrow
N_{1}^{\prime}=H_{1}N_{0}.$

Since $H_{1}$ is normal in $N_{1}^{\prime},$ $N_{1}^{\prime}=N_{0}H_{1}$ and
$H_{1}$ acts trivially on $Z_{1}^{\prime}.$ ∎

Note that these reductions occur precisely because the principal bundles
corresponding to the $G$ actions are partially trivialized by the existence of
the equivariant fibration. Since $Z^{\prime\prime}_{0}$ is a reduction of the
$G$ action on $X_{0}$ and $Z^{\prime}_{1}$ is a reduction for $X_{1}$ there
are, by Proposition 1, equivalences of categories and hence a pull-back
functor $\phi^{\\#}$

(4.7)
$\textstyle{\operatorname{\mathfrak{B}un}_{G}(X_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{\mathfrak{B}un}_{N_{0}}(Z^{\prime\prime}_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigoplus_{\pi\in
H^{\prime}_{0}/N_{0}}\operatorname{\mathfrak{B}un}_{\widehat{Q}_{0}(\pi)}(Y_{0}(\pi))}$$\textstyle{\operatorname{\mathfrak{B}un}_{G}(X_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{*}}$$\textstyle{\operatorname{\mathfrak{B}un}_{N^{\prime}_{1}}(Z^{\prime}_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{\\#}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\bigoplus_{\sigma\in
H^{\prime}_{1}/N_{1}^{\prime}}\operatorname{\mathfrak{B}un}_{\widehat{Q}_{1}^{\prime}(\sigma)}(Y_{1}(\sigma))}$$\scriptstyle{\phi^{\\#}}$

which we proceed to describe more explicitly.

The pull-back operation on the right in (4.7) is the composite of pull-back
for $N_{1}^{\prime}$-equivariant bundles and restriction to the Borel
reduction to $N_{0}$-equivariant bundles. It follows that it can also written
as the composite of restriction, from $N_{1}^{\prime}$-equivariant to
$N_{0}$-equivariant bundles on $Z_{1}^{\prime}$ followed by pull-back of
$N_{0}$-equivariant bundles. The functoriality of reduction to twisted bundles
shows that the second step is just pull-back of twisted bundles so we
concentrate on the first step which can be understood in terms of ‘branching
maps’.

For a compact Lie group and closed subgroup $H_{0}\subset H_{1}$ consider the
bundle over the product of the sets of unitary irreducibles

(4.8)
$\textstyle{\tau\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{1}^{\prime}\times
H_{0}^{\prime}}$

with fibre at $(\sigma,\pi)$ the space $\hom_{H_{0}}(\pi,\sigma)$ of
$H_{0}$-equivariant linear maps from the representation space $M_{0}(\pi)$ of
$\pi$ to the representation space $M_{1}(\sigma)$ of $\sigma.$ The support of
this bundle is proper as a relation.

For simplicity we will denote $N^{\prime}_{1}$ in the discussion by $N_{1}.$
Thus suppose as above that $H_{1}\subset N_{1}$ is a normal subgroup of a
compact Lie group, $H_{0}\subset H_{1},$ and $N_{0}\subset N_{1}$ are closed
subgroups with $H_{0}$ normal in $N_{0}$ and $N_{1}=H_{1}\cdot N_{0}.$ As a
subgroup $N_{0}\subset N_{1}$ acquires a central extension
$\check{N}_{0}(\sigma)\subset\check{N}_{1}(\sigma)$ from the central extension
of $N_{1}$ associated to the orbit of $\sigma\in H_{1}^{\prime}.$ The
tautological bundle $M_{1}^{\prime}$ over $H_{1}^{\prime}$ therefore has an
action of $\check{N}_{0}(\sigma)$ when restricted to $N_{0}\sigma.$ In
consequence the bundle $\tau$ has an action of
$\widehat{N}_{0}(\pi)\times\check{N}_{1}(\sigma)$ over $N_{0}\sigma\times
N_{1}\pi.$

######  Proposition 9.

If $X_{1}$ is a compact manifold with smooth action by $N_{1}$ with fixed
isotropy group $H_{1}$ and $H_{0},$ $N_{0}$ are as above then for each pair
$(\sigma,\pi)\in H_{1}^{\prime}\times H_{0}$ there is a ‘branching map’

(4.9)
$\tau:\operatorname{\mathfrak{B}un}(Y(\sigma);\alpha(\sigma))\longrightarrow\operatorname{\mathfrak{B}un}(Y(\pi);\alpha(\pi))$

from $\widehat{N}_{1}(\sigma)$-equivariant bundles over $(X_{1}\times
N_{1}\sigma)/N_{1}$ to $\widehat{N}_{0}$-equivariant bundles over
$(X_{1}\times N_{0}\pi)/N_{0}$ given by tensor product with $\tau$ pulled back
to $X_{1}.$

######  Corollary 10.

The pull-back map on the right in (4.7) is the composite

(4.10) $\phi^{\\#}=\phi^{*}\circ\tau$

where $\phi^{*}$ is the pull-back of bundle gerbe modules covering the pull-
back of principal bundles.

For the Cartan model for equivariant cohomology, as noted in (3.15), reduction
corresponds to an homotopy equivalence to the complex of equivariant forms on
the reduction. Thus under the multiple reductions on $X_{0},$

(4.11) $({\mathcal{C}}^{\infty}(X_{0};\Lambda^{*}\otimes
S(\mathfrak{g}^{*}))^{G}\longrightarrow({\mathcal{C}}^{\infty}(Z^{\prime\prime}_{0};\Lambda^{*}\otimes
S(\mathfrak{n}_{0}^{*}))^{N_{0}}$

where $\mathfrak{n}_{0}$ is the Lie algebra of $N_{0}$ which normalizes the
isotropy group $H_{0}.$ The quotient group $Q=N_{0}/H_{0}$ acts freely on
$Z^{\prime\prime}_{0}$ and (3.16) gives the further homotopy equivalence

(4.12) $({\mathcal{C}}^{\infty}(Z^{\prime\prime}_{0};\Lambda^{*}\otimes
S(\mathfrak{n}_{0}^{*}))^{N_{0}}\longrightarrow{\mathcal{C}}^{\infty}(Y_{0};\Lambda^{*}\otimes
S_{H_{0}})$

to the deRham complex with coefficients in the flat bundle $S_{H_{0}}$ over
$Y_{0}.$

Similarly in the base the successive reductions and ‘twisted reduction’ give
homotopy equivalences

(4.13) $({\mathcal{C}}^{\infty}(X_{1};\Lambda^{*}\otimes
S(\mathfrak{g}^{*}))^{G}\longrightarrow({\mathcal{C}}^{\infty}(Z^{\prime}_{1};\Lambda^{*}\otimes
S(\mathfrak{n}_{1}^{*}))^{N^{\prime}_{1}}\longrightarrow{\mathcal{C}}^{\infty}(Y_{1};\Lambda^{*}\otimes
S_{H_{1}})$

to the deRham complex with coefficients in $S_{H_{1}}$ which has typical fibre
$S(\mathfrak{n}^{*}_{1}).$ Since $Q=N^{\prime}_{1}/H_{1}=N_{0}/H_{0},$ the
actions on $S(\mathfrak{n}_{0}^{*})$ and $\mathfrak{n}_{1}^{*}$ are consistent
with restriction, induced by the inclusion of $N_{0}\hookrightarrow
N_{1}^{\prime}$ and hence there is a natural pull-back and restriction map
covering $\phi:$

(4.14) $(S_{H_{1}})_{\phi(y_{0})}\longrightarrow(S_{H_{0}})_{y_{0}},\ y_{0}\in
Y_{0}$

and hence an augmented, smooth, pull-back map on ‘deRham’ sections

(4.15) $\phi^{\\#}:{\mathcal{C}}^{\infty}(Y_{1};\Lambda^{*}\otimes
S_{H_{1}})\longrightarrow{\mathcal{C}}^{\infty}(Y_{0};\Lambda^{*}\otimes
S_{H_{0}}).$

######  Proposition 11.

Pull-back and iterated reduction gives a commutative diagram

(4.16) $\textstyle{({\mathcal{C}}^{\infty}(X_{0};\Lambda^{*}\otimes
S(\mathfrak{g}^{*}))^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{C}}^{\infty}(Y_{0};\Lambda^{*}\otimes
S_{H_{0}})}$$\textstyle{({\mathcal{C}}^{\infty}(X_{1};\Lambda^{*}\otimes
S(\mathfrak{g}^{*}))^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{*}}$$\textstyle{{\mathcal{C}}^{\infty}(Y_{1};\Lambda^{*}\otimes
S_{H_{1}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{\\#}}$

where the horizontal maps are homotopy equivalences.

Pull-back for the forms corresponding to delocalized equivariant cohomology
behaves similarly. The quotient bundle $Q$ acts on both $H^{\prime}_{1}$ and
$H^{\prime}_{0}.$ As in the case of a general equivariant bundle the
tautological bundle $M_{1}^{\prime}$ of $H_{1}$-representations over
$Z_{1}^{\prime}\times H_{1}^{\prime}$ decomposes into a bundle over the
product with $H_{0}^{\prime}$

(4.17) $M^{\prime}_{10}\longrightarrow Z_{1}^{\prime}\times
H_{1}^{\prime}\times H_{0}^{\prime}$

with proper support, such that $M^{\prime}_{10}\otimes M^{\prime}_{0}$ pushes
forward off $H_{0}^{\prime}$ to $M_{1}^{\prime}$ over $Z_{1}^{\prime}\times
H_{1}^{\prime}.$ Restricting to a $Q$-orbit $o_{1}(\sigma)\subset
H_{1}^{\prime}$ the support of $M^{\prime}_{10}$ is restricted to the finite
number of orbits $o_{0}(\pi)\subset H_{0}^{\prime}$ of
$H_{0}$-subrepresentations of $\sigma.$ The bundle $M^{\prime}_{1}$ over
$Z_{1}^{\prime}\times o_{1}(\sigma)$ has an action of the corresponding
central extension $\check{Q}(\sigma)$ and following the argument above
$M^{\prime}_{10}$ has an action by the central extension
$\check{Q}(\pi,\sigma)$ which has the property

(4.18) $\check{Q}(\pi,\sigma)\otimes\check{Q}(\sigma)=\check{Q}(\pi)$

with the tensor product referring to the circle bundles giving the central
extensions.

##  5\. Reduced models

Each of the three equivariant cohomology theories, K-theory, delocalized (by
definition) and Cartan, has a model over the iterated space resolving the
quotient by the group action and at this level the behavior of the Chern
character can be seen rather directly.

Not surprisingly, the most complicated of these is equivariant K-theory. Each
of the strata, $X_{\alpha},$ of the normal resolution, has Borel reduction to
a principal bundle with total space $Z_{\alpha}$ and base $Y_{\alpha}$ for the
action of $Q_{\alpha}=N(H_{\alpha})/H_{\alpha}$ where $H_{\alpha}$ is a choice
of isotropy group. We need to choose the isotropy groups consistently to
define the pull-back maps so we proceed categorically, allowing all possible
choices of isotropy group. Once a choice is made the spaces

(5.1) $Y_{\alpha}(\pi)=Z_{\alpha}(\pi)/N(H_{\alpha}),\ \pi\in
H_{\alpha}^{\prime},\ X_{\alpha}(\pi)=X_{\alpha}\times o(\pi),\
o(\pi)=N(H_{\alpha})\pi$

are defined for each orbit of $N(H_{\alpha})$ acting on the category of
irreducible unitary representations $H^{\prime}_{\alpha}.$ Then the reduced
version of an equivariant bundle consists of a finite collection of twisted
bundles for the Dixmier-Douady classes, $a_{\alpha}(\pi),$ of
$Z_{\alpha}(\pi)$ over the corresponding $Y_{\alpha}(\pi).$ As discussed
above, these are bundles gerbes over the $X_{\alpha}(\pi)$ as the total space
of a lifting bundle gerbe given by the free action of $Q_{\alpha}(\pi)$ and
the central extension $\widehat{Q}_{\alpha}(\pi)$ arising from the action of
$N(H_{\alpha})$ on $o(\pi).$ Thus for each stratum we have a category of
reduced equivariant bundles

(5.2)
$\operatorname{\mathfrak{B}un}_{\alpha}=\bigoplus_{H^{\prime}_{\alpha}/N(H_{\alpha})}\operatorname{\mathfrak{B}un}(Y_{\alpha}(\pi),a_{\alpha}(\pi)).$

This behaves naturally under change of choices.

For each $\beta<\alpha$ there is a boundary hypersurface $H_{\beta,\alpha}$ of
$X_{\beta}$ with fibration
$\phi_{\beta,\alpha}:H_{\beta,\alpha}\longrightarrow X_{\alpha}.$ Proposition
4.9 describes the corresponding augmented pull-back map

(5.3)
$\phi_{\beta,\alpha}^{\\#}:\operatorname{\mathfrak{B}un}_{\alpha}\longrightarrow\operatorname{\mathfrak{B}un}_{\beta}\big{|}_{H_{\beta,\alpha}}.$

Then the ‘chain space’ for K-theory consists of
$W_{\alpha}\in\operatorname{\mathfrak{B}un}_{\alpha}$ for each $\alpha$ such
that for all $\alpha<\beta$

(5.4)
$W_{\alpha}\big{|}_{H_{\beta,\alpha}}\simeq\phi^{\\#}_{\beta,\alpha}W_{\beta}$

where the implied isomorphisms must compose under iteration. The collection of
reduced bundles is then a category and the reduced K-theory
$K_{\operatorname{red}}(Y_{*})$ is the corresponding Grothendieck group; of
course, despite the notation, this depends on more information than just the
quotient spaces.

######  Proposition 12.

The equivariant K-theory of $X$ is naturally isomorphic to the reduced
K-theory $K_{\operatorname{red}}(Y_{*})$ of the resolution of the quotient.

###### Proof.

This follows by application of the reduction results above. ∎

For equivariant cohomology there is a closely related ‘reduced’ Cartan model.
In the same setting as above, let $C(H_{\alpha})$ be the space of
$H_{\alpha}$-invariant polynomials on the dual of the Lie algebra of
$H_{\alpha}.$ The normalizer $N(H_{\sigma})$ acts on this space by
conjugation, with the action descending to $Q_{\alpha}$ and locally trivial.
Lifting $C(H_{\alpha})$ to a trivial bundle over $X_{\alpha}$ this action
defines a flat bundle $C_{\alpha}$ over the base $Y_{\alpha}$ of the $G$
action. Then the reduced Cartan model consists of forms

(5.5) $u_{\alpha}\in{\mathcal{C}}^{\infty}(Y_{\alpha};\Lambda^{*}\otimes
C_{\alpha})$

over each $Y_{\alpha}$ with values in $C_{\alpha}$ but connected by (reduced)
pull-back maps as in (4.16),

(5.6) $u_{\alpha}\big{|}_{H_{\beta,\alpha}}=\phi_{\beta,\alpha}^{*}u_{\beta}.$

Again by repeated application if the reduction results above we arrive at

######  Proposition 13.

The equivariant cohomology of $X$ is naturally isomorphic to the reduced
Cartan cohomology which is the deRham cohomology of forms (5.5), (5.6).

Finally we _define_ delocalized equivariant cohomology by replacing the Cartan
space $C(H_{\alpha})$ by the real representation ring of $H_{\alpha},$
$\mathcal{R}(H_{\alpha})=\mathcal{I}(H_{\alpha})\otimes\mathbb{R},$ to which
it is isomorphic in the connected case. The action of $N(H_{\alpha})$ by
conjugation on $H_{\alpha}$ induces an equivariant action on
$\mathcal{I}(H_{\alpha})$ and hence on $\mathcal{R}(H_{\alpha})$ and this
defines a flat bundle $R_{\alpha}$ over $Y_{\alpha}.$ Then the admissible
‘delocalized’ forms are

(5.7)
$v_{\alpha}\in{\mathcal{C}}^{\infty}(Y_{\alpha};\Lambda^{*}\otimes\mathcal{R}_{\alpha})$

connected by the pull-back maps augmented by branching

(5.8) $v_{\alpha}\big{|}_{H_{\beta,\alpha}}=\phi_{\beta,\alpha}^{*}v_{\beta}.$

The delocalized cohomology $H^{*}_{G,\mathrm{dl}}(X)$ is defined to be the
deRham cohomology of this complex of forms. In this sense there is no
reduction theorem but a definition. It is very natural to expect the
possibility a sheaf-theoretic formulation directly over the original space.

The surjective character maps $\mathcal{R}(H)\longrightarrow C(H)$ for the
isotropy groups induce surjective ‘localization’ maps from the forms (5.7),
(5.8) to the Cartan forms (5.5), (5.6) through relaxation of the local
coefficients from $\mathcal{R}_{\alpha}$ to $C_{\alpha}.$ This therefore
descends to a surjective map

(5.9) $\operatorname{loc}:H^{*}_{G,\mathrm{dl}}(X)\longrightarrow
H^{*}_{G}(X).$

To define the equivariant Chern character as a homomorphism giving a
commutative triangle

(5.10)
---
$\textstyle{K_{G}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Ch}_{G,\mathrm{dl}}}$$\scriptstyle{\operatorname{Ch}_{G}}$$\textstyle{H^{\operatorname{even}}_{G,\mathrm{dl}}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{loc}}$$\textstyle{H^{\operatorname{even}}_{G}(X)}$

we start with an appropriate notion of ‘reduced connection’ on the bundles
(5.3), (5.4).

Each bundle $W_{\alpha}$ can be equipped with a connection which is invariant
under the action of $\widehat{N}_{\alpha}.$ The Chern character is then an
even closed form on $X_{\alpha}$ which is invariant under the action of
$\widehat{N}_{\alpha},$ since the center acts trivially on forms thus is a
true form on the base $Y_{\alpha}.$ The same is true for the
$\check{N}_{\alpha}$ action on $M^{\prime}$ as a bundle over $X_{\alpha}$ and
the sum over the fibres of the product of these closed forms is an element of
the space (5.7). The collection of these forms satisfies (5.8) provided the
connections are chosen consistently. This defines the Chern character (5.10)
with commutativity following from the lifting of the standard definition of
the equivariant Chern character, see for example [3].

##  6\. Atiyah-Hirzebruch isomorphism

######  Theorem 14.

For any smooth action by a compact Lie group on a compact manifold the
delocalized equivariant Chern character defines an isomorphism

(6.1)
$\operatorname{Ch}_{G,\mathrm{dl}}:K_{G}(X)\otimes\mathbb{R}\longrightarrow
H^{*}_{G,\mathrm{dl}}(X).$

The proof follows the same lines as in [6]. The delocalized equivariant
cohomology is naturally $\mathbb{Z}_{2}$-graded (not $\mathbb{Z}$-graded
because of the Chern character factors in the augmented pull-back maps). Odd
K-theory is defined in the usual way by suspension, i.e. as the null space of
the restriction morphism

(6.2) $K_{G,\operatorname{red}}(\mathbb{S}\times Y_{*})\longrightarrow
K_{G,\operatorname{red}}(\\{0\\}\times Y_{*}).$

Consider the ‘pruning’ of the isotropy tree. Thus let $A$ be an ordered
collection of the indices, so

(6.3) $\beta<\alpha,\ \beta\in P\Longrightarrow\alpha\in P.$

In each cohomology theory the notion of triviality on the $X_{\alpha}$ for
$\alpha\in P$ is well-defined giving relative versions

(6.4) $K^{*}_{G,\operatorname{red}}(Y_{*};P),\
H^{*}_{G,\mathrm{dl}}(Y_{*};P)\text{ and }H^{*}_{G}(Y_{*};P).$

In all three cohomology theories there is an excision sequence corresponding
to two ordered sets $P$ and $P^{\prime}=P\cup\\{\gamma\\}.$ Namely for
(reduced) equivariant K-theory

(6.5)
$\textstyle{K^{0}_{G,\operatorname{red}}(Y_{*};P^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K^{0}_{G,\operatorname{red}}(Y_{*};P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K^{0}_{G,\operatorname{red}}(Y_{\gamma};P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K^{1}_{G,\operatorname{red}}(Y_{\gamma};P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K^{1}_{G,\operatorname{red}}(Y_{*};P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K^{1}_{G,\operatorname{red}}(Y_{*};P^{\prime}).\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

The delocalized Chern character gives an exact complex

(6.6)
---
$\textstyle{K^{0}_{G,\operatorname{red}}(Y_{*};P^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Ch}}$$\textstyle{K^{0}_{G,\operatorname{red}}(Y_{*};P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Ch}}$$\textstyle{K^{0}_{G,\operatorname{red}}(Y_{\gamma};P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Ch}}$$\textstyle{H^{0}_{\mathrm{dl}}(Y_{*};P^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}_{\mathrm{dl}}(Y_{*};P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}_{\mathrm{dl}}(Y_{\gamma};P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{\mathrm{dl}}(Y_{\gamma};P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{\mathrm{dl}}(Y_{*};P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{\mathrm{dl}}(Y_{*};P^{\prime}).\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K^{1}_{G,\operatorname{red}}(Y_{\gamma};P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Ch}}$$\textstyle{K^{1}_{G,\operatorname{red}}(Y_{*};P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Ch}}$$\textstyle{K^{1}_{G,\operatorname{red}}(Y_{*};P^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Ch}}$

This complex remains exact when tensored with $\mathbb{R}.$ We proceed by
induction over $P$ starting from the case that $P$ contains all elements
except the minimal one corresponding to the open isotropy type. The top right
and lower left Chern character maps are isomorphism as discussed above and by
induction the other two corner Chern maps are isomorphisms. The fives lemma
shows the central maps to be isomorphism, so the induction continues. The case
that $P=\emptyset$ is the Atiyah-Hirzebruch-Baum-Brylinski-MacPherson
isomorphism.

## References

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# Real effective potentials for phase transitions in models with extended
scalar sectors ††thanks: Presented at Matter to the Deepest 2023 international
workshop

Károly Seller  Institute for Theoretical Physics, ELTE Eötvös Loránd
University, Pázmány Péter sétány 1/A, H-1117 Budapest, Hungary

###### Abstract

The effective potential is a widely used phenomenological tool to investigate
phase transitions occurring in the early Universe at finite temperature. In
the standard perturbative treatment the potential becomes complex in some
region of the background field values due to the non-convex nature of the
classical potential in models with spontaneous symmetry breaking. The
imaginary part renders the minimization of the potential impossible when at
finite temperature the absolute minimum is in the complex region. In this talk
we introduce a simple method to calculate an effective potential that is fully
real based on the optimized perturbation theory scheme. We apply the method
for models that extend the Standard Model with an additional singlet scalar.

##  1 Introduction

The effective potential defined via Legendre transformation in quantum field
theories is the effective action evaluated for homogeneous field
configurations. By definition the exact effective potential is a convex and
real function of the background field values. While the exact potential may be
obtained via non-perturbative methods, in phenomenological studies we use the
loop expansion of the defining path integral and truncate the obtained series
at some finite order.

The loop-expanded effective potential can be split into a sum of the classical
potential and loop corrections in which the quantum effects include vacuum
($T=0$) and finite temperature contributions. In a general theory with
multiple scalar fields with background field values $v_{k}$, the one-loop
effective potential can be written as

$V^{[1]}_{\mathrm{eff}}\big{(}\\{v_{k}\\},T\big{)}=V_{\mathrm{cl}}\big{(}\\{v_{k}\\}\big{)}+\sum_{i\in\mathbf{P}}\left[V^{(1)}_{i}\big{(}\\{v_{k}\\}\big{)}+V^{(1)}_{\mathrm{T},\,i}\big{(}\\{v_{k}\\},T\big{)}\right]\,.$
(1)

Here $\\{v_{k}\\}$ is the set of background field values, $T$ is the
temperature, and ${\bf P}$ denotes the set of particles coupled to the scalar
fields. The vacuum quantum corrections are in general UV and IR divergent and
they need to be renormalized using standard renormalization techniques. The IR
divergence arises due to the presence of massless Goldstone modes, and at one
loop order it is removed by considering the pole mass parametrization scheme.

In the SM, the effective potential is a function of a single vacuum
expectation value $v$ of the Brout-Englert-Higgs (BEH) field. At zero
temperature the finite one-loop effective potential is given in Ref. [1].
Concerning the thermal evolution of $v$ the interesting parts are the scalar
contributions

$V^{[1]}_{\rm
SM,eff}(v)\supset\frac{1}{64\pi^{2}}\left[m_{h}^{4}(v)\ln\frac{m_{h}^{2}(v)}{M_{h}^{2}}+3m_{G}^{4}(v)\ln\frac{m_{G}^{2}(v)}{M_{h}^{2}}\right]\,.$
(2)

Here $M_{h}^{2}$ is the Higgs boson mass and the background dependent scalar
masses are given by $m_{h}^{2}(v)=\mu^{2}+3\lambda v^{2}$ and
$m_{G}^{2}(v)=\mu^{2}+\lambda v^{2}$, where the square mass parameter is
$\mu^{2}=-M_{h}^{2}/2<0$. Note that the regularization scale dependence was
removed by a finite wave function renormalization factor that fixed the
residue of the Higgs boson propagator at the pole mass to unity.

The one-loop SM effective potential $V^{[1]}_{\rm SM,eff}(v)$ is complex for
field values $v<v_{0}$, where $v_{0}=\sqrt{-\mu^{2}/\lambda}$ is the minimum
(physical point) of the potential. The reason for the appearance of imaginary
parts is due to the scalar contributions of Eq. (2) that are complex when
$m_{h}^{2}(v)<0$ or $m_{G}^{2}(v)<0$.

At high temperature the SM relaxes to its symmetric phase where the vacuum
expectation value of the BEH field vanishes and the effective potential
becomes convex. However, in the perturbative treatment the vacuum part of the
effective potential close to the origin is complex as indicated above. Adding
thermal contributions does not remove the imaginary part (in fact it makes
things worse) thus at finite temperature the potential cannot be minimised.
Consequently, the effective potential obtained by the standard perturbative
treatment cannot be used to study phase transitions at finite temperature.

In this contribution, I introduce the optimized perturbation theory (OPT)
scheme of Ref. [2] as a simple tool to obtain an effective potential that is
real for all values of the background field. I use it to obtain a real
effective potential at zero temperature in the SM and in the SM extended by
one singlet scalar field. Finally, I show finite temperature contributions and
calculate the critical temperatures in the superweak extension of the SM [3].
This contribution is based on Ref. [1].

##  2 Optimized perturbation theory

The OPT scheme of Ref. [2] is an established calculational tool in quantum
field theory that introduces a modified perturbative series. The method
preserves Ward identities and renormalizability even at higher orders in the
loop expansion as it preserves the symmetries of the Lagrangian. Additionally,
the OPT method can be used to obtain an effective potential that remains real
for all values of the background fields even at higher temperatures where
minimization in the standard approach is hindered by the appearing imaginary
parts at $v<v_{0}$.

###  2.1 Parametrization of the SM

In the OPT scheme we introduce a new squared mass parameter $m^{2}$ that we
assume to be positive. By reorganization of the Lagrangian terms we write the
original Lagrangian potential in the form

$\mathcal{L}\supset\mathcal{V}[\phi]\equiv\mathcal{V}_{\mathrm{OPT}}[\phi]=m^{2}|\phi|^{2}+\lambda|\phi|^{4}+(\mu^{2}-m^{2})|\phi|^{2}\,,$
(3)

where we treat the last term as a counterterm that only contributes at loop
level. We may interpret $m^{2}$ as the tree level value of the mass squared
parameter. With the shifted mass parameter the classical potential is convex
and consequently the scalar masses are real. Since the Lagrangian is unchanged
it follows that the exact result is unchanged, in particular it is independent
of $m^{2}$.

The one-loop effective potential of the SM at zero temperature is

$V_{\mathrm{SM,OPT}}^{[1]}(v;\mu^{2},m^{2})=\underbrace{V_{\mathrm{cl}}(v;m^{2})+\hbox
to0.0pt{$\displaystyle\overbrace{\phantom{\frac{\mu^{2}-m^{2}}{2}v^{2}+V^{(1)}(v;m^{2})}}^{\displaystyle{\rm
one-loop}}$\hss}\frac{\mu^{2}-m^{2}}{2}v^{2}}_{\displaystyle=V_{\rm
cl}(v;\mu^{2})}+V^{(1)}(v;m^{2})\,.$ (4)

The effective potential at any finite order of perturbation theory will depend
on three unknown parameters, $\mu^{2},m^{2},$ and $\lambda$. The dependence on
$m^{2}$ only appears in the one-loop correction.

The parametrization of the OPT potential is done by using the standard
parametrization conditions for the minimum of the potential and the mass of
the Higgs boson,

$\left.\frac{\partial V_{\mathrm{SM,OPT}}^{[1]}(v;\mu^{2},m^{2})}{\partial
v}\right|_{v=v_{0}}=0\,,\quad\left.\frac{\partial^{2}V_{\mathrm{SM,OPT}}^{[1]}(v;\mu^{2},m^{2})}{\partial
v^{2}}\right|_{v=v_{0}}=M_{h}^{2}\,.$ (5)

These two conditions fix two out of the three parameters.

The third condition is called the principle of least sensitivity that requires
that the potential in the physical point (minimum) at fixed order is least
dependent on the choice of $m^{2}$,

$\left.\frac{\partial V_{\mathrm{SM,OPT}}^{(1)}(v;\mu^{2},m^{2})}{\partial
m^{2}}\right|_{v=v_{0}}=0\,.$ (6)

There is no guarantee that these three conditions have real solutions for the
parameters such that $m^{2}>0$. However, it so happens that for the SM the
system is solvable and we find

$m^{2}=69~{}094.6{\rm
GeV}^{2},\quad\lambda=0.12~{}861,\quad\mu^{2}=-8~{}847.85{\rm GeV}^{2}\,.$ (7)

Since $m^{2}>0$ this potential is fully real for all values of the vacuum
expectation value $v$.

###  2.2 Parametrization of singlet scalar extended SM

Figure 1: Parametrization of the SM extended with a singlet complex scalar
field. In the figure we used $M_{s}=260~{}$GeV and $w_{0}=10v_{0}$, with the
regularization scale fixed at the top quark mass, $Q=M_{t}$.

In particle physics models that involve multiple scalar fields the effective
potential is a function of multiple vacuum expectation values. Thankfully, the
generalization of the OPT scheme introduced above is trivial to models with
any number of scalar fields. The general prescription is to introduce shifted
mass parameters $m_{\alpha}^{2}$ for all $\mu_{\alpha}^{2}<0$ present in the
Lagrangian.

One of the simplest extensions is to add one singlet complex scalar field to
the SM with vacuum expectation value $w\neq 0$. Assuming a $Z_{2}$ symmetric
potential and non-vanishing mixing between the scalar fields, one has:

$V_{\rm
OPT}^{[1]}(v,w;\mu^{2}_{\phi},\mu^{2}_{\chi},m^{2}_{\phi},m^{2}_{\chi})=V_{\rm
cl}(v,w;\mu^{2}_{\phi},\mu^{2}_{\chi})+V^{(1)}(v,w;m^{2}_{\phi},m^{2}_{\chi})\,.$
(8)

Similarly to the SM case, the classical potential and the counterterms combine
to form the non-convex classical potential evaluated at
$\mu_{\phi,\chi}^{2}<0$, while the one-loop contributions are all evaluated
with $m_{\phi,\chi}^{2}>0$.

The parametrization conditions are easily generalized to the case of multiple
scalar fields. In the SM extended with a singlet scalar there are in total
seven parameters:
$\mu^{2}_{\phi},\mu^{2}_{\chi},m^{2}_{\phi},m^{2}_{\chi},\lambda_{\phi},\lambda_{\chi},$
and $\lambda^{\prime}\,.$ Two conditions are used to fix the minimum of the
potential:

$\left.\frac{\partial V_{\mathrm{OPT}}^{[1]}(v,w)}{\partial
v}\right|_{v=v_{0},w=w_{0}}=\left.\frac{\partial
V_{\mathrm{OPT}}^{[1]}(v,w)}{\partial w}\right|_{v=v_{0},w=w_{0}}=0\,.$ (9)

The condition to fix the scalar masses is modified, as the second derivative
of the potential is now a symmetric matrix. We set the eigenvalues of this
curvature matrix to be equal to the scalar masses ($v_{i}=(v,w)_{i}$),

$\left.\mathbf{M}^{2}_{ij}=\partial_{v_{i}}\partial_{v_{j}}V_{\mathrm{OPT}}^{[1]}\right|_{v=v_{0},w=w_{0}}\longrightarrow\mathrm{diag}(M_{h}^{2},M_{s}^{2})\,.$
(10)

The last two conditions are the principle of minimum sensitivity enforced on
both $m_{\phi}^{2}$ and $m_{\chi}^{2}$ parameters:

$\left.\frac{\partial V_{\mathrm{OPT}}^{[1]}(v,w)}{\partial
m_{\phi}^{2}}\right|_{v=v_{0},w=w_{0}}=\left.\frac{\partial
V_{\mathrm{OPT}}^{[1]}(v,w)}{\partial
m_{\chi}^{2}}\right|_{v=v_{0},w=w_{0}}=0\,.$ (11)

The six conditions of Eqs. (9)–(11) fix all but one parameter of the one-loop
effective potential. We choose the scalar mixing parameter $\lambda^{\prime}$
as the free paramater in terms of which we study the solution to the
parametrization conditions for various values of $M_{s}^{2}$ and $w_{0}$.

An example parametrization is shown in Fig. 1 for a singlet scalar mass of
$M_{s}=260~{}$GeV and vacuum expectation value $w_{0}=10v_{0}$. The scalar
mixing parameter $\lambda^{\prime}$ is constrained by the mass difference of
the scalar bosons to a symmetric interval around $\lambda^{\prime}=0$, the
excluded region is shown in red. For any $\lambda^{\prime}$ there are two
solutions for each parameter: this merely reflects our ability to choose one
or the other scalar boson as the Higgs boson. The physical branch, containing
the red and green points in Fig. 1 is chosen based on the requirement that the
mass of the Higgs boson depends mostly on the vacuum expectation value of the
BEH field. In the figure, we also see that a solution with both
$m_{\phi,\chi}^{2}>0$ is not always possible. This is an artefact of the
choice of regularization scale $Q$ and could be easily removed by increasing
$Q$. In practice, the larger the new scalar mass, the larger the
regularization scale we should require in order to maintain a real potential.

##  3 Phase transitions

Figure 2: Critical temperatures of phase transitions in the superweak model.
In the left panel we see the temperatures where the SM BEH field acquires a
non-zero vacuum expectation value. In the right panel we see the temperatures
where the new singlet complex scalar field acquires a non-zero vacuum
expectation value.

In this section, we apply the OPT scheme to calculate critical temperatures of
phase transitions in the superweak model. The finite temperature contributions
to the potential do not introduce any new parameters, and we simply
parametrize the potential at zero temperature as described in the previous
section.

The thermal contribution of Eq. (1) in the OPT scheme has the general form

$V_{\mathrm{T,OPT}}^{(1)}\Big{(}\\{v_{k}\\},\\{m_{\alpha}^{2}\\},T\Big{)}=\frac{T^{4}}{2\pi^{2}}\sum_{i\in\mathbf{P}}n_{i}J^{(i)}_{\pm}\Big{(}m^{2}_{i}\big{(}\\{v_{k}\\},\\{m_{\alpha}^{2}\\}\big{)},T\Big{)}\,,$
(12)

where the fermionic or bosonic functions $J_{\pm}(m^{2},T)$ are given in e.g.
Ref. [4]. Most importantly, they are real only if the particle masses are
real. This is achieved by the OPT parametrization and at first glance there is
no new imaginary part introduced by the OPT thermal corrections. We mention,
however, that for $\lambda^{\prime}<0$ the thermal mass of scalar fields can
become imaginary at high temperatures due to $M^{2}_{\rm
scalar}\supset\lambda^{\prime}T^{2}<0$. However, this result is related to the
model definitions and not to the breakdown of perturbation theory, hence it is
expected that the OPT scheme does not fix this.

In the superweak model the spectrum of the SM is extended with a new singlet
scalar field and we may parametrize it as shown in the previous section. In
this model, the singlet scalar field acts as a Higgs boson in the dark sector,
giving mass to a new gauge boson $Z^{\prime}$ and to the right-handed
neutrinos. It follows that both the SM BEH field and the singlet scalar field
necessarily acquire non-vanishing vacuum expectation values. In general, the
two scalar fields acquire vacuum expectation values at different temperatures
that depend on the parameters of the model. An example of the parameter
dependence of the critical temperatures for $M_{s}=300~{}$GeV is shown in Fig.
2.

##  4 Conclusions

In this contribution, I have demonstrated how the optimized perturbation
theory scheme can be used to calculate effective potentials in models with
extended scalar sectors. This method could be used to derive a potential that
is real for all values of the background field and for all temperatures. I
applied the formalism to the SM extended by a singlet complex scalar, and
calculated the critical temperatures of a two-step phase transition in the
superweak model. The method could equally be used for more complicated models
as well.

## References

  * [1] K. Seller, Z. Szép and Z. Trócsányi, “Real effective potentials for phase transitions in models with extended scalar sectors,” JHEP 04 (2023), 096, [arXiv:2301.07961 [hep-ph]].
  * [2] S. Chiku and T. Hatsuda, _Optimized perturbation theory at finite temperature_ , _Phys. Rev. D_ 58 (1998) 076001 [hep-ph/9803226].
  * [3] Z. Trócsányi, “Super-weak force and neutrino masses,” Symmetry 12 (2020) no.1, 107, [arXiv:1812.11189 [hep-ph]].
  * [4] P. B. Arnold and O. Espinosa, “The Effective potential and first order phase transitions: Beyond leading-order,” Phys. Rev. D 47 (1993), 3546 [erratum: Phys. Rev. D 50 (1994), 6662], [arXiv:hep-ph/9212235 [hep-ph]].