Relative Equivariant Localization

• Relative equivariant localization is a method that recovers global invariants by isolating data on a distinguished relative locus such as fixed points, minimal strata, or centers of stratification.
• It employs exact sequences, support or completion arguments, and corrective terms like Euler classes or residues to annihilate complementary contributions and reconstruct the global invariant.
• The approach spans various fields including topology, algebraic geometry, and operator theory, providing a unified framework for analyzing symmetry in equivariant settings.

Searching arXiv for relevant papers on “relative equivariant localization” across the main mathematical usages. Relative equivariant localization denotes a family of localization principles in which a global equivariant invariant is recovered from data attached to a distinguished relative locus: minimal orbit-type strata, fixed-point stacks, centers of a $\Theta$-stratification, a closed invariant subset, or a fiber over a semisimple conjugacy parameter. The invariant being localized varies substantially across the literature: the equivariant Lyusternik–Schnirelmann category, endomorphism rings of motives, factorization homology, coarse indices, Hochschild and cyclic homology, or supersymmetric action functionals. What remains common is the passage from global equivariant geometry to a smaller relative object together with an exactness, completion, or deformation mechanism that kills the complementary contribution after localization (Fontaine et al., 2017, Calmès et al., 2016, Butson, 2020, Chen et al., 2020, Chen, 2017, Genolini et al., 2024).

1. Core pattern of relative localization

In the broadest sense, equivariant localization relates an invariant of a $G$-object to invariants supported on loci where the symmetry is more rigid. In the topological setting of equivariant LS-category, the relevant loci are the minimal orbit-type strata in the closure order on the orbit-type stratification (Fontaine et al., 2017). In factorization homology, the relative pair is $(X,X^G)$, and localization is expressed by the vanishing of the relative factorization homology after inverting suitable parameters (Butson, 2020). In equivariant coarse index theory, the pair is $(X,Y)$, or more generally $(M_1,M_2,Y_1,Y_2)$, and the relative class is supported near the closed invariant subsets $Y_i$ (Chen et al., 2020). In derived-loop and cyclic-homological localization, the relative parameter is a semisimple class $[z]\in G//G$, and completion at $[z]$ identifies invariants of $X/G$ with those of the $z$-fixed stack (Chen, 2017). In non-abelian localization for $G$0-stratified stacks, the centers $G$1 of the strata replace the fixed loci, and the global $G$2-theoretic class is reconstructed from those centers with inverse Euler-class corrections (Halpern-Leistner, 28 Sep 2025).

A plausible common implication is that “relative” does not merely mean “with respect to a subgroup.” In these works it marks a localization datum: a pair, a stratification, a completion parameter, or a boundary condition. The mechanism is then one of three types. First, there is an exact or triangulated comparison, such as a six-term exact sequence or an exact triangle. Second, there is a support or completion argument that annihilates the complement after localization. Third, there is an explicit residue, Euler-class, or normal-bundle correction that reconstructs the original invariant from the localized pieces.

2. Stratified localization in topology and equivariant category

For a proper $G$3-manifold $G$4, the equivariant Lyusternik–Schnirelmann category $G$5 is the least number of $G$6-categorical open subsets required to cover $G$7. The relevant stratification is by modified orbit-type strata

$G$8

with strict partial order

$G$9

Minimal strata are closed, and a $(X,X^G)$0-tubular open subset intersecting a minimal stratum intersects at most one minimal stratum and retracts onto an orbit contained in that stratum. This yields the lower bound

$(X,X^G)$1

where $(X,X^G)$2 indexes the minimal orbit-type strata. If $(X,X^G)$3 admits a minimal $(X,X^G)$4-tubular cover, then the localization formula becomes an equality:

$(X,X^G)$5

For a minimal stratum $(X,X^G)$6, one further has

$(X,X^G)$7

The localization is therefore relative to the orbit-type stratification rather than to fixed points alone (Fontaine et al., 2017).

The toric case provides the model example. Every symplectic toric manifold admits a minimal $(X,X^G)$8-tubular cover, its minimal strata are isolated fixed points, and each contributes $(X,X^G)$9. Hence

$(X,Y)$0

This recovers, for example, $(X,Y)$1 and $(X,Y)$2 (Fontaine et al., 2017).

A related but cohomological localization appears for $(X,Y)$3-contact manifolds. If a torus $(X,Y)$4 acts preserving the contact form $(X,Y)$5, then for $(X,Y)$6 one has the ABBV-type formula

$(X,Y)$7

where the critical components $(X,Y)$8 of the contact moment map replace the fixed-point components. When $(X,Y)$9 is a regular value, the corresponding Witten-type limit and Jeffrey–Kirwan residue formula express integrals on the contact quotient $(M_1,M_2,Y_1,Y_2)$0 in terms of these critical-set contributions (Casselmann et al., 2017).

3. Algebraic, motivic, and categorical forms

In the motivic setting, relative equivariant localization is formulated by passing from $(M_1,M_2,Y_1,Y_2)$1-equivariant motives to a $(M_1,M_2,Y_1,Y_2)$2-equivariant category whose morphisms lie in the image of restriction. For a split reductive group $(M_1,M_2,Y_1,Y_2)$3 with maximal torus $(M_1,M_2,Y_1,Y_2)$4, the category of relative equivariant motives

$(M_1,M_2,Y_1,Y_2)$5

is the categorical image of the forgetful functor $(M_1,M_2,Y_1,Y_2)$6. This relative category is linked to parabolic Demazure modules through the realization

$(M_1,M_2,Y_1,Y_2)$7

and the endomorphism ring of the relative motive embeds as

$(M_1,M_2,Y_1,Y_2)$8

Theorem 4.1 and Corollary 4.2 identify direct-sum decompositions of the split motive $(M_1,M_2,Y_1,Y_2)$9 in $Y_i$0, of the same object in $Y_i$1, and of the motive of the versal flag variety $Y_i$2 in the non-equivariant category, with Rost nilpotence providing the lifting of idempotents and isomorphisms (Calmès et al., 2016).

A different algebraic realization appears in derived-loop and cyclic homology. For a smooth quotient stack $Y_i$3 and a semisimple class $Y_i$4, completion at $Y_i$5 gives

$Y_i$6

Consequently,

$Y_i$7

and similarly for $Y_i$8, $Y_i$9, and $[z]\in G//G$0. The periodic theory satisfies the Atiyah–Segal-type completion formula

$[z]\in G//G$1

with the identity class $[z]\in G//G$2 yielding the $[z]\in G//G$3-periodic derived de Rham cohomology of $[z]\in G//G$4 itself (Chen, 2017).

The most explicit non-abelian formulation replaces fixed loci by centers of a $[z]\in G//G$5-stratification. For a derived stack $[z]\in G//G$6 over $[z]\in G//G$7 with centers $[z]\in G//G$8, the functors $[z]\in G//G$9 and $[z]$0 preserve highest weight cycles and induce

$[z]$1

together with the localization identity

$[z]$2

In the quasi-smooth case this becomes a virtual $[z]$3-theoretic non-abelian localization formula in which the global index is the sum of center contributions corrected by the inverse Euler class of the virtual normal complex (Halpern-Leistner, 28 Sep 2025).

4. Exact triangles, bulk–boundary terms, and field-theoretic analogues

For torus actions on smooth algebraic varieties, equivariant factorization homology admits a direct relative formulation. If $[z]$4 is the fixed-locus inclusion and $[z]$5 is a $[z]$6-equivariant factorization algebra, then

$[z]$7

becomes an isomorphism after inverting a finite set of homogeneous elements $[z]$8. Equivalently,

$[z]$9

After tensoring with the fraction field, one obtains the fixed-component decomposition

$X/G$0

and the Atiyah–Bott identity

$X/G$1

The exact triangle

$X/G$2

makes the relative term explicit (Butson, 2020).

In supersymmetric supergravity, the relative locus is often the bulk fixed set together with the conformal boundary. In $X/G$3, $X/G$4 Euclidean gauged supergravity, Killing-spinor bilinears produce $X/G$5-closed polyforms for the R-symmetry vector $X/G$6, and the on-shell action decomposes as

$X/G$7

The fixed-point term is given by bulk residues over nuts and bolts, while the boundary term is

$X/G$8

In the canonical equivariant scheme, supersymmetric holographic renormalization cancels the boundary contribution, so that the full gravitational free energy reduces to the fixed-point expression

$X/G$9

The resulting UV–IR relations express holonomies and boundary mass deformations in terms of fixed-point residues without solving the bulk equations explicitly (Genolini et al., 2024).

A closely related construction in even-dimensional supergravity uses equivariantly closed polyforms $z$0 satisfying

$z$1

after which Berline–Vergne–Atiyah–Bott localization computes on-shell actions, black-hole entropies, and central charges from the fixed locus of the R-symmetry flow. When the manifold has boundary and $z$2 has no fixed points on the boundary, the bulk fixed-point contributions survive while the boundary terms are canceled by holographic counterterms (Genolini et al., 2023).

5. Operator-theoretic and $z$3-theoretic localization

For proper actions of discrete groups on complete manifolds, the relative equivariant coarse index isolates the contribution of closed invariant subsets. If $z$4 is relative equivariant coarse index data and the curvature operators are uniformly positive outside neighborhoods of $z$5, then each operator $z$6 defines a localized equivariant coarse index

$z$7

and the relative index satisfies

$z$8

This is the equivariant refinement of Roe’s relative coarse index theorem. Under cocompactness assumptions, traces on localized Roe algebras yield a relative $z$9-index theorem as well (Chen et al., 2020).

In circle-equivariant $G$00-theory, localization is controlled by the representation ring

$G$01

For a smooth compact $G$02-manifold $G$03 with fixed set $G$04, if $G$05 vanishes on the finite isotropy set outside $G$06, then the restriction

$G$07

is an isomorphism, and there is also a relative statement for invariant pairs:

$G$08

The same localization mechanism underlies the equivariant Lefschetz formula, which identifies the $G$09-valued Lefschetz index with the module trace on equivariant $G$10-theory (Emerson, 2010).

Localization algebras provide an operator-theoretic Poincaré–Hopf theorem. For a proper cocompact isometric action of a countable discrete group $G$11 on an even-dimensional manifold $G$12, with non-degenerate $G$13-equivariant vector field $G$14, Witten deformation localizes the equivariant $G$15-homology class of the Euler operator $G$16 near the zero set of $G$17. If $G$18 represent the $G$19-orbits of zeros, then

$G$20

Applying the higher-index map gives

$G$21

The paper treats the absolute case and then outlines the corresponding relative localization-algebra construction for pairs $G$22 (Liu et al., 2024).

6. Scope, hypotheses, and terminological variation

The phrase “relative equivariant localization” is therefore context-dependent. In some papers, “relative” refers to a relative category or restriction image, as in $G$23 for motives (Calmès et al., 2016). In others, it refers to a pair, such as $G$24 in factorization homology (Butson, 2020) or $G$25 in coarse index theory (Chen et al., 2020). It may also refer to completion at a relative parameter $G$26 (Chen, 2017), to localization relative to the orbit-type stratification (Fontaine et al., 2017), or to bulk–boundary cancellation in supergravity (Genolini et al., 2024).

The hypotheses are correspondingly rigid. Proper smooth actions are essential in the LS-category and coarse-index settings because they supply tube theorems, slice models, or equivariant Roe-algebra functoriality (Fontaine et al., 2017, Chen et al., 2020). Semisimplicity of $G$27 and smoothness of $G$28 are central in cyclic-homological localization because Luna’s slice theorem and boundedness arguments are used to compare completed loop spaces (Chen, 2017). Quasi-smoothness and cohomological properness of centers are what turn non-abelian localization into a clean virtual $G$29-theoretic index formula (Halpern-Leistner, 28 Sep 2025). In supergravity, the existence of an R-symmetry Killing vector and of the relevant Killing-spinor bilinears is what makes the polyforms $G$30-closed (Genolini et al., 2024).

Failure of the relative cover or support condition is a recurrent obstruction. A minimal $G$31-tubular cover does not exist for all proper $G$32-manifolds, so the LS localization formula may reduce only to a lower bound (Fontaine et al., 2017). In nonabelian groups, loop-space localization is generally false without completion at $G$33 (Chen, 2017). In factorization homology, the isomorphism appears only after inverting a denominator set determined by stabilizer Lie subalgebras (Butson, 2020).

A further variation appears in equivariant stable homotopy theory, where relative localization means Bousfield or finite localization with respect to a thick subcategory of genuine $G$34-spectra. There the issue is not fixed-point reconstruction but preservation of $G$35, $G$36, or $G$37 structures, with closure under admissible indexed norms providing the decisive criterion. In particular, Bousfield localization with respect to an ordinary spectrum, viewed with trivial $G$38-action, preserves equivariant commutative ring spectra (Hill, 2017).

Taken together, these formulations show that relative equivariant localization is best understood as a methodological schema rather than a single theorem: one isolates a relative locus where equivariant structure simplifies, proves that the complement becomes negligible after localization, completion, or deformation, and reconstructs the global invariant from the localized contributions with the appropriate Euler, residue, or boundary correction.