Jeffrey–Kirwan Residue Prescription

• The Jeffrey–Kirwan residue prescription is a method that computes equivariant integrals by reducing them to combinatorial and algebraic residue problems using hyperplane arrangements.
• It leverages Gröbner basis algorithms and chamber decompositions to systematically extract local contributions in high-dimensional localization and symplectic reduction.
• Its applications span symplectic geometry, mirror symmetry, and quantum field theory, offering practical tools for computing invariants in complex geometric settings.

The Jeffrey–Kirwan residue prescription is a foundational framework in symplectic geometry, algebraic geometry, and mathematical physics for computing equivariant integrals, intersection numbers, and volumes of symplectic quotients. It defines a systematic algebraic–combinatorial method to extract the contributions of residues associated with rational functions arising from localization formulas, especially in settings involving group actions, moduli problems, or polytopes. The prescription formalizes and unifies multi-dimensional residue computations by using the geometry of hyperplane arrangements and chamber decompositions, as well as connecting to developments in equivariant localization and mirror symmetry.

1. Algebraic Formulation and Gröbner Basis Algorithm

The Jeffrey–Kirwan residue can be interpreted algebraically as a linear functional on a finite-dimensional space of homogeneous rational functions or polynomials. For a given configuration of covectors $\alpha = [\alpha_1, \dots, \alpha_n]$ in a real vector space $V$ and a fixed regular element $\epsilon \in V^*$ (ensuring that $\epsilon$ lies in the cone generated by a basis of $V^*$), the residue prescription is entirely encoded in the membership of a polynomial $P$ in a homogeneous ideal $I_{\alpha, \epsilon} \subset \mathbb{R}[V]$. This ideal is generated by products of linear forms corresponding to hyperplanes that separate the chamber containing $\epsilon$ from other regions defined by the $\alpha_i$.

The algorithm to compute the Jeffrey–Kirwan residue using Gröbner bases proceeds as follows:

1. Determine the Ideal: Identify the ideal $I_{\alpha, \epsilon}$ so that $P \in I_{\alpha, \epsilon}$ if and only if the JK-residue of $P$ vanishes.
2. Select a Test Basis: Choose a subset $J$ so that $\{\alpha_j : j \in J\}$ forms a basis of $V^*$ and $$\epsilon \in \operatorname{Cone}(\{\alpha_j : j \in J\})$$. Construct a nonzero "test" polynomial $A = \prod_{j\in J} \alpha_j^{d_j}$.
3. Gröbner Basis Computation: For $I_{\alpha, \epsilon}$, compute a Gröbner basis $G$ in a chosen monomial order (e.g., lexicographic).
4. Division to Normal Form: For the polynomial $P$ and test polynomial $A$, compute their normal forms $NI(P), NI(A)$ modulo $G$.
5. Residue Formula: The residue is recovered via:

$$\text{JKRes}^\epsilon \frac{P(x) e^{\epsilon(x)}}{\prod_{i=1}^r Q_i(x)} dx = \frac{NI_{I_{\alpha,\epsilon}}(P)}{NI_{I_{\alpha,\epsilon}}(A)} \cdot \det \left[(\alpha_j, \alpha_j)\right]_{j \in J}$$

where the Gram determinant arises from change of variables.

This approach, paralleling the Cattani–Dickenstein algorithm for Grothendieck residues, exploits the equivalence (in the finite-dimensional quotient $\mathbb{R}[V]/I$) between polynomial data and residue computations. The method is well suited to implementation in computer algebra systems (e.g., Maple, Macaulay2) (Szilágyi, 2012).

2. Geometric Meaning and Localization Principles

The geometrical underpinning of the JK-residue is its emergence in localization formulas for equivariant cohomology and symplectic reduction. Given a compact symplectic manifold $M$ with Hamiltonian $G$-action and momentum map $\mu: M \to \mathfrak{g}^*$, the Marsden–Weinstein reduced space $M // G = \mu^{-1}(0)/G$ often inherits a rich structure, possibly singular or of orbifold type.

Localization formulas in this context (e.g., Atiyah–Bott–Berline–Vergne) reduce global equivariant integrals to sums over contributions from fixed-point loci of torus actions, with weights determined by isotropy representations. In nonabelian or noncompact cases, this principle persists via symplectic cutting (Lerman, Jeffrey–Kogan): the noncompact quotient is recovered as an open subset of a compact "cut space," and residue computations reduce to local data indexed by fixed-point data or solutions to stability conditions. Only poles with "full generating" weight sets corresponding to appropriate cones contribute to the residue (Szilágyi, 2013).

In the most general noncompact/critical setting, the JK prescription is extended via equivariant Jeffrey–Kirwan residues and formal integration, and applies to both Hamiltonian and hyperkähler quotients.

3. Residue Prescriptions and Hyperplane Arrangements

Technically, the JK-residue is a prescription for associating to a meromorphic form

$\omega = \frac{h(z) dz}{g_1(z) \cdots g_R(z)}$

a linear functional on the cohomology associated with the arrangement of affine hyperplanes $\{g_k=0\}$ in $V_\mathbb{C}$. The possible poles correspond to intersections indexed by complete flags in $V^*$. The prescription depends on a regularity vector (or "stability vector," e.g., $\xi$ or $\epsilon$) and the orientation of the arrangement: the JK residue at a pole associated to a subset $\sigma$ of hyperplanes is nonzero only if the stability vector lies in $\operatorname{Cone}(\sigma)$, and is then given up to sign and normalization by $1/|\det(Q_1,\dots,Q_n)|$ for weights $Q_i$ (Nakamura, 2015, Ontani et al., 2021).

This chamber-dependent selection mechanism ensures that only residues with orientation compatible with the chosen stability vector contribute, generalizing the Cauchy residue theorem to higher-dimensional, nonisolated, and combinatorially rich singular sets.

4. Connections to Combinatorics, Representation Theory, and Mirror Symmetry

The Jeffrey–Kirwan residue prescription has deep combinatorial and representation-theoretic ramifications.

• Crystal Melting and Quiver Gauge Theories: In supersymmetric quiver gauge theories, evaluation of flavoured partition functions and Witten indices is governed by a sum over fixed points corresponding to singularities of the integrand, naturally interpreted as "atoms" in a crystal. The allowed configurations are constructed via the no-overlap rule, and the mushrooming structure is encoded in double quiver Yangians—new algebraic structures whose representations are governed by the data of the JK residue (Bao et al., 6 Jan 2025).
• Donaldson–Thomas Invariants and Log Calabi–Yau: For quivers without loops, the abelianized JK residue of explicit meromorphic forms computes Donaldson–Thomas (DT) type invariants. In specific cases such as complete bipartite quivers, these residues are matched with theta functions of the Gross–Hacking–Keel mirror family to a log Calabi–Yau surface, revealing a bridge between localization-residue techniques and mirror symmetry (Ontani et al., 2021).
• Amplituhedron Volume and Positive Geometry: In the amplituhedron program, the JK-residue provides the contour prescription for the canonical volume forms, connecting triangulations, secondary polytopes, and combinatorial structures of positive geometries relevant for scattering amplitudes in planar $\mathcal{N}=4$ super Yang–Mills (Ferro et al., 2018).
• Flag Varieties and K-theory: The iterated residue (and the JK prescription) underpin calculations for stable Grothendieck polynomials and degeneracy loci, offering algorithmic tools and insights into cancellation phenomena such as alternating signs in $K$-theoretic structure constants (Allman et al., 2014).

5. Relations to Singularities, Cohomology, and Morse Theory

The analytical underpinnings of the Jeffrey–Kirwan residue are closely tied to the stationary phase principle and resolution of singularities, as applied to oscillatory integrals in the presence of group actions. For cotangent bundles of $G$-manifolds, the leading asymptotic coefficients in expansion of oscillatory integrals can be interpreted as residues, with the critical set determined by vanishing of the momentum map. Even in cases where the reduced space is singular, intersection cohomology pushforwards can be expressed via the JK-residue evaluated on specific data from the ambient manifold (Ramacher, 2013, Konstantis et al., 2016).

Furthermore, the Morse–Bott stratification (via the norm-square of the moment map) provides a geometric stratification framework in which equivariant Euler classes and residue prescriptions organize the algebraic and topological computation of invariants, extending even to basic cohomology of contact and $K$-contact manifolds (Casselmann, 2016).

6. Extensions, Algorithms, and Integral Cohomology

Current developments further refine the topological and homological context supporting the Jeffrey–Kirwan residue formula. The extension of Kirwan surjectivity and formality theorems from rational to integral coefficients (by inverting the orders of all finite stabilizers) provides the necessary topological background to justify the residue localization procedures at the level of complex cobordism MU-module spectra. This ensures that the residue prescription is robust even in the presence of torsion or non-freeness, with the splitting and formality properties assured in the complex-oriented setting (Pomerleano et al., 8 Oct 2024).

Algorithmically, the Gröbner basis method allows one to compute residues efficiently, by reducing to division and normal form computation in polynomial rings. Such automation makes it practical to apply the residue prescription to high-dimensional and geometrically intricate localization problems (Szilágyi, 2012).

7. Generalizations and Recent Innovations

Recent advances generalize the JK-residue by introducing stability conditions for hyperplane arrangements motivated by total positivity, allowing for a minimal and canonical selection of contributing flags in the iterated residue expansion of integrals with singularities along affine hyperplanes. This approach identifies the relevant poles using leading principal minors and sign conditions, and reiterates the close analogy with, but operational difference from, the chamber-based JK prescription (O'Desky, 22 Aug 2024).

Moreover, residue formulas have been successfully adapted to novel contexts, such as 2d $\mathcal{N}=(0,1)$ gauge theories—where the prescription is modified by quadratic superpotential data and encompasses the Jeffrey–Kirwan residue as a special case—enabling exact computation of elliptic genera and phase diagrams in minimal supersymmetric field theories (Bao et al., 9 Aug 2025).

---

The Jeffrey–Kirwan residue prescription thus functions as a central algebraic and geometric tool for converting global geometric and physical invariants into computable sums of local data, governed by precise combinatorial, algebraic, and topological rules. Its enduring impact spans intersection theory, symplectic geometry, supersymmetric localization, representation theory, mirror symmetry, and beyond.