Jeffrey-Kirwan Residue Prescription

• The Jeffrey-Kirwan residue prescription is a multidimensional residue operation defined for rational differential forms, utilizing a chamber-dependent evaluation to compute intersection pairings and topological invariants.
• It plays a crucial role in symplectic and equivariant geometry by enabling efficient calculations of volumes, intersection numbers, and cohomological invariants in both toric and non-compact settings.
• Algorithmic approaches, often employing Gröbner bases, underpin its systematic evaluation, advancing computations in quantum field theory, geometry, and enumerative invariants.

The Jeffrey-Kirwan (JK) residue prescription is a multidimensional residue operation at the heart of modern equivariant localization, intersection theory for symplectic quotients, and localization computations in both mathematics and physics. Formulated originally in the context of intersection pairings for Hamiltonian quotients, and later applied broadly in geometry, representation theory, and quantum field theory, the JK residue provides a systematic method to select and evaluate residues at singularities of rational forms, encoding both combinatorial and geometric data such as chamber structures and moment maps. Its geometric interpretations, algorithmic realizations (notably via Gröbner bases), and extensions to non-reductive, equivariant, and even non-compact settings have enabled major advances in cohomological computations and partition function evaluations.

1. Core Definition and Principle

The JK residue is constructed for rational differential forms of the type

$$\omega = \frac{P(x)\, dx_1 \wedge \dots \wedge dx_r}{Q_1(x) \cdots Q_n(x)}$$

where each $Q_i$ is a linear form and $P(x)$ is a homogeneous polynomial, on a real (or complexified) vector space $V$ of dimension $r$. One fixes an ordered basis and a generic covector $\eta \in V^*$, which determines a chamber structure within $V$.

Poles of the form correspond to the vanishing of sets of $r$ linearly independent $Q_i$'s, with associated charge vectors $\{Q_{i_1},\ldots, Q_{i_r}\}$. The JK residue at $x_*$—a solution of $Q_{i_k}(x_*)=0$ for $k=1,\dots,r$—with respect to $\eta$ is defined as

$$\mathrm{JKRes}_{\{Q_{i_1},...,Q_{i_r}\},\,\eta} (\omega) = \begin{cases} \frac{1}{|\det(Q_{i_1},...,Q_{i_r})|} P(x_*) & \text{if } \eta \in \mathrm{Cone}(Q_{i_1},...,Q_{i_r}) \ 0 & \text{otherwise} \end{cases}$$

The full residue is the sum over all such candidate poles, weighted as above.

The dependence on $\eta$ reflects the choice of chamber in equivariant geometry (e.g., the choice of polarization, or physical phase), leading to wall-crossing phenomena when $\eta$ is varied.

2. Algorithmic Computation via Gröbner Bases

An efficient computational approach for evaluating JK residues, particularly in intersection theory and symplectic volumes, utilizes Gröbner bases. The method—detailed explicitly in (Szilágyi, 2012)—proceeds as follows:

1. Identify the homogeneous ideal $I$: Construct $I \subset \mathbb{R}[V]$ (or $\mathbb{C}[V]$) of homogeneous polynomials of degree $n-r$. The ideal is generated by products of the form $H^+(x) Q_i(x)$, with $H^+$ corresponding to open half-spaces determined by subsets of $\{Q_i\}$ containing the chamber vector $\epsilon$.
2. Select a test polynomial $A$: Choose $A = \prod_{j\in J} Q_j(x)$, for a subset $J$ such that $\{a_j: j\in J\}$ is a basis of $V^*$ and $\epsilon\in \mathrm{Cone}(a_j: j\in J)$, ensuring the JK residue of $A$ is non-zero.
3. Compute the Gröbner basis $G$ for $I$.
4. Reduce $P$ and $A$ modulo $I$: Find normal forms: $P = Q + N_I(P)$ and $A = Q' + N_I(A)$.
5. Evaluate the JK residue:

$$\mathrm{JKRes}^\Lambda \frac{P(x) e^{\epsilon(x)} dx}{\prod_{i=1}^n Q_i(x)} = \frac{N_I(P)}{N_I(A)} \cdot \frac{1}{\det[(a_i, a_j)]_{i,j \in J}}$$

This algorithm matches that of Cattani-Dickenstein for Grothendieck residues, though essential differences arise from the real (chamber-based) nature of the JK residue and the associated determinant structures.

3. Geometric and Cohomological Applications

The JK residue prescription provides the backbone for computing:

• Intersection numbers on toric and symplectic quotients: The residue computes the pairing of an equivariant class with the fundamental homology class, generalizing to Kähler and even hyperkähler quotients (Szilágyi, 2012, Fisher, 2016).
• Volumes and push-forwards: Volumes of symplectic quotients and pushforwards in equivariant cohomology reduce to JK residue computations (Zielenkiewicz, 2015). In the context of homogeneous spaces, this connects to classical iterated residue formulas for projective spaces and Grassmannians:

$$\int_{G/P} \alpha(\mathcal{R}) = \mathrm{Res}_{\mathfrak{z}=\infty} (\cdots)$$

where residues are taken with respect to auxiliary variables naturally interpreted as characters of maximal tori.

• Non-reductive GIT quotients: The moment map and JK residue construction has been extended to non-reductive quotients where the group $H$ includes an internally graded unipotent radical. Under a "well-adapted" linearisation (with semistability equaling stability), Betti numbers and cohomology rings of $X/\!/H$ are expressed in terms of those for torus quotients, with intersection pairings given as sums of iterated residues in the JK sense (Bérczi et al., 2019).

4. Extensions: Equivariant and Non-Compact Settings

The JK residue formalism admits robust generalizations:

• Equivariant residue: For spaces with additional torus symmetry, the equivariant (parameter-dependent) JK residue is defined for integrals involving extra parameters ("flavor" or "mass" variables), e.g.,

$$\operatorname{EqRes}_x (F(x, s)) = \sum_{\text{poles } V} \mathrm{JKRes}_v F(v, s) dv$$

This is instrumental in non-compact localization and yields equivariant cohomology rings for moduli spaces such as $\operatorname{Hilb}^n(\mathbb{C}^2)$ (Szilágyi, 2013).

• Singular and wall-crossing phenomena: The chamber dependence of $\eta$ underpins wall-crossing; residues at infinity and boundary contributions arise naturally and are essential in analyses of dualities and phase transitions in gauge theories (Bullimore et al., 2020, Ashok et al., 2019).

5. Combinatorial and Algebraic Structures

JK residue computations exhibit deep combinatorial and algebraic patterns:

• Crystal melting models and quiver algebras: In the calculus of flavoured Witten indices, each fixed point selected by the JK prescription corresponds to a "crystal configuration." This leads to the construction of double quiver Yangians and algebras whose operators act on crystal states by adding or removing atoms, encapsulating BPS state counting and the entire fixed-point data of partition functions (Bao et al., 6 Jan 2025).
• Graphical distinction rules: For Nekrasov integrals in $BCD$ gauge groups, graphical and box-diagram criteria derived from the structure of charge vectors specify which poles contribute under the chosen $\eta$, systematizing the evaluation and ensuring correct combinatorial factors (Nakamura, 2015).
• Residue correspondence and secondary polytopes: In certain applications, such as the amplituhedron, the JK residue structures correspond precisely to the secondary polytope of all triangulations—each chamber (i.e., each $\eta$) giving a different triangulation or decomposition of the amplitude (Ferro et al., 2018).

6. Residue Formulas in Physical Computations

JK residue methods provide rigorous frameworks for:

• Supersymmetric localization: In quantum field theory, contour integral representations for partition functions or indices (2d/3d GLSMs, 4d instanton partition functions, elliptic genera) are evaluated via the JK prescription, often reducing multidimensional residues to algorithmic computations (Closset et al., 2015, Bao et al., 9 Aug 2025). The generalized JKG residue for (0,2) GLSMs extends to non-linear divisors in the localization locus.
• DT/PT vertex and wall-crossing: For curve counting on toric Calabi-Yau 4-folds, both Donaldson–Thomas and Pandharipande–Thomas vertices are computed via contour integrals where the JK residue with different choices of $\eta$ (e.g., $(1,1,\ldots,1)$ for DT, $(-1,-1,\ldots,-1)$ for PT) selects different contributing poles, corresponding to physically distinct moduli configurations (Kimura et al., 16 Aug 2025).

7. Broader Impact and Outlook

The JK residue prescription has become a standard and unifying tool across diverse domains:

• It enables explicit and efficient computation of topological invariants, intersection pairings, and volumes of highly structured moduli spaces, including those inaccessible to traditional methods.
• Its chamber- and polarization-dependence encodes wall-crossing, duality, and symmetry breaking phenomena critical in mathematical physics.
• Algorithmic realizations using Gröbner bases, symbolic computation, and statistical mechanical analogs (e.g., crystal melting) bridge the gap between theoretical formalism and computable outputs, allowing implementation in computer algebra systems.
• Its adaptability to equivariant, non-compact, and non-reductive contexts ensures ongoing relevance in both pure mathematics (algebraic geometry, representation theory) and quantum field theory, with applications ranging from instanton counting and non-abelian localization to mirror symmetry and motivic invariants.

Through its versatile and robust formulation, the Jeffrey-Kirwan residue prescription continues to provide the foundational algebraic and combinatorial infrastructure underlying modern equivariant localization, geometry of quotients, and enumerative invariants in both mathematics and physics.