Origami Partition Function in Gauge Theories

• Origami Partition Function is a universal formalism that unifies partition functions from gauge theories, quantum algebras, and moduli-theoretic enumerative invariants.
• It is constructed through equivariant integration over moduli spaces, with contributions localized by multi-dimensional Young diagrams describing D-brane configurations.
• The framework connects BPS/CFT dualities, DT/PT invariants, and quantum q-Langlands, driving advances in algebraic geometry, representation theory, and integrable systems.

The origami partition function is a universal formalism unifying partition functions of gauge theories, quantum algebras, and moduli-theoretic enumerative invariants associated with intersecting brane systems and quiver varieties. It is constructed via equivariant integration over moduli spaces arising from gauge-theoretic, algebraic-geometric, and quantum algebraic input, and it governs a wide array of phenomena—ranging from BPS/CFT and quantum q-Langlands dualities to the generation of higher-dimensional Donaldson–Thomas and Pandharipande–Thomas invariants. Origami partition functions encode the enumerative geometry of instantons and their multi-dimensional generalizations, relate to conformal blocks of quiver W-algebras, and are naturally expressed in terms of contour integrals, lattice partition sums, and combinatorics of multi-dimensional Young diagrams (Kimura et al., 2024, Kimura et al., 2023, Nekrasov, 2017, Jiang, 25 Dec 2025).

1. Physical and Algebraic Construction

The origami partition function is defined for supersymmetric gauge theories engineered by brane systems in toric Calabi–Yau fourfolds, typically of the form $\mathbb{C}^4$. The key input is a system of D0-branes bound to higher-dimensional D(2$p$)-branes wrapping various $\mathbb{C}^p$ subspaces. The low-energy effective theory on D0-branes is a 1d $\mathcal{N}=2$ quiver quantum mechanics, whose BPS ground states are counted by the Witten index—the instanton partition function. The contribution at instanton charge $k$ is given by a Jeffrey–Kirwan–localized contour integral over the Cartan of the gauge group, with the integrand constructed from products of “vector multiplet” and “matter multiplet” contributions (Kimura et al., 2023, Noshita, 11 Feb 2025, Nekrasov, 2017): $$Z_k = \frac{1}{k!} \oint \prod_{I=1}^k \frac{dx_I}{2\pi i x_I} \; Z_{\text{vector}}(x_I) \, Z_{\text{matter}}(x_I;\text{flavors})$$ This formalism localizes the partition function to fixed points labeled by multi-dimensional Young diagrams, with weights determined by the combinatorics of how D0-branes are bound to the various D(2$p$)-branes. The algebraic structure is captured by a free boson Fock space, with vertex operators and screening currents corresponding to the various brane types (D0, D2, D4, D6, D8) and their intersections, and the operator product expansions reproduce the quantum Yang–Baxter structure of quiver W-algebras (Kimura et al., 2024, Kimura et al., 2023).

2. Contour Integral, Combinatorics, and Shell Formula

The origami partition function after localization becomes a sum over colored Young diagrams of various dimensions. For a system labeled by brane data $\vec{\mathcal{A}}$ (e.g., D2, D4, D6, D8), and for fixed equivariant parameters, the function is: $$Z_{\textrm{origami}} = \sum_{\vec{\mathbf{Y}}} q^{|\vec{\mathbf{Y}}|} \prod_{\text{color pairs}} W_{\text{interaction}}(\mathbf{Y}_{\mathcal{A}}, \mathbf{Y}_{\mathcal{B}})$$ In geometric terms, poles of the integrand (or fixed points of the torus action) are labeled by collections of Young diagrams (2d, 3d, 4d) indexed by brane flavors and positions. The shell formula expresses these weights as products over the "shell" $\mathcal{S}(\mathbf{Y})$ of a given Young diagram, encoding the combinatorics of how boxes can be added or removed: $$\mathcal{J}(x|\mathbf{Y}) = \prod_{\boldsymbol{y}\,\in\,\mathcal{S}(\mathbf{Y})} \sh(x-\mathcal{X}(\boldsymbol{y}))^{Q_{\mathbf{Y}}(\boldsymbol{y})}$$ where $Q_{\mathbf{Y}}(\boldsymbol{y})$ is a signed charge and $\mathcal{X}(\boldsymbol{y})$ is determined by equivariant and flavor parameters (Jiang, 25 Dec 2025). This unifies the representation of partition functions for instantons (plane partitions, solid partitions, etc.) and higher-rank quiver gauge theories.

3. W-Algebra, Vertex Functions, and Quantum q-Langlands

The partition function admits a “conformal block” realization associated with quiver W-algebras. This structure arises when the partition function is viewed as a correlator of screening currents and vertex operators: $$Z_{\textrm{origami}} = \langle 0 |\, \prod_{i} T_i(x_i) \, |0 \rangle$$ with each $T_i(x)$ corresponding to a $qq$-character operator (vector, Fock, MacMahon, or solid representation depending on brane type). For the D2 system, the block is given by a contour integral of screening operators around a D8 insertion, with an explicit stable envelope weight encoding the chamber dependence. For quiver varieties (Nakajima), the origami partition function coincides with the $K$-theoretic quasimap vertex function, matching residues with the sum over reverse partitions and reproducing the vortex expansion (Kimura et al., 2024).

The chamber dependence in the ordering of screening currents is algebraically implemented via elliptic $R$-matrices, and the partition function solves the elliptic $q$-KZ equation under suitable zero-weight conditions, encoding the $q$-Langlands correspondence at the quantum and even double affine level.

4. Moduli Space and Geometric Interpretation

Geometrically, the origami partition function is defined over moduli spaces cut out by higher-dimensional ADHM-type equations and realized as zero loci of isotropic sections of certain quadratic vector bundles (Oh–Thomas theory). For the 4d-ADHM quiver, the moduli space $M_{Q_4}(r_A, n)$ is specified by vanishing of a section $s$ with $q(s,s)=0$. The partition function is defined as the integral over the virtual cycle $[M]^{\text{vir}}$ produced by the relevant obstruction theory: $$Z_{r_A}(q) = \sum_{n\geq 0} q^n \int_{[M(r_A, n)]^{\text{vir}}} 1$$ or in $K$-theory by the equivariant Euler characteristic of the virtual structure sheaf (Arbesfeld et al., 1 Feb 2026). At the level of torus fixed points, the function becomes a sum over partitions (combinatorially matching the localization formula above), and the fixed-point signs are canonical (always $+1$ in the relevant orientation). Dimensional reduction of the origami moduli space recovers the ADHM spaces of lower-dimensional instantons.

5. Representation Theory and Operator Algebra

The operator algebra of the origami system is governed by the representation theory of quantum toroidal $\mathfrak{gl}_1$ and quiver W-algebras. The $qq$-characters corresponding to D2, D4, D6, and D8 branes realize vector, Fock, MacMahon, and solid representations, respectively. Fusion operations build higher-dimensional $qq$-characters from lower ones:

• D2 $\to$ D4: Fock $\sim$ fusion of screenings,
• D4 $\to$ D6: MacMahon $\sim$ further fusion,
• D6 $\to$ D8: infinite fusion to solid.

Quadratic relations among $qq$-characters manifest the noncommutative structure of the quiver W-algebra and encode the quantum integrability, BPS/CFT correspondence, and Bethe/Gauge correspondences (Kimura et al., 2023, Noshita, 11 Feb 2025).

6. Applications: DT/PT Invariants, Integrable Systems, and Beyond

Origami partition functions realize a broad family of enumerative invariants, including higher-leg Donaldson–Thomas/Pandharipande–Thomas vertices, and provide operator definitions of Donaldson–Thomas $qq$-characters with general boundary conditions (Kimura et al., 2024). Gluing of D8, D6, D4, and D2 $qq$-characters along patches, faces, edges, and rods in toric Calabi–Yau fourfolds produces the full DT/PT generating series.

In the analytic domain, the origami partition function, via its twisted and orbifolded versions, encodes the full spectral problem of elliptic double Calogero–Moser systems, with the coefficients of the associated characteristic polynomials realized as differential operators on partition functions (Chen et al., 2019).

Integrality, pole-freeness, and the full symmetry structure—manifest in the shell formula and the compactness of the underlying moduli space—further make the origami partition function a universal object in quantum geometry and representation theory (Arbesfeld et al., 1 Feb 2026, Jiang, 25 Dec 2025, Nekrasov, 2017).

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Table: Key Aspects of the Origami Partition Function

Aspect	Mathematical Structure	Physical/Enumerative Significance
Combinatorics	Multi-dimensional partitions, shell formula	Localization to BPS/instanton fixed points
Operator algebra	W-algebra, quantum toroidal $\mathfrak{gl}_1$, $qq$-characters	Duality, integrability, symmetries
Moduli space geometry	4d-ADHM quiver, isotropic section, Oh–Thomas cycles	Cohomological/K-theoretic invariants
Representation theory	Fock, MacMahon, solid representations	BPS/CFT, Bethe/Gauge, $q$-Langlands
Spectral/analytic realization	Characteristic polynomials, spectral curves	Integrable systems, Dyson–Schwinger
Enumerative theory	Quasimaps, stable envelopes, DT/PT vertices	Donaldson–Thomas, Pandharipande–Thomas

The origami partition function thus synthesizes quantum field theory, algebraic geometry, and representation theory, serving as a master generating function for a wide class of BPS invariants, dualities, and spectral problems in mathematics and physics (Kimura et al., 2024, Kimura et al., 2023, Jiang, 25 Dec 2025, Kimura et al., 2024, Arbesfeld et al., 1 Feb 2026).