Symmetric Butterfly Quiver

• The symmetric butterfly quiver is a structured, five-vertex symmetric quiver defined by its unique adjacency matrix and wiring diagram origins from type A2 and C2 Weyl groups.
• It underpins quiver diagonalization by enabling the factorization of motivic Donaldson–Thomas invariants into quantum dilogarithms, affirming integrality and positivity.
• Its applications span quantum cluster algebras and 3D integrable systems, providing explicit constructions for tetrahedron and reflection operators and insights into categorification.

A symmetric butterfly quiver is a canonical and highly structured example of a symmetric quiver, distinguished by its utility in the study of motivic Donaldson–Thomas invariants, quiver diagonalization, quantum cluster algebras, and the representation theory of quantum integrable systems. Recent literature has established its central role in quiver diagonalization, the categorification of algebraic invariants, and constructions of solutions to multidimensional quantum Yang–Baxter equations, including the tetrahedron and reflection equations. The symmetric butterfly quiver arises naturally via the combinatorics of wiring diagrams for the longest elements in type $A_2$ and $C_2$ Weyl groups, and it encapsulates the minimal complexity required to access rich algebraic and representation-theoretic phenomena.

1. Definition and Core Structure

The “symmetric butterfly quiver” can refer to several closely related quiver graphs, depending on context. In symmetric quiver theory as per Jankowski–Kucharski–Longhi–Noshchenko–Sułkowski, the butterfly quiver is the five-vertex undirected, loopless quiver with adjacency matrix: $$A = \begin{pmatrix} 0 & 1 & 1 & 0 & 0\ 1 & 0 & 0 & 1 & 0\ 1 & 0 & 0 & 0 & 1\ 0 & 1 & 0 & 0 & 0\ 0 & 0 & 1 & 0 & 0 \end{pmatrix}$$ Here node 1 (central) connects symmetrically to nodes 2 and 3 (wings), which themselves connect to nodes 4 and 5 (tips), respectively; all arrow pairs are reciprocated ($A_{ij}=A_{ji}$), with $A_{ii}=0$ (Jankowski et al., 2022).

In quantum cluster algebra and quantum integrable systems contexts, the symmetric butterfly (SB) quiver arises from wiring diagrams for the reduced expression of the longest Weyl group element, e.g., $w_0(A_2)=s_2s_1s_2$ for type $A_2$, mapping to a ten-vertex graph comprising the three crossings and five domain faces of the diagram (Inoue et al., 2024). For the type $C$ setting, the SB quiver generalizes to eleven vertices with a weighted, skew-symmetrizable exchange matrix designed for quantum cluster algebraic constructions (Inoue et al., 26 Dec 2025).

2. Quiver Diagonalization and Motivic Invariant Computation

The symmetric butterfly quiver is the canonical nontrivial example for the quiver diagonalization machinery. This process successively “unlinks” pairs of arrows (i.e., removes bidirectional edges and introduces new nodes with loops) until all off-diagonal entries vanish, resulting in a diagonal quiver $Q^{(\infty)}$ with only loops. For the butterfly quiver, four such unlinking steps introduce four new one-loop nodes (nodes 6–9), yielding $Q^{(\infty)}$ with adjacency matrix $\mathrm{diag}(0,0,0,0,0\,|\,1,1,1,1)$, where the first five are the original nodes and the remaining four are new (Jankowski et al., 2022).

The generating function for motivic Donaldson–Thomas (DT) invariants of the original quiver, $P_Q(\mathbf{x},q)$, is equated (under explicit change of variables) to the product of “loop quiver” building blocks

$$P_{Q^{(\infty)}}(\mathbf{y},q) = \prod_{j \in Q^{(\infty)}_0} P_{C^{(\infty)}_{jj}\textrm{-loop}}(y_j, q)$$

where $P_{0\text{-loop}}(y,q)=1/(y;q^2)_\infty$ and $P_{1\text{-loop}}(y,q)=(qy;q^2)_\infty$ (Jankowski et al., 2022).

This diagonalization guarantees factorization into quantum dilogarithms (“motivic expansion”) and gives manifest integrality and positivity of DT invariants for symmetric quivers, extending known $m$-loop combinatorics to all symmetric cases.

3. Quantum Cluster Algebra, Wiring Diagrams, and 3D Integrable Systems

Quantum cluster algebra theory associates to the symmetric butterfly quiver an exchange matrix $B$ and quantum $Y$-seed algebra generated by invertible $Y_i$ with commutation relation $Y_iY_j = q^{2b_{ij}} Y_jY_i$. Cluster mutations proceed via explicit quantum dilogarithm transformations, enabling the construction of automorphisms and representations that realize solutions to higher-dimensional quantum Yang–Baxter equations.

For $A_2$, the SB quiver arises from the wiring diagram of the reduced word $s_2s_1s_2$ and attaches vertices both to crossings and faces, providing the minimal quiver for constructing tetrahedron $R$-operators via quantum cluster mutations and quantum dilogarithms. Explicit operator representations are available in both coordinate and momentum diagonalizations, and the modular double formalism allows direct connection to the Faddeev quantum dilogarithm and the BMS–Sergeev 3D $R$-matrix (Inoue et al., 2024).

For type $C$ settings, the SB quiver is embedded in a “wiring diagram with a mirror,” yielding an eleven-vertex structure and weighted commutation data required for constructing both tetrahedral ($R$) and reflecting ($K$) operators. By appropriate sequence of quantum cluster mutations and Baker–Campbell–Hausdorff analysis, one derives solutions to the 3D reflection equation, generalizing the tetrahedron equation to boundary integrable cases (Inoue et al., 26 Dec 2025). The $R$ and $K$-operators here are products of four and ten quantum dilogarithms, respectively.

4. Categorification and Koszul Algebras for the Butterfly Quiver

To any symmetric quiver $Q$, including the butterfly, one associates the supercommutative quadratic algebra $\mathcal{A}_Q$ generated by potentially infinite sets $e_{i,k}$ (bundled into formal series $e_i(z)$), modulo:

• Super-commutativity (which becomes commutativity here, as $m_{ii}=0$)
• Vanishing of “small derivatives” along edges: $e_i(z)\,\frac{d^p}{dz^p}e_j(z) = 0$ for $0 \le p < m_{ij}$.

In the butterfly case with $m_{ij}=1$ for four bidirectional edges, the defining relation specializes to $e_i(z)\,e_j(z)=0$ for those edges, reflecting the combinatorics of the butterfly adjacency (Dotsenko et al., 2024).

This algebra $\mathcal{A}_{Q_\mathrm{butterfly}}$ is proven to be Koszul, following a reduction via iterated unlinking steps to $m$-loop quiver cases, whose algebras are also Koszul. Koszulness here not only categorifies quiver diagonalization but, through the results of [MR4499100], implies positivity and integrality for all corresponding motivic DT invariants.

5. Factorization, Invariants, and Combinatorial Interpretation

The motivic generating function for the butterfly quiver, once diagonalized, exhibits factorization into products of quantum dilogarithms, with exponents $\Omega_{\mathbf{d},s}$ that enumerate BPS and LMOV (Labastida–Mariño–Ooguri–Vafa) invariants: $$P_Q(\mathbf{x},q) = \prod_{\mathbf{d} \neq 0}\prod_{s \in \mathbb{Z}} \left( x_1^{d_1}\cdots x_5^{d_5}q^s; q^2 \right)_\infty^{\Omega_{\mathbf{d},s}}$$ or refined as

$$P_Q = \prod_{\mathbf{d} \neq 0} \Psi_q((-q^{1/2})^{w(\mathbf{d})} x^{\mathbf{d}})^{\Omega_\mathbf{d}(q)}, \quad w(\mathbf{d}) = \mathbf{d} \cdot A \cdot \mathbf{d}$$

with $\Psi_q$ the quantum dilogarithm (Jankowski et al., 2022).

For the butterfly quiver, the nonzero $\Omega$ occur in degree 1 (corresponding to the five original nodes, with value $-1$) and degree 2 (the four new 1-loop nodes, with value $+1$); higher-degree terms vanish below total degree 2. By the DP-loop decomposition, all such invariants are Laurent polynomials in $q$ with positive integer coefficients.

The combinatorial Lyndon-word and crystal models known for $m$-loop quivers generalize to this context, giving explicit combinatorial formulas for all invariants associated with the symmetric butterfly (Jankowski et al., 2022).

6. Connections to Open BPS Counting and Physical Applications

Upon embedding the symmetric butterfly quiver into an open string or 3d $\mathcal{N}=2$ physical setup, each node variable $x_i$ is interpreted as a monomial in spectral parameters $(\lambda, a, q)$ corresponding to open D-brane or M-theory configurations. The open BPS (LMOV) generating function becomes

$$N(\lambda, a, q) = \sum_{\mathbf{d}, s} \Omega_{\mathbf{d},s}\, \lambda^{\sum d_i n_i} a^{\sum d_i a_i} q^s$$

where the 1-loop nodes correspond to multi-brane configurations, and 0-loop nodes correspond to single-brane (negative) contributions (Jankowski et al., 2022).

This directly recovers the BPS state counting for "two-wing" geometries in open topological string theory, with the symmetry and combinatorial structure of the butterfly quiver ensuring agreement with LMOV invariant predictions.

7. Integrability, Modular Double, and Representation-theoretic Universality

Solutions constructed with the symmetric butterfly quiver in quantum cluster algebra produce not just solutions to the tetrahedron equation but the full 3D reflection equation (boundary integrability), embedding and unifying families of 3D $R$-matrices, including those of Kapranov–Voevodsky (quantized coordinate ring), Bazhanov–Mangazeev–Sergeev (quantum geometry), Kuniba–Matsuike–Yoneyama (quantum six-vertex), and Inoue–Kuniba–Terashima (Fock–Goncharov quiver) (Inoue et al., 2024, Inoue et al., 26 Dec 2025). The modular double construction embeds the compact quantum dilogarithm $\Psi_q$ and its noncompact Faddeev analog $\Phi_b$, ensuring analytic and spectral completeness across broad integrable and representation-theoretic settings.

The inclusion and degeneration to simpler quiver solutions (e.g., Fock–Goncharov) are admitted via explicit algebraic limits, confirming the universality of the symmetric butterfly quiver construction as a connective point for quantum algebraic, combinatorial, and physical frameworks.

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Key References: (Jankowski et al., 2022, Inoue et al., 2024, Inoue et al., 26 Dec 2025, Dotsenko et al., 2024)