arXiv subjects

Cyril Closset

Publications and source records attributed to Cyril Closset.

At least 19 recordsLinked to original sources

arXiv2026

Three-dimensional TQFTs from Argyres--Douglas theories via the 3d/3d correspondence

We study the twisted dimensional reduction of 4d $\mathcal{N}=2$ Argyres--Douglas theories via the 3d/3d correspondence, focussing on $(A_1, A_{2n})$ theories realised as the class-$\mathcal{S}$ theory $T[\mathcal{C}]$ for a curve $\mathcal{C}$ with one irregular puncture. We argue the resulting 3d $\mathcal{N}=4$ rank-0 superconformal field theory (SCFT) arises as the IR fixed point of a Dimofte--Gaiotto--Gukov (DGG) abelian Chern--Simons-matter (ACSM) theory $T[M_3^{(k)}]$ for the lens space $M_3^{(k)}=L(2n+3,2k)$, obtained by fibering $\mathcal{C}$ over the circle with twist $k \in \mathbb{Z}_{2n+3}^\times$. The triangulation of $M_3^{(k)}$, derived from the 4d BPS spectrum, determines the ACSM theory including its superpotential. $T[M_3^{(k)}]$ flows to the expected SCFT when the monopole superpotential is near-maximal in a precise sense, while the maximal superpotential gives $T_A[M_3^{(k)}]$, which flows directly to a non-unitary TQFT, the topological $A$-twist of the SCFT. Geometrically, the SCFT and TQFT points are related by shifting conical singularities between edges of the triangulation, and we propose a correspondence between singular edges and SCFT points predicting when the SCFT is a unitary TQFT. The TQFTs from $M_3^{(k)}$ can support 2d vertex operator algebras (VOAs) on their holomorphic boundary, including the Schur-sector VOA of the 4d SCFT for $k=1$. For $n=k=1$, our construction reproduces the minimal 3d $\mathcal{N}=4$ SCFT of Gang and Yamazaki from $L(5,2)$, where $T_A[M_3^{(k)}]$ is the Yang--Lee TQFT supporting $M(2,5)$ on its boundary. We check our proposals in detail, reconstructing the modular structure of the IR TQFTs from partition functions on Seifert manifolds, and matching phases to determine VOA central charges $c_{\text{2d}} \bmod 8$ from the ACSM data, including the gravitational Chern--Simons level.

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arXiv2026

Galois Covers of Calabi-Yau Quivers and BPS State Counting

BPS quivers are central to our understanding of BPS states in 4d $\mathcal{N}=2$ supersymmetric field theories and of D-branes at Calabi-Yau threefold singularities. The two subjects are deeply interrelated through geometric engineering in Type II string theory, where a CY$_3$ quiver, also known as a 5d BPS quiver, describes fractional branes at a threefold singularity ${\bf X}$. We study the Galois cover ${\skew{2}\tilde Q}\rightarrow Q$ of any BPS quiver $Q$ by a finite abelian group $\mathbb{G}$, leading to a covering quiver ${\skew{2}\tilde Q}$. The Galois cover is determined by a $\mathbb{G}$-grading of the arrows of the quiver $Q$, which can be understood as an orbifolding procedure. In particular, if $Q$ is a CY$_3$ quiver for ${\bf X}$, then the Galois cover $\skew{2}\tilde Q$ is the CY$_3$ quiver for the orbifold singularity ${\bf X}/\mathbb{G}$. We explore such Galois covering procedures in the language of supersymmetric quiver quantum mechanics, in terms of fixed loci under $\mathbb{G}$ actions on moduli spaces of quiver representations, and in terms of homomorphisms between the Kontsevich-Soibelman algebras of $Q$ and ${\skew{2}\tilde Q}$. Our main result is an explicit covering formula for the BPS invariants of 4d $\mathcal{N}=2$ field theories, wherein the rational BPS invariant $\bar{\Omega}^Q(\gamma)$ of $Q$ is expressed as a sum of BPS invariants of $\skew{2}\tilde Q$. We derive this formula in various special cases, which include the case when $\gamma$ is a primitive charge vector, the case of general charge vectors for quivers without loops, and the case of CY$_3$ quivers for some simple geometries such as the conifold or local del Pezzo surfaces. The general formula is presented as a conjecture that can be verified in many examples.

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arXiv2026

Schubert line defects in 3d GLSMs, part II: Partial flag manifolds and parabolic quantum polynomials

We construct Schubert line defects in the 3d $\mathcal{N}=2$ supersymmetric gauged linear sigma model (GLSM) with target space a partial flag manifold $X={\rm Fl}({\boldsymbol{k}};n)$, generalizing our construction for complete flag manifolds given in a companion paper arXiv:2512.19802 (part I). In the context of the 3d GLSM/quantum K-theory correspondence, the Schubert line defects are constructed as 1d $\mathcal{N}=2$ supersymmetric gauge theories coupled to the 3d field theory, and they flow to objects supported on Schubert varieties $X_w \subseteq X$ in the quantum K-theory. The flavored Witten index of the 1d defect is expected to compute the Chern character of $[\mathcal{O}_w]$ -- more precisely, it gives us a polynomial representative of the Schubert class in the quantum K-theory ring. We give strong evidence for this claim by showing in examples that the Witten indices of Schubert defects indeed reproduce a recently-defined set of polynomials that represent the Schubert classes in the Whitney presentation, which we call the parabolic Whitney polynomials. Moreover, upon using the quantum ring relations, we can convert these polynomials into seemingly new polynomials in the Toda presentation, which we call the parabolic quantum Grothendieck polynomials. These new polynomials specialize to known polynomials in various limits, including to the quantum Grothendieck polynomials in the case of the complete flag. In the 2d limit, our construction also realizes the Schubert classes $[X_w]$ in the quantum cohomology ring of the partial flag manifold, and the parabolic quantum Grothendieck polynomials then reduce to previously known parabolic quantum Schubert polynomials.

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arXiv2025

Schubert line defects in 3d GLSMs, part I: Complete flag manifolds and quantum Grothendieck polynomials

We construct new half-BPS line defects in 3d $\mathcal{N}=2$ supersymmetric quiver gauge theories whose Higgs branches are complete flag manifolds $X = {\rm Fl}(n)$. Upon circle compactification, the bulk theory flows to a non-linear sigma model (NLSM) with target space $X$ and the line defects flow to objects supported on Schubert varieties $X_w \subseteq X$. These Schubert line defects form an important basis of the quantum K-theory of $X$. They are realized as $\mathcal{N}=2$ supersymmetric quantum mechanics (SQM) quivers coupled to the 3d gauge theory. We show that the insertion of the Schubert line defect restricts the target space of the 3d gauged linear sigma model (GLSM) to the Schubert variety $X_w$, with the 1d degrees of freedom physically realizing a Bott--Samelson resolution of $X_w$. Moreover, we verify in examples that the 1d flavored Witten index of the quiver SQM reproduces the (equivariant) Chern character of the structure sheaf $\mathcal{O}_{X_w}$ as a (double) quantum Grothendieck polynomial, generalizing previous results for $X$ a Grassmannian manifold. Our construction thus provides a more direct realization of the 3d GLSM/quantum K-theory correspondence for complete flag manifolds. Finally, in the small-circle limit, we obtain a 0d-2d coupled system that realizes the Schubert classes $[X_w]$ in the quantum cohomology ring of $X$.

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arXiv2025

Dualities and trialities in $\mathcal{N}=2$ supersymmetric gauged quantum mechanics

We study new Seiberg-like dualities for 1d $\mathcal{N}=2$ supersymmetric gauge theories -- that is, supersymmetric gauged quantum mechanics -- with unitary gauge group and (anti)fundamental matter in chiral and fermi multiplets, and with non-zero Fayet--Iliopoulos parameter. Similarly to its higher-dimensional analogues, this 1d Seiberg duality is an infrared duality: the supersymmetric ground states of the dual gauge theories match exactly. We provide strong evidence for the dualities, including the matching of the flavoured Witten indices, a Higgs-branch derivation in terms of dual Grassmannian manifolds, and a detailed study of the Coulomb-branch ground states in the abelian case. We study how the supersymmetric ground states, in either dual description, depend on the sign of the Fayet--Iliopoulos parameter, and we explore the corresponding wall-crossing phenomenon. For some special values of the discrete parameters defining the unitary gauge theory, the dualities, combined with trivial wall-crossing, enhances to a triality. This includes, as a special case, the dimensional reduction to 1d of the 2d $\mathcal{N}=(0,2)$ Gadde--Gukov--Putrov triality.

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arXiv2025

The 3d $A$-model and generalised symmetries, Part I: bosonic Chern-Simons theories

The 3d $A$-model is a two-dimensional approach to the computation of supersymmetric observables of three-dimensional $\mathcal{N}=2$ supersymmetric gauge theories. In principle, it allows us to compute half-BPS partition functions on any compact Seifert three-manifold (as well as of expectation values of half-BPS lines thereon), but previous results focussed on the case where the gauge group $\widetilde G$ is a product of simply-connected and/or unitary gauge groups. We are interested in the more general case of a compact gauge group $G=\widetilde G/\Gamma$, which is obtained from the $\widetilde G$ theory by gauging a discrete one-form symmetry. In this paper, we discuss in detail the case of pure $\mathcal{N}=2$ Chern-Simons theories (without matter) for simple groups $G$. When $G=\widetilde G$ is simply-connected, we demonstrate the exact matching between the supersymmetric approach in terms of Seifert fibering operators and the 3d TQFT approach based on topological surgery in the infrared Chern-Simons theory $\widetilde G_k$, including through the identification of subtle counterterms that relate the two approaches. We then extend this discussion to the case where the Chern-Simons theory $G_k$ can be obtained from $\widetilde G_k$ by the condensation of abelian anyons which are bosonic. Along the way, we revisit the 3d $A$-model formalism by emphasising its 2d TQFT underpinning.

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arXiv2024

One-form symmetries and the 3d $\mathcal{N}=2$ $A$-model: Topologically twisted indices and CS theories

We study three-dimensional $\mathcal{N}=2$ supersymmetric Chern-Simons-matter gauge theories with a one-form symmetry in the $A$-model formalism on $\Sigma_g\times S^1$. We explicitly compute expectation values of topological line operators that implement the one-form symmetry. This allows us to compute the topologically twisted index on the closed Riemann surface $\Sigma_g$ for any real compact gauge group $G$ as long as the ground states are all bosonic. All computations are carried out in the effective $A$-model on $\Sigma_g$, whose $S^1$ ground states are the so-called Bethe vacua. We discuss how the 3d one-form symmetry acts on the Bethe vacua, and also how its 't Hooft anomaly constrains the vacuum structure. In the special case of the $\text{SU}(N)_K$ $\mathcal{N}=2$ Chern-Simons theory, we obtain results for the $(\text{SU}(N)/\mathbb Z_r)^{\theta}_K$ $\mathcal{N}=2$ Chern-Simons theories, for all non-anomalous $\mathbb Z_r \subseteq \mathbb Z_N$ subgroups of the centre of the gauge group, and with a $\mathbb Z_r$ $\theta$-angle turned on. In the special cases with $N$ even, $\frac{N}{r}$ odd and $\frac{K}{r}$ even, we find a mixed 't Hooft anomaly between gravity and the $\mathbb Z_r^{(1)}$ one-form symmetry of the $\text{SU}(N)_K$ theory, and the infrared 3d TQFT after gauging is spin. In all cases, we count the Bethe states and the higher-genus states in terms of refinements of Jordan's totient function. This counting gives us the twisted indices if and only if the infrared 3d TQFT is bosonic. Our results lead to precise conjectures about integrality of indices, which appear to have a strong number-theoretic flavour. Note: this paper directly builds upon unpublished notes by Brian Willett from 2020.

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arXiv2024

Reading between the rational sections: Global structures of 4d $\mathcal{N}=2$ KK theories

We study how the global structure of rank-one 4d $\mathcal{N}=2$ supersymmetric field theories is encoded into global aspects of the Seiberg-Witten elliptic fibration. Starting with the prototypical example of the $\mathfrak{su}(2)$ gauge theory, we distinguish between relative and absolute Seiberg-Witten curves. For instance, we discuss in detail the three distinct absolute curves for the $SU(2)$ and $SO(3)_\pm$ 4d $\mathcal{N}=2$ gauge theories. We propose that the $1$-form symmetry of an absolute theory is isomorphic to a torsion subgroup of the Mordell-Weil group of sections of the absolute curve, while the full defect group of the theory is encoded in the torsion sections of a so-called relative curve. We explicitly show that the relative and absolute curves are related by isogenies (that is, homomorphisms of elliptic curves) generated by torsion sections -- hence, gauging a one-form symmetry corresponds to composing isogenies between Seiberg-Witten curves. We apply this approach to Kaluza-Klein (KK) 4d $\mathcal{N}=2$ theories that arise from toroidal compactifications of 5d and 6d SCFTs to four dimensions, uncovering an intricate pattern of 4d global structures obtained by gauging discrete $0$-form and/or $1$-form symmetries. Incidentally, we propose a 6d BPS quiver for the 6d M-string theory on $\mathbb{R}^4\times T^2$.

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arXiv2023

New results on 3d $\mathcal{N}=2$ SQCD and its 3d GLSM interpretation

In this note, we review some new results we recently obtained about the infrared physics of 3d $\mathcal{N}=2$ SQCD with a unitary gauge group, in particular in the presence of a non-zero Fayet-Iliopoulos parameter and with generic values of the Chern-Simons levels. We review the 3d GLSM (also known as 3d A-model) approach to the computation of the 3d $\mathcal{N}=2$ twisted chiral ring of half-BPS lines. For particular values of the Chern-Simons levels, this twisted chiral ring has a neat interpretation in terms of the quantum K-theory (QK) of the Grassmannian manifold. We propose a new set of line defects of the 3d gauge theory, dubbed Grothendieck lines, which represent equivariant Schubert classes in the QK ring. In particular, we show that double Grothendieck polynomials, which represent the equivariant Chern characters of the Schubert classes, arise physically as Witten indices of certain quiver supersymmetric quantum mechanics. We also explain two distinct ways how to compute K-theoretic enumerative invariants using the 3d GLSM approach. \textit{This review article is a contribution to the proceedings of the GLSM@30 conference, which was held in May 2023 at the Simons Center for Geometry and Physics.}

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arXiv2023

Grothendieck lines in 3d $\mathcal{N}=2$ SQCD and the quantum K-theory of the Grassmannian

We revisit the 3d GLSM computation of the equivariant quantum K-theory ring of the complex Grassmannian from the perspective of line defects. The 3d GLSM onto $X={\rm Gr}(N_c, n_f)$ is a circle compactification of the 3d $\mathcal{N}=2$ supersymmetric gauge theory with gauge group $U(N_c)_{k, k+l N_c}$ and $n_f$ fundamental chiral multiplets, for any choice of the Chern-Simons levels $(k,l)$ in the `geometric window'. For $k=N_c-\frac{n_f}{2}$ and $l=-1$, the twisted chiral ring generated by the half-BPS lines wrapping the circle has been previously identified with the quantum K-theory ring QK$_T(X)$. We identify new half-BPS line defects in the UV gauge theory, dubbed Grothendieck lines, which flow to the structure sheaves of the (equivariant) Schubert varieties of $X$. They are defined by coupling $\mathcal{N}=2$ supersymmetric gauged quantum mechanics of quiver type to the 3d GLSM. We explicitly show that the 1d Witten index of the defect worldline reproduces the Chern characters for the Schubert classes, which are written in terms of double Grothendieck polynomials. This gives us a physical realisation of the Schubert-class basis for QK$_T(X)$. We then use 3d $A$-model techniques to explicitly compute QK$_T(X)$ as well as other K-theoretic enumerative invariants such as the topological metric. We also consider the 2d/0d limit of our 3d/1d construction, which gives us local defects in the 2d GLSM, the Schubert defects, that realise equivariant quantum cohomology classes.

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arXiv2023

On the Witten index of 3d $\mathcal{N}=2$ unitary SQCD with general CS levels

We consider unitary SQCD, a three-dimensional $\mathcal{N}=2$ supersymmetric Chern-Simons-matter theory consisting of one $U(N_c)_{k, k+l N_c}$ vector multiplet coupled to $n_f$ fundamental and $n_a$ antifundamental chiral multiplets, where $k$ and $l$ parameterise generic CS levels for $U(N_c)=(SU(N_c)\times U(1))/\mathbb{Z}_{N_c}$. We study the moduli space of vacua of this theory with $n_a=0$, for generic values of the parameters $N_c, k, l, n_f$ and with a non-zero Fayet-Ilopoulos parameter turned on. We uncover a rich pattern of vacua including Higgs, topological and hybrid phases. This allows us to derive a closed-form formula for the flavoured Witten index of unitary SQCD for any $n_f\neq n_a$, generalising previously known results for either $l=0$ or $n_f=n_a$. Finally, we analyse the vacuum structure of recently proposed infrared-dual gauge theories and we match vacua across the dualities, thus providing intricate new checks of those dualities. Incidentally, we also discuss a seemingly new level/rank duality for pure CS theories with $U(N)\times U(N')$ gauge group.

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arXiv2023

Twisted indices, Bethe ideals and 3d $\mathcal{N}=2$ infrared dualities

We study the topologically twisted index of 3d $\mathcal{N}=2$ supersymmetric gauge theories with unitary gauge groups. We implement a Gröbner basis algorithm for computing the $Σ_g\times S^1$ index explicitly and exactly in terms of the associated Bethe ideal, which is defined as the algebraic ideal associated with the Bethe equations of the corresponding 3d $A$-model. We then revisit recently discovered infrared dualities for unitary SQCD with gauge group $U(N_c)_{k, k +l N_c}$ with $l\neq 0$, namely the Nii duality that generalises the Giveon-Kutasov duality, the Amariti-Rota duality that generalises the Aharony duality, and their further generalisations in the case of arbitrary numbers of fundamental and antifundamental chiral multiplets. In particular, we determine all the flavour Chern-Simons contact terms needed to make these dualities work. This allows us to check that the twisted indices of dual theories match exactly. We also initiate the study of the Witten index of unitary SQCD with $l\neq 0$.

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arXiv2022

Partition Functions and Fibering Operators on the Coulomb Branch of 5d SCFTs

We study 5d $\mathcal{N}=1$ supersymmetric field theories on closed five-manifolds $\mathcal{M}_5$ which are principal circle bundles over simply-connected Kähler four-manifolds, $\mathcal{M}_4$, equipped with the Donaldson-Witten twist. We propose a new approach to compute the supersymmetric partition function on $\mathcal{M}_5$ through the insertion of a fibering operator, which introduces a non-trivial fibration over $\mathcal{M}_4$, in the 4d topologically twisted field theory. We determine the so-called Coulomb branch partition function on any such $\mathcal{M}_5$, which is conjectured to be the holomorphic `integrand' of the full partition function. We precisely match the low-energy effective field theory approach to explicit one-loop computations, and we discuss the effect of non-perturbative 5d BPS particles in this context. When $\mathcal{M}_4$ is toric, we also reconstruct our Coulomb branch partition function by appropriately gluing Nekrasov partition functions. As a by-product of our analysis, we provide strong new evidence for the validity of the Lockhart-Vafa formula for the five-sphere partition function.

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arXiv2022

The $U$-plane of rank-one 4d $\mathcal{N}=2$ KK theories

The simplest non-trivial 5d superconformal field theories (SCFT) are the famous rank-one theories with $E_n$ flavour symmetry. We study their $U$-plane, which is the one-dimensional Coulomb branch of the theory on $\mathbb{R}^4 \times S^1$. The total space of the Seiberg-Witten (SW) geometry -- the $E_n$ SW curve fibered over the $U$-plane -- is described as a rational elliptic surface with a singular fiber of type $I_{9-n}$ at infinity. A classification of all possible Coulomb branch configurations, for the $E_n$ theories and their 4d descendants, is given by Persson's classification of rational elliptic surfaces. We show that the global form of the flavour symmetry group is encoded in the Mordell-Weil group of the SW elliptic fibration. We study in detail many special points in parameters space, such as points where the flavour symmetry enhances, and/or where Argyres-Douglas and Minahan-Nemeschansky theories appear. In a number of important instances, including in the massless limit, the $U$-plane is a modular curve, and we use modularity to investigate aspects of the low-energy physics, such as the spectrum of light particles at strong coupling and the associated BPS quivers. We also study the gravitational couplings on the $U$-plane, matching the infrared expectation for the couplings $A(U)$ and $B(U)$ to the UV computation using the Nekrasov partition function.

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arXiv2021

Coulomb and Higgs Branches from Canonical Singularities, Part 1: Hypersurfaces with Smooth Calabi-Yau Resolutions

Compactification of M-theory and of IIB string theory on threefold canonical singularities gives rise to superconformal field theories (SCFTs) in 5d and 4d, respectively. The resolutions and deformations of the singularities encode salient features of the SCFTs and of their moduli spaces. In this paper, we build on Part 0 of this series (arXiv:2007.15600) and further explore the physics of SCFTs arising from isolated hypersurface singularities. We study in detail these canonical isolated hypersurface singularities that admit a smooth Calabi-Yau (crepant) resolution. Their 5d and 4d physics is discussed and their 3d reduction and mirrors (the magnetic quivers) are determined in many cases. As an explorative tool, we provide a Mathematica code which computes key quantities for any canonical isolated hypersurface singularity, including the 5d rank, the 4d Coulomb branch spectrum and central charges, higher-form symmetries in 4d and 5d, and crepant resolutions.

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arXiv2021

Coulomb and Higgs Branches from Canonical Singularities: Part 0

Five- and four-dimensional superconformal field theories with eight supercharges arise from canonical threefold singularities in M-theory and Type IIB string theory, respectively. We study their Coulomb and Higgs branches using crepant resolutions and deformations of the singularities. We propose a relation between the resulting moduli spaces, by compactifying the theories to 3d, followed by 3d $\mathcal{N}=4$ mirror symmetry and an $S$-type gauging of an abelian flavor symmetry. In particular, we use this correspondence to determine the Higgs branch of some 5d SCFTs and their magnetic quivers from the geometry. As an application of the general framework, we observe that singularities that engineer Argyres-Douglas theories in Type IIB also give rise to rank-0 5d SCFTs in M-theory. We also compute the higher-form symmetries of the 4d and 5d SCFTs, including the one-form symmetries of generalized Argyres-Douglas theories of type $(G, G')$.

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arXiv2021

5d and 4d SCFTs: Canonical Singularities, Trinions and S-Dualities

Canonical threefold singularities in M-theory and Type IIB string theory give rise to superconformal field theories (SCFTs) in 5d and 4d, respectively. In this paper, we study canonical hypersurface singularities whose resolutions contain residual terminal singularities and/or 3-cycles. We focus on a certain class of `trinion' singularities which exhibit these properties. In Type IIB, they give rise to 4d $\mathcal{N}=2$ SCFTs that we call $D_p^b(G)$-trinions, which are marginal gaugings of three SCFTs with $G$ flavor symmetry. In order to understand the 5d physics of these trinion singularities in M-theory, we reduce these 4d and 5d SCFTs to 3d $\mathcal{N}=4$ theories, thus determining the electric and magnetic quivers (or, more generally, quiverines). In M-theory, residual terminal singularities give rise to free sectors of massless hypermultiplets, which often are discretely gauged. These free sectors appear as `ugly' components of the magnetic quiver of the 5d SCFT. The 3-cycles in the crepant resolution also give rise to free hypermultiplets, but their physics is more subtle, and their presence renders the magnetic quiver `bad'. We propose a way to redeem the badness of these quivers using a class $\mathcal{S}$ realization. We also discover new S-dualities between different $D_p^b(G)$-trinions. For instance, a certain $E_8$ gauging of the $E_8$ Minahan-Nemeschansky theory is S-dual to an $E_8$-shaped Lagrangian quiver SCFT.

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arXiv2020

On 5d SCFTs and their BPS quivers. Part I: B-branes and brane tilings

We study the spectrum of BPS particles on the Coulomb branch of five-dimensional superconformal field theories (5d SCFTs) compactified on a circle. By engineering these theories in M-theory on ${\mathbf X} \times S^1 $, for ${\mathbf X}$ an isolated Calabi-Yau threefold singularity, we naturally identify the BPS category of the 5d theory on a circle with the derived category of coherent sheaves on a resolution of ${\mathbf X}$. It follows that the BPS spectrum can be studied in terms of 5d BPS quivers, which are the fractional-brane quivers for the singularity ${\mathbf X}$. 5d BPS quivers generalize the well-studied 4d BPS quivers for 4d $\mathcal{N}{=}2$ gauge theories that can be obtained from ${\mathbf X}$ in so-called geometric engineering limits. We study the interplay between 4d and 5d BPS quivers in detail. We particularly focus on examples when ${\mathbf X}$ is a toric singularity, in which case the 5d BPS quiver is given in terms of a brane tiling. For instance, the well-studied $Y^{p,q}$ brane tiling gives a 5d BPS quiver for the $SU(p)_q$ 5d gauge theory. We present a conjecture about the structure of the BPS spectra of a wide class of models, which we test in the simple case of the 5d $SU(2)_0$ theory (more precisely, the $E_1$ SCFT). We also argue that 5d UV dualities can be realized in terms of mutation sequences on the BPS quivers, which are in turn interpreted as autoequivalences of the BPS category.

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