We discuss a general procedure to obtain 1/2 BPS partition functions for generic N=1 quiver gauge theories. These functions count the gauge invariant operators (bosonic and fermionic), charged under all the global symmetries (mesonic and baryonic), in the chiral ring of a given quiver gauge theory. In particular we discuss the inclusion of the spinor degrees of freedom in the partition functions.
Using recent results on BPS quiver theory, we develop a group theoretical method to describe the quiver mutations encoding the quantum mechanical duality relating the spectra of distinct quivers. We illustrate the method by computing the BPS spectrum of the infinite weak chamber of some examples of N=2 supersymmetric gauge models without and with quark hypermultiplets.
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 gen…
We count Higgs "phase" BPS states of general non-Abelian quiver, possibly with loops, by mapping the problem to its Abelian, or toric, counterpart and imposing Weyl invariance later. Precise Higgs index computation is particularly important for quivers with superpotentials; the Coulomb "phase" index is recently shown to miss important BPS states, dubbed intrinsic Higgs states or quiver invariants. We demonstrate how the refined Higgs index is naturally decomposed to a sum over partitions of the charge. We conjecture, and show in simple cases, that this decomposition expresses the Higgs index a…
We explore BPS quivers for D=5 theories, compactified on a circle and geometrically engineered over local Calabi-Yau 3-folds, for which many of known machineries leading to (refined) indices fail due to the fine-tuning of the superpotential. For Abelian quivers, the counting reduces to a geometric one, but the technically challenging $L^2$ cohomology proved to be essential for sensible BPS spectra. We offer a mathematical theorem to remedy the difficulty, but for non-Abelian quivers, the cohomology approach itself fails because the relevant wavefunctions are inherently gauge-theoretical. For t…
We study the basic features of BPS quiver mutations in 4D $\mathcal{N}=2$ supersymmetric quantum field theory with $G=ADE$ gauge symmetries.\ We show, for these gauge symmetries, that there is an isotropy group $\mathcal{G}_{Mut}^{G}$ associated to a set of quiver mutations capturing information about the BPS spectra. In the strong coupling limit, it is shown that BPS chambers correspond to finite and closed groupoid orbits with an isotropy symmetry group $\mathcal{G}_{strong}^{G}$ isomorphic to the discrete dihedral groups $Dih_{2h_{G}}$ contained in Coxeter$(G) $ with $% h_{G}$ the Coxeter n…
We show that generic three-dimensional $\mathcal N=2$ quiver super Chern-Simons-matter theories admit Bogomol'nyi-Prasad-Sommerfield (BPS) Drukker-Trancanelli (DT) type Wilson loops. We investigate both Wilson loops along timelike infinite straight lines in Minkowski spacetime and circular Wilson loops in Euclidean space. In Aharnoy-Bergman-Jafferis-Maldacena theory, we find that generic BPS DT type Wilson loops preserve the same number of supersymmetries as Gaiotto-Yin type Wilson loops. There are several free parameters for generic BPS DT type Wilson loops in the construction, and supersymme…
We study systematically the BPS spectra of N=2 SYM coupled to half--hypermultiplets, the basic example being E_7 SYM coupled to a half--hyper in the 56 irrepr. In order to do this, we determine the BPS quivers with superpotential of such N=2 models using a new technique we introduce. The computation of the BPS spectra in the various chambers is then reduced to the Representation Theory of the resulting quivers. We use the quiver description to study the BPS spectrum at both strong and weak coupling. The following models are discussed in detail: SU(6) SYM coupled to a half 20, SO(12) SYM couple…
We study the Seberg-Witten geometry of 5d ${\cal N}=1$ pure Yang-Mills theories compactified on a circle. The concept of the holonomy saddle implies that there are multiple 4d limits of interacting Seiberg-Witten theories from a single 5d theory, and we explore this in the simplest case of pure $SU(N)$ theories. The compactification leads to $N$ copies of locally indistinguishable 4d pure $SU(N)$ Seiberg-Witten theories in the infrared, glued together in a manner dictated by the Chern-Simons level. We show how this picture naturally builds the 5d BPS quivers which agree with the D0 probe dynam…
We present a survey of the computation of the BPS spectrum of a general four-dimensional N=2 supersymmetric gauge theory in terms of the Representation Theory of quivers with superpotential. We focus on SYM with a general gauge group G coupled to standard matter in arbitrary representations of G (consistent with a non--positive beta--function). The situation is particularly tricky and interesting when the matter consists of an odd number of half-hypermultiplets: we describe in detail SU(6) SYM coupled to half a 20, SO(12) SYM coupled to half a 32, and E_7 SYM coupled to half a 56.
For D=4 BPS state construction, counting, and wall-crossing thereof, quiver quantum mechanics offers two alternative approaches, the Coulomb phase and the Higgs phase, which sometimes produce inequivalent counting. The authors have proposed, in arXiv:1205.6511, two conjectures on the precise relationship between the two, with some supporting evidences. Higgs phase ground states are naturally divided into the Intrinsic Higgs sector, which is insensitive to wall-crossings and thus an invariant of quiver, plus a pulled-back ambient cohomology, conjectured to be an one-to-one image of Coulomb phas…
We review the problem of BPS state counting described by the generalized quiver matrix model of ADHM type. In four dimensions the generating function of the counting gives the Nekrasov partition function and we obtain generalization in higher dimensions. By the localization theorem, the partition function is given by the sum of contributions from the fixed points of the torus action, which are labeled by partitions, plane partitions and solid partitions. The measure or the Boltzmann weight of the path integral can take the form of the plethystic exponential. Remarkably after integration the pa…
We construct new families of 1/4 BPS Wilson loops in circular quiver $\mathcal N=4$ superconformal Chern-Simons-matter (SCSM) theories in three dimensions. They are defined as the holonomy of superconnections that contain non-trivial couplings to scalar and fermions, and cannot be reduced to block-diagonal matrices. Consequently, the new operators cannot be written in terms of double-node Wilson loops, as the ones considered so far in the literature. For particular values of the couplings the superconnection becomes block-diagonal and we recover the known fermionic 1/4 and 1/2 BPS Wilson loops…
We study the Coulomb branches of the rank-one 4d $\mathcal{N} = 2$ quantum field theories, including the KK theories obtained from the circle compactification of the 5d $\mathcal{N}= 1$ $E_n$ Seiberg theories. The focus is set on the relation between their Seiberg-Witten geometries and rational elliptic surfaces, with more attention being given to the modular surfaces, which we completely classify using the classification of subgroups of the modular group ${\rm PSL}(2,\mathbb{Z})$. We derive closed-form expressions for the modular functions for all congruence and some of the non-congruence sub…
The BPS spectrum of certain N=2 supersymmetric field theories can be determined algebraically by studying the representation theory of BPS quivers. We introduce methods based on BPS quivers to study line defects. The presence of a line defect opens up a new BPS sector: framed BPS states can be bound to the defect. The defect can be geometrically described in terms of laminations on a curve. To a lamination we associate certain elements of the Leavitt path algebra of the BPS quiver and use them to compute the framed BPS spectrum. We also provide an alternative characterization of line defects b…
We define "BPS graphs" on punctured Riemann surfaces associated with $A_{N-1}$ theories of class $\mathcal{S}$. BPS graphs provide a bridge between two powerful frameworks for studying the spectrum of BPS states: spectral networks and BPS quivers. They arise from degenerate spectral networks at maximal intersections of walls of marginal stability on the Coulomb branch. While the BPS spectrum is ill-defined at such intersections, a BPS graph captures a useful basis of elementary BPS states. The topology of a BPS graph encodes a BPS quiver, even for higher-rank theories and for theories with cer…
We exactly evaluate the partition function (index) of N=4 supersymmetric quiver quantum mechanics in the Higgs phase by using the localization techniques. We show that the path integral is localized at the fixed points, which are obtained by solving the BRST equations, and D-term and F-term conditions. We turn on background gauge fields of R-symmetries for the chiral multiplets corresponding to the arrows between quiver nodes, but the partition function does not depend on these R-charges. We give explicit examples of the quiver theory including a non-coprime dimension vector. The partition fun…
In this paper we construct and classify novel Drukker-Trancanelli (DT) type BPS Wilson loops along infinite straight lines and circles in $\mathcal N=2,3$ quiver superconformal Chern-Simons-matter theories, Aharony-Bergman-Jafferis-Maldacena (ABJM) theory, and $\mathcal N=4$ orbifold ABJM theory. Generally we have four classes of Wilson loops, and all of them preserve the same supersymmetries as the BPS Gaiotto-Yin (GY) type Wilson loops. There are several free complex parameters in the DT type BPS Wilson loops, and for two classes of Wilson loops in ABJM theory and $\mathcal N=4$ orbifold ABJ…
Three dimensional supersymmetric field theories have large moduli spaces of circular Wilson loops preserving a fixed set of supercharges. We simplify previous constructions of such Wilson loops and amend and clarify their classification. For a generic quiver gauge theory we identify the moduli space as a quotient of $C^m$ for some $m$ by an appropriate symmetry group. These spaces are quiver varieties associated to a cover of the original quiver or a subquiver thereof. This moduli space is generically singular and at the singularities there are large degeneracies of operators which seem differ…
In this paper we study the BPS state counting in the geometry of local obstructed curve with normal bundle O+O(-2). We find that the BPS states have a framed quiver description. Using this quiver description along with the Seiberg duality and the localization techniques, we can compute the BPS state indices in different chambers dictated by stability parameter assignments. This provides a well-defined method to compute the generalized Donaldson-Thomas invariants. This method can be generalized to other affine ADE quiver theories.
BPS sector in $\mathcal{N}=2$, four-dimensional toric quiver gauge theories has previously been studied using crystal melting model and dimer model. We introduce the Mahler measure associated to statistical dimer model to study large $N$ limit of these quiver gauge theories. In this limit, generating function of BPS states in a general toric quiver theory is studied and entropy, growth rate of BPS states and free energy of the quiver are obtained in terms of the Mahler measure. Moreover, a possible third-order phase transition in toric quivers is discussed. Explicit computations of profile fun…
We show the vertex operator formalism for the quiver gauge theory partition function and the $qq$-character of highest-weight module on quiver, both associated with the integral over the quiver variety.
We compute R-charges of the BPS-monopole operators in $\mathcal{N} = 3$ $\widehat{ADE}$ Chern-Simons quiver gauge theories, along the lines of the work of Benna, Klebanov and Klose in \cite{bkk}. These theories have a weakly coupled UV completion in terms of $\mathcal{N}=3$ supersymmetric Chern-Simons Yang-Mills theories. In the UV limit the monopole operators are well approximated by classical solutions. We construct classical BPS and anti-BPS monopole solutions to these theories which preserve $\frac{1}{3}$ supersymmetry all along the RG flow. We compute the $SU(2)_R$ charges in these backgr…
A large class of N=2 quantum field theories admits a BPS quiver description and the study of their BPS spectra is then reduced to a representation theory problem. In such theories the coupling to a line defect can be modelled by framed quivers. The associated spectral problem characterises the line defect completely. Framed BPS states can be thought of as BPS particles bound to the defect. We identify the framed BPS degeneracies with certain enumerative invariants associated with the moduli spaces of stable quiver representations. We develop a formalism based on equivariant localization to com…
We study the BPS spectra of N=2 complete quantum field theories in four dimensions. For examples that can be described by a pair of M5 branes on a punctured Riemann surface we explain how triangulations of the surface fix a BPS quiver and superpotential for the theory. The BPS spectrum can then be determined by solving the quantum mechanics problem encoded by the quiver. By analyzing the structure of this quantum mechanics we show that all asymptotically free examples, Argyres-Douglas models, and theories defined by punctured spheres and tori have a chamber with finitely many BPS states. In al…
We explore the relationship between four-dimensional N=2 quantum field theories and their associated BPS quivers. For a wide class of theories including super-Yang-Mills theories, Argyres-Douglas models, and theories defined by M5-branes on punctured Riemann surfaces, there exists a quiver which implicitly characterizes the field theory. We study various aspects of this correspondence including the quiver interpretation of flavor symmetries, gauging, decoupling limits, and field theory dualities. In general a given quiver describes only a patch of the moduli space of the field theory, and a ke…
We develop a method to build new 5D $\mathcal{N}=1$ gauge models based on Sasaki-Einstein manifolds $Y^{p,q}.$ These models extend the standard 5D ones having a unitary SU$\left( p\right) _{q}$ gauge symmetry based on $% Y^{p,q}$. Particular focus is put on the building of a gauge family with symplectic SP$\left( 2r,\mathbb{R}\right) $ symmetry. These super QFTs are embedded in M-theory compactified on folded toric Calabi-Yau threefolds $% \hat{X}(Y^{2r,0})$ constructed from conical $Y^{2r,0}$. By using outer-automorphism symmetries of 5D $\mathcal{N}=1$\textbf{\ }BPS quivers with unitary SU$\…
A class of half-BPS boundary conditions for $A_{K-1}$ circular quivers
Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）；Dipartimento di Fisica, Università di Trieste（的里雅斯特大学物理系）；INFN, Sezione di Trieste（意大利国家核物理研究所的里雅斯特分部）
We introduce a class of new algebras, the shifted quiver Yangians, as the BPS algebras for type IIA string theory on general toric Calabi-Yau three-folds. We construct representations of the shifted quiver Yangian from general subcrystals of the canonical crystal. We derive our results via equivariant localization for supersymmetric quiver quantum mechanics for various framed quivers, where the framings are determined by the shape of the subcrystals. Our results unify many known BPS state counting problems, including open BPS counting, non-compact D4-branes, and wall crossing phenomena, simply…
In this paper we study the relation between pyramid partitions with a general empty room configuration (ERC) and the BPS states of D-branes on the resolved conifold. We find that the generating function for pyramid partitions with a length n ERC is exactly the same as the D6/D2/D0 BPS partition function on the resolved conifold in particular Kaehler chambers. We define a new type of pyramid partition with a finite ERC that counts the BPS degeneracies in certain other chambers. The D6/D2/D0 partition functions in different chambers were obtained by applying the wall crossing formula. On the oth…
We analyze the fermionic wigging of 1/2-BPS (electric) extremal black hole attractors in N=2, D=5 ungauged Maxwell-Einstein supergravity theories, by exploiting anti-Killing spinors supersymmetry transformations. Regardless of the specific data of the real special geometry of the manifold defining the scalars of the vector multiplets, and differently from the D=4 case, we find that there are no corrections for the near--horizon attractor value of the scalar fields; an analogous result also holds for 1/2-BPS (magnetic) extremal black string. Thus, the attractor mechanism receives no fermionic c…
We discuss new solutions in massive Type IIA supergravity with AdS$_3\times$S$^2$ factors, preserving ${\cal N}=(0,4)$ SUSY. We propose a duality with a precise family of quivers that flow to ${\cal N}=(0,4)$ fixed points at low energies. These quivers consist on two families of linear quivers coupled by matter fields. Physical observables such as the central charges provide stringent checks of the proposed duality. A formal mapping is presented connecting our backgrounds with those dual to six dimensional ${\cal N}=(1,0)$ CFTs, suggesting the existence of a flow across dimensions between the …
We study the moduli space volume of BPS vortices in quiver gauge theories on compact Riemann surfaces. The existence of BPS vortices imposes constraints on the quiver gauge theories. We show that the moduli space volume is given by a vev of a suitable cohomological operator (volume operator) in a supersymmetric quiver gauge theory, where BPS equations of the vortices are embedded. In the supersymmetric gauge theory, the moduli space volume is exactly evaluated as a contour integral by using the localization. Graph theory is useful to construct the supersymmetric quiver gauge theory and to deri…
在非对易空间R^{2n}_theta x G/H上构造了Yang-Mills方程的显式BPS解和非BPS解.给定一个G-表示，通过对G/H上的一个特殊丛进行扭曲，得到了R^{2n}_theta x G/H上的一个G-等变联络的G-等变U（k）丛.这些空间上的U（k）Donaldson-Uhlenbeck-Yau方程在R^{2n}_θ上的特殊规范规范理论中化为涡旋型方程。Seiberg-Witten方程是一个特殊的例子。非对易BPS组态的部分等距，这是从一个等变的Atiyah-Bott-Shapiro建设制定。它们可以被解释为充满空间的膜-反膜系统中的D 0-膜。
The quiver Yangian, an infinite-dimensional algebra introduced recently in arXiv:2003.08909, is the algebra underlying BPS state counting problems for toric Calabi-Yau three-folds. We introduce trigonometric and elliptic analogues of quiver Yangians, which we call toroidal quiver algebras and elliptic quiver algebras, respectively. We construct the representations of the shifted toroidal and elliptic algebras in terms of the statistical model of crystal melting. We also derive the algebras and their representations from equivariant localization of three-dimensional $\mathcal{N}=2$ supersymmetr…
We study the discrete flows generated by the symmetry group of the BPS quivers for Calabi-Yau geometries describing five dimensional superconformal quantum field theories on a circle. These flows naturally describe the BPS particle spectrum of such theories and at the same time generate bilinear equations of q-difference type which, in the rank one case, are q-Painlevé equations. The solutions of these equations are shown to be given by grand canonical topological string partition functions which we identify with $τ$-functions of the cluster algebra associated to the quiver. We exemplify our c…
We argue how to identify supersymmetric quiver quantum mechanics description of BPS states, which arise in string theory in brane systems representing knots. This leads to a surprising relation between knots and quivers: to a given knot we associate a quiver, so that various types of knot invariants are expressed in terms of characteristics of a moduli space of representations of the corresponding quiver. This statement can be regarded as a novel type of categorification of knot invariants, and among its various consequences we find that Labastida-Mariño-Ooguri-Vafa (LMOV) invariants of a knot…
We investigate the connections between flavored quivers, dimer models, and BPS pyramids for generic toric Calabi-Yau threefolds from various perspectives. We introduce a purely field theoretic definition of both finite and infinite pyramids in terms of quivers with flavors. These pyramids are associated to the counting of BPS invariants for generic toric Calabi-Yau threefolds. We discuss how cluster transformations provide an efficient recursive method for computing pyramid partition functions and show that the recursion is equivalent to the multidimensional octahedron recurrence. Transitions …
We introduce and explore the relation between quivers and 3-manifolds with the topology of the knot complement. This idea can be viewed as an adaptation of the knots-quivers correspondence to Gukov-Manolescu invariants of knot complements (also known as $F_K$ or $\hat{Z}$). Apart from assigning quivers to complements of $T^{(2,2p+1)}$ torus knots, we study the physical interpretation in terms of the BPS spectrum and general structure of 3d $\mathcal{N}=2$ theories associated to both sides of the correspondence. We also make a step towards categorification by proposing a $t$-deformation of all …
The BPS sector of $\mathcal{N}=2$, $4d$ toric quiver gauge theories, and its corresponding D6-D2-D0 branes on Calabi-Yau threefolds, have been previously studied using integrable lattice models such as the crystal melting model and the dimer model. The asymptotics of the BPS sector, in the large N limit, can be studied using the Mahler measure theory, \cite{Zah}. In this work, we consider the class of isoradial quivers and study their thermodynamical observables and phase structure. Building on our previous results, and using the relation between the Mahler measure and hyperbolic 3-manifolds, …
We study the Gauge/Bethe correspondence for two-dimensional $\mathcal{N}=(2,2)$ supersymmetric quiver gauge theories associated with toric Calabi-Yau three-folds, whose BPS algebras have recently been identified as the quiver Yangians. We start with the crystal representations of the quiver Yangian, which are placed at each site of the spin chain. We then construct integrable models by combining the single-site crystals into crystal chains by a coproduct of the algebra, which we determine by a combination of representation-theoretical and gauge-theoretical arguments. For non-chiral quivers, we…
We study the defect groups of $D_p^b(G)$ theories using geometric engineering and BPS quivers. In the simple case when $b=h^\vee (G)$, we use the BPS quivers of the theory to see that the defect group is compatible with a known Maruyoshi-Song flow. To extend to the case where $b\neq h^\vee (G)$, we use a similar Maruyoshi-Song flow to conjecture that the defect groups of $D_p^b(G)$ theories are given by those of $G^{(b)}[k]$ theories. In the cases of $G=A_n, \;E_6, \;E_8$ we cross check our result by calculating the BPS quivers of the $G^{(b)}[k]$ theories and looking at the cokernel of their …
In this note we review some of the uses of framed quivers to study BPS invariants of Donaldson-Thomas type. We will mostly focus on non-compact Calabi-Yau threefolds. In certain cases the study of these invariants can be approached as a generalized instanton problem in a six dimensional cohomological Yang-Mills theory. One can construct a quantum mechanics model based on a certain framed quiver which locally describes the theory around a generalized instanton solution. The problem is then reduced to the study of the moduli spaces of representations of these quivers. Examples include the affine…
We study the BPS particle spectrum of five-dimensional superconformal field theories (SCFTs) on $\mathbb{R}^4\times S^1$ with one-dimensional Coulomb branch, by means of their associated BPS quivers. By viewing these theories as arising from the geometric engineering within M-theory, the quivers are naturally associated to the corresponding local Calabi-Yau threefold. We show that the symmetries of the quiver, descending from the symmetries of the Calabi-Yau geometry, together with the affine root lattice structure of the flavor charges, provide equations for the Kontsevich-Soibelman wall-cros…
We study the refined and unrefined crystal/BPS partition functions of D6-D2-D0 brane bound states for all toric Calabi-Yau threefolds without compact 4-cycles and some non-toric examples. They can be written as products of (generalized) MacMahon functions. We check our expressions and use them as vacuum characters to study the gluings. We then consider the wall crossings and discuss possible crystal descriptions for different chambers. We also express the partition functions in terms of plethystic exponentials. For $\mathbb{C}^3$ and tripled affine quivers, we find their connections to nilpote…
