arXiv subjects

Marcus Sperling

Publications and source records attributed to Marcus Sperling.

At least 37 records · Page 2Linked to original sources

arXiv2022

Balanced B and D-type orthosymplectic quivers -- Magnetic quivers for product theories

We investigate orthosymplectic quivers that take the shape of D-type and B-type Dynkin diagrams. The D-type orthosymplectic quivers explored here contain a balanced "fork", i.e., a balanced subquiver with a D-type bifurcation, whereas the B-type orthosymplectic quivers are obtained by folding the D-type quivers. The Coulomb branches of these quivers are products of two moduli spaces. In the second part, the relevant orthosymplectic quivers are shown to emerge as magnetic quivers for brane configurations involving ON$^0$ planes. Notably, the appearance of ON$^0$ plane clarifies the product nature of the theories in question. The derivation leads to the analysis of magnetic quivers from branes systems with intersecting Op, O(p+2), and ON$^0$ planes.

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arXiv2022

Magnetic quivers and line defects -- On a duality between 3d N=4 unitary and orthosymplectic quivers

Supersymmetric Sp(k) quantum chromodynamics with 8 supercharges in space-time dimensions 3 to 6 can be realised by two different Type II brane configurations in the presence of orientifolds. Consequently, two types of magnetic quivers describe the Higgs branch of the Sp(k) SQCD theory. This is a salient example of a general phenomenon: a given hyper-Kahler Higgs branch may admit several magnetic quiver constructions. It is then natural to wonder if these different magnetic quivers, which are described by 3d N=4 theories, are dual theories. In this work, the unitary and orthosymplectic magnetic quiver theories are subjected to a variety of tests, providing evidence that they are IR dual to each other. For this, sphere partition function and supersymmetric indices are compared. Also, we study half BPS line defects and find interesting regularities from the viewpoints of exact results, brane configurations, and one-form symmetry.

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arXiv2021

E-string Quantum Curve

In this work we study the quantisation of the Seiberg-Witten curve for the E-string theory compactified on a two-torus. We find that the resulting operator expression belongs to the class of elliptic quantum curves. It can be rephrased as an eigenvalue equation with eigenvectors corresponding to co-dimension 2 defect operators and eigenvalues to co-dimension 4 Wilson surfaces wrapping the elliptic curve, respectively. Moreover, the operator we find is a generalised version of the van Diejen operator arising in the study of elliptic integrable systems. Although the microscopic representation of the co-dimension 4 defect only furnishes an $\mathrm{SO}(16)$ flavour symmetry in the UV, we find an enhancement in the IR to representations in terms of affine $E_8$ characters. Finally, using the Nekrasov-Shatashvili limit of the E-string BPS partition function, we give a path integral derivation of the quantum curve.

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arXiv2021

Elliptic Quantum Curves of 6d SO(N) theories

We discuss supersymmetric defects in 6d $\mathcal{N}=(1,0)$ SCFTs with $\mathrm{SO}(N_c)$ gauge group and $N_c-8$ fundamental flavors. The codimension 2 and 4 defects are engineered by coupling the 6d gauge fields to charged free fields in four and two dimensions, respectively. We find that the partition function in the presence of the codimension 2 defect on $\mathbb{R}^4\times \mathbb{T}^2$ in the Nekrasov-Shatashvili limit satisfies an elliptic difference equation which quantizes the Seiberg-Witten curve of the 6d theory. The expectation value of the codimension 4 defect appearing in the difference equation is an even (under reflection) degree $N_c$ section over the elliptic curve when $N_c$ is even, and an odd section when $N_c$ is odd. We also find that RG-flows of the defects and the associated difference equations in the 6d $\mathrm{SO}(2N+1)$ gauge theories triggered by Higgs VEVs of KK-momentum states provide quantum Seiberg-Witten curves for $\mathbb{Z}_2$ twisted compactifications of the 6d $\mathrm{SO}(2N)$ gauge theories.

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arXiv2021

Fibre-base duality of 5d KK theories

We study circle compactifications of 6d superconformal field theories giving rise to 5d rank 1 and rank 2 Kaluza-Klein theories. We realise the resulting theories as M-theory compactifications on local Calabi-Yau 3-folds and match the prepotentials from geometry and field theory. One novelty in our approach is that we include explicit dependence on bare gauge couplings and mass parameters in the description which in turn leads to an accurate parametrisation of the prepotential including all parameters of the field theory. We find that the resulting geometries admit "fibre-base" duality which relates their six-dimensional origin with the purely five-dimensional quantum field theory interpretation. The fibre-base duality is realised simply by swapping base and fibre curves of compact surfaces in the local Calabi-Yau which can be viewed as the total space of the anti-canonical bundle over such surfaces. Our results show that such swappings precisely occur for surfaces with a zero self-intersection of the base curve and result in an exchange of the 6d and 5d pictures.

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arXiv2021

Branes, Quivers, and the Affine Grassmannian

Brane systems provide a large class of gauge theories that arise in string theory. This paper demonstrates how such brane systems fit with a somewhat exotic geometric object, called the affine Grassmannian. This gives a strong motivation to study physical aspects of the affine Grassmannian. Explicit quivers are presented throughout the paper, and a quiver addition algorithm to generate the affine Grassmannian is introduced. An important outcome of this study is a set of quivers for new elementary slices.

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arXiv2020

Magnetic Lattices for Orthosymplectic Quivers

For any gauge theory, there may be a subgroup of the gauge group which acts trivially on the matter content. While many physical observables are not sensitive to this fact, the identification of the precise gauge group becomes crucial when the magnetic spectrum of the theory is considered. This question is addressed in the context of Coulomb branches for $3$d $\mathcal{N}=4$ quiver gauge theories, which are moduli spaces of dressed monopole operators. Since monopole operators are characterized by their magnetic charge, the identification of the gauge group is imperative for the determination of the magnetic lattice. It is well-known that the gauge group of unframed unitary quivers is the product of all unitary nodes in the quiver modded out by the diagonal $\mathrm{U}(1)$ acting trivially on the matter representation. This reasoning generalises to the notion that a choice of gauge group associated to a quiver is given by the product of the individual nodes quotiented by any subgroup that acts trivially on the matter content. For unframed (unitary-) orthosymplectic quivers composed of $\mathrm{SO}(\textrm{even})$, $\mathrm{USp}$, and possibly $\mathrm{U}$ gauge nodes, the maximal subgroup acting trivially is a diagonal $\mathbb{Z}_2$. For unframed unitary quivers with a single $\mathrm{SU}(N)$ node it is $\mathbb{Z}_N$. We use this notion to compute the Coulomb branch Hilbert series of many unitary-orthosymplectic quivers. Examples include nilpotent orbit closures of the exceptional E-type algebras and magnetic quivers that arise from brane physics. This includes Higgs branches of theories with 8 supercharges in dimensions $4$, $5$, and $6$. A crucial ingredient in the calculation of exact refined Hilbert series is the alternative construction of unframed magnetic quivers from resolved Slodowy slices, whose Hilbert series can be derived from Hall-Littlewood polynomials.

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arXiv2020

S-fold magnetic quivers

Magnetic quivers and Hasse diagrams for Higgs branches of rank $r$ 4d $\mathcal{N}=2$ SCFTs arising from $\mathbb{Z}_{\ell}$ $\mathcal{S}$-fold constructions are discussed. The magnetic quivers are derived using three different methods: 1) Using clues like dimension, global symmetry, and the folding parameter $\ell$ to guess the magnetic quiver. 2) From 6d $\mathcal{N}=(1,0)$ SCFTs as UV completions of 5d marginal theories, and specific FI deformations on their magnetic quiver, which is further folded by $\mathbb{Z}_{\ell}$. 3) From T-duality of Type IIA brane systems of 6d $\mathcal{N}=(1,0)$ SCFTs and explicit mass deformation of the resulting brane web followed by $\mathbb{Z}_{\ell}$ folding. A choice of the ungauging scheme, either on a long node or on a short node, yields two different moduli spaces related by an orbifold action, thus suggesting a larger set of SCFTs in four dimensions than previously expected.

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arXiv2020

Elliptic Quantum Curves of Class $\mathcal{S}_k$

Quantum curves arise from Seiberg-Witten curves associated to 4d $\mathcal{N}=2$ gauge theories by promoting coordinates to non-commutative operators. In this way the algebraic equation of the curve is interpreted as an operator equation where a Hamiltonian acts on a wave-function with zero eigenvalue. We find that this structure generalises when one considers torus-compactified 6d $\mathcal{N}=(1,0)$ SCFTs. The corresponding quantum curves are elliptic in nature and hence the associated eigenvectors/eigenvalues can be expressed in terms of Jacobi forms. In this paper we focus on the class of 6d SCFTs arising from M5 branes transverse to a $\mathbb{C}^2/\mathbb{Z}_k$ singularity. In the limit where the compactified 2-torus has zero size, the corresponding 4d $\mathcal{N}=2$ theories are known as class $\mathcal{S}_k$. We explicitly show that the eigenvectors associated to the quantum curve are expectation values of codimension 2 surface operators, while the corresponding eigenvalues are codimension 4 Wilson surface expectation values.

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arXiv2020

Magnetic Quivers from Brane Webs with O5 Planes

Magnetic quivers have led to significant progress in the understanding of gauge theories with 8 supercharges at UV fixed points. For a given low-energy gauge theory realised via a Type II brane construction, there exist magnetic quivers for the Higgs branches at finite and infinite gauge coupling. Comparing these moduli spaces allows to study the non-perturbative effects when transitioning to the fixed point. For 5d $\mathcal{N}=1$ SQCD, 5-brane webs have been an important tool for deriving magnetic quivers. In this work, the emphasis is placed on 5-brane webs with orientifold 5-planes which give rise to 5d theories with orthogonal or symplectic gauge groups. For this set-up, the magnetic quiver prescription is derived and contrasted against a unitary magnetic quiver description extracted from an O$7^-$ construction. Further validation is achieved by a derivation of the associated Hasse diagrams. An important class of families considered are the orthogonal exceptional $E_n$ families ($-\infty < n \leq 8$), realised as infinite coupling Higgs branches of $\mathrm{Sp}(k)$ gauge theories with fundamental matter. In particular, the moduli spaces are realised by a novel type of magnetic quivers, called unitary-orthosymplectic quivers.

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arXiv2020

Magnetic quivers for rank 1 theories

Magnetic quivers and Hasse diagrams for Higgs branches of rank 1 $4d$ $\mathcal{N}=2$ SCFTs are provided. These rank 1 theories fit naturally into families of higher rank theories, originating from higher dimensions, which are addressed.

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arXiv2020

Magnetic Quivers, Higgs Branches, and 6d N=(1,0) Theories -- Orthogonal and Symplectic Gauge Groups

M5 branes on a D-type ALE singularity display various phenomena that introduce additional massless degrees of freedom. The M5 branes are known to fractionate on a D-type singularity. Whenever two fractional M5 branes coincide, tensionless strings arise. Therefore, these systems do not admit a low-energy Lagrangian description. Focusing on the 6-dimensional N=(1,0) world-volume theories on the M5 branes, the vacuum moduli space has two branches were either the scalar fields in the tensor multiplet or the scalars in the hypermultiplets acquire a non-trivial vacuum expectation value. As suggested in previous work, the Higgs branch may change drastically whenever a BPS-string becomes tensionless. Recently, magnetic quivers have been introduced with the aim to capture all Higgs branches over any point of the tensor branch. In this paper, the formalism is extended to Type IIA brane configurations involving O6 planes. Since the 6d N=(1,0) theories are composed of orthosymplectic gauge groups, the derivation rules for the magnetic quiver in the presence of O6 planes have to be conjectured. This is achieved by analysing the 6d theories for a single M5 brane on a D-type singularity and deriving the magnetic quivers for the finite and infinite gauge coupling Higgs branch from a brane configuration. The validity of the proposed derivation rules is underpinned by deriving the associated Hasse diagram. For multiple M5 branes, the approach of this paper provides magnetic quivers for all Higgs branches over any point of the tensor branch. In particular, an interesting infinite gauge coupling transition is found that is related to the SO(8) non-Higgsable cluster.

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arXiv2020

4d N=1 from 6d D-type N=(1,0)

Compactifications of 6d N=(1,0) SCFTs give rise to new 4d N=1 SCFTs and shed light on interesting dualities between such theories. In this paper we continue exploring this line of research by extending the class of compactified 6d theories to the D-type case. The simplest such 6d theory arises from D5 branes probing D-type singularities. Equivalently, this theory can be obtained from an F-theory compactification using "-2"-curves intersecting according to a D-type quiver. Our approach is two-fold. We start by compactifying the 6d SCFT on a Riemann surface and compute the central charges of the resulting 4d theory by integrating the 6d anomaly polynomial over the Riemann surface. As a second step, in order to find candidate 4d UV Lagrangians, there is an intermediate 5d theory that serves to construct 4d domain walls. These can be used as building blocks to obtain torus compactifications. In contrast to the A-type case, the vanishing of anomalies in the 4d theory turns out to be very restrictive and constraints the choices of gauge nodes and matter content severely. As a consequence, in this paper one has to resort to non-maximal boundary conditions for the 4d domain walls. However, the comparison to the 6d theory compactified on the Riemann surface becomes less tractable.

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arXiv2019

The Higgs Mechanism -- Hasse Diagrams for Symplectic Singularities

We explore the geometrical structure of Higgs branches of quantum field theories with 8 supercharges in 3, 4, 5 and 6 dimensions. They are symplectic singularities, and as such admit a decomposition (or foliation) into so-called symplectic leaves, which are related to each other by transverse slices. We identify this foliation with the pattern of partial Higgs mechanism of the theory and, using brane systems and recently introduced notions of magnetic quivers and quiver subtraction, we formalise the rules to obtain the Hasse diagram which encodes the structure of the foliation. While the unbroken gauge symmetry and the number of flat directions are obtainable by classical field theory analysis for Lagrangian theories, our approach allows us to characterise the geometry of the Higgs branch by a Hasse diagram with symplectic leaves and transverse slices, thus refining the analysis and extending it to non-Lagrangian theories. Most of the Hasse diagrams we obtain extend beyond the cases of nilpotent orbit closures known in the mathematics literature. The geometric analysis developed in this paper is applied to Higgs branches of several Lagrangian gauge theories, Argyres-Douglas theories, five dimensional SQCD theories at the conformal fixed point, and six dimensional SCFTs.

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arXiv2019

Magnetic Quivers, Higgs Branches, and 6d N=(1,0) Theories

The physics of M5 branes placed near an M9 plane on an A-type ALE singularity exhibits a variety of phenomena that introduce additional massless degrees of freedom. There are tensionless strings whenever two M5 branes coincide or whenever an M5 brane approaches the M9 plane. These systems do not admit a low-energy Lagrangian description so new techniques are desirable to shed light on the physics of these phenomena. The 6-dimensional N=(1,0) world-volume theory on the M5 branes is composed of massless vector, tensor, and hyper multiplets, and has two branches of the vacuum moduli space where either the scalar fields in the tensor or hyper multiplets receive vacuum expectation values. Focusing on the Higgs branch of the low-energy theory, previous works suggest the conjecture that a new Higgs branch arises whenever a BPS-string becomes tensionless. Consequently, a single theory admits a multitude of Higgs branches depending on the types of tensionless strings in the spectrum. The two main phenomena discrete gauging and small E8 instanton transition can be treated in a concise and effective manner by means of Coulomb branches of 3-dimensional N=4 gauge theories. In this paper, a formalism is introduced that allows to derive a novel object from a brane configuration, called the magnetic quiver. The main features are as follows: (i) the 3d Coulomb branch of the magnetic quiver yields the Higgs branch of the 6d system, (ii) all discrete gauging and E8 instanton transitions have an explicit brane realisation, and (iii) exceptional symmetries arise directly from brane configurations. The formalism facilitates the description of Higgs branches at finite and infinite gauge coupling as spaces of dressed monopole operators.

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arXiv2018

Resolutions of nilpotent orbit closures via Coulomb branches of 3-dimensional N=4 theories

The Coulomb branches of certain 3-dimensional N=4 quiver gauge theories are closures of nilpotent orbits of classical or exceptional algebras. The monopole formula, as Hilbert series of the associated Coulomb branch chiral ring, has been successful in describing the singular hyper-Kähler structure. By means of the monopole formula with background charges for flavour symmetries, which realises real mass deformations, we study the resolution properties of all (characteristic) height two nilpotent orbits. As a result, the monopole formula correctly reproduces (i) the existence of a symplectic resolution, (ii) the form of the symplectic resolution, and (iii) the Mukai flops in the case of multiple resolutions. Moreover, the (characteristic) height two nilpotent orbit closures are resolved by cotangent bundles of Hermitian symmetric spaces and the unitary Coulomb branch quiver realisations exhaust all the possibilities.

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arXiv2018

Discrete quotients of 3-dimensional N = 4 Coulomb branches via the cycle index

The study of Coulomb branches of 3-dimensional N=4 gauge theories via the associated Hilbert series, the so-called monopole formula, has been proven useful not only for 3-dimensional theories, but also for Higgs branches of 5 and 6-dimensional gauge theories with 8 supercharges. Recently, a conjecture connected different phases of 6-dimensional Higgs branches via gauging of a discrete global $S_n$ symmetry. On the corresponding 3-dimensional Coulomb branch, this amounts to a geometric $S_n$-quotient. In this note, we prove the conjecture on Coulomb branches with unitary nodes and, moreover, extend it to Coulomb branches with other classical groups. The results promote discrete $S_n$-quotients to a versatile tool in the study of Coulomb branches.

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arXiv2018

Intersecting branes, Higgs sector, and chirality from $\mathcal{N}=4$ SYM with soft SUSY breaking

We consider $SU(N)$ $\mathcal{N}=4$ super Yang-Mills with cubic and quadratic soft SUSY breaking potential, such that the global $SU(4)_R$ is broken to $SU(3)$ or further. As shown recently, this set-up supports a rich set of non-trivial vacua with the geometry of self-intersecting $SU(3)$ branes in 6 extra dimensions. The zero modes on these branes can be interpreted as 3 generations of bosonic and chiral fermionic strings connecting the branes at their intersections. Here, we uncover a large class of exact solutions consisting of branes connected by Higgs condensates, leading to Yukawa couplings between the chiral fermionic zero modes. Under certain decoupling conditions, the backreaction of the Higgs on the branes vanishes exactly. The resulting physics is that of a spontaneously broken chiral gauge theory on branes with fluxes. In particular, we identify combined brane plus Higgs configurations which lead to gauge fields that couple to chiral fermions at low energy. This turns out to be quite close to the Standard Model and its constructions via branes in string theory. As a by-product, we construct a $G_2$-brane solution corresponding to a squashed fuzzy coadjoint orbit of $G_2$.

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