arXiv subjects

Craig Lawrie

Publications and source records attributed to Craig Lawrie.

At least 19 recordsLinked to original sources

arXiv2026

Algebro-geometric bootstrapping from OPE decoupling

We conjecture that decoupling relations in the operator product expansion of a 4d $\mathcal{N}=2$ superconformal field theory (SCFT) are encoded by an algebro-geometric object: a bifiltered affine scheme. We demonstrate how this scheme reproduces the Macdonald index (thus the Schur index) as well as the Higgs branch. Although the associated scheme typically admits continuous deformations, we find that a geometric extremization principle uniquely fixes these moduli, thereby providing a possible geometric route toward a classification of 4d $\mathcal{N}=2$ SCFTs.

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arXiv2025

Index from a point

We propose an algebro-geometric interpretation of the Schur and Macdonald indices of four-dimensional $\mathcal{N}=2$ superconformal field theories (SCFTs). We conjecture that there exists an affine scheme $X$, which reduces to the Higgs branch as a variety, such that the Hilbert series of the (appropriately-graded) arc space of its polynomial ring $J_\infty(\mathbb{C}[X])$ encodes the indices. Distinct local descriptions of a (singular) point correspond to distinct choices of $X$, giving rise to families of $\mathcal{N}=2$ SCFTs each without a Higgs branch. These local descriptions directly translate into nilpotency relations in the operator product expansions. We test our conjecture across a variety of (generalized) Argyres--Douglas theories.

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arXiv2025

Discrete Gauging of 6d SCFTs and Wreathed 3d $\mathcal{N}=4$ Quivers

We study the Higgs branch moduli space of certain 6d $(1,0)$ SCFTs after gauging their $\mathbb{Z}_2$ Green-Schwarz automorphism. We explain how to read the flavor symmetry of such SCFTs directly from the 6d construction, and we confirm the expectation by computing the Coulomb branch Hilbert series of their $\mathbb{Z}_2$-wreathed 3d $\mathcal{N}=4$ magnetic quiver. To perform the latter computation, we explicitly introduce a methodology to determine such Hilbert series for $\mathbb{Z}_2$-wreathed orthosymplectic quivers.

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arXiv2024

A landscape of 4d N=1 SCFTs with a=c

We study a landscape of four-dimensional $\mathcal{N}=1$ superconformal field theories (SCFTs) with identical central charges. These theories are obtained by renormalization group flows triggered by supersymmetry-preserving superpotential deformations of the $\mathcal{N}=1$ gauging of the flavor symmetry of a collection of $\mathcal{N}=2$ $\mathcal{D}_p(G)$ Argyres--Douglas SCFTs. In this work, we focus on the fixed points in the landscape of the $SU(3)$ gauging of three copies of the $\mathcal{D}_2(SU(3)) = H_2$ theory together with an adjoint-valued chiral multiplet. We catalogue the network of $a = c$ fixed points, and, along the way, we find a variety of dualities and instances of supersymmetry enhancement.

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arXiv2024

A Pathway to Decay and Fission of Orthosymplectic Quiver Theories

We present an algorithm to extract the Coulomb branch Hasse diagram of orthosymplectic 3d $\mathcal{N}=4$ quiver gauge theories. The algorithm systematically predicts all descendant theories arising from Coulomb branch Higgsing, thereby detailing the stratification of the symplectic singularity defined by the initial Coulomb branch. Leveraging the Lie algebra isomorphism $\mathfrak{su}(4) \cong \mathfrak{so}(6)$, we validate our algorithm via the 3d mirror of 4d theories of class $\mathcal{S}$ of such type. This comparison involves moduli spaces that admit both orthosymplectic and unitary quiver realisations, the latter being well-understood via standard techniques such as Decay and Fission. Higgsing on the Coulomb branch of the 3d mirror or magnetic quiver translates to Higgs branch renormalization group flows of the corresponding higher-dimensional SCFTs. Thus, we benchmark our method via Higgsing 6d $\mathcal{N}=(1,0)$ D-type orbi-instanton theories, predicting novel Higgsing patterns involving products of interacting fixed points, and class $\mathcal{S}$ theories of type $\mathfrak{so}(2N)$, demonstrating Higgsing to products of theories of types specified by Levi subalgebras of $\mathfrak{so}(2N)$.

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arXiv2024

Detecting Homeomorphic 3-manifolds via Graph Neural Networks

Motivated by the enumeration of the BPS spectra of certain 3d $\mathcal{N}=2$ supersymmetric quantum field theories, obtained from the compactification of 6d superconformal field theories on three-manifolds, we study the homeomorphism problem for a class of graph-manifolds using Graph Neural Network techniques. Utilizing the JSJ decomposition, a unique representation via a plumbing graph is extracted from a graph-manifold. Homeomorphic graph-manifolds are related via a sequence of von Neumann moves on this graph; the algorithmic application of these moves can determine if two graphs correspond to homeomorphic graph-manifolds in super-polynomial time. However, by employing Graph Neural Networks (GNNs), the same problem can be addressed, at the cost of accuracy, in polynomial time. We build a dataset composed of pairs of plumbing graphs, together with a hidden label encoding whether the pair is homeomorphic. We train and benchmark a variety of network architectures within a supervised learning setting by testing different combinations of two convolutional layers (GEN, GCN, GAT, NNConv), followed by an aggregation layer and a classification layer. We discuss the strengths and weaknesses of the different GNNs for this homeomorphism problem.

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arXiv2024

Generalized symmetry constraints on deformed 4d (S)CFTs

We explore the consequence of generalized symmetries in four-dimensional $\mathcal{N}=1$ superconformal field theories. First, we classify all possible supersymmetric gauge theories with a simple gauge group that have a nontrivial one-form symmetry and flows to a superconformal field theory. Upon identifying unbroken discrete zero-form symmetries from the ABJ anomaly, we find that many of these theories have mixed zero-form/one-form 't Hooft anomalies. Then we classify the relevant deformations of these SCFTs that preserve the anomaly. From this mixed anomaly together with the anomalies of the discrete zero-form symmetries, we find obstructions for the relevant deformations of these SCFTs to flow to a trivially gapped phase. We also study non-Lagrangian SCFTs formed by gauging copies of Argyres-Douglas theories and constrain their deformations. In particular, we explore a new duality between the diagonal gauging of two $\mathcal{D}_3(SU(N))$ theories and $SU(N)$ gauge theory with two adjoints. We also repeat our analysis for a host of non-supersymmetric gauge theories having nontrivial one-form symmetry including examples that appear to flow to Bank-Zaks type CFTs.

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arXiv2024

The Higgs Branch of 6d (1,0) SCFTs & LSTs with DE-type SUSY Enhancement

We detail the Higgs branches of 6d $(1,0)$ superconformal field theories (SCFTs) and little string theories (LSTs) that exhibit supersymmetry-enhancing Higgs branch renormalization group flows to the 6d $(2,0)$ SCFTs and LSTs of type DE. Generically, such theories are geometrically engineered in F-theory via a configuration of $(-2)$-curves, arranged in an (affine) DE-type Dynkin diagram, and supporting special unitary gauge algebras; this describes the effective field theory on the tensor branch of the SCFT. For the Higgsable to D-type $(2,0)$ SCFTs/LSTs, there generically also exists a Type IIA brane description, involving a Neveu--Schwarz orientifold plane, which allows for the derivation of a magnetic quiver for the Higgs branch. These are 3d $\mathcal{N}=4$ unitary-orthosymplectic quivers whose Coulomb branch is isomorphic to the Higgs branch of the 6d theories. From this magnetic quiver, together with an extended quiver subtraction algorithm that we explain, the foliation structure of the Higgs branch as a symplectic singularity is unveiled. For this class of 6d SCFTs, we observe a simple rule, which we refer to as "slice subtraction," to read off the transverse slice in the foliation from the tensor branch. Based on this slice subtraction observation, we conjecture the transverse slices in the Higgsable to E-type $(2,0)$ Hasse diagram, where the SCFTs lack any known magnetic quiver for their Higgs branches.

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arXiv2023

The Bestiary of 6d (1,0) SCFTs: Nilpotent Orbits and Anomalies

Many six-dimensional $(1,0)$ SCFTs are known to fall into families labelled by nilpotent orbits of certain simple Lie algebras. For each of the three infinite series of such families, we show that the anomalies for the continuous zero-form global symmetries of a theory labelled by a nilpotent orbit $O$ of $\mathfrak{g}$ can be determined from the anomalies of the theory associated to the trivial nilpotent orbit (the parent theory), together with the data of $O$. In particular, knowledge of the tensor branch field theory is bypassed completely. We show that the known anomalies, previously determined from the geometric/atomic construction, are reproduced by analyzing the Nambu--Goldstone modes inside of the moment map associated to the $\mathfrak{g}$ flavor symmetry of the parent SCFT. This provides further evidence for the physics underlying the labelling of the SCFTs by nilpotent orbits. We remark on some consequences, such as the reinterpretation of the 6d $a$-theorem for such SCFTs in terms of group theory.

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arXiv2023

Holographic duals of Higgsed $\mathcal{D}_p^b(BCD)$

We construct the AdS$_5$ holographic duals to all non-Lagrangian 4d $\mathcal{N}=2$ superconformal field theories of Argyres--Douglas type, namely, $\mathcal{D}_p^{\,b}(G)$, arising from class $\mathcal{S}$ of classical type involving irregular punctures of regular semi-simple type. The 11d supergravity duals contain an internal space of the form of a fibered product of a disc with a squashed and fibered four-sphere and includes orbifold projections which depend on the type of twist lines/outer-automorphism twists in the class $\mathcal{S}$ theory. We verify the holographic duality by determining and matching the anomalies (including the central charges $a$ and $c$ and the flavor central charges) at leading and subleading orders. The Higgs branch of the conformal field theory is described via Higgsing by a nilpotent orbit of a classical Lie algebra; we find the exact closed form formulae for the central charges for every Higgsing. We prove that in the supergravity duals, constraints on the type of partitions associated to allowable Higgsings are enforced by the consistency condition known as the t-rule.

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arXiv2023

The Higgs Branch of Heterotic LSTs: Hasse Diagrams and Generalized Symmetries

We study the Higgs branches of the 6d $(1,0)$ little string theories that live on the worldvolume of NS5-branes probing an ADE-singularity in the heterotic $E_8 \times E_8$ and $\mathrm{Spin}(32)/\mathbb{Z}_2$ string theories. On the $E_8 \times E_8$ side, such LSTs are obtained via fusion of orbi-instanton SCFTs. For the $\mathbb{C}^2/\mathbb{Z}_K$ orbifolds, we determine a magnetic quiver for the Higgs branch from the alternative Type IIA brane system engineering the LST; we show that the magnetic quiver obtained in this way is the same as the Coulomb gauging of the 3d mirrors associated to the orbi-instanton building blocks. Using quiver subtraction, we determine the Hasse diagram of Higgs branch RG flows between the LSTs, and we analyze how the structure constants of the generalized global symmetries vary along the edges of the Hasse diagram. From the Hasse diagram of the Higgs branch, we are immediately able to identify LSTs with the same T-duality-invariant properties, and thus to propose candidate T-dual pairs. We perform a similar analysis of the Higgs branch Hasse diagram and putative T-dual families for particular $E_6$-orbifold LSTs by taking advantage of a duality between a rank zero orbi-instanton theory and a rank one conformal matter theory.

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arXiv2023

Super-Spin Chains for 6D SCFTs

Nearly all 6D superconformal field theories (SCFTs) have a partial tensor branch description in terms of a generalized quiver gauge theory consisting of a long one-dimensional spine of quiver nodes with links given by conformal matter; a strongly coupled generalization of a bifundamental hypermultiplet. For theories obtained from M5-branes probing an ADE singularity, this was recently leveraged to extract a protected large R-charge subsector of operators, with operator mixing controlled at leading order in an inverse large R-charge expansion by an integrable spin $s$ Heisenberg spin chain, where $s$ is determined by the $\mathfrak{su}(2)_{R}$ R-symmetry representation of the conformal matter operator. In this work, we show that this same structure extends to the full superconformal algebra $\mathfrak{osp}(6,2|1)$. In particular, we determine the corresponding Bethe ansatz equations which govern this super-spin chain, as well as distinguished subsectors which close under operator mixing. Similar considerations extend to 6D little string theories (LSTs) and 4D $\mathcal{N} = 2$ SCFTs with the same generalized quiver structures.

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arXiv2023

Intermediate Defect Groups, Polarization Pairs, and Non-invertible Duality Defects

Within the framework of relative and absolute quantum field theories (QFTs), we present a general formalism for understanding polarizations of the intermediate defect group and constructing non-invertible duality defects in theories in $2k$ spacetime dimensions with self-dual gauge fields. We introduce the polarization pair, which fully specifies absolute QFTs as far as their $(k-1)$-form defect groups are concerned, including their $(k-1)$-form symmetries, global structures (including discrete $θ$-angle), and local counterterms. Using the associated symmetry TFT, we show that the polarization pair is capable of succinctly describing topological manipulations, e.g., gauging $(k-1)$-form global symmetries and stacking counterterms, of absolute QFTs. Furthermore, automorphisms of the $(k-1)$-form charge lattice naturally act on polarization pairs via their action on the defect group; they can be viewed as dualities between absolute QFTs descending from the same relative QFT. Using this formalism, we present a prescription for building non-invertible symmetries of absolute QFTs. A large class of known examples, e.g., non-invertible defects in 4D $\mathcal{N}=4$ super-Yang--Mills, can be reformulated via this prescription. As another class of examples, we identify and investigate in detail a family of non-invertible duality defects in 6D superconformal field theories (SCFTs), including from the perspective of the symmetry TFT derived from Type IIB string theory.

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arXiv2023

Emergent N=4 supersymmetry from N=1

We discover a four-dimensional $\mathcal{N}=1$ supersymmetric field theory that is dual to the $\mathcal{N}=4$ super Yang-Mills theory with gauge group $SU(2n+1)$ for each $n$. The dual theory is constructed through the diagonal gauging of the $SU(2n+1)$ flavor symmetry of three copies of a strongly-coupled superconformal field theory (SCFT) of Argyres-Douglas type. We find that this theory flows in the infrared to a strongly-coupled $\mathcal{N}=1$ SCFT that lies on the same conformal manifold as $\mathcal{N}=4$ super Yang-Mills with gauge group $SU(2n+1)$. Our construction provides a hint on why certain $\mathcal{N}=1,2$ SCFTs have identical central charges ($a=c$).

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arXiv2023

Infinitely many 4d N=1 SCFTs with a=c

We study a rich set of four-dimensional $\mathcal{N}=1$ superconformal field theories (SCFTs) with both central charges identical: $a = c$. We construct them via the diagonal $\mathcal{N}=1$ gauging of the flavor symmetry $G$ of a collection of $\mathcal{N}=2$ Argyres--Douglas theories of type $\mathcal{D}_p(G)$, with or without additional adjoint chiral multiplets. In this way, we construct infinitely-many theories that flow to interacting SCFTs with $a = c$ in the infrared. Finally, we briefly highlight the features of the SCFTs without $a = c$ that arise from generalizing this construction.

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arXiv2022

Isomorphisms of 4d N=2 SCFTs from 6d

There exist 4d $\mathcal{N}=2$ SCFTs in class $\mathcal{S}$ which have different constructions as punctured Riemann surfaces, but which nevertheless appear to describe the same physics. Some of these class $\mathcal{S}$ theories have an alternative construction as torus-compactifications of 6d $(1,0)$ SCFTs. We demonstrate that the 6d SCFTs are isomorphic. Each 6d SCFT in question can be obtained from a parent 6d SCFT by Higgs branch renormalization group flow, and the parent theory possesses a discrete symmetry under which the relevant Higgs branch flows are exchanged. The existence of this discrete symmetry, which may be embedded in an enhanced continuous symmetry, proves that the original pair of class $\mathcal{S}$ theories are, in fact, isomorphic.

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arXiv2022

Distinguishing 6d (1,0) SCFTs: an extension to the geometric construction

We provide a new extension to the geometric construction of 6d $(1,0)$ SCFTs that encapsulates Higgs branch structures with identical global symmetry but different spectra. In particular, we find that there exist distinct 6d $(1,0)$ SCFTs that may appear to share their tensor branch description, flavor symmetry algebras, and central charges. For example, such subtleties arise for the very even nilpotent Higgsing of $(\mathfrak{so}_{4k}, \mathfrak{so}_{4k})$ conformal matter; we propose a method to predict at which conformal dimension the Higgs branch operators of the two theories differ via augmenting the tensor branch description with the Higgs branch chiral ring generators of the building block theories. Torus compactifications of these 6d $(1,0)$ SCFTs give rise to 4d $\mathcal{N}=2$ SCFTs of class $\mathcal{S}$ and the Higgs branch of such 4d theories are captured via the Hall--Littlewood index. We confirm that the resulting 4d theories indeed differ in their spectra in the predicted conformal dimension from their Hall--Littlewood indices. We highlight how this ambiguity in the tensor branch description arises beyond the very even nilpotent Higgsing of $(\mathfrak{so}_{4k}, \mathfrak{so}_{4k})$ conformal matter, and hence should be understood for more general classes of 6d $(1,0)$ SCFTs.

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arXiv2022

Two 6d origins of 4d SCFTs: class $\mathcal{S}$ and 6d (1,0) on a torus

We consider all 4d $\mathcal{N}=2$ theories of class $\mathcal{S}$ arising from the compactification of exceptional 6d $(2,0)$ SCFTs on a three-punctured sphere with a simple puncture. We find that each of these 4d theories has another origin as a 6d $(1,0)$ SCFT compactified on a torus, which we check by identifying and comparing the central charges and the flavor symmetry. Each 6d theory is identified with a complex structure deformation of $(\mathfrak{e}_n,\mathfrak{e}_n)$ minimal conformal matter, which corresponds to a Higgs branch renormalization group flow. We find that this structure is precisely replicated by the partial closure of the punctures in the class $\mathcal{S}$ construction. We explain how the plurality of origins makes manifest some aspects of 4d SCFTs, including flavor symmetry enhancements and determining if it is a product SCFT. We further highlight the string theoretic basis for this identification of 4d theories from different origins via mirror symmetry.

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