arXiv subjects

Mario Martone

Publications and source records attributed to Mario Martone.

At least 19 recordsLinked to original sources

arXiv2024

Allowed Coulomb branch scaling dimensions of four-dimensional $\mathcal{N} = 2$ SCFTs

A basic datum of a rank-$r$ $\mathcal{N}{=}2$ superconformal field theory (SCFT) is the $r$-tuple of its Coulomb branch scaling dimensions, i.e., the scaling dimensions of a set of special protected scalar operators whose vevs generate the coordinate ring of the Coulomb branch of the theory. It is well known that when the coordinate ring is freely generated these scaling dimensions can only take values in a small set of rational numbers. But there are further constraints on which $r$-tuples of these numbers can appear. The main aim of this work is to clarify what these are. Along the way we also compute explicitly the $r$-tuples of allowed scaling dimensions for theories of ranks $r = 2, 3, 4$.

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arXiv2024

Free field realizations for rank-one SCFTs

In this paper, we construct the associated vertex operator algebras for all $\mathcal{N}=2$ superconformal field theories of rank one. We give a uniform presentation through free-field realizations, which turns out to be a particularly suitable framework for this task. The elementary building blocks of the construction are dictated by the low energy degrees of freedom on the Higgs branch, which are well understood for rank-one theories. We further analyze the interplay between Higgs and Coulomb data on the moduli space of vacua, which tightly constrain the overall structure of the free field realizations. Our results suggest a plausible bottom-up classification scheme for low-rank SCFTs incorporating vertex algebra techniques.

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arXiv2023

Genus 2 Seiberg-Witten curves for rank 2 N=4 superYang-Mills theories

We determine new genus 2 Seiberg-Witten curves for four dimensional rank 2 absolute N=4 superYang-Mills theories using the automorphism twist approach. The conformal manifolds of these curves agree with those predicted by S-duality orbits of global structures, and we use this to identify which of the two S-duality orbits of the $so(5) \simeq sp(4)$ superYang-Mills theory the genus-2 curve corresponds to. We also compare the curves to earlier constructions of Seiberg-Witten curves for these theories as spectral curves of integrable systems. These spectral curves have genus greater than the rank, and so only give a Coulomb branch geometry upon projection to a sublattice of the homology lattice of the curves. We show how to determine the correct sublattice projection, and find that the integrable system curves do not apply to our theories.

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arXiv2023

Simplifying the Type $A$ Argyres-Douglas Landscape

A well-established organisational principle for Argyres--Douglas-type $\mathcal{N}=2$ superconformal field theories in four dimensions is to characterise such theories by the data defining a(n irregular) Hitchin system on $\mathbb{CP}^1$. The dictionary between Hitchin system data and various features of the corresponding SCFT has been studied extensively, but the overall structure of the resulting space of SCFTs still appears quite complicated. In this work, we systematically delineate a variety of simplifications that arise within this class of constructions due to several large classes of isomorphisms between SCFTs associated with inequivalent Hitchin system data (and their exactly marginal gaugings). We restrict to the most studied class of theories, namely the type $A$ theories without outer automorphism twists.

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arXiv2023

High-temperature expansion of the Schur index and modularity

High-temperature ($q\to1$) asymptotics of 4d superconformal indices of Lagrangian theories have been recently analyzed up to exponentially suppressed corrections. Here we use RG-inspired tools to extend the analysis to the exponentially suppressed terms in the context of Schur indices of $N=2$ SCFTs. In particular, our approach explains the curious patterns of logarithms (polynomials in $1/\log q$) found by Dedushenko and Fluder in their numerical study of the high-temperature expansion of rank-$1$ theories. We also demonstrate compatibility of our results with the conjecture of Beem and Rastelli that Schur indices satisfy finite-order, possibly twisted, modular linear differential equations (MLDEs), and discuss the interplay between our approach and the MLDE approach to the high-temperature expansion. The expansions for $q$ near roots of unity are also treated. A byproduct of our analysis is a proof (for Lagrangian theories) of rationality of the conformal dimensions of all characters of the associated VOA, that mix with the Schur index under modular transformations.

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arXiv2023

Taken by Surprise: Contrast effect for Similarity Scores

Accurately evaluating the similarity of object vector embeddings is of critical importance for natural language processing, information retrieval and classification tasks. Popular similarity scores (e.g cosine similarity) are based on pairs of embedding vectors and disregard the distribution of the ensemble from which objects are drawn. Human perception of object similarity significantly depends on the context in which the objects appear. In this work we propose the $\textit{surprise score}$, an ensemble-normalized similarity metric that encapsulates the contrast effect of human perception and significantly improves the classification performance on zero- and few-shot document classification tasks. This score quantifies the surprise to find a given similarity between two elements relative to the pairwise ensemble similarities. We evaluate this metric on zero/few shot classification and clustering tasks and typically find 10-15 % better performance compared to raw cosine similarity. Our code is available at https://github.com/MeetElise/surprise-similarity.

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arXiv2022

The rank-2 classification problem III: curves with additional automorphisms

This is the third in a series of papers which outlines an approach to the classification of $\mathcal{N}{=}2$ superconformal field theories at rank 2 via the study of their Coulomb branch geometries. Here we use the fact that the encoding of a Coulomb branch geometry as a Seiberg-Witten curve and 1-form enjoys a large reparametrisation invariance. While there is always a unique way to fix this invariance such that the curve and 1-form are single-valued over the Coulomb branch -- the "canonical frame" of the curve used in the first two papers in this series -- there are other useful frames in which the curve is single-valued but the 1-form is allowed to be multi-valued. In these frames, which we call "automorphism frames", the 1-form is periodic up to an automorphism twist. We argue that the multi-valuedness of the automorphism frame can simplify the computational complexity of finding new consistent scale invariant solutions. We demonstrate this in an example by using the automorphism frame to construct for the first time a genus 2 Seiberg-Witten curve for the $\mathcal{N}{=}4$ SU(3) superYang-Mills theory, a solution that is hard to find by other approaches.

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arXiv2022

The rank 2 classification problem II: mapping scale-invariant solutions to SCFTs

This is the second of a series of papers outlining an approach to the classification of $\mathcal{N}{=}2$ superconformal field theories at rank 2 via a systematic analysis of their Coulomb branches, mathematically described by special K\"ahler scale invariant geometries. Here we describe how to make the translation between geometry and field theory data. We apply this strategy to the special K\"ahler geometries found in the first paper of the series where we made strong simplifying assumptions on the form of the solutions. Remarkably, we find that our bottom-up classification strategy pays off even in this simplified setup. All scale invariant solutions in the first paper of the series have an interpretation as the Coulomb branch of at least one $\mathcal{N}{=}2$ superconformal field theory, many matching what is already known. But we also predict the existence of six new rank 2 theories, including an entire new set closed under renormalization group flow. We characterise and discuss in detail each one of these new $\mathcal{N}{=}2$ superconformal field theories. To our knowledge none of them have known string theoretic or higher dimensional realisations.

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arXiv2022

The rank 2 classification problem I: scale invariant geometries

In this first of a series of three papers we outline an approach to classifying 4d $\mathcal{N}{=}2$ superconformal field theories at rank 2. The classification of allowed scale invariant $\mathcal{N}=2$ Coulomb branch geometries of dimension (or rank) greater than one is a famous open problem whose solution will greatly constrain the space of $\mathcal{N}{=}2$ superconformal field theories. At rank 2 the problem is equivalent to finding all possible genus 2 Seiberg-Witten curves and 1-forms satisfying a special K\"ahler condition. This is tractable because regular genus 2 Riemann surfaces can be uniformly described as binary-sextic plane curves, and the Seiberg-Witten curves are families of such curves varying meromorphically over the two-dimensional base. There are also solutions consisting of families of degenerate genus-2 Riemann surfaces given by a bouquet of two elliptic curves which are described by a different set of curves. In this paper we set up and carry out the analysis of the generic case, i.e., those whose typical fiber is a regular genus-2 Riemann surface with no extended automorphism, and find the complete answer for polynomial coefficients.

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arXiv2022

Dirac pairings, one-form symmetries and Seiberg-Witten geometries

The Coulomb phase of a quantum field theory, when present, illuminates the analysis of its line operators and one-form symmetries. For 4d $\mathcal{N}=2$ field theories the low energy physics of this phase is encoded in the special K\"ahler geometry of the moduli space of Coulomb vacua. We clarify how the information on the allowed line operator charges and one-form symmetries is encoded in the special K\"ahler structure. We point out the important difference between the lattice of charged states and the homology lattice of the abelian variety fibered over the moduli space, which, when principally polarized, is naturally identified with a choice of the lattice of mutually local line operators. This observation illuminates how the distinct S-duality orbits of global forms of $\mathcal{N}=4$ theories are encoded geometrically.

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arXiv2022

Exceptional moduli spaces for exceptional $\mathcal{N}=3$ theories

It is expected on general grounds that the moduli space of 4d $\mathcal{N}=3$ theories is of the form $\mathbb{C}^{3r}/\Gamma$, with $r$ the rank and $\Gamma$ a crystallographic complex reflection group (CCRG). As in the case of Lie algebras, the space of CCRGs consists of several infinite families, together with some exceptionals. To date, no 4d $\mathcal{N}=3$ theory with moduli space labelled by an exceptional CCRG (excluding Weyl groups) has been identified. In this work we show that the 4d $\mathcal{N}=3$ theories proposed in \cite{Garcia-Etxebarria:2016erx}, constructed via non-geometric quotients of type-$\mathfrak{e}$ 6d (2,0) theories, realize nearly all such exceptional moduli spaces. In addition, we introduce an extension of this construction to allow for twists and quotients by outer automorphism symmetries. This gives new examples of 4d $\mathcal{N}=3$ theories going beyond simple S-folds.

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arXiv2022

Snowmass White Paper on SCFTs

Superconformal field theories (SCFTs) occupy a central role in the study of many aspects of quantum field theory. In this white paper for the Snowmass process we give a brief overview of aspects of SCFTs in $3\leq D \leq 6$ space-time dimensions, including classification efforts and some of the vast current research trends on the physical and mathematical structures generated by this rich class of physical theories.

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arXiv2022

Needles in a haystack: An algorithmic approach to the classification of 4d $\mathcal{N}=2$ SCFTs

There is a well-known map from 4d $\mathcal{N}=2$ superconformal field theories (SCFTs) to 2d vertex operator algebras (VOAs). The 4d Schur index corresponds to the VOA vacuum character, and must be a solution with integral coefficients of a modular differential equation. This suggests a classification program for 4d $\mathcal{N}=2$ SCFTs that starts with modular differential equations and proceeds by imposing all known constraints that follow from the 4d $\to$ 2d map. This program becomes fully algorithmic once one specifies the $\mathrm{\textit{order}}$ of the modular differential equation and the $\mathrm{\textit{rank}}$ (complex dimension of the Coulomb branch) of the $\mathcal{N}=2$ theory. As a proof of concept, we apply the algorithm to the study of rank-two $\mathcal{N}=2$ SCFTs whose Schur indices satisfy a fourth-order untwisted modular differential equation. Scanning over a large number of putative cases, only 15 satisfy all of the constraints imposed by our algorithm, six of which correspond to known 4d SCFTs. More sophisticated constraints can be used to argue against the existence of the remaining nine cases. Altogether, this indicates that our knowledge of such rank-two SCFTs is surprisingly complete.

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arXiv2021

Magnetic quivers for rank 2 theories

In this note we construct magnetic quivers for the known rank-2 four dimensional $\mathcal{N}=2$ superconformal field theories. For every rank-1 theory one can find a unitary magnetic quiver; we observe that this is no longer possible at rank 2. Our list of magnetic quivers necessarily includes orthosymplectic quivers, in addition to unitary ones, of both the simply and non-simply laced variety. Using quiver subtraction, one can compute Higgs branch Hasse diagrams and compare with the results obtained via other methods finding nearly perfect agreement.

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arXiv2021

The Characteristic Dimension of Four-dimensional $\mathcal{N} = 2$ SCFTs

In this paper we introduce the characteristic dimension of a four dimensional $\mathcal{N}=2$ superconformal field theory, which is an extraordinary simple invariant determined by the scaling dimensions of its Coulomb branch operators. We prove that only nine values of the characteristic dimension are allowed, $-\infty$, 1 ,6/5, 4/3, 3/2, 2, 3, 4, and 6, thus giving a new organizing principle to the vast landscape of 4d $\mathcal N=2$ SCFTs. Whenever the characteristic dimension differs from 1 or 2, only very constrained special K\"ahler geometries (i.e. isotrivial, diagonal and rigid) are compatible with the corresponding set of Coulomb branch dimensions and extremely special, maximally strongly coupled, BPS spectra are allowed for the theories which realize them. Our discussion applies to superconformal field theories of arbitrary rank, i.e. with Coulomb branches of any complex dimension. Along the way, we predict the existence of new $\mathcal{N}=3$ theories of rank two with non-trivial one-form symmetries.

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arXiv2021

On the compactification of 5d theories to 4d

We study general properties of the mapping between 5$d$ and 4$d$ superconformal field theories (SCFTs) under both twisted circle compactification and tuning of local relevant deformation and CB moduli. After elucidating in generality when a 5$d$ SCFT reduces to a 4$d$ one, we identify nearly all $\mathcal{N}=1$ 5$d$ SCFT parents of rank-2 4$d$ $\mathcal{N}=2$ SCFTs. We then use this result to map out the mass deformation trajectories among the rank-2 theories in 4$d$. This can be done by first understanding the mass deformations of the 5$d$ $\mathcal{N}=1$ SCFTs and then map them to 4$d$. The former task can be easily achieved by exploiting the fact that the 5$d$ parent theories can be obtained as the strong coupling limit of Lagrangian theories, and the latter by understanding the behavior under compactification. Finally we identify a set of general criteria that 4$d$ moduli spaces of vacua have to satisfy when the corresponding SCFTs are related by mass deformations and check that all our RG-flows satisfy them. Many of the mass deformations we find are not visible from the corresponding complex integrable systems.

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arXiv2021

A new rank-2 Argyres-Douglas theory

We provide evidence for the existence of a new strongly-coupled four dimensional $\mathcal{N}=2$ superconformal field theory arising as a non-trivial IR fixed point on the Coulomb branch of the mass-deformed superconformal Lagrangian theory with gauge group $G_2$ and four fundamental hypermultiplets. Notably, our analysis proceeds by using various geometric constraints to bootstrap the data of the theory, and makes no explicit reference to the Seiberg-Witten curve. We conjecture a corresponding VOA and check that the vacuum character satisfies a linear modular differential equation of fourth order. We also propose an identification with existing class $\mathcal{S}$ constructions.

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arXiv2021

Testing our understanding of SCFTs: a catalogue of rank-2 $\mathcal{N}$=2 theories in four dimensions

In this paper we begin mapping out the space of rank-2 $\mathcal{N}=2$ superconformal field theories (SCFTs) in four dimensions. This represents an ideal set of theories which can be potentially classified using purely quantum field-theoretic tools, thus providing a precious case study to probe the completeness of the current understanding of SCFTs, primarily derived from string theory constructions. Here, we collect and systematize a large amount of field theoretic data characterizing each theory. We also provide a detailed description of each case and determine the theories' Coulomb, Higgs and Mixed branch stratification. The theories naturally organize themselves into series connected by RG flows but which have gaps suggesting that our current understanding is not complete.

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