arXiv subjects

Jonathan J. Heckman

Publications and source records attributed to Jonathan J. Heckman.

At least 73 records · Page 4Linked to original sources

arXiv2020

Geometric Approach to 3D Interfaces at Strong Coupling

We study 4D systems in which parameters of the theory have position dependence in one spatial direction. In the limit where these parameters jump, this can lead to 3D interfaces supporting localized degrees of freedom. A priori, this sort of position dependence can occur at either weak or strong coupling. Demanding time-reversal invariance for $U(1)$ gauge theories with a duality group $Γ\subset SL(2,\mathbb{Z})$ leads to interfaces at strong coupling which are characterized by the real component of a modular curve specified by $Γ$. This provides a geometric method for extracting the electric and magnetic charges of possible localized states. We illustrate these general considerations by analyzing some 4D $\mathcal{N} = 2$ theories with 3D interfaces. These 4D systems can also be interpreted as descending from a six-dimensional theory compactified on a three-manifold generated by a family of Riemann surfaces fibered over the real line. We show more generally that 6D superconformal field theories compactified on such spaces also produce trapped matter by using the known structure of anomalies in the resulting 4D bulk theories.

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arXiv2020

Geometric Unification of Higgs Bundle Vacua

Higgs bundles are a central tool used to study a range of intersecting brane systems in string compactifications. Solutions to the internal gauge theory equations of motion for the corresponding worldvolume theories of branes give rise to different low energy effective field theories. This has been heavily used in the study of M-theory on local $G_2$ spaces and F-theory on local elliptically fibered Calabi-Yau fourfolds. In this paper we show that the 3D $\mathcal{N} = 1$ effective field theory defined by M-theory on a local $Spin(7)$ space unifies the Higgs bundle data associated with 4D $\mathcal{N} = 1$ M- and F-theory vacua. This 3D system appears as an interface with finite thickness between different 4D vacua. We develop the general formalism of M-theory on such local $Spin(7)$ spaces, and build explicit interpolating solutions. This provides a complementary local gauge theory analysis of a recently proposed approach to constructing $Spin(7)$ spaces from generalized connected sums.

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arXiv2020

S-folds, String Junctions, and 4D $\mathcal{N} = 2$ SCFTs

S-folds are a non-perturbative generalization of orientifold 3-planes which figure prominently in the construction of 4D $\mathcal{N} = 3$ SCFTs and have also recently been used to realize examples of 4D $\mathcal{N} = 2$ SCFTs. In this paper we develop a general procedure for reading off the flavor symmetry experienced by D3-branes probing 7-branes in the presence of an S-fold. We develop an S-fold generalization of orientifold projection which applies to non-perturbative string junctions. This procedure leads to a different 4D flavor symmetry algebra depending on whether the S-fold supports discrete torsion. We also show that this same procedure allows us to read off admissible representations of the flavor symmetry in the associated 4D $\mathcal{N} = 2$ SCFTs. Furthermore this provides a prescription for how to define F-theory in the presence of S-folds with discrete torsion.

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arXiv2020

More on the Matter of 6D SCFTs

M5-branes probing an ADE singularity lead to 6D SCFTs with (1,0) supersymmetry. On the tensor branch, the M5-branes specify domain walls of a 7D Super Yang-Mills theory with gauge group G of ADE-type, thus providing conformal matter for a broad class of generalized quiver theories. Additionally, these theories have G x G flavor symmetry, and a corresponding Higgs branch. In this note we use the F-theory realization of these theories to calculate the scaling dimension of the operator parameterizing seven-brane recombination, i.e. motion of the stack of M5-branes off of the orbifold singularity. In all cases with an interacting fixed point, we find that this operator has scaling dimension at least six, and defines a marginal irrelevant deformation.

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arXiv2020

T-Branes, String Junctions, and 6D SCFTs

Recent work on 6D superconformal field theories (SCFTs) has established an intricate correspondence between certain Higgs branch deformations and nilpotent orbits of flavor symmetry algebras associated with T-branes. In this paper, we return to the stringy origin of these theories and show that many aspects of these deformations can be understood in terms of simple combinatorial data associated with multi-pronged strings stretched between stacks of intersecting 7-branes in F-theory. This data lets us determine the full structure of the nilpotent cone for each semi-simple flavor symmetry algebra, and it further allows us to characterize symmetry breaking patterns in quiver-like theories with classical gauge groups. An especially helpful feature of this analysis is that it extends to "short quivers" in which the breaking patterns from different flavor symmetry factors are correlated.

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arXiv2020

T-Branes and $G_2$ Backgrounds

Compactification of M- / string theory on manifolds with $G_2$ structure yields a wide variety of 4D and 3D physical theories. We analyze the local geometry of such compactifications as captured by a gauge theory obtained from a three-manifold of ADE singularities. Generic gauge theory solutions include a non-trivial gauge field flux as well as normal deformations to the three-manifold captured by non-commuting matrix coordinates, a signal of T-brane phenomena. Solutions of the 3D gauge theory on a three-manifold are given by a deformation of the Hitchin system on a marked Riemann surface which is fibered over an interval. We present explicit examples of such backgrounds as well as the profile of the corresponding zero modes for localized chiral matter. We also provide a purely algebraic prescription for characterizing localized matter for such T-brane configurations. The geometric interpretation of this gauge theory description provides a generalization of twisted connected sums with codimension seven singularities at localized regions of the geometry. It also indicates that geometric codimension six singularities can sometimes support 4D chiral matter due to physical structure "hidden" in T-branes.

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arXiv2020

General Prescription for Global U(1)'s in 6D SCFTs

We present a general prescription for determining the global U(1) symmetries of six-dimensional superconformal field theories (6D SCFTs). We use the quiver-like gauge theory description of the tensor branch to identify candidate U(1) symmetries which can act on generalized matter. The condition that these candidate U(1)'s are free of Adler-Bell-Jackiw (ABJ) anomalies provides bottom-up constraints for U(1)'s. This agrees with the answer obtained from symmetry breaking patterns induced by Higgs branch flows. We provide numerous examples illustrating the details of this proposal. In the F-theory realization of these theories, some of these symmetries originate from deformations of non-abelian flavor symmetries localized on a component of the discriminant, while others come from an additional generator of the Mordell-Weil group. We also provide evidence that some of these global U(1)'s do not arise from gauge symmetries, as would happen in taking a decoupling limit of a model coupled to six-dimensional supergravity.

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arXiv2019

Fine Tuning, Sequestering, and the Swampland

We conjecture and present evidence that any effective field theory coupled to gravity in flat space admits at most a finite number of fine tunings, depending on the amount of supersymmetry and spacetime dimension. In particular, this means that there are infinitely many non-trivial correlations between the allowed deformations of a given effective field theory in the gravitational context. Fine tuning of parameters allows us to obtain some consistent CFTs in the IR limit of gravitational theories. Related to finiteness of fine tunings, we conjecture that except for a finite number of CFTs, the rest cannot be consistently coupled to gravity and belong to the swampland. Moreover, we argue that even though matter sectors coupled to gravity may sometimes be partially sequestered, there is an irreducible level of mixing between them, correlating and coupling infinitely many operators between these sectors.

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arXiv2019

F-theory and the Classification of Little Strings

Little string theories (LSTs) are UV complete non-local 6D theories decoupled from gravity in which there is an intrinsic string scale. In this paper we present a systematic approach to the construction of supersymmetric LSTs via the geometric phases of F-theory. Our central result is that all LSTs with more than one tensor multiplet are obtained by a mild extension of 6D superconformal field theories (SCFTs) in which the theory is supplemented by an additional, non-dynamical tensor multiplet, analogous to adding an affine node to an ADE quiver, resulting in a negative semidefinite Dirac pairing. We also show that all 6D SCFTs naturally embed in an LST. Motivated by physical considerations, we show that in geometries where we can verify the presence of two elliptic fibrations, exchanging the roles of these fibrations amounts to T-duality in the 6D theory compactified on a circle.

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arXiv2019

Nilpotent Networks and 4D RG Flows

Starting from a general $\mathcal{N} = 2$ SCFT, we study the network of $\mathcal{N} = 1$ SCFTs obtained from relevant deformations by nilpotent mass parameters. We also study the case of flipper field deformations where the mass parameters are promoted to a chiral superfield, with nilpotent vev. Nilpotent elements of semi-simple algebras admit a partial ordering connected by a corresponding directed graph. We find strong evidence that the resulting fixed points are connected by a similar network of 4D RG flows. To illustrate these general concepts, we also present a full list of nilpotent deformations in the case of explicit $\mathcal{N} = 2$ SCFTs, including the case of a single D3-brane probing a $D$- or $E$-type F-theory 7-brane, and 6D $(G,G)$ conformal matter compactified on a $T^2$, as described by a single M5-brane probing a $D$- or $E$-type singularity. We also observe a number of numerical coincidences of independent interest, including a collection of theories with rational values for their conformal anomalies, as well as a surprisingly nearly constant value for the ratio $a_{\mathrm{IR}} / c_{\mathrm{IR}}$ for the entire network of flows associated with a given UV $\mathcal{N} = 2$ SCFT. The $\texttt{arXiv}$ submission also includes the full dataset of theories which can be accessed with a companion $\texttt{Mathematica}$ script.

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arXiv2019

Pixelated Dark Energy

We study the phenomenology of a recent string construction with a quantum mechanically stable dark energy. A mild supersymmetry protects the vacuum energy but also allows $O(10 - 100)$ TeV scale superpartner masses. The construction is holographic in the sense that the 4D spacetime is generated from "pixels" originating from five-branes wrapped over metastable five-cycles of the compactification. The cosmological constant scales as $Λ\sim 1/N$ in the pixel number. An instability in the construction leads to cosmic expansion. This also causes more five-branes to wind up in the geometry, leading to a slowly decreasing cosmological constant which we interpret as an epoch of inflation followed by (pre-)heating when a rare event occurs in which the number of pixels increases by an order one fraction. The sudden appearance of radiation triggers an exponential increase in the number of pixels. Dark energy has a time varying equation of state with $w_a=-3Ω_{m,0}(1+w_0)/2$, which is compatible with current bounds, and could be constrained further by future data releases. The pixelated nature of the Universe also implies a large-$l$ cutoff on the angular power spectrum of cosmological observables with $l_{\rm max} \sim O(N)$. We also use this pixel description to study the thermodynamics of de Sitter space, finding rough agreement with effective field theory considerations.

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arXiv2019

F-theory and Dark Energy

Motivated by its potential use as a starting point for solving various cosmological constant problems, we study F-theory compactified on the warped product $\mathbb{R}_{\text{time}} \times S^3 \times Y_{8}$ where $Y_{8}$ is a $Spin(7)$ manifold, and the $S^3$ factor is the target space of an $SU(2)$ Wess--Zumino--Witten (WZW) model at level $N$. Reduction to M-theory exploits the abelian duality of this WZW model to an $S^3 / \mathbb{Z}_N$ orbifold. In the large $N$ limit, the untwisted sector is captured by 11D supergravity. The local dynamics of intersecting 7-branes in the $Spin(7)$ geometry is controlled by a Donaldson--Witten twisted gauge theory coupled to defects. At late times, the system is governed by a 1D quantum mechanics system with a ground state annihilated by two real supercharges, which in four dimensions would appear as "$\mathcal{N} = 1/2$ supersymmetry" on a curved background. This leads to a cancellation of zero point energies in the 4D field theory but a split mass spectrum for superpartners of order $Δm_\text{4D} \sim \sqrt{M_\text{IR} M_\text{UV}}$ specified by the IR and UV cutoffs of the model. This is suggestively close to the TeV scale in some scenarios. The classical 4D geometry has an intrinsic instability which can produce either a collapsing or expanding Universe, the latter providing a promising starting point for a number of cosmological scenarios. The resulting 1D quantum mechanics in the time direction also provides an appealing starting point for a more detailed study of quantum cosmology.

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arXiv2018

Towards Exotic Matter and Discrete Non-Abelian Symmetries in F-theory

We present a prescription in F-theory for realizing matter in "exotic" representations of product gauge groups. For 6D vacua, bifundamental hypermultiplets are engineered by starting at a singular point in moduli space which includes 6D superconformal field theories coupled to gravity. A deformation in Higgs branch moduli space takes us to a weakly coupled gauge theory description. In the corresponding elliptically fibered Calabi--Yau threefold, the minimal Weierstrass model parameters $(f,g,Δ)$ vanish at collisions of the discriminant at least to order $(4,6,12)$, but with sufficiently high order of tangency to ensure the existence of T-brane deformations to a weakly coupled gauge theory with exotic bifundamentals. We present explicit examples including bifundamental hypermultiplets of $\mathfrak{e}_7 \times \mathfrak{su}_2$ and $\mathfrak{e}_6 \times \mathfrak{su}_3$, each of which have dual heterotic orbifold descriptions. Geometrically, these matter fields are delocalized across multiple points of an F-theory geometry. Symmetry breaking with such representations can be used to produce high dimension representations of simple gauge groups such as the four-index symmetric representation of $\mathfrak{su}_2$ and the three-index symmetric representation of $\mathfrak{su}_3$, and after further higgsing can yield discrete non-abelian symmetries.

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arXiv2018

Fission, Fusion, and 6D RG Flows

We show that all known 6D SCFTs can be obtained iteratively from an underlying set of UV progenitor theories through the processes of "fission" and "fusion." Fission consists of a tensor branch deformation followed by a special class of Higgs branch deformations characterized by discrete and continuous homomorphisms into flavor symmetry algebras. Almost all 6D SCFTs can be realized as fission products. The remainder can be constructed via one step of fusion involving these fission products, whereby a single common flavor symmetry of decoupled 6D SCFTs is gauged and paired with a new tensor multiplet at the origin of moduli space, producing an RG flow "in reverse" to the UV. This leads to a streamlined labeling scheme for all known 6D SCFTs in terms of a few pieces of group theoretic data. The partial ordering of continuous homomorphisms $\mathfrak{su}(2) \rightarrow \mathfrak{g}_{\text{flav}}$ for $\mathfrak{g}_{\text{flav}}$ a flavor symmetry also points the way to a classification of 6D RG flows.

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arXiv2018

4D Gauge Theories with Conformal Matter

One of the hallmarks of 6D superconformal field theories (SCFTs) is that on a partial tensor branch, all known theories resemble quiver gauge theories with links comprised of 6D conformal matter, a generalization of weakly coupled hypermultiplets. In this paper we construct 4D quiverlike gauge theories in which the links are obtained from compactifications of 6D conformal matter on Riemann surfaces with flavor symmetry fluxes. This includes generalizations of super QCD with exceptional gauge groups and quarks replaced by 4D conformal matter. Just as in super QCD, we find evidence for a conformal window as well as confining gauge group factors depending on the total amount of matter. We also present F-theory realizations of these field theories via elliptically fibered Calabi-Yau fourfolds. Gauge groups (and flavor symmetries) come from 7-branes wrapped on surfaces, conformal matter localizes at the intersection of pairs of 7-branes, and Yukawas between 4D conformal matter localize at points coming from triple intersections of 7-branes. Quantum corrections can also modify the classical moduli space of the F-theory model, matching expectations from effective field theory.

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arXiv2018

Punctures and Dynamical Systems

With the aim of better understanding the class of 4D theories generated by compactifications of 6D superconformal field theories (SCFTs), we study the structure of N = 1 supersymmetric punctures for class S_Gamma theories, namely the 6D SCFTs obtained from M5-branes probing an ADE singularity. For M5-branes probing a C^2 / Z_k singularity, the punctures are governed by a dynamical system in which evolution in time corresponds to motion to a neighboring node in an affine A-type quiver. Classification of punctures reduces to determining consistent initial conditions which produce periodic orbits. The study of this system is particularly tractable in the case of a single M5-brane. Even in this "simple" case, the solutions exhibit a remarkable level of complexity: Only specific rational values for the initial momenta lead to periodic orbits, and small perturbations in these values lead to vastly different late time behavior. Another difference from half BPS punctures of class S theories includes the appearance of a continuous complex "zero mode" modulus in some puncture solutions. The construction of punctures with higher order poles involves a related set of recursion relations. The resulting structures also generalize to systems with multiple M5-branes as well as probes of D- and E-type orbifold singularities.

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arXiv2018

6D Fractional Quantum Hall Effect

We present a 6D generalization of the fractional quantum Hall effect involving membranes coupled to a three-form potential in the presence of a large background four-form flux. The low energy physics is governed by a bulk 7D topological field theory of abelian three-form potentials with a single derivative Chern-Simons-like action coupled to a 6D anti-chiral theory of Euclidean effective strings. We derive the fractional conductivity, and explain how continued fractions which figure prominently in the classification of 6D superconformal field theories correspond to a hierarchy of excited states. Using methods from conformal field theory we also compute the analog of the Laughlin wavefunction. Compactification of the 7D theory provides a uniform perspective on various lower-dimensional gapped systems coupled to boundary degrees of freedom. We also show that a supersymmetric version of the 7D theory embeds in M-theory, and can be decoupled from gravity. Encouraged by this, we present a conjecture in which IIB string theory is an edge mode of a 10+2-dimensional bulk topological theory, thus placing all twelve dimensions of F-theory on a physical footing.

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arXiv2017

T-Branes at the Limits of Geometry

Singular limits of 6D F-theory compactifications are often captured by T-branes, namely a non-abelian configuration of intersecting 7-branes with a nilpotent matrix of normal deformations. The long distance approximation of such 7-branes is a Hitchin-like system in which simple and irregular poles emerge at marked points of the geometry. When multiple matter fields localize at the same point in the geometry, the associated Higgs field can exhibit irregular behavior, namely poles of order greater than one. This provides a geometric mechanism to engineer wild Higgs bundles. Physical constraints such as anomaly cancellation and consistent coupling to gravity also limit the order of such poles. Using this geometric formulation, we unify seemingly different wild Hitchin systems in a single framework in which orders of poles become adjustable parameters dictated by tuning gauge singlet moduli of the F-theory model.

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