Coulomb Branch Research Articles

Factorisation and holomorphic blocks in 4d

We study N=1 theories on Hermitian manifolds of the form M^4=S^1xM^3 with M^3 a U(1) fibration over S^2, and their 3d N=2 reductions. These manifolds admit an Heegaard-like decomposition in solid tori D^2xT^2 and D^2xS^1. We prove that when the 4d and 3d anomalies are cancelled the matrix integrands in the Coulomb branch partition functions can be factorised in terms of 1-loop factors on D^2xT^2 and D^2xS^1 respectively. By evaluating the Coulomb branch matrix integrals we show that the 4d and 3d partition functions can be expressed as sums of products of 4d and 3d holomorphic blocks.

Instanton corrections to the effective action of N = 4 $$ \mathcal{N}=4 $$ SYM

We compute the one-instanton effective action of $$ \mathcal{N}=4 $$ super Yang-Mills theory with gauge group Sp(2N). The result can be written in a very compact and manifestly supersymmetric form involving an integral over the superspace of an irrational function of the $$ \mathcal{N}=4 $$ on-shell superfields. In the Coulomb branch, the instanton corrects both the MHV and next-to-next-MHV higher derivative terms D 4 F 2n+2 and F 2n+4. We confirm at the non-perturbative level the non-renormalization theorems for MHV F 2n+2 terms that are expected to receive perturbative corrections only at n-loops. We compute also the one and two-loop corrections to the D 4 F 4 term and show that its completion under $$ \mathrm{S}\mathrm{L}\left(2,\mathrm{\mathbb{Z}}\right) $$ duality is consistent with the one-instanton results of U(2) gauge group.

On the ground state wave function of matrix theory

We propose an explicit construction of the leading terms in the asymptotic expansion of the ground state wave function of BFSS SU(N ) matrix quantum mechanics. Our proposal is consistent with the expected factorization property in various limits of the Coulomb branch, and involves a different scaling behavior from previous suggestions. We comment on some possible physical implications.

Hilbert series for theories with Aharony duals

The algebraic structure of moduli spaces of 3d N=2 supersymmetric gauge theories is studied by computing the Hilbert series which is a generating function that counts gauge invariant operators in the chiral ring. These U(N_c) theories with N_f flavors have Aharony duals and their moduli spaces receive contributions from both mesonic and monopole operators. In order to compute the Hilbert series, recently developed techniques for Coulomb branch Hilbert series in 3d N=4 are extended to 3d N=2. The Hilbert series computation leads to a general expression of the algebraic variety which represents the moduli space of the U(N_c) theory with N_f flavors and its Aharony dual theory. A detailed analysis of the moduli space is given, including an analysis of the various components of the moduli space.

The Hilbert series of 3d Yang–Mills theories with vectorlike matter

This paper presents a formula for the Hilbert series that counts gauge invariant chiral operators in 3d Yang–Mills theories with vectorlike matter and no Chern–Simons interactions. The formula counts ’t Hooft monopole operators dressed by gauge invariants of a residual gauge theory of massless fields in the monopole background, which is determined by the Higgs mechanism. The sum over magnetic charges is restricted due to instanton effects that partially lift the classical Coulomb branch. The formalism is applied to unitary and symplectic gauge theories with fundamental matter, reproducing old results for the moduli space of vacua and the chiral ring, without resorting to any further effective superpotential on the moduli space.

Exact coefficients for higher dimensional operators with sixteen supersymmetries

We consider constraints on higher dimensional operators for supersymmetric effective field theories. In four dimensions with maximal supersymmetry and SU(4) R-symmetry, we demonstrate that the coefficients of abelian operators F^n with MHV helicity configurations must satisfy a recursion relation, and are completely determined by that of F^4. As the F^4 coefficient is known to be one-loop exact, this allows us to derive exact coefficients for all such operators. We also argue that the results are consistent with the SL(2,Z) duality symmetry. Breaking SU(4) to Sp(4), in anticipation for the Coulomb branch effective action, we again find an infinite class of operators whose coefficient that are determined exactly. We also consider three-dimensional N=8 as well as six-dimensional N=(2,0),(1,0) and (1,1) theories. In all cases, we demonstrate that the coefficient of dimension-six operator must be proportional to the square of that of dimension-four.

Brane polarization is no cure for tachyons

Anti-M2 and anti-D3 branes placed in regions with charges dissolved in fluxes have a tachyon in their near-horizon region, which causes these branes to repel each other. If the branes are on the Coulomb branch this tachyon gives rise to a runaway behavior, but when the branes are polarized into five-branes this tachyon only appears to lower the energy of the polarized branes, without affecting its stability. We analyze brane polarization in the presence of a brane-brane-repelling tachyon and show that when the branes are polarized along the direction of the tachyon the polarized shell is unstable. This implies that tachyons cannot be cured by brane polarization and indicates that, at least in a certain regime of parameters, anti-D3 branes polarized into NS5 branes at the bottom of the Klebanov-Strassler solution have an instability.

N = 2 $$ \mathcal{N}=2 $$ Argyres-Douglas theories, N = 1 $$ \mathcal{N}=1 $$ SQCD and Seiberg duality

We revisit the study of singular points in the Coulomb branch of N = 2 SQCD in four dimensions with gauge group SU(N) and odd number of flavors. For certain choices of the mass parameters these vacua are not lifted by a mass term for the chiral multiplet in the adjoint representation. By using recent results about the M5 brane description of N = 1 theories we study the resulting vacua and argue that the low-energy effective theory has a simple Lagrangian description involving a free chiral multiplet in the adjoint representation of the flavor symmetry group, a system somewhat reminiscent o f the standard low-energy pion description of the real-world QCD. This fact is quite remarkable in view of the fact that the underlying N = 2 SCFT (the Argyres-Douglas systems) are strongly-coupled non-local theories of quarks and monopoles.

Dessins d’enfants in N = 2 $$ \mathcal{N}=2 $$ generalised quiver theories

We study Grothendieck's dessins d'enfants in the context of the $\mathcal{N}=2$ supersymmetric gauge theories in $\left(3+1\right)$ dimensions with product $SU\left(2\right)$ gauge groups which have recently been considered by Gaiotto et al. We identify the precise context in which dessins arise in these theories: they are the so-called ribbon graphs of such theories at certain isolated points in the Coulomb branch of the moduli space. With this point in mind, we highlight connections to other work on trivalent dessins, gauge theories, and the modular group.

Topological charges in 2DN=(2,2)theories and massive BPS states

We study how charges of global symmetries that are manifest in the ultra-violet definition of a theory are realized as topological charges in its infra-red effective theory for two-dimensional theories with $\mathcal{N}=(2,2)$ supersymmetry. We focus on the charges that the states living on $S^1$ carry. The central charge---or BPS masses---of the supersymmetry algebra play a crucial role in making this correspondence precise. We study two examples: $U(1)$ gauge theories with chiral matter, and world-volume theories of "dynamical surface operators" of 4d $\mathcal{N}=2$ gauge theories. In the former example, we show that the flavor charges of the theory are realized as topological winding numbers in the effective theory on the Coulomb branch. In the latter, we show that there is a one-to-one correspondence between topological charges of the effective theory of the dynamical surface operator and the electric, magnetic, and flavor charges of the 4d gauge theory. We also examine the topologically charged massive BPS states on $S^1$ and discover that the massive BPS spectrum is sensitive to the radius of the circle in the simplest theory---the free theory of a periodic twisted chiral field. We clarify this behavior by showing that the massive BPS spectrum on $S^1$, unlike the BPS ground states, cannot be identified as elements of a cohomology.

Moduli dynamics as a predictive tool for thermal maximally supersymmetric Yang-Mills at large N

Maximally supersymmetric (p+1)-dimensional Yang-Mills theory at large N and finite temperature, with possibly compact spatial directions, has a rich phase structure. Strongly coupled phases may have holographic descriptions as black branes in various string duality frames, or there may be no gravity dual. In this paper we provide tools in the gauge theory which give a simple and unified picture of the various strongly coupled phases, and transitions between them. Building on our previous work we consider the effective theory describing the moduli of the gauge theory, which can be computed precisely when it is weakly coupled far out on the Coulomb branch. Whilst for perturbation theory naive extrapolation from weak coupling to strong gives little information, for this moduli theory naive extrapolation from its weakly to its strongly coupled regime appears to encode a surprising amount of information about the various strongly coupled phases. We argue it encodes not only the parametric form of thermodynamic quantities for these strongly coupled phases, but also certain transcendental factors with a geometric origin, and allows one to deduce transitions between the phases. We emphasise it also gives predictions for the behaviour of a large class of local operators in these phases.

The equivariant A-twist and gauged linear sigma models on the two-sphere

We study two-dimensional $\mathcal{N}=(2,2)$ supersymmetric gauged linear sigma models (GLSM) on the $\Omega$-deformed sphere, $S^2_\Omega$, which is a one-parameter deformation of the $A$-twisted sphere. We provide an exact formula for the $S^2_\Omega$ supersymmetric correlation functions using supersymmetric localization. The contribution of each instanton sector is given in terms of a Jeffrey-Kirwan residue on the Coulomb branch. In the limit of vanishing $\Omega$-deformation, the localization formula greatly simplifies the computation of $A$-twisted correlation functions, and leads to new results for non-abelian theories. We discuss a number of examples and comment on the $\epsilon_\Omega$-deformation of the quantum cohomology relations. Finally, we present a complementary Higgs branch localization scheme in the special case of abelian gauge groups.

On the central charge extension of the N = 4 $$ \mathcal{N}=4 $$ SYM spin chain

In this paper it is argued that the central charge extension of the Coulomb branch of $$ \mathcal{N}=4 $$ SYM theory appears as a limit of Beisert’s central charge extension of the planar $$ \mathcal{N}=4 $$ spin chain in the presence of boundaries. These boundaries are interpreted as D-branes that source the central charge and are realized as giant gravitons and dual giant gravitons in the AdS dual. The BPS states that correspond to short representations of the centrally extended algebra on the spin chain can stop from existing when they cross walls of stability that depend on the position of the branes. These walls can be understood easily at weak coupling in the SU(2) sector.

Holographic entanglement entropy and the internal space

We elaborate on the role of extremal surfaces probing the internal space in AdS/CFT. Extremal surfaces in AdS quantify the "geometric" entanglement between different regions in physical space for the dual CFT. This, however, is just one of many ways to split a given system into subsectors, and extremal surfaces in the internal space should similarly quantify entanglement between subsectors of the theory. For the case of AdS$_5\times$S$^5$, their area was interpreted as entanglement entropy between U(n) and U(m) subsectors of U(n+m) N=4 SYM. Making this proposal precise is subtle for a number of reasons, the most obvious being that from the bulk one usually has access to gauge-invariant quantities only, while a split into subgroups is inherently gauge variant. We study N=4 SYM on the Coulomb branch, where some of the issues can be mitigated and the proposal can be sharpened. Continuing back to the original AdS$_5\times$S$^5$ geometry, we obtain a modified proposal, based on the relation of the internal space to the R-symmetry group.

The landscape of M-theory compactifications on seven-manifolds with G2 holonomy

We study the physics of globally consistent four-dimensional $\mathcal{N}=1$ supersymmetric M-theory compactifications on $G_2$ manifolds constructed via twisted connected sum; there are now perhaps fifty million examples of these manifolds. We study a rich example that exhibits $U(1)^3$ gauge symmetry and a spectrum of massive charged particles that includes a trifundamental. Applying recent mathematical results to this example, we compute membrane instanton corrections to the superpotential and spacetime topology change in a compact model; the latter include both the (non-isolated) $G_2$ flop and conifold transitions. The conifold transition spontaneously breaks the gauge symmetry to $U(1)^2$, and associated field theoretic computations of particle charges make correct predictions for the topology of the deformed $G_2$ manifold. We discuss physical aspects of the abelian $G_2$ landscape broadly, including aspects of Higgs and Coulomb branches, membrane instanton corrections, and some general aspects of topology change.

R^{3} index for four-dimensional (N)=2 field theories.

In theories with N=2 supersymmetry on R^{3,1}, supersymmetric bound states can decay across walls of marginal stability in the space of Coulomb branch parameters, leading to discontinuities in the BPS indices Ω(γ,u). We consider a supersymmetric index I which receives contributions from 1/2-BPS states, generalizing the familiar Witten index Tr(-1)^{F}e^{-βH}. We expect I to be smooth away from loci where massless particles appear, thanks to contributions from the continuum of multiparticle states. Taking inspiration from a similar phenomenon in the hypermultiplet moduli space of N=2 string vacua, we conjecture a formula expressing I in terms of the BPS indices Ω(γ,u), which is continuous across the walls and exhibits the expected contributions from single particle states at large β. This gives a universal prediction for the contributions of multiparticle states to the index I. This index is naturally a function on the moduli space after reduction on a circle, closely related to the canonical hyperkähler metric and hyperholomorphic connection on this space.

Complete graphs, Hilbert series, and the Higgs branch of the 4d [formula omitted][formula omitted] SCFTs

The strongly interacting 4d N=2 SCFTs of type (An,Am) are the simplest examples of models in the (G,G′) class introduced by Cecotti, Neitzke, and Vafa in arXiv:1006.3435. These systems have a known 3d N=4 mirror only when n+1 divides m+1. By 4d/2d correspondence, we show that in this case these systems have a nontrivial global flavor symmetry group, and, therefore, a non-trivial Higgs branch. As an application of the methods of arXiv:1309.2657, we then compute the refined Hilbert series of the Coulomb branch of the 3d mirror for the simplest models in the series. This equals the refined Hilbert series of the Higgs branch of the (An,Am) SCFT, providing interesting information about the Higgs branch of these non-lagrangian theories.

The Cheshire cap

A key role in black hole dynamics is played by the inner horizon; most of the entropy of a slightly nonextremal charged or rotating black hole is carried there, and the covariant entropy bound suggests that the rest lies in the region between the inner and outer horizon. An attempt to match this onto results of the microstate geometries program suggests that a `Higgs branch' of underlying long string states of the configuration space realizes the degrees of freedom on the inner horizon, while the `Coulomb branch' describes the inter-horizon region and beyond. Support for this proposal comes from an analysis of the way singularities develop in microstate geometries, and their close analogy to corresponding structures in fivebrane dynamics. These singularities signal the opening up of the long string degrees of freedom of the theory, which are partly visible from the geometry side. A conjectural picture of the black hole interior is proposed, wherein the long string degrees of freedom resolve the geometrical singularity on the inner horizon, yet are sufficiently nonlocal to communicate information to the outer horizon and beyond.

Conformal quivers and melting molecules

Quiver quantum mechanics describes the low energy dynamics of a system of wrapped D-branes. It captures several aspects of single and multicentered BPS black hole geometries in four-dimensional $$ \mathcal{N} $$ = 2 supergravity such as the presence of bound states and an exponential growth of microstates. The Coulomb branch of an Abelian three node quiver is obtained by integrating out the massive strings connecting the D-particles. It allows for a scaling regime corresponding to a deep AdS2 throat on the gravity side. In this scaling regime, the Coulomb branch is shown to be an SL(2, ℝ) invariant multi-particle superconformal quantum mechanics. Finally, we integrate out the strings at finite temperature — rather than in their ground state — and show how the Coulomb branch ‘melts’ into the Higgs branch at high enough temperatures. For scaling solutions the melting occurs for arbitrarily small temperatures, whereas bound states can be metastable and thus long lived. Throughout the paper, we discuss how far the analogy between the quiver model and the gravity picture, particularly within the AdS2 throat, can be taken.

Argyres-Douglas theories and S-duality

We generalize S-duality to N=2 superconformal field theories (SCFTs) with Coulomb branch operators of non-integer scaling dimension. As simple examples, we find minimal generalizations of the S-dualities discovered in SU(2) gauge theory with four fundamental flavors by Seiberg and Witten and in SU(3) gauge theory with six fundamental flavors by Argyres and Seiberg. Our constructions start by weakly gauging diagonal SU(2) and SU(3) flavor symmetry subgroups of two copies of a particular rank-one Argyres-Douglas theory (along with sufficient numbers of hypermultiplets to guarantee conformality of the gauging). As we explore the resulting conformal manifold of the SU(2) SCFT, we find an action of S-duality on the parameters of the theory that is reminiscent of Spin(8) triality. On the other hand, as we explore the conformal manifold of the SU(3) theory, we find that an exotic rank-two SCFT emerges in a dual SU(2) description.