Moduli Space Of Vacua Research Articles

Holography, brane intersections and six-dimensional SCFTs

We study supersymmetric intersections of NS5-, D6- and D8-branes in type IIA string theory. We focus on the supergravity description of this system and identify a “near horizon” limit in which we recover the recently classified supersymmetric seven-dimensional AdS solutions of massive type IIA supergravity. Using a consistent truncation to seven-dimensional gauged supergravity we construct a universal supersymmetric deformation of these AdS vacua. In the holographic dual six-dimensional (1,0) superconformal field theory this deformation describes a universal RG flow on the tensor branch of the vacuum moduli space triggered by a vacuum expectation value for a protected scalar operator of dimension four.

On 4d rank-one N = 3 $$ \mathcal{N}=3 $$ superconformal field theories

We study the properties of 4d N=3 superconformal field theories whose rank is one, i.e. those that reduce to a single vector multiplet on their moduli space of vacua. We find that the moduli space can only be of the form C^3/Z_k for k=1,2,3,4,6, and that the supersymmetry automatically enhances to N=4 for k=1,2. In addition, we determine the central charges a and c in terms of k, and construct the associated 2d chiral algebras, which turn out to be exotic N=2 supersymmetric W-algebras.

Search for a Minimal N=1 Superconformal Field Theory in 4D.

We discuss a candidate for a minimal interacting four-dimensional N=1 superconformal field theory. The model contains a chiral primary operator u satisfying the chiral ring relation u^{2}=0, and its scaling dimension is Δ(u)=1.5. The model is derived by turning on a N=1 preserving deformation of N=2 A_{2} Argyres-Douglas theory. The central charges are given by (a,c)=(263/768,271/768)≃(0.342,0.353). There is no moduli space of vacua, no flavor symmetry, and the chiral ring is finite.

Gauge theories and dessins d’enfants: beyond the torus

Dessin d'Enfants on elliptic curves are a powerful way of encoding doubly-periodic brane tilings, and thus, of four-dimensional supersymmetric gauge theories whose vacuum moduli space is toric, providing an interesting interplay between physics, geometry, combinatorics and number theory. We discuss and provide a partial classification of the situation in genera other than one by computing explicit Belyi pairs associated to the gauge theories. Important also is the role of the Igusa and Shioda invariants that generalise the elliptic $j$-invariant.

Reduced Chern-Simons quiver theories and cohomological 3-algebra models

We study the BPS spectrum and vacuum moduli spaces in dimensional reductions of Chern-Simons-matter theories with N>=2 supersymmetry to zero dimensions. Our main example is a matrix model version of the ABJM theory which we relate explicitly to certain reduced 3-algebra models. We find the explicit maps from Chern-Simons quiver matrix models to dual IKKT matrix models. We address the problem of topologically twisting the ABJM matrix model, and along the way construct a new twist of the IKKT model. We construct a cohomological matrix model whose partition function localizes onto a moduli space specified by 3-algebra relations which live in the double of the conifold quiver. It computes an equivariant index enumerating framed BPS states with specified R-charges which can be expressed as a combinatorial sum over certain filtered pyramid partitions.

Numerical elimination and moduli space of vacua

We propose a new computational method to understand the vacuum moduli space of (supersymmetric) field theories. By combining numerical algebraic geometry (NAG) and elimination theory, we develop a powerful, efficient, and parallelizable algorithm to extract important information such as the dimension, branch structure, Hilbert series and subsequent operator counting, as well as variation according to coupling constants and mass parameters. We illustrate this method on a host of examples from gauge theory, string theory, and algebraic geometry.

On the structure of quadrilateral brane tilings

Brane tilings provide the most general framework in string and M-theory for matching toric Calabi-Yau singularities probed by branes with superconformal fixed points of quiver gauge theories. The brane tiling data consists of a bipartite tiling of the torus which encodes both the classical superpotential and gauge-matter couplings for the quiver gauge theory. We consider the class of tilings which contain only tiles bounded by exactly four edges and present a method for generating any tiling within this class by iterating combinations of certain graph-theoretic moves. In the context of D3-branes in IIB string theory, we consider the effect of these generating moves within the corresponding class of supersymmetric quiver gauge theories in four dimensions. Of particular interest are their effect on the superpotential, the vacuum moduli space and the conditions necessary for the theory to reach a superconformal fixed point in the infrared. We discuss the general structure of physically admissible quadrilateral brane tilings and Seiberg duality in terms of certain composite moves within this class.

MODELING M-THEORY VACUA VIA GAUGED S-DUALITY

We construct a model of M-theory vacua using gauged S-duality and the Chan–Paton symmetries by introducing an infinite number of open string charges. In the Bechi–Rouet–Stora–Tyutin formalism, the local description of the gauged S-duality on its moduli space of vacua is fully determined by one physical state condition on the vacua. We introduce the string probe of the spatial degrees of freedom and define the increment of the cosmic time. The dimensionality of space–time and the gauge group of the low energy effective theory originate in the symmetries (with or without their breakdown) in our model. This modeling leads to the derived category formulation of the quantum mechanical world including gravity and to the concept of a nonlinear potential of gauged and affinized S-duality which specifies the morphism structure of this derived category.

Dimensional Reduction Over the Quantum Sphere and Non-Abelian q-Vortices

We extend equivariant dimensional reduction techniques to the case of quantum spaces which are the product of a Kaehler manifold M with the quantum two-sphere. We work out the reduction of bundles which are equivariant under the natural action of the quantum group SU_q(2), and also of invariant gauge connections on these bundles. The reduction of Yang-Mills gauge theory on the product space leads to a q-deformation of the usual quiver gauge theories on M. We formulate generalized instanton equations on the quantum space and show that they correspond to q-deformations of the usual holomorphic quiver chain vortex equations on M. We study some topological stability conditions for the existence of solutions to these equations, and demonstrate that the corresponding vacuum moduli spaces are generally better behaved than their undeformed counterparts, but much more constrained by the q-deformation. We work out several explicit examples, including new examples of non-abelian vortices on Riemann surfaces, and q-deformations of instantons whose moduli spaces admit the standard hyper-Kaehler quotient construction.

$ \mathcal{N} = 1 $ curves for trifundamentals

We study the Coulomb phase of \( \mathcal{N} = 1 \) SU(2)3 gauge theory coupled to one trifundamental field, and generalizations thereof. The moduli space of vacua is always one-dimensional with multiple unbroken U(1) fields. We find that the \( \mathcal{N} = 1 \) Seiberg-Witten curve which encodes the U(1) couplings is given by the double cover of a Riemann surface branched at the poles and the zeros of a meromorphic function.

Fractional vortices and lumps

We study what might be called fractional vortices, vortex configurations with the minimum winding from the viewpoint of their topological stability, but which are characterized by various notable substructures in the transverse energy distribution. The fractional vortices occur in diverse Abelian or non-Abelian generalizations of the Higgs model. The global and local features characterizing these are studied, and we identify the two crucial ingredients for their occurrence---the vacuum degeneracy leading to nontrivial vacuum moduli $\mathcal{M}$, and the BPS nature of the vortices. Fractional vortices are further classified into two kinds. The first type of such vortices appear when $\mathcal{M}$ has orbifold ${\mathbb{Z}}_{n}$ singularities; the second type occurs in systems in which the vacuum moduli space $\mathcal{M}$ possesses either a deformed geometry or some singularity. These general features are illustrated with several concrete models.

Phases of M2-brane theories

We investigate different toric phases of 2+1 dimensional quiver gauge theories arising from M2-branes probing toric Calabi-Yau 4 folds. A brane tiling for each toric phase is presented. We apply the `forward algorithm' to obtain the toric data of the mesonic moduli space of vacua and exhibit the equivalence between the vacua of different toric phases of a given singularity. The structures of the Master space, the mesonic moduli space, and the baryonic moduli space are examined in detail. We compute the Hilbert series and use them to verify the toric dualities between different phases. The Hilbert series, R-charges, and generators of the mesonic moduli space are matched between toric phases.

𝒩 = 8 superconformal gauge theories andM2 branes

Based on recent developments, in this letter we find 2+1 dimensional gauge theories with scale invariance and N=8 supersymmetry. The gauge theories are defined by a Lagrangian and are based on an infinite set of 3-algebras, constructed as an extension of ordinary Lie algebras. Recent no-go theorems on the existence of 3-algebras are circumvented by relaxing the assumption that the invariant metric is positive definite. The gauge group is non compact, and its maximally compact subgroup can be chosen to be any ordinary Lie group, under which the matter fields are adjoints or singlets. The theories are parity invariant and do not admit any tunable coupling constant. In the case of SU(N) the moduli space of vacua contains a branch of the form (R^8)^N/S_N. These properties are expected for the field theory living on a stack of M2 branes.

D-instanton derivation of multi-fermionF-terms in supersymmetric QCD

We investigate effects of field theory instantons by considering D-instantons in a suitable D3-brane background. In supersymmetric QCD with SU(N_c) gauge group with N_f=N_c flavors, the moduli space of vacua is deformed by instantons. This effect can be described by the chiral interactions which are called multi-fermion F-terms. We derive these chiral interaction terms as D-instanton effects in the presence of D3-branes. For SU(2), the obtained result agrees with the previous result worked out by Beasley and Witten [hep-th/0409149]. We also explicitly work out those for the case of the symplectic gauge group, and show that they describe the deformation of the moduli space.

Strings on conifolds from strong coupling dynamics, part I

A method to solve various aspects of the strong coupling expansion of the superconformal field theory duals of AdS5 ? X geometries from first principles is proposed. The main idea is that at strong coupling the configurations that dominate the low energy dynamics of the field theory compactified on a three sphere are given by certain non-trivial semi-classical configurations in the moduli space of vacua. We show that this approach is self-consistent and permits one to express most of the dynamics in terms of an effective = 4 SYM dynamics. This has the advantage that some degrees of freedom that move the configurations away from moduli space can be treated perturbatively, unifying the essential low energy dynamics of all of these theories. We show that with this formalism one can compute the energies of strings in the BMN limit in the Klebanov-Witten theory from field theory considerations, matching the functional form of results found using AdS geometry. This paper also presents various other technical results for the semiclassical treatment of superconformal field theories.

Strings on conifolds from strong coupling dynamics: quantitative results

Three quantitative features of string theory on AdS_5 x X_5, for any (quasi)regular Sasaki-Einstein X_5, are recovered exactly from an expansion of field theory at strong coupling around configurations in the moduli space of vacua. These configurations can be thought of as a generalized matrix model of (local) commuting matrices. First, we reproduce the spectrum of scalar Kaluza-Klein modes on X_5. Secondly, we recover the precise spectrum of BMN string states, including a nontrivial dependence on the volume of X_5. Finally, we show how the radial direction in global AdS_5 emerges universally in these theories by exhibiting states dual to AdS giant gravitons.

Comments on multiple M2-branes

Recently a three-dimensional field theory was derived that is consistent with all the symmetries expected of the worldvolume action for multiple M2-branes. In this note we examine several physical predictions of this model and show that they are in agreement with expected M2-brane dynamics. In particular, we discuss the quantization of the Chern-Simons coefficient, the vacuum moduli space, a massive deformation leading to fuzzy three-sphere vacua, and a possible large n limit. In this large n limit, the fuzzy funnel solution correctly reproduces the mass of an M5-brane.

Resolved Calabi–Yau cones and flows from [formula omitted] superconformal field theories

We discuss D3-branes on cohomogeneity-three resolved Calabi–Yau cones over L a b c spaces, for which a 2-cycle or 4-cycle has been blown up. In terms of the dual quiver gauge theory, this corresponds to motion along the non-mesonic, or baryonic, directions in the moduli space of vacua. In particular, a dimension-two and/or dimension-six scalar operator gets a vacuum expectation value. These resolved cones support various harmonic ( 2 , 1 ) -forms which reduce the ranks of some of the gauge groups either by a Seiberg duality cascade or by Higgsing. We also discuss higher-dimensional resolved Calabi–Yau cones. In particular, we obtain square-integrable ( 2 , 2 ) -forms for eight-dimensional cohomogeneity-four Calabi–Yau metrics.

Counting BPS operators in the chiral ring of $\cal N=2$ supersymmetric gauge theories or $\cal N=2$ braine surgery

This note is presenting the generating functions which count the BPS operators in the chiral ring of a $\cal N=2$ quiver gauge theory that lives on N D3-branes probing an ALE singularity. The difficulty in this computation arises from the fact that this quiver gauge theory has a moduli space of vacua that splits into many branches — the Higgs, the Coulomb, and mixed branches. As a result, there can be operators which explore those different branches and the counting gets complicated by having to deal with such operators while avoiding over or under counting. The solution to this problem turns out to be very elegant and is presented in this note. Some surprises with “surgery” of generating functions arises.

Coupling Constant Dependence of the Kaluza-Klein Spectrum in Five-Dimensional SQCD on S1

We investigate the Kaluza-Klein (KK) spectrum of N=1 supersymmetric gauge theory compactified on a circle. We concentrate on a model with the gauge group SU(2) and four massless matter fields in the fundamental representation. We derive the exact mass formula of KK modes using Seiberg-Witten theory. From the mass formula and the D3-brane probe realization, we determine the spectrum of the KK modes of the matter fields and gauge fields. We find that the lightest KK state of the gauge fields is stable in the entire vacuum moduli space, while the lightest KK state of the matter fields decays more readily than other KK states in one region of the moduli space. This region becomes small as we decrease the five-dimensional gauge coupling constant g 5 , and vanishes in the limit g 5 → 0. This result continuously connects the known KK spectrum in the weak coupling limit and that in the strong coupling limit.