Moduli Space Of Vacua Research Articles

Coulomb and Higgs phases of G2-manifolds

Ricci flat manifolds of special holonomy are a rich framework as models of the extra dimensions in string/M-theory. At special points in vacuum moduli space, special kinds of singularities occur and demand a physical interpretation. In this paper we show that the topologically distinct G2-holonomy manifolds arising from desingularisations of codimension four orbifold singularities due to Joyce and Karigiannis correspond physically to Coulomb and Higgs phases of four dimensional gauge theories. The results suggest generalisations of the Joyce-Karigiannis construction to arbitrary ADE-singularities and higher order twists which we explore in detail in explicitly solvable local models. These models allow us to derive an isomorphism between moduli spaces of Ricci flat metrics on these non-compact G2-manifolds and flat ADE-connections on compact flat 3-manifolds which we establish explicitly for SU(n).

Covariantly constant anomalies on conformal manifolds

Operators with integer scaling dimensions in even-dimensional conformal field theories exhibit well-known type-B Weyl anomalies. In general, these anomalies depend non-trivially on exactly marginal couplings. We study the corresponding fully covariantised anomaly functional on conformal manifolds in several examples. We show that a natural consequence of the Wess-Zumino consistency condition is that the anomalies are covariantly constant with respect to the exactly marginal couplings. The argument is general and applies even when the conformal symmetry is spontaneously broken on moduli spaces of vacua.

Exploring the orthosymplectic zoo

We study the Higgs branch of the SCFT limit of 5d SO(6) and SO(8) gauge theory with hypermultiplets in the spinor and vector representations. In the case of SO(6) gauge theories, we contrast the magnetic quivers obtained with those of SU(4) gauge theory with hypermultiplets in the fundamental and second rank antisymmetric representations. Since SU(4) gauge theories admit several different values of the Chern-Simons level, we make some observations about how to distinguish those theories from the brane webs of the SO(6) theories. In the case of SO(8) gauge theories, we use SO(8) triality to propose (naively) inequivalent magnetic quivers, which will turn out to have the same moduli spaces of vacua, at least locally around their most singular loci. We encounter several interesting new phenomena occurring in the magnetic quivers, such as hypermultiplets between neighbouring symplectic gauge nodes and matter in two-index representations of unitary nodes. We also give a prescription for computing the local Coulomb branch Hilbert series for quivers involving bad USp(2) gauge nodes.

Model of Persistent Breaking of Discrete Symmetry.

We show there exist UV-complete field-theoretic models in general dimension, including 2+1, with the spontaneous breaking of a global symmetry, which persists to the arbitrarily high temperatures. Our example is a conformal vector model with the O(N)×Z_{2} symmetry at zero temperature. Using conformal perturbation theory we establish Z_{2} symmetry is broken at finite temperature for N>17. Similar to recent constructions of [N. Chai et al., Phys. Rev. D 102, 065014 (2020).PRVDAQ2470-001010.1103/PhysRevD.102.065014, 2N. Chai et al., Phys. Rev. Lett. 125, 131603 (2020).PRLTAO0031-900710.1103/PhysRevLett.125.131603], in the infinite N limit our model has a nontrivial conformal manifold, a moduli space of vacua, which gets deformed at finite temperature. Furthermore, in this regime the model admits a persistent breaking of O(N) in 2+1 dimensions, therefore providing another example where the Coleman-Hohenberg-Mermin-Wagner theorem can be bypassed.

The bi-conical vector model at 1/N

We study finite N aspects of the O(m) × O(N − m) vector model with quartic interactions in general 2 ≤ d ≤ 6 spacetime dimensions. This model has recently been shown [1, 2] to display the phenomenon of persistent symmetry breaking at a perturbative Wilson-Fisher-like fixed point in d = 4 − ϵ dimensions. The large rank limit of the biconical model displays a conformal manifold and a moduli space of vacua. We find a set of three double trace scalar operators that are respectively irrelevant, relevant and marginal deformations of the conformal manifold in general d. We calculate the anomalous dimensions of the single and multi-trace scalar operators to the first sub-leading order in the large rank expansion. The anomalous dimension of the marginal operator does not vanish in general, indicating that the conformal manifold is lifted at finite N . In the case of equal ranks we are able to derive explicitly the scaling dimensions of various operators as functions of only d.

New mathcal{N} = 2 superconformal field theories from mathcal{S} -folds

We study the four-dimensional mathcal{N} = 2 superconformal field theories that describe D3-branes probing the recently constructed mathcal{N} = 2 mathcal{S} -folds in F-theory. We introduce a novel, infinite class of superconformal field theories related to mathcal{S} -fold theories via partial Higgsing. We determine several properties of both the mathcal{S} -fold models and this new class of theories, including their central charges, Coulomb branch spectrum, and moduli spaces of vacua, by bringing to bear an array of field-theoretical techniques, to wit, torus-compactifications of six-dimensional mathcal{N} = (1, 0) theories, class mathcal{S} technology, and the SCFT/VOA correspondence.

Chiral rings, Futaki invariants, plethystics, and Gr\xf6bner bases

We study chiral rings of 4d mathcal{N} = 1 supersymmetric gauge theories via the notion of K-stability. We show that when using Hilbert series to perform the computations of Futaki invariants, it is not enough to only include the test symmetry information in the former’s denominator. We discuss a way to modify the numerator so that K-stability can be correctly determined, and a rescaling method is also applied to simplify the calculations involving test configurations. All of these are illustrated with a host of examples, by considering vacuum moduli spaces of various theories. Using Gröbner basis and plethystic techniques, many non-complete intersections can also be addressed, thus expanding the list of known theories in the literature.

3d mirrors of the circle reduction of twisted A2N theories of class S

Mirror symmetry has proven to be a powerful tool to study several properties of higher dimensional superconformal field theories upon compactification to three dimensions. We propose a quiver description for the mirror theories of the circle reduction of twisted A2N theories of class S in four dimensions. Although these quivers bear a resemblance to the star-shaped quivers previously studied in the literature, they contain unitary, symplectic and special orthogonal gauge groups, along with hypermultiplets in the fundamental representation. The vacuum moduli spaces of these quiver theories are studied in detail. The Coulomb branch Hilbert series of the mirror theory can be matched with that of the Higgs branch of the corresponding four dimensional theory, providing a non-trivial check of our proposal. Moreover various deformations by mass and Fayet-Iliopoulos terms of such quiver theories are investigated. The fact that several of them flow to expected theories also gives another strong support for the proposal. Utilising the mirror quiver description, we discover a new supersymmetry enhancement renormalisation group flow.

Magnetic quivers, Higgs branches, and 6d mathcal{N} = (1, 0) theories \u2014 orthogonal and symplectic gauge groups

M5 branes on a D-type ALE singularity display various phenomena that introduce additional massless degrees of freedom. The M5 branes are known to fractionate on a D-type singularity. Whenever two fractional M5 branes coincide, tensionless strings arise. Therefore, these systems do not admit a low-energy Lagrangian description. Focusing on the 6-dimensional mathcal{N} = (1, 0) world-volume theories on the M5 branes, the vacuum moduli space has two branches were either the scalar fields in the tensor multiplet or the scalars in the hypermultiplets acquire a non-trivial vacuum expectation value. As suggested in previous work, the Higgs branch may change drastically whenever a BPS-string becomes tensionless. Recently, magnetic quivers have been introduced with the aim to capture all Higgs branches over any point of the tensor branch. In this paper, the formalism is extended to Type IIA brane configurations involving O6 planes. Since the 6d mathcal{N} = (1, 0) theories are composed of orthosymplectic gauge groups, the derivation rules for the magnetic quiver in the presence of O6 planes have to be conjectured. This is achieved by analysing the 6d theories for a single M5 brane on a D-type singularity and deriving the magnetic quivers for the finite and infinite gauge coupling Higgs branch from a brane configuration. The validity of the proposed derivation rules is underpinned by deriving the associated Hasse diagram. For multiple M5 branes, the approach of this paper provides magnetic quivers for all Higgs branches over any point of the tensor branch. In particular, an interesting infinite gauge coupling transition is found that is related to the SO(8) non-Higgsable cluster.

Skyrme and Faddeev models in the low-energy limit of 4d Yang–Mills–Higgs theories

Firstly, we consider U(Nc) Yang–Mills gauge theory on R3,1 with Nf>Nc flavours of scalar fields in the fundamental representation of U(Nc). The moduli space of vacua is the Grassmannian manifold Gr(Nc,Nf). It is shown that for strong gauge coupling this 4d Yang–Mills–Higgs theory reduces to the Faddeev sigma model on R3,1 with Gr(Nc,Nf) as target. Its action contains the standard two-derivative sigma-model term as well as the four-derivative Skyrme-type term, which stabilizes solutions against scaling. Secondly, we consider a Yang–Mills–Higgs model with Nf=2Nc and a Higgs potential breaking the flavour group U(Nf)=U(2Nc) to U+(Nc)×U−(Nc), realizing the simplest A2⊕A2-type quiver gauge theory. The vacuum moduli space of this model is the group manifold Uh(Nc) which is the quotient of U+(Nc)×U−(Nc) by its diagonal subgroup. When the gauge coupling constant is large, this 4d Yang–Mills–Higgs model reduces to the Skyrme sigma model on R3,1 with Uh(Nc) as target. Thus, both the Skyrme and the Faddeev model arise as effective field theories in the infrared of Yang–Mills–Higgs models.

Magnetic quivers, Higgs branches and 6d mathcal{N} = left(1,kern0.5em 0right) theories

The physics of M5 branes placed near an M9 plane on an A-type ALE singularity exhibits a variety of phenomena that introduce additional massless degrees of freedom. There are tensionless strings whenever two M5 branes coincide or whenever an M5 brane approaches the M9 plane. These systems do not admit a low-energy Lagrangian description so new techniques are desirable to shed light on the physics of these phenomena. The 6-dimensional mathcal{N} = left(1,kern0.5em 0right) world-volume theory on the M5 branes is composed of massless vector, tensor, and hyper multiplets, and has two branches of the vacuum moduli space where either the scalar fields in the tensor or hyper multiplets receive vacuum expectation values. Focusing on the Higgs branch of the low-energy theory, previous works suggest the conjecture that a new Higgs branch arises whenever a BPS-string becomes tensionless. Consequently, a single theory admits a multitude of Higgs branches depending on the types of tensionless strings in the spectrum. The two main phenomena discrete gauging and small E8instanton transition can be treated in a concise and effective manner by means of Coulomb branches of 3-dimensional mathcal{N} = 4 gauge theories. In this paper, a formalism is introduced that allows to derive a novel object from a brane configuration, called the magnetic quiver. The main features are as follows: (i) the 3d Coulomb branch of the magnetic quiver yields the Higgs branch of the 6d system, (ii) all discrete gauging and E8 instanton transitions have an explicit brane realisation, and (iii) exceptional symmetries arise directly from brane configurations. The formalism facilitates the description of Higgs branches at finite and infinite gauge coupling as spaces of dressed monopole operators.

A Physical Origin for Singular Support Conditions in Geometric Langlands Theory

We explain how the nilpotent singular support condition introduced into the geometric Langlands conjecture by Arinkin and Gaitsgory arises naturally from the point of view of N = 4 supersymmetric gauge theory. We define what it means in topological quantum field theory to restrict a category of boundary conditions to the full subcategory of objects compatible with a fixed choice of vacuum, both in functorial field theory and in the language of factorization algebras. For B-twisted N = 4 gauge theory with gauge group G, the moduli space of vacua is equivalent to h*/W , and the nilpotent singular support condition arises by restricting to the vacuum 0 in h*/W. We then investigate the categories obtained by restricting to points in larger strata, and conjecture that these categories are equivalent to the geometric Langlands categories with gauge symmetry broken to a Levi subgroup, and furthermore that by assembling such for the groups GL_n for all positive integers n one finds a hidden factorization structure for the geometric Langlands theory.

Confinement in 3d mathcal{N} = 2 Spin(N) gauge theories with vector and spinor matters

We present various confinement phases in three-dimensional mathcal{N} = 2 Spin(N) gauge theories with vector and spinor matters. The quantum Coulomb branch in the moduli space of vacua is drastically modified when the rank of the gauge group and the matter contents are changed. In many examples, the Coulomb branch is one- or two-dimensional but its interpretation varies. In some examples, the Coulomb branch becomes three-dimensional and we need to introduce a “dressed” Coulomb branch operator.

The Infrared Fixed Points of 3d $\mathcal{N}=4$ $USp(2N)$ SQCD Theories

We derive the algebraic description of the Coulomb branch of 3d \mathcal{N}=4𝒩=4USp(2N)USp(2N) SQCD theories with N_fNf fundamental hypermultiplets and determine their low energy physics in any vacuum from the local geometry of the moduli space, identifying the interacting SCFTs which arise at singularities and possible extra free sectors. The SCFT with the largest moduli space arises at the most singular locus on the Coulomb branch. For N_f > 2NNf>2N (good theories) it sits at the origin of the conical variety as expected. For N_f =2NNf=2N we find two separate most singular points, from which the two isomorphic components of the Higgs branch of the UV theory emanate. The SCFTs sitting at any of these two vacua have only odd dimensional Coulomb branch generators, which transform under an accidental SU(2)SU(2) global symmetry. We provide a direct derivation of their moduli spaces of vacua, and propose a Lagrangian mirror theory for these fixed points. For 2 \leq N_f < 2N2≤Nf<2N the most singular locus has one or two extended components, for N_fNf odd or even, and the low energy theory involves an interacting SCFT of one of the above types, plus free twisted hypermultiplets. For N_f=0,1Nf=0,1 the Coulomb branch is smooth. We complete our analysis by studying the low energy theory at the symmetric vacuum of theories with N < N_f \le 2NN<Nf≤2N, which exhibits a local Seiberg-like duality.

Curious aspects of three-dimensional mathcal{N}=1 SCFTs

We study the dynamics of certain 3d mathcal{N}=1 time reversal invariant theories. Such theories often have exact moduli spaces of supersymmetric vacua. We propose several dualities and we test these proposals by comparing the deformations and supersymmetric ground states. First, we consider a theory where time reversal symmetry is only emergent in the infrared and there exists (nonetheless) an exact moduli space of vacua. This theory has a dual description with manifest time reversal symmetry. Second, we consider some surprising facts about mathcal{N}=2 U(1) gauge theory coupled to two chiral superfields of charge 1. This theory is claimed to have emergent SU(3) global symmetry in the infrared. We propose a dual Wess-Zumino description (i.e. a theory of scalars and fermions but no gauge fields) with manifest SU(3) symmetry but only mathcal{N}=1 supersymmetry. We argue that this Wess-Zumino model must have enhanced supersymmetry in the infrared. Finally, we make some brief comments about the dynamics of mathcal{N}=1 SU(N) gauge theory coupled to Nf quarks in a time reversal invariant fashion. We argue that for Nf< N there is a moduli space of vacua to all orders in perturbation theory but it is non-perturbatively lifted.

Exact results in 3d mathcal{N} = 2 Spin(7) gauge theories with vector and spinor matters

We study three-dimensional mathcal{N} = 2 Spin(7) gauge theories with NS spinorial matters and with Nf vectorial matters. The quantum Coulomb branch on the moduli space of vacua is one- or two-dimensional depending on the matter contents. For particular values of (Nf, NS ), we find s-confinement phases and derive exact superpotentials. The 3d dynamics of Spin(7) is connected to the 4d dynamics via KK-monopoles. Along the Higgs branch of the Spin(7) theories, we obtain 3d mathcal{N} = 2 G2 or SU(4) theories and some of them lead to new s-confinement phases. As a check of our analysis we compute superconformal indices for these theories.

Low-energy dynamics of 3d mathcal{N} = 2 G2 supersymmetric gauge theory

We study a three-dimensional mathcal{N} = 2 supersymmetric G2 gauge theory with and without fundamental matters. We find that a classical Coulomb branch of the moduli space of vacua is partly lifted by monopole-instantons and the quantum Coulomb moduli space would be described by a complex one-dimensional space. Depending on the number of the matters in a fundamental representation, the low-energy dynamics of the theory shows various phases like s-confinement or quantum merging of the Coulomb and the Higgs branches. We also investigate superconformal indices as an independent check of our analysis.

Green-Schwarz automorphisms and 6D SCFTs

All known interacting 6D superconformal field theories (SCFTs) have a tensor branch which includes anti-chiral two-forms and a corresponding lattice of string charges. Automorphisms of this lattice preserve the Dirac pairing and specify discrete global and gauge symmetries of the 6D theory. In this paper we compute this automorphism group for 6D SCFTs. This discrete data determines the geometric structure of the moduli space of vacua. Upon compactification, these automorphisms generate Seiberg-like dualities, as well as additional theories in discrete quotients by the 6D global symmetries. When a perturbative realization is available, these discrete quotients correspond to including additional orientifold planes in the string construction.

Deformation Quantization and Superconformal Symmetry in Three Dimensions

We investigate the structure of certain protected operator algebras that arise in three-dimensional {mathcal{N}=4} superconformal field theories. We find that these algebras can be understood as a quantization of (either of) the half-BPS chiral ring(s). An important feature of this quantization is that it has a preferred basis in which the structure constants of the quantum algebra are equal to the OPE coefficients of the underlying superconformal theory. We identify several nontrivial conditions that the quantum algebra must satisfy in this basis. We consider examples of theories for which the moduli space of vacua is either the minimal nilpotent orbit of a simple Lie algebra or a Kleinian singularity. For minimal nilpotent orbits, the quantum algebras (and their preferred bases) can be uniquely determined. These algebras are related to higher spin algebras. For Kleinian singularities the algebras can be characterized abstractly—they are spherical subalgebras of symplectic reflection algebras—but the preferred basis is not easily determined. We find evidence in these examples that for a given choice of quantum algebra (defined up to a certain gauge equivalence), there is at most one choice of canonical basis. We conjecture that this is the case for general {mathcal{N}=4} SCFTs.

The long flow to freedom

Two-dimensional field theories do not have a moduli space of vacua. Instead, it is common that their low-energy behavior is a sigma model with a target space. When this target space is compact its renormalization group flow is standard. When it is non-compact the continuous spectrum of operators can change the qualitative behavior. Here we discuss two-dimensional gauge theories with mathcal{N} = (2, 2) supersymmetry. We focus on two specific theories, for which we argue that they flow to free chiral multiplets at low energies: the U(1) gauge theory with one flavor (two chiral superfields with charges plus and minus one) and a non-zero Fayet-Iliopoulos term, and pure SU(N ) gauge theories. We argue that the renormalization group flow of these theories has an interesting order of limits issue. Holding the position on the target space fixed, the space flattens out under the renormalization group. On the other hand, if we first go to infinity on the target space and then perform the renormalization group, we always have a non-trivial space, e.g. a cone with a deficit angle. We explain how to interpret low-energy dualities between theories with non-compact target spaces. We expect a similar qualitative behavior also for other non-compact sigma models, even when they do not flow to free theories.