Orbifold Equivalence Research Articles

Computational aspects of orbifold equivalence

AbstractIn this paper we study the computational feasibility of an algorithm to prove orbifold equivalence between potentials describing Landau–Ginzburg models. Through a comparison with state-of-the-art results of Gröbner basis computations in cryptology, we infer that the algorithm produces systems of equations that are beyond the limits of current technical capabilities. As such the algorithm needs to be augmented by ‘inspired guesswork’, and we provide examples of applying this approach.

McKay correspondence and orbifold equivalence

We prove that a pair of singularities related by a transformation arising from the McKay correspondence are orbifold equivalent. From this we deduce a McKay type category equivalence for the matrix factorization categories.

Topological deconfinement transition in QCD at finite isospin density

The confinement–deconfinement transition is discussed from topological viewpoints. The topological change of the system is achieved by introducing the dimensionless imaginary chemical potential (θ). Then, the non-trivial free-energy degeneracy becomes the signal of the deconfinement transition and it can be visualized by using the map of the thermodynamic quantities to the circle S1 along θ. To understand this “topological” deconfinement transition at finite real quark chemical potential (μR), we consider the isospin chemical potential (μiso) in the effective model of QCD. The phase diagram at finite μiso is identical with that at finite μR outside of the pion-condensed phase at least in the large-Nc limit via the well-known orbifold equivalence. In the present effective model, the topological deconfinement transition does not show a significant dependence on μiso and then we can expect that this tendency also appears at small μR. Also, the chiral transition and the topological deconfinement transition seems to be weakly correlated. If we will access lattice QCD data for the temperature dependence of the quark number density at finite μiso with θ=π/3, our surmise can be judged.

Orbifold completion of defect bicategories

Orbifolds of two-dimensional quantum field theories have a natural formulation in terms of defects or domain walls. This perspective allows for a rich generalisation of the orbifolding procedure, which we study in detail for the case of topological field theories. Namely, a TFT with defects gives rise to a pivotal bicategory of "world sheet phases" and defects between them. We develop a general framework which takes such a bicategory \mathcal B as input and returns its "orbifold completion" \mathcal B_{\mathrm {orb}} . The completion satisfies the natural properties \mathcal B \subset \mathcal B_{\mathrm {orb}} and (\mathcal B_{\mathrm {orb}})_{\mathrm{orb}} \cong \mathcal B_{\mathrm {orb}} , and it gives rise to various new equivalences and nondegeneracy results. When applied to TFTs, the objects in \mathcal B_{\mathrm {orb}} correspond to generalised orbifolds of the theories in \mathcal B . In the example of Landau–Ginzburg models we recover and unify conventional equivariant matrix factorisations, prove when and how (generalised) orbifolds again produce open/closed TFTs, and give nontrivial examples of new orbifold equivalences.

New Nonperturbative Methods in Quantum Field Theory: From Large-NOrbifold Equivalence to Bions and Resurgence

We present a broad conceptual introduction to some new ideas in nonperturbative quantum field theory (QFT) that have led to progress toward an understanding of quark confinement in gauge theories and, more broadly, toward a nonperturbative continuum definition of QFTs. We first present exact orbifold equivalences of supersymmetric and nonsupersymmetric QFTs in the large-N limit and exact equivalences of large-N theories in infinite volume to large-N theories in finite volume, or even at a single point. We discuss principles by which calculable QFTs are continuously connected to strong-coupling QFTs, allowing understanding of the physics of confinement or the absence thereof. We discuss the role of particular saddle solutions, termed bions, in weak-coupling calculable regimes. The properties of bions motivate an extension of semiclassical methods used to evaluate functional integrals to include families of complex saddles (Picard–Lefschetz theory). This analysis leads us to the resurgence program, which may provide a framework for combining divergent perturbation series with semiclassical instanton and bion/renormalon contributions. This program could provide a nonperturbative definition of the path integral.

Strangely dual orbifold equivalence I

In this brief note we prove orbifold equivalence between two potentials described by strangely dual exceptional unimodular singularities of type $K_{14}$ and $Q_{10}$ in two different ways. The matrix factorizations proving the orbifold equivalence give rise to equations whose solutions are permuted by Galois groups which differ for different expressions of the same singularity.

Dualities from largeNorbifold equivalence in Chern-Simons-matter theories with flavor

We study large-$N$ orbifold equivalences involving three-dimensional $\mathcal{N}=3$ and $\mathcal{N}=4$ supersymmetric quiver Chern-Simons-matter theories. The gravity dual of the $\mathcal{N}=3$ Chern-Simons-matter theory is described by ${\mathrm{AdS}}_{4}\ifmmode\times\else\texttimes\fi{}{\mathcal{M}}_{7}$, where the tri-Sasaki manifold ${\mathcal{M}}_{7}$ is known as the Eschenburg space. We find evidence that a large-$N$ orbifold equivalence for the $\mathcal{N}=4$ case continues from the $M$-theory limit to the weak coupling limit. For the $\mathcal{N}=3$ case, we find consistent large-$N$ equivalences involving a projection changing the nodes of the gauge groups, and also for a projection changing Chern-Simons levels where for the latter projection, the Bogomol'nyi Prasad-Sommerfield monopole operators behave as expected in large-$N$ equivalence. For both cases we show, using the gravity dual, that the critical temperature of the confinement/deconfinement transition does not change and the entropy behaves as expected under the orbifold equivalence. We show that large-$N$ orbifold equivalence changing Chern-Simons levels can be explained using the planar equivalence in the mirror dual.

IIB duals of D = 3 $ \mathcal{N} = 4 $ circular quivers

Abstract We construct the type-IIB AdS4 ⋉ K supergravity solutions which are dual to the three-dimensional $ \mathcal{N} = 4 $ superconformal field theories that arise as infrared fixed points of circular-quiver gauge theories. These superconformal field theories are labeled by a triple $ \left( {\rho, \hat{\rho},L} \right) $ subject to constraints, where ρ and $ \hat{\rho} $ are two partitions of a number N, and L is a positive integer. We show that in the limit of large L the localized five- branes in our solutions are effectively smeared, and these type-IIB solutions are dual to the near-horizon geometry of M-theory M2-branes at a $ {{{{{\mathbb{C}}^4}}} \left/ {{\left( {{Z_k}\times {Z_{\widehat{k}}}} \right)}} \right.} $ orbifold singularity. Our IIB solutions resolve the singularity into localized five-brane throats, without breaking the conformal symmetry. The constraints satisfied by the triple $ \left( {\rho, \hat{\rho},L} \right) $ , together with the enhanced non-abelian flavour symmetries of the superconformal field theories are precisely reproduced by the type-IIB supergravity solutions. As a bonus, we uncover a novel type of “orbifold equivalence” between different quantum field theories and provide quantitative evidence for this equivalence.

A worldsheet extension of $ O\left( {d,d\left| \mathbb{Z} \right.} \right) $

Abstract We study superconformal interfaces between $ \mathcal{N}=\left( {1,1} \right) $ supersymmetric sigma models on tori, which preserve a $ \widehat{u}{(1)^{2d }} $ current algebra. Their fusion is non-singular and, using parallel transport on CFT deformation space, it can be reduced to fusion of defect lines in a single torus model. We show that the latter is described by a semi-group extension of $ O\left( {d,d\left| \mathbb{Q} \right.} \right) $ ), and that (on the level of Ramond charges) fusion of interfaces agrees with composition of associated geometric integral transformations. This generalizes the well-known fact that T-duality can be geometrically represented by Fourier-Mukai transformations. Interestingly, we find that the topological interfaces between torus models form the same semi-group upon fusion. We argue that this semi-group of orbifold equivalences can be regarded as the α′ deformation of the continuous O(d, d) symmetry of classical supergravity.

Holographic realization of large-N c orbifold equivalence with non-zero chemical potential

Recently, it has been suggested that large-Nc orbifold equivalences may be applicable to certain theories with chemical potentials, including QCD, in certain portions of their phase diagram. When valid, such an equivalence offers the possibility of relating large-Nc QCD at non-zero baryon chemical potential, a theory with a complex fermion determinant, to a related theory whose fermion determinant is real and positive. In this paper, we provide a test of this large Nc equivalence using a holographic realization of a supersymmetric theory with baryon chemical potential and a related theory with isospin chemical potential. We show that the two strongly-coupled, large-Nc theories are equivalent in a large region of the phase diagram.

No-Go Theorem for Critical Phenomena in Large-NcQCD

We derive some rigorous results on the chiral phase transition in QCD and QCD-like theories with a large number of colors, N(c), based on the QCD inequalities and the large-N(c) orbifold equivalence. We show that critical phenomena and associated soft modes are forbidden in flavor-symmetric QCD at finite temperature T and finite but not so large quark chemical potential μ for any nonzero quark mass. In particular, the critical point in QCD at a finite baryon chemical potential μ(B)=N(c)μ is ruled out, if the coordinate (T, μ) is outside the pion condensed phase in the corresponding phase diagram of QCD at a finite isospin chemical potential μ(I)=2μ.

Universality of Phases in QCD and QCD-like Theories

We argue that the whole or the part of the phase diagrams of QCD and QCD-like theories should be universal in the large-N_c limit through the orbifold equivalence. The whole phase diagrams, including the chiral phase transitions and the BEC-BCS crossover regions, are identical between SU(N_c) QCD at finite isospin chemical potential and SO(2N_c) and Sp(2N_c) gauge theories at finite baryon chemical potential. Outside the BEC-BCS crossover region in these theories, the phase diagrams are also identical to that of SU(N_c) QCD at finite baryon chemical potential. We give examples of the universality in some solvable cases: (i) QCD and QCD-like theories at asymptotically high density where the controlled weak-coupling calculations are possible, (ii) chiral random matrix theories of different universality classes, which are solvable large-N (large volume) matrix models of QCD. Our results strongly suggest that the chiral phase transition and the QCD critical point at finite baryon chemical potential can be studied using sign-free theories, such as QCD at finite isospin chemical potential, in lattice simulations.

Generating new dualities through the orbifold equivalence: a demonstration in ABJM and four-dimensional quivers

We show that the recently proposed large N equivalence between ABJM theories with Chern-Simons terms of different rank and level, U(N 1)k1 × U(N 1) − k1 and U(N 2) k2 × U(N 2)−k2 , but the same value of N′ = N 1 k 1 = N2k2, can be explained using planar equivalence in the mirror duals. The combination of S-dualities and orbifold equivalence can be applied to other cases as well, with very appealing results. As an example we show that two different quiver theories with k nodes can be easily shown to be Seiberg dual through the orbifold equivalence, but it requires order k 2 steps to give a proof when Seiberg duality is performed node by node.

On a new type of orbifold equivalence and M-theoretic [formula omitted] duality

We consider the large-N limit of N=6U(N)×U(N) superconformal Chern–Simons (ABJM) theory with fixed level k, which is conjectured to be dual to M-theory on AdS4×(S7/Zk) background. We point out that the so-called orbifold equivalence on the gravity side, combined with the AdS4/CFT3 duality, predicts a hitherto unknown type of duality on the gauge theory side. It establishes the equivalence between a class of observables, which are not necessarily protected by supersymmetry, in strongly coupled ABJM theories away from the planar approximation, with different values of k and N but sharing common kN. This limit is vastly different from the planar limit, and hence from the gauge theory point of view the duality is more difficult to explain compared to the previously known analogous equivalence between planar gauge theories, where one can explicitly prove the equivalence diagrammatically using the dominance of the planar diagrams.

Orbifold equivalence for finite density QCD and effective field theory

In the large N c limit, some apparently different gauge theories turn out to be equivalent due to large N c orbifold equivalence. We use effective field theory techniques to explore orbifold equivalence, focusing on the specific case of a recently discovered relation between an SO(2N c ) gauge theory and QCD. The equivalence to QCD has been argued to hold at finite baryon chemical potential, μ B , so long as one deforms the SO(2N c ) theory by certain “double-trace” terms. The deformed SO(2N c ) theory can be studied without a sign problem in the chiral limit, in contrast to SU(N c ) QCD at finite μ B . The purpose of the double-trace deformation in the SO(2N c ) theory is to prevent baryon number symmetry from breaking spontaneously at finite density, which is necessary for the equivalence to large N c QCD to be valid. The effective field theory analysis presented here clarifies the physical significance of double-trace deformations, and strongly supports the proposed equivalence between the deformed SO(2N c ) theory and large N c QCD at finite density.

Orbifold Equivalence and the Sign Problem at Finite Baryon Density

We point out that SO(2N(c)) gauge theory with N(f) fundamental Dirac fermions does not have a sign problem at finite baryon number chemical potential μ(B). One can thus use lattice Monte Carlo simulations to study this theory at finite density. The absence of a sign problem in the SO(2N(c)) theory is particularly interesting because a wide class of observables in the SO(2N(c)) theory coincide with observables in QCD in the large N(c) limit, as we show using the technique of large N(c) orbifold equivalence. We argue that the orbifold equivalence between the two theories continues to hold at finite μ(B) provided one adds appropriate deformation terms to the SO(2N(c)) theory. This opens up the prospect of learning about QCD at finite μ(B) using lattice studies of the SO(2N(c)) theory.

Volume independence in largeNcQCD-like gauge theories

Volume independence in large $\Nc$ gauge theories may be viewed as a generalized orbifold equivalence. The reduction to zero volume (or Eguchi-Kawai reduction) is a special case of this equivalence. So is temperature independence in confining phases. In pure Yang-Mills theory, the failure of volume independence for sufficiently small volumes (at weak coupling) due to spontaneous breaking of center symmetry, together with its validity above a critical size, nicely illustrate the symmetry realization conditions which are both necessary and sufficient for large $\Nc$ orbifold equivalence. The existence of a minimal size below which volume independence fails also applies to Yang-Mills theory with antisymmetric representation fermions [QCD(AS)]. However, in Yang-Mills theory with adjoint representation fermions [QCD(Adj)], endowed with periodic boundary conditions, volume independence remains valid down to arbitrarily small size. In sufficiently large volumes, QCD(Adj) and QCD(AS) have a large $\Nc$ ``orientifold'' equivalence, provided charge conjugation symmetry is unbroken in the latter theory. Therefore, via a combined orbifold-orientifold mapping, a well-defined large $\Nc$ equivalence exists between QCD(AS) in large, or infinite, volume and QCD(Adj) in arbitrarily small volume. Since asymptotically free gauge theories, such as QCD(Adj), are much easier to study (analytically or numerically) in small volume, this equivalence should allow greater understanding of large $\Nc$ QCD in infinite volume.