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Gröbner bases for operads
Get PDFWe define a new monoidal category on collections (shuffle composition). Monoids in this category (shuffle operads) turn out to bring a new insight in the theory of symmetric operads. For this category, we develop the machinery of Gröbner bases for operads, and present operadic versions of Bergman’s Diamond Lemma and Buchberger’s algorithm. This machinery can be applied to study symmetric operads. In particular, we obtain an effective algorithmic version of Hoffbeck’s PBW criterion of Koszulness for (symmetric) quadratic operads
Superspace actions for multiple M2-branes, metric 3-algebras, and their classification
Get PDFWe classify canonical metric 3-algebra structures on matrix algebras and find novel three-dimensional conformally invariant actions in N = 4 projective super-space based on them. These can be viewed as Chern-Simons theories with special matter content and couplings. We explore the relations of these actions with the N = 2 actions based on generalized 3-Lie algebras found earlier and relate them to the original Bagger-Lambert-Gustavsson action
The non-Abelian Interferometer
Get PDFWe consider the tunneling current through a double point-contact Fabry-Pérot interferometer such as used in recent experimental studies of the fractional quantum Hall plateau at filling fraction ν = 5/2. We compare the predictions of several different models of the state of the electrons at this plateau: the Moore-Read, anti-Pfaffian, SU(2)_2 NAF, K = 8 strong pairing, and (3, 3, 1) states. All of these predict the existence of charge e/4 quasiparticles, but the first three are non-Abelian while the last two are Abelian. We give explicit formulas for the scaling of charge e/2 and charge e/4 quasiparticle contributions to the current as a function of temperature, gate voltage and distance between the two point contacts for all three models. Based on these, we analyze several possible explanations of two phenomena reported in Refs. 1,2, namely halving of the period of the observed resistance oscillations with rising temperature and alternation between the same two observed periods at low temperatures as the area of the interference loop is varied with a side gate. We conclude that the most likely explanation is that the observed alternation is due to switching between even and odd numbers of charge e/4 quasiparticles enclosed within the loop as a function of side gate voltage, which is a clear signature of the presence of non-Abelian anyons. However, there are important features of the data which do not have a simple explanation within this picture. We suggest further experiments which could help rule out some possible scenarios. We make the corresponding predictions for future tunneling and interference experiments at the other observed second Landau level fractional quantum Hall states
A Description of Kitaev’s Honeycomb Model with Toric-Code Stabilizers
Get PDFWe present a description of the Kitaev honeycomb lattice model as a BCS type system. A 2-dimensional fermionization procedure is outlined and the derived eigenstates of the system are shown to be Cooper-paired products of toric-code states. We extend our analysis to a torus, giving particular attention to the ground state of the fully periodic vortex-free sector
Freeness Theorems for Operads via Gröbner Bases
Get PDFWe show how to use Gröbner bases for operads to prove various freeness theorems: freeness of certain operads as nonsymmetric operads, freeness of an operad L as a P-module for an inclusion P → L, freeness of a suboperad. This gives new proofs of many known results of this type and helps to prove some new results
Dimensional reduction, monopoles and dynamical symmetry breaking
Get PDFWe consider SU(2)-equivariant dimensional reduction of Yang-Mills-Dirac theory on manifolds of the form M × CP^1 , with emphasis on the effects of non-trivial magnetic flux on CP^1 . The reduction of Yang-Mills fields gives a chain of coupled Yang-Mills-Higgs systems on M with a Higgs potential leading to dynamical symmetry breaking, as a consequence of the monopole fields. The reduction of SU(2)-symmetric fermions gives massless Dirac fermions on M trans-forming under the low-energy gauge group with Yukawa couplings, again as a result of the internal U(1) fluxes. The tower of massive fermionic Kaluza-Klein states also has Yukawa interactions and admits a natural SU(2)-equivariant truncation by replacing CP^1 with a fuzzy sphere. In this approach it is possible to obtain exactly massless chiral fermions in the effective field theory with Yukawa interactions, without any further requirements. We work out the spontaneous symmetry breaking patterns and determine the complete physical particle spectrum in a number of explicit examples
Three Generations on the Quintic Quotient
Get PDFA three-generation SU(5) GUT, that is 3 × (10 + 5) and a single 5–5 pair, is constructed by compactification of the E_8 heterotic string. The base manifold is the Z_5 × Z_5 -quotient of the quintic, and the vector bundle is the quotient of a positive monad. The group action on the monad and its bundle-valued coho-mology is discussed in detail, including topological restrictions on the existence of equivariant structures. This model and a single Z_5 quotient are the complete list of three generation quotients of positive monads on the quintic
A three-generation Calabi-Yau manifold with small Hodge numbers
Get PDFWe present a complete intersection Calabi-Yau manifold Y that has Euler number −72 and which admits free actions by two groups of automorphisms of order 12. These are the cyclic group Z_12 and the non-Abelian dicyclic group Dic_3 . The quotient manifolds have χ = −6 and Hodge numbers (h^11 , h^21 ) = (1, 4). With the standard embedding of the spin connection in the gauge group, Y gives rise to an E_6 gauge theory with 3 chiral generations of particles. The gauge group may be broken further by means of the Hosotani mechanism combined with continuous deformation of the background gauge field. For the non-Abelian quotient we obtain a model with 3 generations with the gauge group broken to that of the standard model. Moreover there is a limit in which the quotients develop 3 conifold points. These singularities may be resolved simultaneously to give another manifold with (h^11 , h^21 ) = (2, 2) that lies right at the tip of the distribution of Calabi–Yau manifolds. This strongly suggests that there is a heterotic vacuum for this manifold that derives from the 3 generation model on the quotient of Y . The manifold Y may also be realised as a hypersurface in the toric variety. The symmetry group does not act torically, nevertheless we are able to identify the mirror of the quotient manifold by adapting the construction of Batyrev
Non-Abelian Yang-Mills–Higgs vortices
Get PDFIn this Letter we present new, genuinely non-Abelian vortex solutions in SU(2) Yang-Mills–Higgs theory with only one isovector scalar field. These non-Abelian solutions branch off their Abelian counterparts (Abrikosov–Nielsen-Olesen vortices) for precise values of the Higgs potential coupling constant λ. For all values of λ, their energies lie below those of the Abelian energy profiles, the latter being logarithmically divergent as λ → ∞. The non-Abelian branches plateau in the limit λ → ∞ and their number increases with λ, this number becoming infinite. For each vorticity, the gaps between the plateauing energy levels become constant. In this limit the non-Abelian vortices are non-interacting and are described by the self-dual vortices of the O(3) sigma model. Hence by continuity, we can expect that they are stable also for finite values of λ