Invariant Vertex Research Articles

A Progressive Approach to Scalar Field Topology.

This article introduces progressive algorithms for the topological analysis of scalar data. Our approach is based on a hierarchical representation of the input data and the fast identification of topologically invariant vertices, which are vertices that have no impact on the topological description of the data and for which we show that no computation is required as they are introduced in the hierarchy. This enables the definition of efficient coarse-to-fine topological algorithms, which leverage fast update mechanisms for ordinary vertices and avoid computation for the topologically invariant ones. We demonstrate our approach with two examples of topological algorithms (critical point extraction and persistence diagram computation), which generate interpretable outputs upon interruption requests and which progressively refine them otherwise. Experiments on real-life datasets illustrate that our progressive strategy, in addition to the continuous visual feedback it provides, even improves run time performance with regard to non-progressive algorithms and we describe further accelerations with shared-memory parallelism. We illustrate the utility of our approach in batch-mode and interactive setups, where it respectively enables the control of the execution time of complete topological pipelines as well as previews of the topological features found in a dataset, with progressive updates delivered within interactive times.

Cubic interaction vertices for continuous-spin fields and arbitrary spin massive fields

Light-cone gauge formulation of relativistic dynamics of a continuous-spin field propagating in the flat space is developed. Cubic interaction vertices of continuous-spin massless fields and totally symmetric arbitrary spin massive fields are studied. We consider parity invariant cubic vertices that involve one continuous-spin massless field and two arbitrary spin massive fields and parity invariant cubic vertices that involve two continuous-spin massless fields and one arbitrary spin massive field. We construct the complete list of such vertices explicitly. Also we demonstrate that there are no cubic vertices describing consistent interaction of continuous-spin massless fields with arbitrary spin massless fields.

Establishing gauge invariant linear response of fermionic superfluids with pair fluctuations: A diagrammatic approach

We present a fully gauge-invariant linear response theory for ultra-cold Fermi gases undergoing BCS-Bose–Einstein Condensation (BEC) crossover with pair fluctuation effects included, especially in the superfluid phase. The gauge invariance is verified by introducing an effective external electromagnetic (EM) field. For pure BCS-type superfluids, the gauge invariance of the linear response theory can be maintained by constructing a full external EM vertex. By consistently including fluctuations of the order parameters as the self-energy effect is included in the quasi-particles, the Ward identity (WI) is satisfied as a consequence of gauge invariance. The construction of a gauge invariant vertex is non-trivial since in the broken symmetry phase the Nambu–Goldstone mode intertwines with the pairing fluctuation. We find that, under a suitable diagrammatic representation, the construction of such a vertex can be greatly simplified and allows us to build a vertex with pseudogap effects and consistent with the WI in the superfluid phase. We focus on a particular approach known as the G0Gt-matrix of pairing fluctuations, but our formalism also works for the G0G0t-matrix theory as well.

Revisiting light stringy states in view of the 750 GeV diphoton excess

We investigate light massive string states that appear at brane intersections. They replicate the massless spectrum in a richer fashion and may be parametrically lighter than standard Regge excitations. We identify the first few physical states and determine their BRST invariant vertex operators. In the supersymmetric case we reconstruct the super-multiplet structure. We then compute some simple interactions, such as the decay rate of a massive scalar or vector into two massless fermions. Finally we suggest an alternative interpretation of the 750 GeV diphoton excess at LHC in terms of a light massive string state, a replica of the Standard Model Higgs.

The Z, Z' → γγγ decays in the minimal 331 model

A complete calculation at one-loop level for the Z, Z' → γγγ decays is presented in the context of the minimal 331 model, which predicts the existence of a new Z' neutral gauge boson, Y++, Y+ charged gauge bosons, new exotic quarks and charged scalars. Bose symmetry is exploited to write a compact and manifest SUC[(3)-invariant vertex function for the γγγV0 coupling. Previous results on the Z → γγγ are reproduced (Br(Z → γγγ) ~ 10−10). It is found that this decay is insensitive to the effects of new heavy particles. This contrasts with the Z' → γγγ decay where the associated branching ratio is of the order of 10−6.

Coarse graining of spin net models: dynamics of intertwiners

Spin foams are models of quantum gravity and therefore quantum space time. A key open issue is to determine the possible continuum phases of these models. Progress on this issue has been prohibited by the complexity of the full four-dimensional models. We consider here simplified analogue models, so called spin nets, that retain the main dynamical ingredient of spin foams, the simplicity constraints. For a certain class of these spin net models we determine the phase diagram and therefore the continuum phases via a coarse graining procedure based on tensor network renormalization. This procedure will also reveal an unexpected fixed point, which turns out to define a new triangulation invariant vertex model.

Fragmentation of Heavy Quarks Using Wave Function at the Origin

In this paper we investigate the fragmentation function for heavy quarkanium using a global potential model. We employ the invariant vertex factor under parity, Lorentz transverse and invariant amplitude scattering for the fragmentation function. In the heavy quark limit we have obtained the S-wave analytical fragmentation functions for the heavy mesons at leading order perturbative QCD. The wave function of the bound state for mesons was set at origin of the framework. The fragmentation function is calculated with respect to the Z and Pt variables for m1+m2 and Mexperimental scales. In order to examine the accuracy of our calculations, the obtained results compared to the BFGW method.

Anomalous conformal currents, shadow fields, and massive AdS fields

In the framework of the gauge invariant approach involving Stueckelberg and auxiliary fields, a totally symmetric arbitrary spin anomalous conformal current and shadow field in flat space-time of dimension greater than or equal to 4 are studied. Gauge invariant differential constraints for such anomalous conformal current and shadow field and realization of global conformal symmetries are obtained. A gauge invariant two-point vertex of the arbitrary spin anomalous shadow field is also obtained. In the Stueckelberg gauge frame, the two-point gauge invariant vertex becomes the standard two-point vertex of CFT. A light-cone gauge two-point vertex of the arbitrary spin anomalous shadow field is derived. The AdS/CFT correspondence for arbitrary spin anomalous conformal current and shadow field and the respective normalizable and non-normalizable modes of massive arbitrary spin AdS field is studied. The AdS field is considered in modified de Donder gauge which simplifies considerably the study of AdS/CFT correspondence. We show that on-shell leftover gauge symmetries of bulk massive field are related to gauge symmetries of boundary anomalous conformal current and shadow field, while the modified de Donder gauge condition for a bulk massive field is related to differential constraints for boundary anomalous conformal current and shadow field.

Gauge invariant approach to low-spin anomalous conformal currents and shadow fields

Conformal low-spin anomalous currents and shadow fields in flat space-time of dimension greater than or equal to four are studied. Gauge invariant formulation for such currents and shadow fields is developed. Gauge symmetries are realized by involving Stueckelberg and auxiliary fields. Gauge invariant differential constraints for anomalous currents and shadow fields and realization of global conformal symmetries are obtained. Gauge invariant two-point vertices for anomalous shadow fields are also obtained. In Stueckelberg gauge frame, these gauge invariant vertices become the standard two-point vertices of CFT. Light-cone gauge two-point vertices of the anomalous shadow fields are derived. AdS/CFT correspondence for anomalous currents and shadow fields and the respective normalizable and non-normalizable solutions of massive low-spin AdS fields is studied. The bulk fields are considered in modified de Donder gauge that leads to decoupled equations of motion. We demonstrate that leftover on-shell gauge symmetries of bulk massive fields correspond to gauge symmetries of boundary anomalous currents and shadow fields, while the modified (Lorentz) de Donder gauge conditions for bulk massive fields correspond to differential constraints for boundary anomalous currents and shadow fields.

Light-cone gauge NSR strings in noncritical dimensions II — Ramond sector

Light-cone gauge superstring theory in noncritical dimensions corresponds to a worldsheet theory with nonstandard longitudinal part in the conformal gauge. The longitudinal part of the worldsheet theory is a superconformal field theory called X^{\pm} CFT. We show that the X^{\pm} CFT combined with the super-reparametrization ghost system can be described by free variables. It is possible to express the correlation functions in terms of these free variables. Bosonizing the free variables, we construct the spin fields and BRST invariant vertex operators for the Ramond sector in the conformal gauge formulation. By using these vertex operators, we can rewrite the tree amplitudes of the noncritical light-cone gauge string field theory, with external lines in the (R,R) sector as well as those in the (NS,NS) sector, in a BRST invariant way.

Gauge invariant two-point vertices of shadow fields, AdS/CFT, and conformal fields

In the framework of gauge invariant Stueckelberg approach, totally symmetric arbitrary spin shadow fields in flat space-time of dimension greater than or equal to four are studied. Gauge invariant two-point vertices for such shadow fields are obtained. We demonstrate that, in Stueckelberg gauge frame, these gauge invariant vertices become the standard two-point vertices of CFT. Light-cone gauge two-point vertices of the shadow fields are also obtained. AdS/CFT correspondence for the shadow fields and the non-normalizable solutions of free massless totally symmetric arbitrary spin AdS fields is studied. AdS fields are considered in a modified de Donder gauge and this simplifies considerably the study of AdS/CFT correspondence. We demonstrate that the bulk action, when it is evaluated on solution of the Dirichlet problem, leads to the two-point gauge invariant vertex of shadow field. Also we shown that the bulk action evaluated on solution of the Dirichlet problem leads to new description of conformal fields. The new description involves Stueckelberg gauge symmetries and gives simple higher-derivative Lagrangian for the conformal arbitrary spin field. In the Stueckelberg gauge frame, our Lagrangian becomes the standard Lagrangian of conformal field. Light-cone gauge Lagrangian of the arbitrary spin conformal field is also obtained.

Prepotential formulation of SU(3) lattice gauge theory

The SU(3) lattice gauge theory is reformulated in terms of SU(3) prepotential harmonic oscillators. This reformulation has enlarged SU(3)⊗U(1)⊗U(1) gauge invariance under which the prepotential operators transform like matter fields. The Hilbert space of SU(3) lattice gauge theory is shown to be equivalent to the Hilbert space of the prepotential formulation satisfying certain color invariant Sp(2, R) constraints. The SU(3) irreducible prepotential operators which solve these Sp(2, R) constraints are used to construct SU(3) gauge invariant Hilbert spaces at every lattice site in terms of SU(3) gauge invariant vertex operators. The electric fields and the link operators are reconstructed in terms of these SU(3) irreducible prepotential operators. We show that all the SU(3) Mandelstam constraints become local and take a very simple form within this approach. We also discuss the construction of all possible linearly independent SU(3) loop states which solve the Mandelstam constraints. The techniques can be easily generalized to SU(N).

DecaysZ→gggandZ′→gggin the minimal 331 model

We perform a complete calculation at the one-loop level for the $Zggg$ and ${Z}^{\ensuremath{'}}ggg$ couplings in the context of the minimal 331 model, which predicts the existence of a new ${Z}^{\ensuremath{'}}$ gauge boson and new exotic quarks. Bose symmetry is exploited to write a compact and manifest $S{U}_{C}(3)$-invariant vertex function for the $Vggg$ ($V=Z$, ${Z}^{\ensuremath{'}}$) coupling. Previous results on the $Z\ensuremath{\rightarrow}ggg$ decay in the standard model are reproduced. It is found that this decay is insensitive to the effects of the new exotic quarks. This in contrast with the ${Z}^{\ensuremath{'}}\ensuremath{\rightarrow}ggg$ decay, which is sensitive to both the standard model and exotic quarks, whose branching ratio is larger than that of the $Z\ensuremath{\rightarrow}ggg$ transition by about a factor of 4.

(NON)TRIVIALITY OF PURE SPINORS AND EXACT PURE SPINOR–RNS MAP

All the BRST-invariant operators in pure spinor formalism in d = 10 can be represented as BRST commutators, such as [Formula: see text] where λ+ is the U(5) component of the pure spinor transforming as [Formula: see text]. Therefore, in order to secure nontriviality of BRST cohomology in pure spinor string theory, one has to introduce "small Hilbert space" and "small operator algebra" for pure spinors, analogous to those existing in RNS formalism. As any invariant vertex operator in RNS string theory can also represented as a commutator V = {Q brst , LV} where L = -4c∂ξξe-2ϕ, we show that mapping [Formula: see text] to L leads to identification of the pure spinor variable λα in terms of RNS variables without any additional nonminimal fields. We construct the RNS operator satisfying all the properties of λα and show that the pure spinor BRST operator ∮λαdα is mapped (up to similarity transformation) to the BRST operator of RNS theory under such a construction.

Beyond the singularity of the 2Dcharged black hole

Two dimensional charged black holes in string theory can be obtained as exact (SL(2,) × U(1)) / U(1) quotient CFTs. The geometry of the quotient is induced from that of the group, and in particular includes regions beyond the black hole singularities. Moreover, wavefunctions in such black holes are obtained from gauge invariant vertex operators in the SL(2,) CFT, hence their behavior beyond the singularity is determined. When the black hole is charged we find that the wavefunctions are smooth at the singularities, and known results of infinite blue shifts at the horizons persist. Unlike the uncharged case, scattering waves prepared beyond the singularity are not fully reflected; part of the wave is transmitted through the singularity. Hence, the physics outside the horizon of a charged black hole is sensitive to conditions set behind the past singularity.

The relationship between skew group algebras and orbifold theory

Let V be a simple vertex operator algebra and let G be a finite automorphism group of V . In [C. Dong, G. Yamskulna, Math. Z., to appear], it was shown that any irreducible V -module is a completely reducible V G -module where V G is the G -invariant vertex operator subalgebra of V . In this paper, we give an alternative proof of this fact using the theory of skew group algebras. We also extend this result to any irreducible g -twisted V -module when g is in the center of G and V is a g -rational vertex operator algebra.

Microscopic string analysis of the D0–D8 brane system and dual R-R states

Using the boundary state formalism, we perform a microscopic string analysis of the interaction between two D-branes and provide a local interpretation for the R-R force in the D0–D8 brane system. To do so, we construct BRST invariant vertex operators for the massless R-R states in the asymmetric picture that are proportional to potentials rather than field strengths. The Hilbert space of such R-R states contains combinations of two vectors that decouple from all physical amplitudes, even in the presence of boundaries. Identifying these vectors, we remove the null states and recover duality relations among R-R potentials. If we specify to the D0–D8 brane system, this mechanism implies that the R-R 1-form state has a non-zero overlap with both the D0-brane and the D8-brane, thus explaining from a local point of view the non-vanishing R-R contribution in the interaction for the D0–D8 brane system and those related to it by duality.

Ghost systems: a vertex algebra point of view

Fermionic and bosonic ghost systems are defined each in terms of a single vertex algebra which admits a one-parameter family of conformal structures. The observation that these structures are related to each other provides a simple way to obtain character formulae for a general twisted module of a ghost system. The U(1) symmetry and its subgroups that underlie the twisted modules also define an infinite set of invariant vertex subalgebras. Their structure is studied in detail from a W algebra point of view with particular emphasis on Z N-invariant subalgebras of the fermionic ghost system.

Scattering of closed string states from a quantized D-particle

By developing an appropriate path-integral formalism, we compute, in bosonic string theory, the disk amplitude for the scattering of closed string states from a D-particle, in which the collective coordinate of the D-particle is fully quantized. As a consequence, the recoil of the D-particle is naturally taken into account. Our result can be readily factorized in the closed string channel to yield the boundary state describing the recoiling D-particle. This turned out to agree with the BRST invariant vertex recently proposed by Ishibashi to the leading order in the derivative expansion, but it will receive corrections in subsequent orders. The advantage of our formalism is that it is extendible to deal with more general processes involving multiple D-particles. A viewpoint regarding our work as describing a dynamical transition of CFT's is also discussed.

One-parameter family of an integrable spl(2|1) vertex model: Algebraic Bethe ansatz and ground state structure

We formulate in terms of the quantum inverse scattering method the exact solution of a spl(2|1) invariant vertex model recently introduced in the literature. The corresponding transfer matrix is diagonalized by using the algebraic (nested) Bethe ansatz approach. The ground state structure is investigated and we argue that a Pokrovsky-Talapov transition is favored for a certain value of the 4-dimensional spl(2|1) parameter.