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We study quantum properties of SU(2|1) supersymmetric (deformed $$ \mathcal{N} $$ N = 4, d = 1 supersymmetric) extension of the superintegrable Smorodinsky-Winternitz system on a complex Euclidian space ℂN. The full set of wave functions is constructed and the energy spectrum is calculated. It is shown that SU(2|1) supersymmetry implies the bosonic and fermionic states to belong to separate energy levels, thus exhibiting the “even-odd” splitting of the spectra. The superextended hidden symmetry operators are also defined and their action on SU(2|1) multiplets of the wave functions is given. An equivalent description of the same system in terms of superconformal SU(2|1, 1) quantum mechanics is considered and a new representation of the hidden symmetry generators in terms of the SU(2|1, 1) ones is found.

We show that, under a natural scaling, the small-time behavior of the logarithmic derivatives of the subelliptic heat kernel on $SU(2)$ converges to their analogues on the Heisenberg group at time 1. Realizing $SU(2)$ as $\mathbb{S}^3$, we then generalize these results to higher-order odd-dimensional spheres equipped with their natural subRiemannian structure, where the limiting spaces are now the higher-dimensional Heisenberg groups., 32 pages, corrected errors in preliminary estimates in Section 2.1

We study supersymmetry breaking deformations of the $\mathcal{N}=1$ 5d fixed point known as $E_1$, the UV completion of $SU(2)$ super-Yang-Mills. The phases of the non-supersymmetric theory can be characterized by Chern-Simons terms involving background $U(1)$ gauge fields, allowing us to identify a phase transition at strong coupling. We propose that this may signify the emergence of a non-trivial, non-supersymmetric CFT in $d=4+1$ dimensions., 18 pages, 2 figures. v2: minor typos fixed, references added

Starting with the braided quantum group SU q ( 2 ) for a complex deformation parameter q we perform the construction of the quotient SU q ( 2 ) / T which serves as a model of a quantum sphere. Then we follow the reasoning of Podleś who for real q classified quantum spaces with the action of SU q ( 2 ) with appropriate spectral properties. These properties can also be expressed in the context of the braided quantum SU q ( 2 ) (with complex q) and we find that they lead to precisely the same family of quantum spaces as found by Podleś for the real parameter | q | .

In this paper we first provide the proof of $SU(2)$ Y-Map convergence. Then, by using $SU(1,1)$ LQG simplicity constraints we define $SU(1,1)$ Y-Map from infinitely differentiable with a compact support functions on $SU(1,1)$ to the functions (not necessarily square integrable) on $SL(2,C)$, and prove its convergence as well., arXiv admin note: text overlap with arXiv:1509.01312

We study the semiclassical limit of a class of invariant tensors for infinite-dimensional unitary representations of $\mathrm{SL}(2,\mathbb{C})$ of the principal series, corresponding to generalized Clebsch-Gordan coefficients with $n\geq3$ legs. We find critical configurations of the quantum labels with a power-law decay of the invariants. They describe 3d polygons that can be deformed into one another via a Lorentz transformation. This is defined viewing the edge vectors of the polygons are the electric part of bivectors satisfying a (frame-dependent) relation between their electric and magnetic parts known as $\gamma$-simplicity in the loop quantum gravity literature. The frame depends on the SU(2) spin labelling the basis elements of the invariants. We compute a saddle point approximation using the critical points and provide a leading-order approximation of the invariants. The power-law is universal if the SU(2) spins have their lowest value, and $n$-dependent otherwise. As a side result, we provide a compact formula for $\gamma$-simplicity in arbitrary frames. The results have applications to the current EPRL model, but also to future research aiming at going beyond the use of fixed time gauge in spin foam models., Comment: 29 pages and 3 figures. v2:minor corrections and a subsection added to match published version

The generalized SU(2) Proca theory (GSU2P) is a variant of the well known generalized Proca theory where the vector field belongs to the Lie algebra of the SU(2) group of global transformations under which the action is made invariant. New interesting possibilities arise in this framework because of the existence of new interactions of purely non-Abelian character and new configurations of the vector field resulting in spatial spherical symmetry and the cosmological dynamics being driven by the propagating degrees of freedom. We study the two-dimensional phase space of the system that results when the cosmic triad configuration is employed in the Friedmann-Lemaitre-Robertson-Walker background and find an attractor curve whose attraction basin both covers almost all the allowed region and does not include a Big-Bang singularity. Such an attractor curve corresponds to a primordial inflationary solution that has the following characteristic properties: 1.) it is a de Sitter solution whose Hubble parameter is regulated by a generalized version of the SU(2) group coupling constant, 2.) it is constant-roll including, as a limiting case, the slow-roll variety, 3.) a number of e-folds $N > 60$ is easily reached, 4.) it has a graceful exit into a radiation dominated period powered by the canonical kinetic term of the vector field and the Einstein-Hilbert term. The free parameters of the action are chosen such that the tensor sector of the theory is the same as that of general relativity at least up to second-order perturbations, thereby avoiding the presence of ghost and Laplacian instabilities in the tensor sector as well as making the gravity waves propagate at light speed. This is a proof of concept of the interesting properties we could find in this scenario when the coupling constants be replaced by general coupling functions and more terms be discovered in the GSU2P., LaTeX file in WileYMSP-template style, 28 pages, 20 figures. Version published in Annalen der Physik. Dedicated to Carlos J. Quimbay Herrera, Marta A. Losada Falk, and David H. Lyth

We study the effects of addition of the Chern-Simons (CS) term in the minimal Yang-Mills (YM) matrix model composed of two $2\ifmmode\times\else\texttimes\fi{}2$ matrices with $SU(2)$ gauge and $SO(2)$ global symmetry. We obtain the Hamiltonian of this system in appropriate coordinates and demonstrate that its dynamics is sensitive to the values of both the CS coupling, $\ensuremath{\kappa}$, and the conserved conjugate momentum, ${p}_{\ensuremath{\phi}}$, associated to the $SO(2)$ symmetry. We examine the behavior of the emerging chaotic dynamics by computing the Lyapunov exponents and plotting the Poincar\'e sections as these two parameters are varied and, in particular, find that the largest Lyapunov exponents evaluated within a range of values of $\ensuremath{\kappa}$ are above what is computed at $\ensuremath{\kappa}=0$, for $\ensuremath{\kappa}{p}_{\ensuremath{\phi}}l0$. We also give estimates of the critical exponents for the Lyapunov exponent as the system transits from the chaotic to nonchaotic phase with ${p}_{\ensuremath{\phi}}$ approaching to a critical value.

We compute the large N limit of the partition function of the Euclidean Yang--Mills measure with structure group SU(N) or U(N) on all closed compact surfaces, orientable or not, excepted for the sphere and the projective plane. This limit is finite as opposed to the case of the sphere and presumably the projective plane. We expect that the results we present might give an insight towards the master field on these surfaces., 25 pages, 4 figures

Non-Abelian gauge field plays an important role in high-energy physics through Yang-Mill theory, in condensed-matter physics, for instance, with anomalous quantum Hall effect, and in quantum information. In its SU(2) symmetry version, Hamiltonians with non-Abelian gauge field reduce to spin-orbit coupling-like systems, taking the general form: H = p 2 /2m+ A.p /m , where m is the mass of the particle, p the momentum and A the non-Abelian gauge field with 2x2 matrix components, decomposed on SU(2) Lie group generators. These class of Hamiltonians can be simulate with ultracold gas [1] using various methods [2] . The topological properties of spin-orbit-coupled system have been explore analysing the energy bands in 2D Fermionic system [3] . Here, the eigenenergies have momentum dependence, leading to Rabi flopping with periodic change of quasi-momentum, even if the non-Abelian gauge field is homogenous. This oscillation is similar to relativistic Zitterbewegung predicted in Dirac equation and simulated in ultracold system [4] .

New types of 3D solutions for the classic Yang–Mills equations in the Faddeev–Niemi reformulation are found. In a particular case, these solutions describe 3D vortices.

The perturbative expansion of the Chern–Simons path integral predicts a formula of the asymptotic expansion of the quantum invariant of a 3-manifold. When $$q=\exp (2 \pi \sqrt{-1}/N)$$ , there have been some researches where the asymptotic expansion of the quantum $$\mathrm{SU}(2)$$ invariant is presented by a sum of contributions from $$\mathrm{SU}(2)$$ flat connections whose coefficients are square roots of the Reidemeister torsions. When $$q=\exp (4 \pi \sqrt{-1}/N)$$ , it is conjectured recently that the quantum $$\mathrm{SU}(2)$$ invariant of a closed hyperbolic 3-manifold M is of exponential order of N whose growth is given by the complex volume of M. The first author showed in the previous work that this conjecture holds for the hyperbolic 3-manifold $$M_p$$ obtained from $$S^3$$ by p surgery along the figure-eight knot. From the physical viewpoint, we use the (formal) saddle point method when $$q=\exp (4 \pi \sqrt{-1}/N)$$ , while we have used the stationary phase method when $$q=\exp ({2 \pi \sqrt{-1}}/N)$$ , and these two methods give quite different resulting formulas from the mathematical viewpoint. In this paper, we show that a square root of the Reidemeister torsion appears as a coefficient in the semi-classical approximation of the asymptotic expansion of the quantum $$\mathrm{SU}(2)$$ invariant of $$M_p$$ at $$q = \exp (4 \pi \sqrt{-1}/N)$$ . Further, when $$q = \exp (4 \pi \sqrt{-1}/N)$$ , we show that the semi-classical approximation of the asymptotic expansion of the quantum $$\mathrm{SU}(2)$$ invariant of some Seifert 3-manifolds M is presented by a sum of contributions from some of $$\mathrm{SL}_2 {\mathbb C}$$ flat connections on M, and square roots of the Reidemeister torsions appear as coefficients of such contributions.

Classical strings propagating in $AdS_{3}\times S^{3}\times T^{4}$ supported with Neveu-Schwarz-Neveu-Schwarz flux are described by a Wess-Zumino-Witten model. In this note, we study the emergence of their semiclassical $SU(2)$ spectrally flowed sectors as the Landau-Lifshitz limit of the underlying quantum spin chain. We consider the propagator in the coherent state picture, and find that the time interval is discretized proportionally to the lattice spacing. In the Landau-Lifshitz limit, where both time and space become continuous, we derive a path integral representation of the propagator for each spectrally flowed sector. We prove that the arbitrariness of the global phase of coherent states is mapped to the gauge freedom of the $B$-field in the classical action. We show that higher order corrections in the Landau-Lifshitz limit are suppressed by inverse powers of the 't Hooft coupling., 10 pages, Latex. v2: Published version. v3: Acknowledgement added

A method for the decomposition of data functions sampled on a finite fragment of rectangular lattice is described. The symmetry of a square lattice in a 2-dimensional real Euclidean space is either given by the semisimple Lie group $$SU(2)\times SU(2)$$ or equivalently by the Lie algebra $$A_1\times A_1$$, or by the simple Lie group O(5) or its Lie algebra called $$C_2$$ or equivalently $$B_2$$. In this paper we consider the first of these possibilities which is applied to data which is given in 2 orthogonal directions—hence the method is a concatenation of two 1-dimensional cases. The asymmetry we underline here is a different density of discrete data points in the two orthogonal directions which cannot be studied with the simple Lie group symmetry.

In this paper we investigate different ways of deriving the A-cycle period as a series in instanton counting parameter $q$ for ${\cal N}=2$ SYM with up to four antifundamental hypermultiplets in NS limit of $\Omega$ background. We propose a new method for calculating the period and demonstrate its efficiency by explicit calculations. The new way of doing instanton counting is more advantageous compared to known standard techniques and allows to reach substantially higher order terms with less effort. This approach is applied for the pure case as well as for the case with several hypermultiplets. We also investigate a numerical method for deriving the $A$-cycle period valid for arbitrary values of $q$. Analyzing large $q$ asymptotic we get convincing agreement with an analytic expression deduced from a conjecture by Alexei Zamolodchikov in a different context., Comment: 28 pages, 6 figures, some clarifications and citations added, published version

The spectral properties of a set of local gauge (BRST) invariant composite operators are investigated in theSU(2) Yang–Mills–Higgs model with a single Higgs field in the fundamental representation, quantized in the ’t Hooft$$R_{\xi }$$Rξ-gauge. These operators can be thought of as a BRST invariant version of the elementary fields of the theory, the Higgs and gauge fields, with which they share a gauge independent pole mass. The two-point correlation functions of both BRST invariant composite operators and elementary fields, as well as their spectral functions, are investigated at one-loop order. It is shown that the spectral functions of the elementary fields suffer from a strong unphysical dependence from the gauge parameter$$\xi $$ξ, and can even exhibit positivity violating behaviour. In contrast, the BRST invariant local operators exhibit a well defined positive spectral density.

Coherent-state superpositions are of great importance for many quantum subjects, ranging from foundational to technological, e.g., from tests of collapse models to quantum metrology. Here we explore various aspects of these states, related to the connection between sub-Planck structures present in their Wigner function and their sensitivity to displacements (ultimately determining their metrological potential). We review this for the usual Heisenberg-Weyl algebra associated to a harmonic oscillator, and extend it to find analogous results for the $\mathfrak{su}(2)$ algebra, typically associated with angular momentum. In particular, in the Heisenberg-Weyl case, we identify phase-space structures with support smaller than the Planck action in both Schr\"{o}dinger-cat-state mixtures and superpositions, the latter known as compass states. However, as compared to coherent states, compass states are shown to have $\sqrt{N}$-enhanced sensitivity against displacements in all phase-space directions ($N$ is the average number of quanta), whereas cat states and cat mixtures show such enhanced sensitivity only for displacements in specific directions. We then show that these same properties apply for analogous SU(2) states provided (i) coherent states are restricted to the equator of the sphere that plays the role of phase space for this group, (ii) we associate the role of the Planck action to the size of SU(2) coherent states in such a sphere, and (iii) we associate the role of $N$ with the total angular momentum.

We provide a closed-form expression for the motivic Kontsevich-Soibelman invariant for M theory in the background of the toric Calabi-Yau threefold KF0. This encodes the refined Bogomol'nyi-Prasad-Sommerfield spectrum of SU(2) 5D N=1 Yang-Mills theory on S1×R4, corresponding to rank-zero Donaldson-Thomas invariants for KF0, anywhere on the Coulomb branch., Physical Review Letters, 126 (21), ISSN:0031-9007, ISSN:1079-7114

The Stern&ndash, Gerlach experiment is notoriously counter-intuitive. The official theory is that the spin of a fermion remains always aligned with the magnetic field. Its directions are thus quantized: It can only be spin-up or spin-down. However, that theory is based on mathematical errors in the way it (mis)treats spinors and group theory. We present here a mathematically rigorous theory for a fermion in a magnetic field, which is no longer counter-intuitive. It is based on an understanding of spinors in SU(2) which is only Euclidean geometry. Contrary to what Pauli has been reading into the Stern&ndash, Gerlach experiment, the spin directions are not quantized. The new corrected paradigm, which solves all conceptual problems, is that the fermions precess around the magnetic-field just as Einstein and Ehrenfest had conjectured. Surprisingly, this leads to only two energy states, which should be qualified as precession-up and precession-down rather than spin-up and spin-down. Indeed, despite the presence of the many different possible angles &theta, between the spin axis s and the magnetic field B, the fermions can only have two possible energies m0c2&plusmn, &mu, B. The values &plusmn, B thus do not correspond to the continuum of values &minus, &middot, B Einstein and Ehrenfest had conjectured. The energy term V=&minus, B is a macroscopic quantity. It is a statistical average over a large ensemble of fermions distributed over the two microscopic states with energies &plusmn, B, and as such not valid for individual fermions. The two fermion states with energy &plusmn, B are not potential-energy states. We also explain the mathematically rigorous meaning of the up and down spinors. They represent left-handed and right-handed reference frames, such that now everything is intuitively clear and understandable in simple geometrical terms. The paradigm shift does not affect the Pauli principle.

We study a $SU(2)$-symmetric spin-${3}/{2}$ system on a bipartite lattice close to the antiferromagnetic $SU(4)$-symmetric point, which can be described by the $CP^{3}$ model with a perturbation breaking the symmetry from $SU(4)$ down to $SU(2)$ and favoring the N\'eel ordering. We show that the effective theory of the perturbed model is not the usual $O(3)$ nonlinear sigma model (NLSM), but rather the $O(3)\times O(2)$ NLSM. We show that in the presence of perturbation, the topological charge $q$ of the $CP^{3}$ field is connected to the $O(3)$-NLSM type topological charge of the spin texture $Q$ (defined in a usual way via the unit N\'eel vector) by the relation $q=3Q$, thus under the influence of the perturbation unit-charge skyrmions of $CP^{3}$ model bind into triplets. We also show that in the general spin-$S$ case, symmetry breaking from $SU(2S+1)$ to $SU(2)$ results in the general relation $2S Q_{O(3)}=q_{CP^{2S}}$ between $CP^{2S}$ and $O(3)$ charges, so one can expect $2S$-multiplet binding of skyrmions., Comment: 8 pages

In this study, we investigate the Monopole-Antimonopole Pair (MAP) solution in the SU(2)×U(1) Weinberg- Salam theory with ¢-winding number, n = 3 for bifurcation phenomena. At n = 3, the ‘t Hooft-Polyakov monopole merges with antimonopole to form a vortex ring. Other than fundamental solution, two new bifurcating full vortex ring solution branches were found when Higgs coupling constant, λ reaches a critical value λc. The two new branches possess higher energies than the fundamental solutions. These solutions behave differently from the vortex ring configuration we studied in SU(2) Georgi-Glashow theory. We investigated on the total energy E, vortex ring diameter dρ, and magnetic dipole moment µm, for 0 ≤ λ ≤49.

We investigate, whereas the SU(4)/SU(2) model, is viable for describing strong interactions. The existing problems of confinement and asymptotic freedom, two phenomena that can not be described by the SU(3) model, might indicate that we need something "larger" than the SU(3) model. The considered coset SU(4)/SU(2) or the Stiefel manifold V^2(C_4) contains an su(3) algebra, plus additional degrees of freedom that resembes the Feddev-Poppov concept. The richer structure of this coset, give us enough room, to seek for new phenomena, as its dimensionality is 12. The consideration of the SU(4)/SU(2) model flavors firstly the unification of nuclear fields (strong and weak) in an SU(4) model.

We discuss the deconfinement and the $\mathcal{C}\mathcal{P}$-breaking phase transitions at $\ensuremath{\theta}=\ensuremath{\pi}$ in Yang-Mills theories. The 't Hooft anomaly matching prohibits the confined phase with $\mathcal{C}\mathcal{P}$ symmetry and requires ${T}_{\mathrm{dec}}(\ensuremath{\theta}=\ensuremath{\pi})\ensuremath{\le}{T}_{\mathrm{CP}}$, where ${T}_{\mathrm{dec}}(\ensuremath{\theta}=\ensuremath{\pi})$ and ${T}_{\mathrm{CP}}$ denote the deconfinement and the $\mathcal{C}\mathcal{P}$-restoration temperatures, respectively, at $\ensuremath{\theta}=\ensuremath{\pi}$. We analytically study these two phase transitions in softly broken $\mathcal{N}=1$ supersymmetric Yang-Mills theories on small ${\mathbb{R}}^{3}\ifmmode\times\else\texttimes\fi{}{S}^{1}$ with the periodic boundary condition for gluinos. For most gauge groups except SU(2) in this model, we find that the inequality is saturated, so deconfinement and $\mathcal{C}\mathcal{P}$ restoration occur simultaneously. We demonstrate special features of the SU(2) gauge theory: there is a finite window of two temperatures, ${T}_{\mathrm{dec}}(\ensuremath{\pi})l{T}_{\mathrm{CP}}$, which indicates the existence of a novel $\mathcal{C}\mathcal{P}$-broken deconfined phase. We also discuss an implication of the novel phase for domain walls and their junctions.

$Q\bar{Q}$ potential is studied in the SU(2) gauge theory. Based on the nonlinear gauge of the Curci–Ferrari type, the possibility of a gluon condensation $\langle A_{\mu}^+A_{\mu}^-\rangle$ in a low-energy region has been considered at the one-loop level. Instead of the magnetic monopole condensation, this condensation makes classical gluons massive, and can yield a linear potential. We show that this potential consists of the Coulomb plus linear part and an additional part. Comparing with the Cornell potential, we study this confining potential in detail, and find that the potential has two implicit scales: $r_c$ and $\tilde{R}_0$. The meanings of these scales are clarified. We also show that the Cornell potential that fits well to this confining potential is obtained by taking these scales into account.

In this work we demonstrate new BCH-like relations involving the generators of the su(1, 1), su(2) and so(2, 1) Lie algebras. We use our results to obtain in a straightforward way the composition of an arbitrary number of elements of the corresponding Lie groups. In order to make a self-consistent check of our results, as a first application we recover the non-trivial composition law of two arbitrary squeezing operators. As a second application, we show how our results can be used to compute the time evolution operator of physical systems described by time-dependent hamiltonians given by linear combinations of the generators of the aforementioned Lie algebras., 7 pages

The Lie algebra of the Lorentz group O(3,3) admits two types of SU(2) &times, SU(2) subalgebras: a standard form based on spatial rotation generators and a second form based on temporal rotation generators. The units of measurement for the conserved quantity due to invariance under temporal rotations are investigated and found to be the same units of measure as the Planck constant. The breaking of time reversal symmetry is considered and found to affect the chiral properties of a temporal SU(2) &times, SU(2) algebra. Finally, the symmetry between algebras is explored and pairs of algebras are found to be related by SU(2) &times, U(1) symmetry, while a group of three algebras are related by SO(4) symmetry.

We study the phase diagram of a one-dimensional version of the Kitaev spin-1/2 model with an extra ``$\Gamma$-term", using analytical, density matrix renormalization group and exact diagonalization methods. Two intriguing phases are found. In the gapless phase, although the exact symmetry group of the system is discrete, the low energy theory is described by an emergent SU(2)$_1$ Wess-Zumino-Witten (WZW) model. On the other hand, the spin-spin correlation functions exhibit SU(2) breaking prefactors, even though the exponents and the logarithmic corrections are consistent with the SU(2)$_1$ predictions. A modified nonabelian bosonization formula is proposed to capture such exotic emergent ``partial" SU(2) symmetry. In the ordered phase, there is numerical evidence for an $O_h\rightarrow D_4$ spontaneous symmetry breaking., Comment: 34 pages, 18 figures

We construct a general form for an F-theory Weierstrass model over a general base giving a 6D or 4D supergravity theory with gauge group (SU(3) × SU(2) × U(1))/ℤ6 and generic associated matter, which includes the matter content of the standard model. The Weierstrass model is identified by unHiggsing a model with U(1) gauge symmetry and charges q

We perform the momentum-space quantization of a spin-less particle moving on the SU(2) group manifold, that is, the three-dimensional sphere S-3, by using a non-canonical method entirely based on symmetry grounds. To achieve this task, non-standard (contact) symmetries are required as already shown in a previous article where the configuration-space quantization was given. The Hilbert space in the momentum space representation turns out to be made of the subset of oscillatory solutions of the Helmholtz equation in four dimensions. The most relevant result is the fact that both the scalar product and the generalized Fourier transform between configuration and momentum spaces deviate notably from the naively expected expressions, the former exhibiting now a non-trivial kernel, under a double integral, traced back to the non-trivial topology of the phase space, even though the momentum space as such is flat. In addition, momentum space itself appears directly as the carrier space of an irreducible representation of the symmetry group, and the Fourier transform as the unitary equivalence between two unitary irreducible representations., The authors thank the Spanish Ministerio de Ciencia, Innovacion y Universidades for financial support (FIS2017-84440-C2-2-P) and K B Wolf and G Garrigos for useful discussions. V A acknowledges financial support from the State Agency for Research of the Spanish MCIU through the 'Center of Excellence Severo Ochoa' award for the Instituto de Astrofisica de Andalucia (SEV-2017-0709). J G acknowledges financial support from the Spanish MICINN (PGC2018-097831-B-I00).

The quantum concurrence of $$\textit{SU}(2) \otimes \textit{SU}(2)$$ spin–parity states is shown to be invariant under $$\textit{SO}(1,3)$$ Lorentz boosts and O(3) rotations when the density matrices are constructed in consonance with the covariant probabilistic distribution of Dirac massive particles. Similar invariance properties are obtained for the quantum purity and for the trace of unipotent density matrix operators. The reported invariance features—obtained in the scope of the $$\textit{SU}(2) \otimes \textit{SU}(2)$$ corresponding to just one of the inequivalent representations enclosed by the $$\textit{SL}(2,{\mathbb {C}})\otimes \textit{SL}(2,{\mathbb {C}})$$ symmetry—set a more universal and kinematical-independent meaning for the quantum entanglement encoded in systems containing not only information about spin polarization but also the correlated information about intrinsic parity. Such a covariant framework is used for computing the Lorentz invariant spin–parity entanglement of spinorial particles coupled to a magnetic field, through which the extensions to more general Poincare classes of spinor interactions are straightforwardly depicted.

We study the fused $SU(2)$ models put forward by Date et al., that are a series of models with arbitrary number of blocks, which is the degree of the polynomial equation obeyed by the Boltzmann weights. We demonstrate by a direct calculation that a version of BMW (Birman--Murakami--Wenzl) algebra is obeyed by five, six and seven blocks models, conjecturing that it is part of the algebra valid for any model with more than two blocks. To establish this conjecture, we assume that a certain ansatz holds for the baxterization of the models. We use the Yang--Baxter equation to describe explicitly the algebra for five blocks, obtaining $19$ additional non--trivial relations. We name this algebra 5--CB (Conformal Braiding) algebra. Our method can be utilized to describe the algebra for any solvable model of this type and for any number of blocks., 20 pages, no figures, Mathematica notebook attached. Added references and corrected some erroneous 5--CB relations

The SU(2) gauge theory with 8 fundamental fermions is studied using unimproved staggered regularization. A phase transition or a crossover at strong coupling, which can be a bulk transition. By using chiral random matrix model we analyze the chiral condensate of this system. We also report the chiral susceptibility and its volume dependence near the transition point.

In quantum theory the commutation relations of pairs of observables play a central role. In the last two chapters we saw the commutation relations of the three components of the angular momentum operators and the three infinitesimal rotations \(R_1, R_2\), and \(R_3\).

Let $M$ be complex projective manifold, and $A$ a positive line bundle on it. Assume that $G=SU(2)$ acts on $M$ in a Hamiltonian manner, with nowhere vanishing moment map, and that this action linearizes to $A$. Then there is an associated unitary representation of $G$ on the associated algebro-geometric Hardy space, and the isotypical components are all finite dimensional. We consider the local and global asymptotic properties of the equivariant projector associated to a weight $k , oldsymbol{ u }$, when $oldsymbol{ u }$ is fixed and $k ightarrow +infty$.

We describe a technique to study the asymptotics of SL(2,C) invariant tensors associated to graphs, with unitary irreps and lowest SU(2) spins, and apply it to the Lorentzian EPRL-KKL (Engle, Pereira, Rovelli, Livine; Kaminski, Kieselowski, Lewandowski) model of quantum gravity. We reproduce the known asymptotics of the 4-simplex graph with a different perspective on the geometric variables and introduce an algorithm valid for any graph. On general grounds, we find that critical configurations are not just Regge geometries, but a larger set corresponding to conformal twisted geometries. These can be either Euclidean or Lorentzian, and include curved and flat 4d polytopes as subsets. For modular graphs, we show that multiple pairs of critical points exist, and there exist critical configurations of mixed signature, Euclidean and Lorentzian in different subgraphs, with no 4d embedding possible., 40 Pages + 5 Appendices. 11 Figures. v2: Refined presentation of the general algorithm, additional minor amendments. v3: paragraph added in section 5 about curved embeddings

Monopole solutions in SU(2) Yang-Mills theory which includes spinor fields described by the nonlinear Dirac equation are obtained. It is demonstrated that the energy spectrum of such a system possesses a global minimum whose appearance is brought about solely by the nonlinear spinor fields. It is shown that the monopole solution obtained differs in principle from the 't~Hooft-Polyakov monopole in that it is topologically trivial., Comment: 7 pages, 4 figures

As a modified gravity theory that introduces new gravitational degrees of freedom, the generalized SU(2) Proca theory (GSU2P for short) is the non-Abelian version of the well-known generalized Proca theory where the action is invariant under global transformations of the SU(2) group. This theory was formulated for the first time in Phys. Rev. D 94 (2016) 084041, having implemented the required primary constraint-enforcing relation to make the Lagrangian degenerate and remove one degree of freedom from the vector field in accordance with the irreducible representations of the Poincar\'e group. It was later shown in Phys. Rev. D 101 (2020) 045008, ibid 045009, that a secondary constraint-enforcing relation, which trivializes for the generalized Proca theory but not for the SU(2) version, was needed to close the constraint algebra. It is the purpose of this paper to implement this secondary constraint-enforcing relation in GSU2P and to make the construction of the theory more transparent. Since several terms in the Lagrangian were dismissed in Phys. Rev. D 94 (2016) 084041 via their equivalence to other terms through total derivatives, not all of the latter satisfying the secondary constraint-enforcing relation, the work was not so simple as directly applying this relation to the resultant Lagrangian pieces of the old theory. Thus, we were motivated to reconstruct the theory from scratch. In the process, we found the beyond GSU2P., Comment: LaTeX file in RevTeX 4.1 style, 22 pages, no figures. v2: minor changes, including the small change in the title, in order to make the discussion clearer and more accurate. The appendix, having become inconclusive after we recognized that Eq. A19 is not as general as was originally thought, has been removed. Version to be published in Physical Review D

The way physics of loop quantum gravity is affected by the underlying quantization ambiguities is an open question. We address this issue in the context of loop quantum cosmology using gauge-covariant fluxes. Consequences are explored for two choices of regularization parameters: $\mu_0$ and $\bar \mu$ in presence of a positive cosmological constant, and two choices of regularizations of the Hamiltonian constraint in loop quantum cosmology: the standard and the Thiemann regularization. We show that novel features of singularity resolution and bounce, occurring due to gauge-covariant fluxes, exist also for Thiemann-regularized dynamics. The $\mu_0$-scheme is found to be unviable as in standard loop quantum cosmology when a positive cosmological constant is included. Our investigation brings out a surprising result that the nature of emergent matter in the pre-bounce regime is determined by the choice of regulator in the Thiemann regularization of the scalar constraint whether or not one uses gauge-covaraint fluxes. Unlike $\bar \mu$-scheme where the emergent matter is a cosmological constant, the emergent matter in $\mu_0$-scheme behaves as a string gas., Comment: 31 pages, 10 figures. References and a figure added. To appear in PRD

We point out an error in our previous work, that affects the claims made for the special Lie groups and for the SO(Nc) groups of lowest dimension, which are therefore retracted.Our earlier results stand for larger-dimension SO(Nc), as well as for Sp(Nc) and SU(Nc) classical Lie groups.

We study the Argyres–Douglas theories realized at the superconformal point in the Coulomb moduli space of N = 2 supersymmetric S U ( 2 ) QCD with N f = 1 , 2 , 3 hypermultiplets in the Nekrasov–Shatashvili limit of the Omega-background. The Seiberg–Witten curve of the theory is quantized in this limit and the periods receive the quantum corrections. By applying the WKB method for the quantum Seiberg–Witten curve, we calculate the quantum corrections to the Seiberg–Witten periods around the superconformal point up to the fourth order in the parameter of the Omega background.

We show that the dual partition function of the pure $\mathcal N=2$ $SU(2)$ gauge theory in the self-dual $\Omega$-background (a) is given by Fredholm determinant of a generalized Bessel kernel and (b) coincides with the tau function associated to the general solution of the Painleve III equation of type $D_8$ (radial sine-Gordon equation). In particular, the principal minor expansion of the Fredholm determinant yields Nekrasov combinatorial sums over pairs of Young diagrams.

We extend the free fermionic classification methodology to the class of standard-like heterotic-string vacua, in which the $SO(10)$ GUT symmetry is broken at the string level to $SU(3)\times SU(2)\times U(1)^2$. The space of GGSO free phase configurations in this case is vastly enlarged compared to the corresponding $SO(6)\times SO(4)$ and $SU(5)\times U(1)$ vacua. Extracting substantial numbers of phenomenologically viable models therefore requires a modification of the classification methods. This is achieved by identifying conditions on the GGSO projection coefficients, which are satisfied at the $SO(10)$ level by random phase configurations, and that lead to three generation models with the $SO(10)$ symmetry broken to the $SU(3)\times SU(2)\times U(1)^2$ subgroup. Around each of these fertile $SO(10)$ configurations, we perform a complete classification of standard-like models, by adding the $SO(10)$ symmetry breaking basis vectors, and scanning all the associated GGSO phases. Following this methodology we are able to generate some $10^7$ three generation Standard-like Models. We present the results of the classification and one exemplary model with distinct phenomenological properties, compared to previous SLM constructions., 37 pages. 3 figures. 9 tables. Minor corrections. Published version

We prove uniform decay estimates in the entire exterior of the Schwarzschild black hole for gauge invariant norms on the Yang–Mills fields valued in the Lie algebra associated to the Lie group S U ( 2 ) . We assume that the initial data are spherically symmetric satisfying a certain Ansatz, and have small energy, which eliminates the stationary solutions which do not decay. In particular, there do not exist any Coulomb type solutions satisfying this Ansatz. We first prove a Morawetz type estimate for the Yang–Mills fields within this setting, using the Yang–Mills equations directly. We then adapt the proof constructed in previous work by the first author to show local energy decay and uniform decay of the L ∞ norm of the middle components in the entire exterior of the Schwarzschild black hole, including the event horizon.

A new super-exceptional embedding construction of the heterotic M5-brane's sigma-model was recently shown to produce, at leading order in the super-exceptional vielbein components, the super-Nambu-Goto (Green-Schwarz-type) Lagrangian for the embedding fields plus the Perry-Schwarz Lagrangian for the free abelian self-dual higher gauge field. Beyond that, further fields and interactions emerge in the model, arising from probe M2- and probe M5-brane wrapping modes. Here we classify the full super-exceptional field content and work out some of its characteristic interactions from the rich super-exceptional Lagrangian of the model. We show that SU(2)xU(1)-valued scalar and vector fields emerge from probe M2- and M5-branes wrapping the vanishing cycle in the A_1-type singularity; together with a pair of spinor fields of U(1)-hypercharge +-1 and each transforming as SU(2) iso-doublets. Then we highlight the appearance of a WZW-type term in the super-exceptional PS-Lagrangian and find that on the electromagnetic field it gives the first-order non-linear DBI-correction, while on the iso-vector scalar field it has the form characteristic of the coupling of vector mesons to pions via the Skyrme baryon current. We discuss how this is suggestive of a form of SU(2)-flavor chiral hadrodynamics emerging on the single (N=1) M5 brane, different from, but akin to, holographic large-$N$ QCD., Comment: 38 pages

The integrability of the\ $\Lambda-$Einstein-nonlinear $SU(2)$ $\sigma$-model with nonvanishing cosmological charge is studied. We apply the method of singularity analysis of differential equations and we show that the equations for the gravitational field are integrable. The first few terms of the solution are presented., Comment: 6 pages, 2 figures, published at EPJC