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We perform a systematic study of integrated four-point functions of half-BPS operators in four-dimensional $\mathcal{N}=4$ super Yang-Mills theory with gauge group $SU(N)$. These observables, defined by a certain spacetime integral of $\langle\mathcal{O}_2\mathcal{O}_2\mathcal{O}_p\mathcal{O}_p\rangle$ where $\mathcal{O}_p$ is a superconformal primary of charge $p$, are known to be computable by supersymmetric localization, yet are non-trivial functions of the complexified gauge coupling $\tau$. We find explicit and remarkably simple results for several classes of these observables, exactly as a function of $N$ and $\tau$. Their physical and formal properties are greatly illuminated upon employing the $SL(2,\mathbb{Z})$ spectral decomposition: in this S-duality-invariant eigenbasis, the integrated correlators are fixed simply by polynomials in the spectral parameter. These polynomials are determined recursively by linear algebraic equations relating different $N$ and $p$, such that all integrated correlators are ultimately fixed in terms of the integrated stress tensor multiplets in the $SU(2)$ theory. Our computations include the full matrix of integrated correlators at low values of $p$, and a certain infinite class involving operators of arbitrary $p$. The latter satisfy an open lattice chain equation for all $N$, reminiscent of the Toda equation obeyed by extremal correlators in $\mathcal{N}=2$ superconformal theories. We compute ensemble averages of these observables and analyze our solutions at large $N$, confirming and predicting features of semiclassical AdS$_5\, \times$ S$^5$ supergravity amplitudes., Comment: 42+24 pages

A dark sector with non-abelian gauge symmetry provides a sound framework to justify stable dark matter (DM) candidates. We consider scalar fields charged under a $SU(N)$ gauge group, and show that the centre of $SU(N)$, the discrete subgroup $Z_N$ also known as $N$-ality, can ensure the stability of scalar DM particles. We analyse in some details two minimal DM models of this class, based on $SU(2)$ and $SU(3)$, respectively. These models have non-trivial patterns of spontaneous symmetry breaking, leading to distinctive phenomenological implications. For the $SU(2)$ model these include a specific interplay of two DM states, with the same interactions but different masses, and several complementary DM annihilation regimes, either within the dark sector or through the Higgs portal. The $SU(3)$ model predicts dark radiation made of a pair of dark photons with a unique gauge coupling, as well as regimes where DM semi-annihilations become dominant and testable., 27 pages, 5 figures

In four-dimensional $\mathcal{N}=4$ super Yang-Mills theory with gauge group $SU(N)$, we present a closed-form solution for a family of integrated four-point functions involving stress tensor multiplet composites of arbitrary R-charge. These integrated correlators are shown to be equivalent to a one-dimensional semi-infinite lattice of harmonic oscillators with nearest-neighbor interactions, evolving over the fundamental domain of $SL(2,\mathbb{Z})$. The solution, exceptionally simple in an $SL(2,\mathbb{Z})$-invariant eigenbasis, is exact in the R-charge $p$, rank $N$, and complexified gauge coupling $\tau$. This permits a systematic and non-perturbative large charge expansion for any $N$ and $\tau$. Especially novel is a double-scaled "gravity regime" in which $p \sim N^2 \gg 1$, holographically dual to a large charge regime of semiclassical type IIB string theory in AdS$_5\, \times$ S$^5$. Our results in this limit provide a holographic computation of integrated semiclassical string amplitudes at arbitrary string coupling, including an emergent string scale with a large charge dressing factor. We compare to extremal correlators in superconformal QCD, for which we predict new genus expansions at large charge scaling with $N$., Comment: 31 pages + appendices

39th International Symposium on Lattice Field Theory, Lattice 2022, Bonn, Germany, 8 Aug 2022 - 13 Aug 2022; Proceedings of Science / International School for Advanced Studies (LATTICE2022), 040 (2022). doi:10.22323/1.430.0040, In this proceeding contribution we discuss how to define canonical momenta for SU(N) lattice gauge theories in the Hamiltonian formalism in a basis where the gauge field operators are diagonal. For an explicit discretisation of SU(2) we construct the momenta and check the violation of the fundamental commutation relations., Published by SISSA, Trieste

We study non-planar corrections in two special $\mathcal N=2$ superconformal $SU(N)$ gauge theories that are planar-equivalent to $\mathcal N=4$ SYM theory: two-nodes quiver model with equal couplings and $\mathcal N=2$ vector multiplet coupled to two hypermultiplets in rank-2 symmetric and antisymmetric representations. We focus on two observables in these theories that admit representation in terms of localization matrix model: free energy on 4-sphere and the expectation value of half-BPS circular Wilson loop. We extend the methods developed in arXiv:2207.11475 to derive a systematical expansion of non-planar corrections to these observables at strong 't Hooft coupling constant $\lambda$. We show that the leading non planar corrections are given by a power series in $\lambda^{3/2}/N^2$ with rational coefficients. Sending $N$ and the coupling constant $\lambda$ to infinity with $\lambda^{3/2}/N^2$ kept fixed corresponds to the familiar double scaling limit in matrix models. We find that in this limit the observables in the two models are related in a remarkably simple way: the free energies differ by the factor of $2$, whereas the Wilson loop expectation values coincide. Surprisingly, these relations hold only at strong coupling, they are not valid in the weak coupling regime. We also discuss a dual string theory interpretation of the leading corrections to the free energy in the double scaling limit suggesting their relation to curvature corrections in type IIB string effective action., Comment: 54 pages. v2: minor clarifications

In the holographic approach to cosmology, cosmological observables are described in terms of correlators of a three-dimensional boundary quantum field theory. As a concrete model, we study the $3D$ massless $SU(N)$ scalar matrix field theory with a $\phi^4$ interaction. On the lattice, the energy-momentum tensor (EMT) in this theory can mix with the operator $\phi^2$. We utilize the Wilson Flow to renormalize the EMT on the lattice, and present numerical results for the mixing coefficient for $N = 2$. Obtaining the renormalized EMT will allow us to make predictions for the CMB power spectra in the regime where the dual QFT is non-perturbative., Comment: 10 pages, 3 figures, presented at the 38th International Symposium on Lattice Field Theory - LATTICE2021 - 26-30 July, 2021, Massachusetts Institute of Technology

We propose a broad class of $d$-dimensional conformal field theories of $SU(N)$ adjoint scalar fields generalising the 4$d$ Fishnet CFT (FCFT) discovered by \"O. G\"urdogan and one of the authors as a special limit of $\gamma$-deformed $\mathcal{N}=4$ SYM theory. In the planar $N\to\infty $ limit the FCFTs are dominated by the ``fishnet" planar Feynman graphs. These graphs are explicitly integrable, as was shown long ago by A. Zamolodchikov. The Zamolodchikov's construction, based on the dual Baxter lattice (straight lines on the plane intersecting at arbitrary slopes) and the star-triangle identities, can serve as a ``loom" for ``weaving" the Feynman graphs of these FCFTs, with certain types of propagators, at any $d$. The Baxter lattice with $M$ different slopes and any number of lines parallel to those, generates an FCFT consisting of $M(M-1)$ fields and a certain number of chiral vertices of different valences with distinguished couplings. These non-unitary, logarithmic CFTs enjoy certain reality properties for their spectrum due to a symmetry similar to the PT-invariance of non-hermitian hamiltonians proposed by C. Bender. We discuss in more detail the theories generated by a loom with $M=2,3,4$, and the generalisation of the loom FCFTs for spinning fields in 4$d$., Comment: 35 pages, 21 figures

Constant temporal backgrounds have become a convenient tool for the discussion of continuum non-abelian gauge theories at finite temperature, as they provide a grasp on the confinement/deconfinement transition. It has been pointed that such backgrounds are better dealt with within so-called Cartan-Weyl color bases, rather than within the standard Cartesian bases. Here, we discuss the specificities of the evaluation of SU($N$) color factors within these bases, extending the discussion in Ref. [1]., 27 pages, 1 figure

A nonperturbative determination of the energy-momentum tensor is essential for understanding the physics of strongly coupled systems. The ability of the Wilson flow to eliminate divergent contact terms makes it a practical method for renormalizing the energy-momentum tensor on the lattice. In this paper, we utilize the Wilson flow to define a procedure to renormalize the energy-momentum tensor for a three-dimensional massless scalar field in the adjoint of $SU(N)$ with a $\varphi^4$ interaction on the lattice. In this theory the energy-momentum tensor can mix with $\varphi^2$ and we present numerical results for the mixing coefficient for the $N=2$ theory., Comment: 29 pages, 9 figures

Journal of high energy physics 06(6), 153 (2021). doi:10.1007/JHEP06(2021)153, Using numerical bootstrap method, we determine the critical exponents of the minimal three-dimensional $ \mathcal{N} $ = 1 Wess-Zumino models with cubic superpotetential $ \mathcal{W}\sim {d}_{ijk}{\Phi}^i{\Phi}^j{\Phi}^k $. The tensor d$_{ijk}$ is taken to be the invariant tensor of either permutation group S$_{N}$, special unitary group SU(N), or a series of groups called F$_{4}$ family of Lie groups. Due to the equation of motion, at the Wess-Zumino fixed point, the operator d$_{ijk}$��$^{j}$��$^{k}$ is a (super)descendant of ��$^{i}$. We observe such super-multiplet recombination in numerical bootstrap, which allows us to determine the scaling dimension of the super-field ���$_{��}$., Published by SISSA, [Trieste]

So far, none of attempts to quantize gravity has led to a satisfactory model that not only describe gravity in the realm of a quantum world, but also its relation to elementary particles and other fundamental forces. Here, we outline the preliminary results for a model of quantum universe, in which gravity is fundamentally and by construction quantic. The model is based on three well motivated assumptions with compelling observational and theoretical evidence: quantum mechanics is valid at all scales, quantum systems are described by their symmetries, universe has infinite independent degrees of freedom. The last assumption means that the Hilbert space of the Universe has SU(N&rarr, &infin, )&cong, areapreservingDiff.(S2) symmetry, which is parameterized by two angular variables. We show that, in the absence of a background spacetime, this Universe is trivial and static. Nonetheless, quantum fluctuations break the symmetry and divide the Universe to subsystems. When a subsystem is singled out as reference&mdash, observer&mdash, and another as clock, two more continuous parameters arise, which can be interpreted as distance and time. We identify the classical spacetime with parameter space of the Hilbert space of the Universe. Therefore, its quantization is meaningless. In this view, the Einstein equation presents the projection of quantum dynamics in the Hilbert space into its parameter space. Finite dimensional symmetries of elementary particles emerge as a consequence of symmetry breaking when the Universe is divided to subsystems/particles, without having any implication for the infinite dimensional symmetry and its associated interaction-percived as gravity. This explains why gravity is a universal force.

Exact field theory dualities can be implemented by duality domain walls such that passing any operator through the interface maps it to the dual operator. This paper describes the S-duality wall of four-dimensional ${\cal N}=2$ SU(N) SQCD with 2N hypermultiplets in terms of fields on the defect, namely three-dimensional ${\cal N}=2$ SQCD with gauge group U(N-1) and 2N flavours, with a monopole superpotential. The theory is self-dual under a duality found by Benini, Benvenuti and Pasquetti, in the same way that T[SU(N)] (the S-duality wall of ${\cal N}=4$ super Yang-Mills) is self-mirror. The domain-wall theory can also be realized as a limit of a USp(2N-2) gauge theory; it reduces to known results for N=2. The theory is found through the AGT correspondence by determining the braiding kernel of two semi-degenerate vertex operators in Toda CFT., v2: correct superpotential to fix symmetries, better introduction and references, 42 pages

We propose two models where a U(1) Peccei-Quinn global symmetry arises accidentally and is respected up to high-dimensional operators, so that the axion solution to the strong CP problem is successful even in the presence of Planck-suppressed operators. One model is SU$(N)$ gauge interactions with fermions in the fundamental and a scalar in the symmetric. The axion arises from spontaneous symmetry breaking to SO$(N)$, that confines at a lower energy scale. Axion quality in the model needs $N \gtrsim 10$. SO bound states and possibly monopoles provide extra Dark Matter candidates beyond the axion. In the second model the scalar is in the anti-symmetric: SU$(N)$ broken to Sp$(N)$ needs even $N \gtrsim 20$. The cosmological DM abundance, consisting of axions and/or super-heavy relics, can be reproduced if the PQ symmetry is broken before inflation (Boltzmann-suppressed production of super-heavy relics) or after (super-heavy relics in thermal equilibrium get partially diluted by dark glue-ball decays)., Comment: 26 pages, 5 figures. v3: extra application mentioned; dilution factor in eq. (27) corrected, qualitative conclusions unchanged

We compute the term $\sim g^3$ in the free energy for a $SU(N)$ gauge theory with nonzero holonomy at nonzero temperature. If the holonomy is generated kinematically by the introduction of gauge invariant sources coupled to Polyakov loops, the contribution of charged (off-diagonal) gluons to the free energy at order $g^3$, ${\cal F}^{\left( 3\rm{:c.g.} \right)}$, is singular: ${\cal F}^{\left( 3\rm{:c.g.} \right)} \neq 0 $ without holonomy, but ${\cal F}^{\left( 3 \rm{:c.g.} \right)}= 0$ when the holonomy is nonzero, even infinitesimally. We show that the absence of the charged gluon contribution is required by gauge invariance alone and is therefore a universal feature., 14 pages

Recently an exact duality between topological string and the spectral theory of operators constructed from mirror curves to toric Calabi-Yau threefolds has been proposed. At the same time an exact quantization condition for the cluster integrable systems associated to these geometries has been conjectured. The consistency between the two approaches leads to an infinite set of constraints for the refined BPS invariants of the toric Calabi-Yau threefolds. We prove these constraints for the $Y^{N,m}$ geometries using the $K$-theoretic blowup equations for $SU(N)$ SYM with generic Chern-Simons invariant $m$., Comment: 26 pages, 3 figures, 2 tables, a few typos corrected

We study superconformal deformations of the $T_\rho^{\hat\rho}[SU(N)]$ theories of Gaiotto-Hanany-Witten, paying special attention to mixed-branch operators with both electrically- and magnetically-charged fields. We explain why all marginal ${\cal N}=2$ operators of an ${\cal N}=4$ CFT$_3$ can be extracted unambiguously from the superconformal index. Computing the index at the appropriate order we show that the mixed moduli in $T_\rho^{\hat\rho}[SU(N)]$ theories are double-string operators transforming in the (Adjoint, Adjoint) representation of the electric and magnetic flavour groups, up to some overcounting for quivers with abelian gauge nodes. We comment on the holographic interpretation of the results, arguing in particular that gauged supergravities can capture the entire moduli space if, in addition to the (classical) parameters of the background solution, one takes also into account the (quantization) moduli of boundary conditions., Comment: 35 pages. v3: published version (more explicit formulas)

Using numerical bootstrap method, we determine the critical exponents of the minimal three-dimensional $N$ = 1 Wess-Zumino models with cubic superpotetential $W\sim d_{ijk}\Phi_i\Phi_j\Phi_k$. The tensor $d_{ijk}$ is taken to be the invariant tensor of either permutation group $S_N$, special unitary group $SU(N)$, or a series of groups called F$_4$ family of Lie groups. Due to the equation of motion, at the Wess-Zumino fixed point, the operator $d_{ijk}\Phi_i\Phi_j\Phi_k$ is a (super)descendant of $\Phi_i$ . We observe such super-multiplet recombination in numerical bootstrap, which allows us to determine the scaling dimension of the super-field $\Phi_i$.

We study the universal critical properties of the QED$_3$-Gross-Neveu-Yukawa model with $N$ flavors of four-component Dirac fermions coupled to a real scalar order parameter at four-loop order in the $\epsilon$ expansion below four dimensions. For $N=1$, the model is conjectured to be infrared dual to the $SU(2)$-symmetric noncompact $\mathbb{C}$P$^1$ model, which describes the deconfined quantum critical point of the N\'eel-valence-bond-solid transition of spin-1/2 quantum antiferromagnets on the two-dimensional square lattice. For $N=2$, the model describes a quantum phase transition between an algebraic spin liquid and a chiral spin liquid in the spin-1/2 kagom\'e antiferromagnet. For general $N$ we determine the order parameter anomalous dimension, the correlation length exponent, the stability critical exponent, as well as the scaling dimensions of $SU(N)$ singlet and adjoint fermion mass bilinears at the critical point. We use Pad\'e approximants to obtain estimates of critical properties in 2+1 dimensions., Comment: 18 pages, 12 figures, 3 tables

Journal of high energy physics 1807(07), 169 (2018). doi:10.1007/JHEP07(2018)169, We compute and analyse the low-lying spectrum of 2+1 dimensional SU(N) Yang-Mills theory on a spatial torus of size l × l with twisted boundary conditions. This paper extends our previous work [1]. In that paper we studied the sector with non-vanishing electric flux and concluded that the energies only depend on the parameters through two combinations: x = λN l/(4π) (with λ the ’t Hooft coupling) and the twist angle $ \tilde{\theta} $ defined in terms of the magnetic flux piercing the two-dimensional box. Here we made a more complete study and we are able to condense our results, obtained by non-perturbative lattice methods, into a simple expression which has important implications for the absence of tachyonic instabilities, volume independence and non-commutative field theory. Then we extend our study to the sector of vanishing electric flux. We conclude that the onset of the would-be large-volume glueball states occurs at an approximately fixed value of x, much before the stringy torelon states have become very massive., Published by Springer, Berlin

We discuss the $SU(3)/[U(1)\times U(1)]$ nonlinear sigma model in 1+1D and, more broadly, its linearized counterparts. Such theories can be expressed as $U(1)\times U(1)$ gauge theories and therefore allow for two topological $\theta$-angles. These models provide a field theoretic description of the $SU(3)$ chains. We show that, for particular values of $\theta$-angles, a global symmetry group of such systems has a 't Hooft anomaly, which manifests itself as an inability to gauge the global symmetry group. By applying anomaly matching, the ground-state properties can be severely constrained. The anomaly matching is an avatar of the Lieb-Schultz-Mattis (LSM) theorem for the spin chain from which the field theory descends, and it forbids a trivially gapped ground state for particular $\theta$-angles. We generalize the statement of the LSM theorem and show that 't Hooft anomalies persist even under perturbations which break the spin-symmetry down to the discrete subgroup $\mathbb Z_3\times\mathbb Z_3\subset SU(3)/\mathbb Z_3$. In addition the model can further be constrained by applying global inconsistency matching, which indicates the presence of a phase transition between different regions of $\theta$-angles. We use these constraints to give possible scenarios of the phase diagram. We also argue that at the special points of the phase diagram the anomalies are matched by the $SU(3)$ Wess-Zumino-Witten model. We generalize the discussion to the $SU(N)/U(1)^{N-1}$ nonlinear sigma models as well as the 't Hooft anomaly of the $SU(N)$ Wess-Zumino-Witten model, and show that they match. Finally the $(2+1)$-dimensional extension is considered briefly, and we show that it has various 't Hooft anomalies leading to nontrivial consequences., Comment: 28 pages, 3 figures;(v2) Sec.VB added, Fig.2 updated, refs updated

We study the universal critical properties of the QED$_3$-Gross-Neveu-Yukawa model with $N$ flavors of four-component Dirac fermions coupled to a real scalar order parameter at four-loop order in the $\epsilon$ expansion below four dimensions. For $N=1$, the model is conjectured to be infrared dual to the $SU(2)$-symmetric noncompact $\mathbb{C}$P$^1$ model, which describes the deconfined quantum critical point of the N\'eel-valence-bond-solid transition of spin-1/2 quantum antiferromagnets on the two-dimensional square lattice. For $N=2$, the model describes a quantum phase transition between an algebraic spin liquid and a chiral spin liquid in the spin-1/2 kagom\'e antiferromagnet. For general $N$ we determine the order parameter anomalous dimension, the correlation length exponent, the stability critical exponent, as well as the scaling dimensions of $SU(N)$ singlet and adjoint fermion mass bilinears at the critical point. We use Pad\'e approximants to obtain estimates of critical properties in 2+1 dimensions.

In this note, I revisit integrals over $\SU(N)$ of the form $ \int DU\, U_{i_1j_1}\cdots U_{i_pj_p}\Ud_{k_1l_1}\cdots \Ud_{k_nl_n}$. While the case $p=n$ is well known, it seems that explicit expressions for $p=n+N$ had not appeared in the literature. Similarities and differences, in particular in the large $N$ limit, between the two cases are discussed, Comment: 1 figure

International audience; We discuss quantum field theories with global SU(N) and O(N) symmetries for which temporal direction is compactified on a circle of size L with periodicity of fields up to a global symmetry transformation, i.e., twisted boundary conditions. Such boundary conditions correspond to an insertion of the global symmetry operator in the partition function. We argue in general and prove in particular for CP(N-1) and O(N) nonlinear sigma models that large-N volume independence holds. Further we show that the CP(N-1) theory is free from the Affleck phase transition confirming the Ünsal-Dunne continuity conjecture.

We extend our recent study on the duality between stringy higher spin theories and free CFTs in the $SU(N)$ adjoint representation to other matrix models namely the free $SO(N)$ and $Sp(N)$ adjoint models as well as the free $U(N)\times U(M)$ bi-fundamental and $O(N)\times O(M)$ bi-vector models. After determining the spectrum of the theories in the planar limit by Polya counting, we compute the one loop vacuum energy and Casimir energy for their respective bulk duals by means of the CIRZ method that we have introduced recently. We also elaborate on possible ambiguities in the application of this method., 37 pages, 7 figures

We study M-theory on a Calabi-Yau fourfold with a smooth surface $S$ of $A_{N-1}$ singularities. The resulting three-dimensional theory has a $\mathcal{N}=2$ $SU(N)$ gauge theory sector, which we obtain from a twisted dimensional reduction of a seven-dimensional $\mathcal{N}=1$ $SU(N)$ gauge theory on the surface $S$. A variant of the Vafa-Witten equations governs the moduli space of the gauge theory, which, for a trivial $SU(N)$ principal bundle over $S$, admits a Coulomb and a Higgs branch. In M-theory these two gauge theory branches arise from a resolution and a deformation to smooth Calabi-Yau fourfolds, respectively. We find that the deformed Calabi-Yau fourfold associated to the Higgs branch requires for consistency a non-trivial four-form background flux in M-theory. The flat directions of the flux-induced superpotential are in agreement with the gauge theory prediction for the moduli space of the Higgs branch. We illustrate our findings with explicit examples that realize the Coulomb and Higgs phase transition in Calabi-Yau fourfolds embedded in weighted projective spaces. We generalize and enlarge this class of examples to Calabi-Yau fourfolds embedded in toric varieties with an $A_{N-1}$ singularity in codimension two., Comment: 33 pages, v2: references added, v3: references added --- version published in CMP

The quantum formalism can be completed by assuming that a density operator can also represent a pure state. An 'extended Bloch representation' (EBR) then results, in which not only states, but also the measurement-interactions can be represented. The Born rule is obtained as an expression of the subjective lack of knowledge about the measurement-interaction that is each time actualized. Entanglement can also be consistently described in the EBR, as it remains compatible with the principle according to which a composite entity exists only if its components also exist, and therefore are in pure states., 11 pages, 3 figures

In this paper we describe a new integration method for the groups $U(N)$ and $SU(N)$, for which we verified numerically that it is polynomially exact for $N\le 3$. The method is applied to the example of 1-dimensional QCD with a chemical potential. We explore, in particular, regions of the parameter space in which the sign problem appears due the presence of the chemical potential. While Markov Chain Monte Carlo fails in this region, our new integration method still provides results for the chiral condensate on arbitrary precision, demonstrating clearly that it overcomes the sign problem. Furthermore, we demonstrate that our new method leads to orders of magnitude reduced errors also in other regions of parameter space., added reference in introduction

(2016)., We compute the topological susceptibility of the SU(N) Yang-Mills theory in the large-N limit with a percent level accuracy. This is achieved by measuring the gradient-flow definition of the susceptibility at three values of the lattice spacing for N=3,4,5,6. Thanks to this coverage of parameter space, we can extrapolate the results to the large-N and continuum limits with confidence. Open boundary conditions are instrumental to make simulations feasible on the finer lattices at the larger N.

In this work we provide for a description of the low-energy physics of interacting multi-species fermions in terms of the bound-states that are stabilized in these systems when a spin gap opens. We argue that, at energies much smaller than the spin gap, these systems are described by a Luttinger liquid of bound-states that depends, on top of the charge stiffness $\nu$ and the charge velocity $u$, on a "Fermi" momentum $P_F$ satisfying $qP_F = Nk_F$ where $q$ is the charge of the bound-state, $N$ the number of species and $k_F$ is the Fermi momentum in the non-interacting limit. We further argue that for generic interactions, generic bound-states are likely to be stabilized. They are associated with emergent, in general non-local, symmetries and are in the number of five. The first two consist of either a charge $q=N$ local $SU(N)$ singlet or a charge $q=N$ bound-state made of two local $SU(p)$ and $SU(N-p)$ singlets. In this case the Fermi momentum $P_F=k_F$ is preserved. The three others have an enhanced Fermi vector $P_F$. The latter are either charge $q=2$ bosonic p-wave and s-wave pairs with $SO(N)$ and $SP(N)$ symmetry and $P_F=Nk_F/2$ or a composite fermion of charge $q=1$ with $P_F=Nk_F$. The instabilities of these Luttinger liquid states towards incompressible phases and their possible topological nature are also discussed., Comment: 46 Pages

We compute the topological susceptibility of the SU(N) Yang-Mills theory in the large-N limit with a percent level accuracy. This is achieved by measuring the gradient-flow definition of the susceptibility at three values of the lattice spacing for N=3,4,5,6. Thanks to this coverage of parameter space, we can extrapolate the results to the large-N and continuum limits with confidence. Open boundary conditions are instrumental to make simulations feasible on the finer lattices at the larger N., Comment: 10 pages, 1 figure

Entangled states are in conflict with a general physical principle which expresses that a composite entity exists if and only if its components also exist, and the hypothesis that pure states represent the actuality of a physical entity, i.e., its 'existence'. A possible way to solve this paradox consists in completing the standard formulation of quantum mechanics, by adding more pure states. We show that this can be done, in a consistent way, by using the extended Bloch representation of quantum mechanics, recently introduced to provide a possible solution to the measurement problem. Hence, with the solution proposed by the extended Bloch representation of quantum mechanics, the situation of entangled states regains full intelligibility., 7 pages

A generalized Bloch sphere, in which the states of a quantum entity of arbitrary dimension are geometrically represented, is investigated and further extended, to also incorporate the measurements. This extended representation constitutes a general solution to the measurement problem, inasmuch it allows to derive the Born rule as an average over hidden-variables, describing not the state of the quantum entity, but its interaction with the measuring system. According to this modelization, a quantum measurement is to be understood, in general, as a tripartite process, formed by an initial deterministic decoherence-like process, a subsequent indeterministic collapse-like process, and a final deterministic purification-like process. We also show that quantum probabilities can be generally interpreted as the probabilities of a first-order non-classical theory, describing situations of maximal lack of knowledge regarding the process of actualization of potential interactions, during a measurement., 44 pages and 9 figures

Using the generating function of SU(n) we find the conjugate state of SU(n) basis and we find in terms of Gel'fand basis of SU(3(n-1)) the representation of the invariants of the Kronecker products of SU(n). We find a formula for the number of the elementary invariants of SU(n). We apply our method to the coupling of SU(3) and we find a new expression of the isoscalar of Wigner symbols ($\lambda 10,\lambda 2 \mu 2; \lambda 3 \mu 3$).