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Relaxation rates in nearly integrable systems usually increase quadratically with the strength of the perturbation that breaks integrability. We show that the relaxation rates can be significantly smaller in systems that are integrable along two intersecting lines in the parameter space. In the vicinity of the intersection point, the relaxation rates of certain observables increase with the fourth power of the distance from this point, whereas for other observables one observes standard quadratic dependence on the perturbation. As a result, one obtains exceedingly long-living prethermalization but with a reduced number of the nearly conserved operators. We show also that such a scenario can be realized in spin ladders., Comment: 12 pages, 8 figures (including Supplementary Material)

Purely ballistic transport is a rare feature even for integrable models. By numerically studying the Heisenberg chain with the power-law exchange, \mbox{$J\propto1/r^\alpha$}, where $r$ is a distance, we show that for spin anisotropy $\Delta \simeq \exp(-\alpha+2)$ the system exhibits a quasiballistic spin transport and the presence of fermionic excitation which do not decay up to extremely long times $\sim10^3/J$. This conclusion is reached on the base of the dynamics of spin domains, the dynamical spin conductivity, inspecting the matrix elements of the spin-current operator, and by the analysis of most conserved operators. Our results smoothly connects two models where fully ballistic transport is present: free particles with nearest-neighbor hopping and the isotropic Haldane-Shastry model.

A fluid-dynamic approach to the diffusion of heavy quarks in the quark--gluon plasma (QGP) is presented. Specifically, we analyze the Fokker-Planck equation for the momentum transport of heavy quarks from a fluid perspective and use a mapping to second-order fluid-dynamics to determine conductivities and relaxation times governing their spatial diffusion. By investigating the relation between the two approaches, we provide new insights concerning the level of local thermalization of charm and bottom quarks inside the expanding QGP. Our results indicate that a fluid-dynamic description of diffusion is feasible for charm quarks at least for the latest stages of the fireball evolution., Comment: 8 pages, 4 figures

We numerically study the relaxation of correlation functions in weakly perturbed integrable XXZ chain. The decay of the spin-current and the energy-current correlations at zero magnetization are well described by single, but quite distinct, relaxation rates governed by the square of the perturbation strength $g$. However, at finite magnetization a single correlation function reveals multiple relaxation rates. The result can be understood in terms of multi-scale relaxation scenario, where various relaxation times are linked with various quantities which are conserved in the reference integrable system. On the other hand, the correlations of non-commuting quantities, being conserved at particular anisotropies $\Delta$, decay non-exponentially with characteristic time scale linear in $g$.

We explore a two-qubit system defined on valley isospins of two electrons confined in a gate-defined double quantum dot created within a MoS$_2$ monolayer flake. We show how to initialize, control, interact and read out such valley qubits only by electrical means using voltages applied to the local planar gates, which are layered on the top of the flake. By demonstrating the two-qubit exchange or readout via the Pauli blockade, we prove that valley qubits in transition-metal-dichalcogenide semiconductors family fulfill the universality criteria and represent a scalable quantum computing platform. Our numerical experiments are based on the tight-binding model for a MoS$_2$ monolayer, which gives single-electron eigenstates that are then used to construct a basis of Slater-determinants for the two-electron configuration space. We express screened electron-electron interactions in this basis by calculating the Coulomb matrix elements using localized Slater-type orbitals. Then we solve the time-dependent Schr\"odinger equation and obtain an exact time-evolution of the two-electron system. During the evolution we simultaneously solve the Poison equation, finding the confinement potential controlled via voltages applied to the gates., Comment: 15 pages, 12 figures

We consider a single electron confined within a quantum wire in a system of two electrostatically-induced QDs defined by nearby gates. The time-varying electric field, of single GHz frequency, perpendicular to the quantum wire, is used to induce the Rashba coupling and enable spin-dependent resonant tunneling of the electron between two adjacent potential wells with fidelity over 99.5%. This effect can be used for the high fidelity all-electrical electron-spin initialization or readout in the spin-based quantum computer. In contrast to other spin initialization methods, our technique can be performed adiabatically without increase in the energy of the electron. Our simulations are supported by a realistic self-consistent time-dependent Poisson-Schroedinger calculations., Comment: 7 pages, 3 figures

The renormalization group plays an essential role in many areas of physics, both conceptually and as a practical tool to determine the long-distance low-energy properties of many systems on the one hand and on the other hand search for viable ultraviolet completions in fundamental physics. It provides us with a natural framework to study theoretical models where degrees of freedom are correlated over long distances and that may exhibit very distinct behavior on different energy scales. The nonperturbative functional renormalization-group (FRG) approach is a modern implementation of Wilson's RG, which allows one to set up nonperturbative approximation schemes that go beyond the standard perturbative RG approaches. The FRG is based on an exact functional flow equation of a coarse-grained effective action (or Gibbs free energy in the language of statistical mechanics). We review the main approximation schemes that are commonly used to solve this flow equation and discuss applications in equilibrium and out-of-equilibrium statistical physics, quantum many-particle systems, high-energy physics and quantum gravity., Comment: v3) Review article, 93 pages + bibliography, 35 figures

This report summarizes the presentations and discussions during the Rapid Reaction Task Force "Dynamics of critical fluctuations: Theory -- phenomenology -- heavy-ion collisions", which was organized by the ExtreMe Matter Institute EMMI and held at GSI, Darmstadt, Germany in April 2019. We address the current understanding of the dynamics of critical fluctuations in QCD and their measurement in heavy-ion collision experiments. In addition, we outline what might be learned from studying correlations in other physical systems, such as cold atomic gases., Comment: 89 pages, 11 figures, report on an ExtreMe Matter Institute EMMI Rapid Reaction Task Force

The aim of presented research is to design a nanodevice based on a gate-defined quantum dot within a MoS$_2$ monolayer in which we confine a single electron. By applying control voltages to the device gates we modulate the confinement potential and force intervalley transitions. The present Rashba spin-orbit coupling additionally allows for spin operations. Moreover, both effects enable the spin-valley SWAP. The device structure is modeled realistically, taking into account feasible dot-forming potential and electric field that controls the Rasha coupling. Therefore, by performing reliable numerical simulations, we show how by electrically controlling the state of the electron in the device, we can obtain single- and two-qubit (thus universal) gates in a spin-valley two-qubit system. Through simulations we investigate possibility of implementation of two qubits \textit{locally}, based on single electron, with an intriguing feature that two-qubit gates are easier to realize than single ones., Comment: 13 pages, 17 figures

Physics Beyond Colliders is an exploratory study aimed at exploiting the full scientific potential of CERN's accelerator complex and its scientific infrastructure in the next two decades through projects complementary to the LHC, HL-LHC and other possible future colliders. These projects should target fundamental physics questions that are similar in spirit to those addressed by high-energy colliders, but that require different types of beams and experiments. A kick-off workshop held in September 2016 identified a number of areas of interest and working groups have been set-up to study and develop these directions. All projects currently under consideration are presented including physics motivation, a brief outline of the experimental set-up and the status of the corresponding beam and detector technological studies. The proposals are also put in context of the worldwide landscape and their implementation issues are discussed., Comment: This document (66 pages, 19 figures) is the summary document of the Physics Beyond Colliders (PBC) study. It follows the PBC mandate and draws on a whole set of documents produced in the context of PBC (https://pbc.web.cern.ch), in particular the reports of the QCD and BSM working groups also available at arXiv:1901.04482 and arXiv:1901.09966

The present document discusses plans for a compact, next-generation multi-purpose detector at the LHC as a follow-up to the present ALICE experiment. The aim is to build a nearly massless barrel detector consisting of truly cylindrical layers based on curved wafer-scale ultra-thin silicon sensors with MAPS technology, featuring an unprecedented low material budget of 0.05% X$_0$ per layer, with the innermost layers possibly positioned inside the beam pipe. In addition to superior tracking and vertexing capabilities over a wide momentum range down to a few tens of MeV/$c$, the detector will provide particle identification via time-of-flight determination with about 20~ps resolution. In addition, electron and photon identification will be performed in a separate shower detector. The proposed detector is conceived for studies of pp, pA and AA collisions at luminosities a factor of 20 to 50 times higher than possible with the upgraded ALICE detector, enabling a rich physics program ranging from measurements with electromagnetic probes at ultra-low transverse momenta to precision physics in the charm and beauty sector., Comment: Input to the 2020 Update of the European Particle Physics Strategy

This report summarises the main findings of the QCD Working Group in the CERN Physics Beyond Colliders Study.

For the understanding of fluctuation measurements in heavy-ion collisions it is crucial to develop quantitatively reliable dynamical descriptions which take the non-perturbative nature of QCD near the phase transition into account. We discuss a novel QCD-assisted transport approach based on non-equilibrium chiral fluid dynamics and the effective action of low energy QCD. In this framework, we study the time-evolution of fluctuation measures of the critical mode, notably the kurtosis, for a non-expanding system. From this, we can estimate the equilibration times of critical mode fluctuations in the QCD phase diagram. These allow us to identify both the phase boundary and the critical region near the QCD critical point., Comment: Quark Matter 2018 Proceedings

Heavy-ion collisions are well described by a dynamical evolution with a long hydrodynamical phase. In this phase the properties of the strongly coupled quark-gluon plasma are reflected in the equation of state (EoS) and the transport coefficients, most prominently by the shear and bulk viscosity over entropy density ratios $\eta$/s(T) and $\zeta$/s(T), respectively. While the EoS is by now known to a high accuracy, the transport coefficients and in particular their temperature and density dependence are not well known from first-principle computations yet, as well as the possible influence they can have once used in hydrodynamical simulations. In this work, the most recent QCD-based parameters are provided as input to the MUSIC framework. A ratio $\eta$/s(T) computed with a QCD based approach is used for the first time \cite{Haas:2013hpa,Christiansen:2014ypa}. The IP-Glasma model is used to describe the initial energy density distribution, and UrQMD for the dilute hadronic phase. Simulations are performed for Pb--Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV, for different centrality intervals. The resulting kinematic distributions of the particles produced in the collisions are compared to data from the LHC, for several experimental observables. The high precision of the experimental results and the broad variety of observables considered allow to critically verify the quality of the description based on first-principle input to the hydrodynamic evolution., Comment: 10 pages, 8 figures

We examine spin-dependent displacement of a single electron, resulting in separation and relocation of the electron wavefunction components, and thus charge parts, corresponding to opposite spins. This separation is induced by a pulse of an electric field which generates varying Rashba type spin-orbit coupling. This mechanism is next implemented in a nanodevice based on a gated quantum dot defined within a quantum nanowire. The electric field pulse is generated by ultrafast changes of voltages, of the order of several hundred mV, applied to nearby gates. The device is modeled realistically with appropriate material parameters and voltages applied to the gates, yielding an accurate confinement potential and Rashba coupling. At the end, we propose a spin-to-charge conversion device, which with an additional charge detector will allow for electron spin state measurement., Comment: 7 pages, 7 figures

We propose a fast and accurate spin initialization method for a single electron trapped in an electrostatic quantum dot. The dot is created in a nanodevice composed of a catalytically grown indium antimonide (InSb) nanowire and nearby gates to which control voltages are applied. Initially we insert a single electron of arbitrary spin into the wire. Operations on spin are performed using the Rashba spin-orbit interaction induced by an electric field. First, a single pulse of voltages applied to lateral gates is used to split the electron wavepacket into two parts with opposite spin orientations. Next, another voltage pulse applied to the remaining gates rotates spins of both parts in opposite directions by $\pi/2$. This way, initially opposite spin parts eventually point in the same direction, along the axis of the quantum wire. We thus set spin in a predefined direction regardless of its initial orientation. This is achieved in time less than $60\,\mathrm{ps}$ without the use of microwaves, photons or external magnetic fields., Comment: 7 pages, 6 figures

We report on broadly based systematic investigations of the modeling components for open heavy-flavor diffusion and energy loss in strongly interacting matter in their application to heavy-flavor observables in high-energy heavy-ion collisions, conducted within an EMMI Rapid Reaction Task Force framework. Initial spectra including cold-nuclear-matter effects, a wide variety of space-time evolution models, heavy-flavor transport coefficients, and hadronization mechanisms are scrutinized in an effort to quantify pertinent uncertainties in the calculations of nuclear modification factors and elliptic flow of open heavy-flavor particles in nuclear collisions. We develop procedures for error assessments and criteria for common model components to improve quantitative estimates for the (low-momentum) heavy-flavor diffusion coefficient as a long-wavelength characteristic of QCD matter as a function of temperature, and for energy loss coefficients of high-momentum heavy-flavor particles., Comment: 78 pages, 29 figures, report on an EMMI Rapid Reaction Task Force; v2: small revision, accepted for publication in Nucl. Phys. A

The aim of presented research is to design a nanodevice, based on a MoS$_2$ monolayer, performing operations on a well-defined valley qubit. We show how to confine an electron in a gate induced quantum dot within the monolayer, and to perform the NOT operation on its valley degree of freedom. The operations are carried out all electrically via modulation of the confinement potential by oscillating voltages applied to the local gates. Such quantum dot structure is modeled realistically. Through these simulations we investigate the possibility of realization of a valley qubit in analogy with a realization of the spin qubit. We accurately model the potential inside the nanodevice accounting for proper boundary conditions on the gates and space-dependent materials permittivity by solving the generalized Poisson's equation. The time-evolution of the system is supported by realistic self-consistent Poisson-Schr\"odinger tight-binding calculations. The tight-binding calculations are further confirmed by simulations within the effective continuum model., Comment: 13 pages, 12 figures

I report on the first application of a novel, generalized Bayesian reconstruction (BR) method for spectral functions to the characterization of QCD constituents. These spectral functions find applications in off-shell kinetics of the quark-gluon plasma and in calculations of transport coefficients. The new BR method is applied to Euclidean propagator data, obtained in Landau gauge on lattices with $N_f=2+1+1$ dynamical flavors by the "twisted mass at finite temperature" (tmfT) collaboration. The deployed reconstruction method is designed for spectral functions that can exhibit positivity violation (opposed to that of hadronic bound states). The transversal and longitudinal gluon spectral functions show a robust structure composed of quasiparticle peak and a negative trough. Characteristic differences between the hadronic and the plasma phase and between the two channels become visible. We obtain the temperature dependence of the transversal and longitudinal gluon masses., Comment: talk given by E.-M. Ilgenfritz at the 11th APCTP-BLTP JINR-PNPI-SPbSU Joint Workshop , Peterhof, 24 -- 28 July 2017, appears in the Proceedings (to be published in PEPAN Letters)

We propose a nanodevice for single-electron spin initialization. It is based on a gated planar semiconductor heterostructure with a quantum well and with potentials generated by voltages applied to local gates. Initially we insert an electron with arbitrary spin into the nanodevice. Next we perform a sequence of spin manipulations, after which the spin is set in a desired direction (e.g., the growth direction). The operations are done all-electrically, do not require any external fields and do not depend on the initial spin direction., Comment: 6 pages, 4 figures

We propose a nanodevice based on a typical planar semiconductor heterostructure with lateral confinement potential created by voltages applied to local electrodes. We show how to obtain near parabolical confinement along the nanodevice, and how to use coherent states of the harmonic oscillator for spatial separation of electron densities corresponding to opposite spin directions. In such a way, an entangled state of Schr\"odinger's cat type is created. We performed simulations of a realistic nanodevice model by numerical solving the time-dependent Schr\"odinger's equation together with simultaneous tracking of the controllable confinement potential via solution of the Poisson's equation at every time step., Comment: 15 pages, 21 figures

We propose an all-electrically controlled nanodevice - a gated semiconductor nanowire - capable of generating a coherent state of a single electron trapped in a harmonic oscillator or superposition of such coherent states - the Schr\"odinger cat state. In the proposed scheme, electron in the ground state of the harmonic potential is driven by resonantly oscillating Rashba spin-orbit coupling. This allows for the creation of the Schr\"odinger cat state with superposition amplitudes depending on the initial electron spin state. Such a method can be used for initialization of a single spin qubit defined in a coherent state. The harmonic confinement potential along the InSb nanowire and the modulation of the Rashba spin-orbit coupling is obtained by proper gating. The results are supported by realistic three-dimensional time-dependent self consistent Poisson-Schr\"odinger calculations., Comment: 6 pages, 4 figures

A novel method and nanodevice are introduced that allows to rotate the single electron spin confined in a gated electrostatic InSb nanowire quantum dot. Proposed method does not require application of any (oscillating or static) external magnetic fields. Our proposal instead employs spatial and time modulation of confining potential induced by electric gates, which, in turn leads to oscillating Rashba type spin-orbit coupling. Moving electron back and forth in such a variable Rashba field allows for realization of spin rotations around two different axes separately without using an external magnetic field. The results are supported by realistic three-dimensional time dependent Poisson-Schr\"{o}dinger calculations for systems and material parameters corresponding to experimentally accessible structures., Comment: 9 pages, 14 figures

We investigate the stability of the Sarma phase in two-component fermion systems in three spatial dimensions. For this purpose we compare strongly-correlated systems with either relativistic or non-relativistic dispersion relation: relativistic quarks and mesons at finite isospin density and spin-imbalanced ultracold Fermi gases. Using a Functional Renormalization Group approach, we resolve fluctuation effects onto the corresponding phase diagrams beyond the mean-field approximation. We find that fluctuations induce a second order phase transition at zero temperature, and thus a Sarma phase, in the relativistic setup for large isospin chemical potential. This motivates the investigation of the cold atoms setup with comparable mean-field phase structure, where the Sarma phase could then be realized in experiment. However, for the non-relativistic system we find the stability region of the Sarma phase to be smaller than the one predicted from mean-field theory. It is limited to the BEC side of the phase diagram, and the unitary Fermi gas does not support a Sarma phase at zero temperature. Finally, we propose an ultracold quantum gas with four fermion species that has a good chance to realize a zero-temperature Sarma phase., Comment: version published in Phys.Lett.B; 10 pages, 5 figures

We investigate the phase structure of spin-imbalanced unitary Fermi gases beyond mean-field theory by means of the Functional Renormalization Group. In this approach, quantum and thermal fluctuations are resolved in a systematic manner. The discretization of the effective potential on a grid allows us to accurately account for both first- and second-order phase transitions that are present on the mean-field level. We compute the full phase diagram in the plane of temperature and spin-imbalance and discuss the existence of other conjectured phases such as the Sarma phase and a precondensation region. In addition, we explain on a qualitative level how we expect that in-situ density images are affected by our findings and which experimental signatures may potentially be used to probe the phase structure., Comment: version published in Phys.Rev.A; 14 pages, 6 figures

A nanodevice capable of separating spins of two electrons confined in a quantum dot formed in a gated semiconductor nanowire is proposed. Two electrons confined initially in a single quantum dot in the singlet state are transformed into the system of two electrons confined in two spatially separated quantum dots with opposite spins. In order to separate the electrons' spins we exploit transitions between the singlet and the triplet state which are induced by resonantly oscillating Rashba spin-obit coupling strength. The proposed device is all electrically controlled and the electron spin separation can be realized within tens of picoseconds. The results are supported by solving numerically quasi-one-dimensional time-dependent Schroedinger equation for two electrons, where the electron-electron correlations are taken into account in the exact manner.

In this paper, several nanodevices which realize basic single heavy hole qubit operations are proposed and supported by time dependent self consistent Poisson-Schr\"{o}dinger calculations using a four band heavy hole-light hole model. In particular we propose a set of nanodevices which can act as Pauli X, Y, Z quantum gates and as a gate that acts similar as a Hadamard gate (i.e. it creates a balanced superposition of basis states but with an additional phase factor) on the heavy hole spin qubit. We also present the design and simulation of a gated semiconductor nanodevice which can realize an arbitrary sequence of all these proposed single quantum logic gates. The proposed devices exploit the self-focusing effect of the hole wave function which allows for guiding the hole along a given path in the form of a stable soliton-like wave packet. Thanks to the presence of the Dresselhaus spin orbit coupling, the motion of the hole along a certain direction is equivalent to the application of an effective magnetic field which induces in turn a coherent rotation of the heavy hole spin. The hole motion and consequently the quantum logic operation is initialized only by weak static voltages applied to the electrodes which cover the nanodevice. The proposed gates allow for an all electric and ultrafast (tens of picoseconds) heavy hole spin manipulation and give the possibility to implement a scalable architecture of heavy hole spin qubits for quantum computation applications., Comment: 13 pages, 9 figures

We present the current status of ongoing efforts to use functional methods, Dyson-Schwinger equations and functional renormalization group equations, for the description of the infrared regime of nonabelian (pure) gauge theories in the Coulomb gauge. In particular, we present a new determination of the color-Coulomb potential with the help of the functional renormalization group that results in an almost linearly rising potential between static color charges at large spatial distances., Comment: 11 pages, 3 figures. Slightly improved version of J. Phys.: Conf. Ser. 287 (2011) 012023

I briefly review results obtained within the variational Hamiltonian approach to Yang-Mills theory in Coulomb gauge and confront them with recent lattice data. The variational approach is extended to non-Gaussian wave functionals including three- and four-gluon kernels in the exponential of the vacuum wave functional and used to calculate the three-gluon vertex. A new functional renormalization group flow equation for Hamiltonian Yang--Mills theory in Coulomb gauge is solved for the gluon and ghost propagator under the assumption of ghost dominance. The results are compared to those obtained in the variational approach., Comment: 6 pages, 9 figures. Invited talk given by H. Reinhardt at "T(r)opical QCD 2010", September 26--October 1, 2010, Cairns, Australia

The participants in this discussion session of the QCHS 9 meeting were each asked the following question: "What would be the most useful piece of information that you could obtain, by whatever means, that would advance your own program, and/or our general understanding of confinement?" This proceedings contains a brief summary of each panel member's contribution to the discussion, provided by the panel members themselves., Comment: 8 pages. Roundtable discussion at the Quark Confinement and Hadron Spectrum 9 conference (Madrid, Aug. 29 - Sep. 3, 2010). To appear in the conference proceedings, published by the AIP. v2: References added. v3: Reference corrected

I review results recently obtained within the Hamiltonian approach to Yang-Mills theory in Coulomb gauge. In particular, I will present results for the ghost and gluon propagators and compare these with recent lattice data. Furthermore, I will give an interpretation of the inverse of the ghost form factor as the dielectric function of the Yang-Mills vacuum. Our ansatz for the vacuum wave functional will be checked by means of functional renormalization group flow equations, which are solved for the gluon energy and the ghost form factor. Finally, we calculate the Wilson loop for the vacuum wave functional obtained from the variational approach, using a Dyson equation., Comment: 12 pages, 20 figures. Invited talk given by H. Reinhardt at the Workshop on QCD Green's Functions, Confinement and Phenomenology, September 2009, Trento

The phase transition to superfluidity and the BCS-BEC crossover for an ultracold gas of fermionic atoms is discussed within a functional renormalization group approach. Non-perturbative flow equations, based on an exact renormalization group equation, describe the scale dependence of the flowing or average action. They interpolate continuously from the microphysics at atomic or molecular distance scales to the macroscopic physics at much larger length scales, as given by the interparticle distance, the correlation length, or the size of the experimental probe. We discuss the phase diagram as a function of the scattering length and the temperature and compute the gap, the correlation length and the scattering length for molecules. Close to the critical temperature, we find the expected universal behavior. Our approach allows for a description of the few-body physics (scattering and molecular binding) and the many-body physics within the same formalism., Comment: 30 pages, 13 figures

A functional renormalisation group study for the BEC-BCS crossover for ultracold gases of fermionic atoms is presented. We discuss the fixed point which is at the origin of universality for broad Feshbach resonances. All macroscopic quantities depend only on one relevant parameter, the concentration a k_F, besides their dependence on the temperature in units of the Fermi energy. In particular, we compute the universal ratio between molecular and atomic scattering length in vacuum. We also present an estimate to which level of accuracy universality holds for gases of Li and K atoms., Comment: 19 pages, 3 figures, to be published in PRA

The functional renormalisation group is used for the BCS-BEC crossover in gases of ultracold fermionic atoms. In a simple truncation, we see how universality and an effective theory with composite bosonic di-atom states emerge. We obtain a unified picture of the whole phase diagram. The flow reflects different effective physics at different scales. In the BEC limit as well as near the critical temperature, it describes an interacting bosonic theory., Comment: 4 pages, 4 figures

We summarise an analysis of the infrared regime of Landau gauge QCD by means of a flow equation approach. The infrared behaviour of gluon and ghost propagators is evaluated. The results provide further evidence for the Kugo-Ojima confinement scenario. We also discuss their relation to results obtained with other functional methods as well as the lattice., Comment: 3 pages, talk given by JMP at 6th Conference on Quark Confinement and the Hadron Spectrum, Villasimius, Sardinia, Italy, 21-25 Sep 2004

In this talk the gauge symmetry for Wilsonian flows in pure Yang-Mills theories is discussed. The background field formalism is used for the construction of a gauge invariant effective action. The symmetries of the effective action under gauge transformations for both the gauge field and the auxiliary background field are separately evaluated. Modified Ward-Takahashi and background field identities are used in my study. Finally it is shown how the symmetry properties of the full theory are restored in the limit where the cut-off is removed., Comment: 6 pages, to appear in the Proceedings of the 2nd Conference on the Exact Renormalization Group, Rome 2000

We apply the ADHM instanton construction to SU(2) gauge theory on T^n x R^(4-n)for n=1,2,3,4. To do this we regard instantons on T^n x R^(4-n) as periodic (modulo gauge transformations) instantons on R^4. Since the R^4 topological charge of such instantons is infinite the ADHM algebra takes place on an infinite dimensional linear space. The ADHM matrix M is related to a Weyl operator (with a self-dual background) on the dual torus tilde T^n. We construct the Weyl operator corresponding to the one-instantons on T^n x R^(4-n). In order to derive the self-dual potential on T^n x R^(4-n) it is necessary to solve a specific Weyl equation. This is a variant of the Nahm transformation. In the case n=2 (i.e. T^2 x R^2) we essentially have an Aharonov Bohm problem on tilde T^2. In the one-instanton sector we find that the scale parameter, lambda, is bounded above, (lambda)^2 tv<4 pi, tv being the volume of the dual torus tilde T^2., Comment: 35 pages, LATeX. New section on Nahm transform included, presentation improved, reference added, to appear in Nuclear Physics B

It is shown that in the two-exponential version of Liouville theory the coefficients of the three-point functions of vertex operators can be determined uniquely using the translational invariance of the path integral measure and the self-consistency of the two-point functions. The result agrees with that obtained using conformal bootstrap methods. Reflection symmetry and a previously conjectured relationship between the dimensional parameters of the theory and the overall scale are derived., Comment: Plain TeX File; 15 Pages

The quantisation of the two-dimensional Liouville field theory is investigated using the path integral, on the sphere, in the large radius limit. The general form of the $N$-point functions of vertex operators is found and the three-point function is derived explicitly. In previous work it was inferred that the three-point function should possess a two-dimensional lattice of poles in the parameter space (as opposed to a one-dimensional lattice one would expect from the standard Liouville potential). Here we argue that the two-dimensionality of the lattice has its origin in the duality of the quantum mechanical Liouville states and we incorporate this duality into the path integral by using a two-exponential potential. Contrary to what one might expect, this does not violate conformal invariance; and has the great advantage of producing the two-dimensional lattice in a natural way., Comment: Plain TeX File; 36 pages

We consider pure Yang Mills theory on the four torus. A set of non-Abelian transition functions is presented which encompass all instanton sectors. It is argued that these transition functions are a convenient starting point for gauge fixing. In particular, we give an extended Abelian projection with respect to the Polyakov loop, where $A_0$ is independent of time and in the Cartan subalgebra. In the non-perturbative sectors such gauge fixings are necessarily singular. These singularities can be restricted to Dirac strings joining monopole and anti-monopole like ``defects''., Comment: 24 pages, 4 figures, Latex 2e, some minor improvements

We calculate within an algebraic Bjorken-Johnson-Low (BJL) method anomalous Schwinger terms of fermionic currents and the Gauss law operator in chiral gauge theories. The current algebra is known to violate the Jacobi identity in an iterative computation. Our method takes the subtleties of the equal-time limit into account and leads to an algebra that fulfills the Jacobi identity. The non-iterative terms appearing in the double commutators can be traced back directly to the projective representation of the gauge group., Comment: 17 pages, latex, no figure

A simple derivation of the free energy and expectation values of Polyakov-loops in $QCD_2$ via path integral methods is given. In the chosen gauge (which can be generalized to 4 dimensions) without Gribov-copies the Fadeev-Popov determinant and the integration over the space component of the gauge field cancel exactly and we are left only with an integration over the zero components of the gauge field in the Cartan sub-algebra. This way the Polyakov-loop operators become Vertex-operators in a simple quantum mechanical model. The number of fermionic zero modes is related to the winding-numbers of $A_0$ in this gauge., Comment: some comments added and some typos corrected, 17 pages, Latex, 2 figures

Within the framework of generalized functions a general consistent definition of double commutators is given. This definition respects the Jacobi identity even if the regularization is removed. The double commutator of fermionic currents is calculated in this limit. We show that BJL--type prescriptions and point--splitting prescriptions for calculating double commutators fail to give correct results in free field theory., Comment: 10 pages,latex, no figure, report-no corrected