Wolfgang Bietenholz

Large N Dynamics of Dimensionally Reduced 4D SU(N) Super Yang-Mills Theory

We perform Monte Carlo simulations of a supersymmetric matrix model, which is obtained by dimensional reduction of 4D SU(N) super Yang-Mills theory. The model can be considered as a four-dimensional counterpart of the IIB matrix model. We extract the space-time structure represented by the eigenvalues of bosonic matrices. In particular we compare the large-N behavior of the space-time extent with the result obtained from a low-energy effective theory. We measure various Wilson loop correlators which represent string amplitudes and we observe a non-trivial universal scaling in N. We also observe that the Eguchi-Kawai equivalence to ordinary gauge theory does hold at least within a finite range of scale. Comparison with the results for the bosonic case clarifies the role of supersymmetry in the large-N dynamics. It does affect the multi-point correlators qualitatively, but the Eguchi-Kawai equivalence is observed even in the bosonic case.

A non-perturbative study of 4d U(1) non-commutative gauge theory — the fate of one-loop instability

Recent perturbative studies show that in 4d non-commutative spaces, the trivial (classically stable) vacuum of gauge theories becomes unstable at the quantum level, unless one introduces sufficiently many fermionic degrees of freedom. This is due to a negative IR-singular term in the one-loop effective potential, which appears as a result of the UV/IR mixing. We study such a system non-perturbatively in the case of pure U(1) gauge theory in four dimensions, where two directions are non-commutative. Monte Carlo simulations are performed after mapping the regularized theory onto a U(N) lattice gauge theory in d = 2. At intermediate coupling strength, we find a phase in which open Wilson lines acquire non-zero vacuum expectation values, which implies the spontaneous breakdown of translational invariance. In this phase, various physical quantities obey clear scaling behaviors in the continuum limit with a fixed non-commutativity parameter θ, which provides evidence for a possible continuum theory. The extent of the dynamically generated space in the non-commutative directions becomes finite in the above limit, and its dependence on θ is evaluated explicitly. We also study the dispersion relation. In the weak coupling symmetric phase, it involves a negative IR-singular term, which is responsible for the observed phase transition. In the broken phase, it reveals the existence of the Nambu-Goldstone mode associated with the spontaneous symmetry breaking.

Monte Carlo studies of the IIB matrix model at large N

The low-energy effective theory of the IIB matrix model developed by H. Aoki et al. is written down explicitly in terms of bosonic variables only. The effective theory is then studied by Monte Carlo simulations in order to investigate the possibility of a spontaneous breakdown of ten-dimensional Lorentz invariance. The imaginary part of the effective action, which causes the so-called sign problem in the simulation, is dropped by hand. The extent of the eigenvalue distribution of the bosonic matrices shows a power-law large-N behavior, consistent with a simple branched-polymer prediction. We observe, however, that the eigenvalue distribution becomes more and more isotropic in the ten-dimensional space-time as we increase N. This suggests that if the spontaneous breakdown of Lorentz invariance really occurs in the IIB matrix model, a crucial role must be played by the imaginary part of the effective action.

Phase diagram and dispersion relation of the noncommutative lambda phi**4 model in d = 3

We present a non-perturbative study of the ?4 model in a three dimensional euclidean space, where the two spatial coordinates are non-commutative. Our results are obtained from numerical simulations of the lattice model, after its mapping onto a dimensionally reduced, twisted hermitean matrix model. In this way we first reveal the explicit phase diagram of the non-commutative ?4 lattice model. We observe that the ordered regime splits into a phase of uniform order and a phase of two stripes of opposite sign, and more complicated patterns. Next we discuss the behavior of the spatial and temporal correlators. From the latter we extract the dispersion relation, which allows us to introduce a dimensionful lattice spacing. To extrapolate to zero lattice spacing and infinite volume we perform a double scaling limit, which keeps the non-commutativity tensor constant. The dispersion relation in the disordered phase stabilizes in this limit, which represents a non-perturbative renormalization. In particular this confirms the existence of a striped phase in the continuum limit, in accordance with a conjecture by Gubser and Sondhi. The extrapolated dispersion relation also exhibits UV/IR mixing as a non-perturbative effect. Finally we add some observations about a Nambu-Goldstone mode in the striped phase, and about the corresponding model in d = 2.

A non-perturbative study of gauge theory on a non-commutative plane

Abstract:We performa non-perturbativestudyofpuregauge theoryin atwo dimensionalnon-commutative (NC) space. On the lattice, it is equivalent to the twisted Eguchi-Kawaimodel, which we simulated at N ranging from 25 to 515. We observe a clear large-N scalingfor the 1- and 2-point function of Wilson loops, as well as the 2-point function of Polyakovlines. The 2-point functions agree with a universal wave function renormalization. Based ona Morita equivalence, the large-N double scaling limit corresponds to the continuum limitof NC gauge theory, so the observed large-N scaling demonstrates the non-perturbativerenormalizability of this NC field theory. The area law for the Wilson loops holds at smallphysical area as in commutative 2d planar gauge theory, but at large areas we find anoscillating behavior instead. In that regime the phase of the Wilson loop grows linearlywith the area. This agrees with the Aharonov-Bohm effect in the presence of a constantmagnetic field, identified with the inverse non-commutativity parameter.Keywords: Non-Commutative Geometry, Matrix Models, Lattice Gauge Field Theories,Field Theories in Lower Dimensions.

Spectral properties of the overlap Dirac operator in QCD

We discuss the eigenvalue distribution of the overlap Dirac operator in quenched QCD on lattices of size 84 , 104 and 124 at β = 5.85 and β = 6. We distinguish the topological sectors and study the distributions of the leading non-zero eigenvalues, which are stereographically mapped onto the imaginary axis. Thus they can be compared to the predictions of random matrix theory applied to the -expansion of chiral perturbation theory. We find a satisfactory agreement, if the physical volume exceeds about (1.2 fm)4 . For the unfolded level spacing distribution we find an accurate agreement with the random matrix conjecture on all volumes that we considered.

Axial correlation functions in the epsilon regime: A Numerical study with overlap fermions

We present simulation results employing overlap fermions for the axial correlation functions in the -regime of chiral perturbation theory. In this regime, finite size effects and topology play a dominant role. Their description by quenched chiral perturbation theory is compared to our numerical results in quenched QCD. We show that lattices with a linear extent L > 1.1 fm are necessary to interpret the numerical data obtained in distinct topological sectors in terms of the -expansion. Such lattices are, however, still substantially smaller than the ones needed in standard chiral perturbation theory. However, we also observe severe difficulties at very low values of the quark mass, in particular in the topologically trivial sector.

Convergence rate and locality of improved overlap fermions

Abstract We construct new Ginsparg–Wilson fermions for QCD by inserting an approximately chiral Dirac operator—which involves ingredients of a perfect action—into the overlap formula. This accelerates the convergence of the overlap Dirac operator by a factor of 5 compared to the standard construction, which inserts the Wilson fermion as a point of departure. Taking into account the effort for treating the improved fermion, we are left with an total computational overhead of about a factor 3. This remaining factor is likely to be compensated by other virtues; here we show that the level of locality is clearly improved, so that the exponent of the correlation decay is doubled. We also show that approximate rotation invariance is drastically improved, but a careful scaling test has to be postponed.

Going chiral: Overlap versus twisted mass fermions

We compare the behavior of overlap fermions, which are chirally invariant, and of Wilson twisted mass fermions at full twist in the approach to the chiral limit. Our quenched simulations reveal that with both formulations of lattice fermions pion masses of (250 MeV) can be reached in practical applications. Our comparison is done at a fixed value of the lattice spacing a 0.123 fm. A number of quantities are measured such as hadron masses, pseudoscalar decay constants and quark masses obtained from Ward identities. We also determine the axial vector renormalization constants in the case of overlap fermions.We compare the behavior of overlap fermions, which are chirally invariant, and of Wilson twisted mass fermions at full twist in the approach to the chiral limit. Our quenched simulations reveal that with both formulations of lattice fermions pion masses of O(250 MeV) can be reached in practical applications. Our comparison is done at a fixed value of the lattice spacing a=0.123 fm. A number of quantities are measured such as hadron masses, pseudoscalar decay constants and quark masses obtained from Ward identities. We also determine the axial vector renormalization constants in the case of overlap fermions.

Probing the fuzzy sphere regularisation in simulations of the 3d λϕ4 model

We regularise the 3d λ4 model by discretising the Euclidean time and representing the spatial part on a fuzzy sphere. The latter involves a truncated expansion of the field in spherical harmonics. This yields a numerically tractable formulation, which constitutes an unconventional alternative to the lattice. In contrast to the 2d version, the radius R plays an independent role. We explore the phase diagram in terms of R and the cutoff, as well as the parameters m2 and λ. Thus we identify the phases of disorder, uniform order and non-uniform order. We compare the result to the phase diagrams of the 3d model on a non-commutative torus, and of the 2d model on a fuzzy sphere. Our data at strong coupling reproduce accurately the behaviour of a matrix chain, which corresponds to the c = 1-model in string theory. This observation enables a conjecture about the thermodynamic limit.