Margarita Garcia Perez, Antonio Gonzalez-Arroyo, Alfonso Sastre
We study the solutions of the Dirac equation in the adjoint representation(gluinos) in the background field of SU(2) unit charge calorons. Our solutions are forced to be antiperiodic in thermal time and would occur naturally in a semiclassical approach to N=1 Super-symmetric Yang-Mills theory at finite temperature.
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We couple fermion fields in the adjoint representation (gluinos) to the SU(2) gauge field of unit charge calorons defined on R^3 x S_1. We compute corresponding zero-modes of the Dirac equation. These are relevant in semiclassical studies of N=1 Super-symmetric Yang-Mills theory. Our formulas, show that, up to a term proportional to the vector potential, the modes can be constructed by different linear combinations of two contributions adding up to the total caloron field strength.
Pure Yang-Mills instantons are considered on S^1 x R^3 -- so-called calorons. The holonomy -- or Polyakov loop around the thermal S^1 at spatial infinity -- is assumed to be a non-centre element of the gauge group SU(n) as most appropriate for QCD applications in the confined phase. It is shown that a charge k caloron can be seen as a collection of nk massive magnetic monopoles each carrying fractional topological charge. This interpretation offers a physically appealing way of introducing monopole degrees of freedom into pure gluodynamics: as constituents of finite temperature instantons. New…
Calorons are finite action solutions to the anti-selfdual Yang-Mills equations on $\mathbb{R}^3\times S^1$. They are generally constructed by the so called Nahm construction. We perform the numerical Nahm transform for the Nahm data of 3-calorons with $C_3$-symmetry, which do not have the monopole limits. Dissimilar to the cases of having monopole limits, we can trace the zero-circumference limit of $S^1$.It is found that the action density of the calorons tends to fade away as $S^1$ shrinks.
We give the analytic result for the fermion zero-mode of the SU(2) calorons with non-trivial holonomy. It is shown that the zero-mode is supported on ONLY ONE of the constituent monopoles. We discuss some of its implications.
We construct the fermion zero-mode for arbitrary charge one SU(n) calorons with non-trivial holonomy, both in the finite temperature context (anti-periodic boundary conditions in time) and in the Kaluza-Klein compactification context (periodic boundary conditions in time). The zero-mode is localised on one of the constituent monopoles and we discuss a relation to the Callias index theorem.
We derive analytic formulas for the zero-modes of the Dirac equation in the adjoint representation in the background field of Q=1 SU(N) calorons. Solutions with various boundary conditions are obtained, including the physically most relevant cases of periodic and antiperiodic ones. The latter are essential ingredients in a semiclassical treatment of finite temperature supersymmetric Yang-Mills theory. A detailed discussion of adjoint zero-modes in several other contexts is also presented.
The full ADHM-Nahm formalism is employed to find exact higher charge caloron solutions with non-trivial holonomy, extended beyond the axially symmetric solutions found earlier. Particularly interesting is the case where the constituent monopoles, that make up these solutions, are not necessarily well-separated. This is worked out in detail for charge 2. We resolve the structure of the extended core, which was previously localized only through the singularity structure of the zero-mode density in the far field limit. We also show that this singularity structure agrees exactly with the abelian c…
We analyze the zero-modes of the Dirac operator in quenched SU(3) gauge configurations at non-zero temperature and compare periodic and anti-periodic temporal boundary conditions for the fermions. It is demonstrated that for the different boundary conditions often the modes are localized at different space-time points and have different sizes. Our observations are consistent with patterns expected for Kraan - van Baal solutions of the classical Yang-Mills equations. These solutions consist of constituent monopoles and the zero-modes are localized on different constituents for different boundar…
Across the finite temperature transition to the quark-gluon plasma, the QCD topological susceptibility decreases sharply. Thus in the high temperature phase the remaining topological objects (possibly calorons) form a weakly interacting dilute gas. The overlap Dirac operator, through its exact zero modes, allows one to measure the net topological charge. We show that separately the number of positively and negatively charged topological objects can also be extracted from the low end of the overlap Dirac spectrum. We find that slightly above the phase transition their number distributions are a…
Starting from our proposed model of the Yang-Mills vacuum based on fractional instantons, we review the intellectual itinerary which has guided part of our scientific activity up to our recent work on adjoint zero-modes for calorons.
Topological objects of $SU(3)$ gluodynamics are studied at the infrared scale near the transition temperature with the help of zero and near-zero modes of the overlap Dirac operator. We construct UV filtered topological charge densities corresponding to three versions of the temporal boundary condition applied to this operator, for which the zero mode is known to be located on corresponding three constituent dyons (antidyons) in the reference case of an analytical (anti)caloron solution. The clustering of the three topological charge densities marks the positions of three types of dyons and an…
We give analytical and numerical solutions for the zero modes of the Dirac operator in topological SU(2) gauge backgrounds at nonzero chemical potential. Continuation from imaginary to real chemical potential is used to systematically derive analytical zero modes for calorons at arbitrary holonomy and, in particular limits, for instantons and dyons (magnetic monopoles). For the latter a spherical ansatz is explored as well. All the zero mode exhibit stronger peaks at the core and negative regions in their densities. We discuss the structure of the corresponding overlap matrix elements. For dis…
We compute the functional determinant for the fluctuations around the most general self-dual configuration with unit topological charge for 4D SU(2) Yang-Mills with one compactified direction. This configuration is called "instanton with non-trivial holonomy" or "Kraan-van-Baal-Lee-Lu caloron". It is a generalization of the usual instantons for the case of non-zero temperature. We extend the earlier results of Diakonov, Gromov, Petrov and Slizovsky to arbitrary values of parameters.
We present an exact normalisable zero-energy chiral fermion solution for abelian BPS dipoles. For a single dipole, this solution is contained within the high temperature limit of the SU(2) caloron with non-trivial holonomy.
The Euclidean formulation of SU(2) Yang-Mills thermodynamics admits periodic, (anti)selfdual solutions to the fundamental, classical equation of motion which possess one unit of topological charge: (anti)calorons. A spatial coarse graining over the central region in a pair of such localised field configurations with trivial holonomy generates an inert adjoint scalar field $ϕ$, effectively describing the pure quantum part of the thermal ground state in the induced quantum field theory. Here we show for the limit of zero holonomy how (anti)calorons associate a temperature independent electric pe…
We argue that the fermionic zero mode in non-trivial gauge field backgrounds must have a zero. We demonstrate this explicitly for calorons where its location is related to a constituent monopole. Furthermore a topological reasoning for the existence of the zero is given which therefore will be present for any non-trivial configuration. We propose the use of this property in particular for lattice simulations in order to uncover the topological content of a configuration.
Gluodynamics and two-flavor QCD at non-zero temperature are studied with the so-called overimproved cooling technique under which caloron solutions may remain stable. We consider topological configurations either at the first occuring stable plateau of topological charge or at the first (anti)selfdual plateau and find the corresponding topological susceptibility at various temperatures on both sides of the thermal transition or crossover. In pure gluodynamics the topological susceptibility drops sharply at the deconfinement temperature while in full QCD it decreases smoothly at temperatures ab…
We perform a semi-classical study of chiral symmetry breaking and of the spectrum of the Dirac operator in QCD with adjoint fermions. For this purpose we calculate matrix elements of the adjoint Dirac operator between instanton zero modes and study their symmetry properties. We present simulations of the instanton ensemble for different numbers of Majorana fermions in the adjoint representation. These simulations provide evidence that instantons lead to gluino condensation in supersymmetric gluodynamics.
We use the fermion zero-modes in the background of multi-caloron solutions with non-trivial holonomy as a probe for constituent monopoles. We find in general indication for an extended structure. However, for well separated constituents these become point-like. We analyse this in detail for the SU(2) charge 2 case, where one is able to solve the relevant Nahm equation exactly, beyond the piecewize constant solutions studied previously. Remarkably the zero-mode density can be expressed in the high temperature limit as a function of the conserved quantities that classify the solutions of the Nah…
Complete wetting is a universal phenomenon associated with interfaces separating coexisting phases. For example, in the pure gluon theory, at $T_c$ an interface separating two distinct high-temperature deconfined phases splits into two confined-deconfined interfaces with a complete wetting layer of confined phase between them. In supersymmetric Yang-Mills theory, distinct confined phases may coexist with a Coulomb phase at zero temperature. In that case, the Coulomb phase may completely wet a confined-confined interface. Finally, at the high-temperature phase transition of gluons and gluinos, …
The dynamical N=1, SU(2) Super Yang-Mills theory is studied on the lattice using a new lattice fermion regulator, domain wall fermions. This formulation even at non-zero lattice spacing does not require fine-tuning, has improved chiral properties and can produce topological zero-mode phenomena. Numerical simulations of the full theory on lattices with the topology of a torus indicate the formation of a gluino condensate which is sustained at the chiral limit. The condensate is non-zero even for small volume and small supersymmetry breaking mass where zero mode effects due to gauge fields with …
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