Topological content ofSU(2)gauge fields belowTc

Abstract

Finite temperature Euclidean SU (2) lattice gauge fields generated in the confinement phase close to the deconfinement phase transition are subjected to cooling. The aim is to identify long-living, almost-classical local excitations which carry (generically non-integer) topological charge. Two kinds of spatial boundary conditions (fixed holonomy and standard periodic boundary conditions) are applied. For the lowest-action almost-classical configurations we find that their relative probability semi-quantitatively agrees for both types of boundary conditions. We find calorons with unit topological charge as well as (anti-)selfdual lumps (BPS-monopoles or dyons) combined in pairs of non-integer (equal or opposite sign) topological charge. For calorons and separated pairs of equal-sign dyons obtained by cooling we have found that (i) the gluon field is well-described by Kraan-van Baal solutions of the Euclidean Yang-Mills field equations and (ii) the lowest Wilsonfermion modes are well-described by analytic solutions of the corresponding Dirac equation. For metastable configurations found at higher action, the multi-center structure can be interpreted in terms of dyons and antidyons, using the gluonic and fermionic indicators as in the dyon-pair case. Additionally, the Abelian monopole structure and field strength correlators between the centers are useful to analyse the configurations in terms of dyonic constituents. We argue that a semi-classical approximation of the non-zero temperature path integral should be built on superpositions of solutions with non-trivial holonomy.

Instantons at work

“…It should be noted that recent lattice simulations support the presence of KvBLL calorons (or, more generally, dyons) near the phase transition point [58,59].…”

Laplacian modes probing gauge fields

“…The monopoles make up the topological charge with the help of the so-called Taubes winding [23]: one of the monopoles performs a full rotation in the unbroken U (1)-subgroup relative to the other monopole when completing a full period in the time-like direction. For a gauge invariant statement one has to connect the field strength at the different monopole cores by a Schwinger line [24]. In the periodic gauge that we use for the calorons [9] put on the lattice, the link variables U µ (x) are static at the P = ½ 2 monopole, while they rotate around the holonomy direction τ 3 at the P = −½ 2 monopole.…”

Calorons and dyons at the thermal phase transition analyzed by overlap fermions

“…We should remind the reader that chirally improved lattice fermions [41,42] (another realization of GinspargWilson [51] fermions) have already been used to analyze unsmeared lattice configurations for the presence of calorons [52]. Before that, unimproved Wilson fermions have been employed [53] for the description of nearly classical calorons and dyons obtained by cooling. What was common to both techniques was inspired by the theoretically known behavior of zero modes of caloron-like configurations [54].…”

“…What was common to both techniques was inspired by the theoretically known behavior of zero modes of caloron-like configurations [54]. Thus, particular emphasis was first given to the zero modes (or the real modes in case of the Wilson-Dirac operator), which must be present in configurations with topological charge Q = 0, and to the effect on them of changing the boundary conditions for the Dirac operator [52,53,55,56]. Confronting the zeromode pattern with the picture revealed by smearing, it became clear [55] that the zero modes are part of the topological structure of a typical Monte Carlo configuration but cannot exhaustively explain it.…”