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T. Schweigler:
"Topological Objects and Chiral Symmetry Breaking in QCD";
Supervisor: M. Faber; Atominstitut, 2012; final examination: 2012-10-03.

In this master thesis, topological objects in SU(2) gauge theory are investigated. Besides investigating the objects in general, I also tried to find out more about their importance for spontaneous chiral symmetry breaking.

The lattice gauge object, whose properties have been mainly investigated, is the spherical vortex \cite{jordan2008tests,hollwieser2010lattice,hollwieser2012critical}. %The spherical vortex is a center vortex with spherical Dirac volume and a very special color structure. Because of this color structure, the object has topological charge $Q_T=1$.
As already mentioned in the cited papers, the spherical vortex seems to be some sort of squeezed instanton. In this document, a very detailed investigation of the spherical vortex is performed. The investigation takes place partly in the continuum and partly on the lattice.

In previous work, only spherical vortices with temporal extent of one lattice unit have been investigated. For such objects, one gets vanishing topological charge but nonvanishing difference $n_- - n_+=1$ of left and right chiral zeromodes. In this document, it is shown that this discrepancy is simply a discretization effect. For growing temporal extent of the vortex, the lattice topological charge approaches 1 and the index theorem $Q_T=n_- - n_+$ is fulfilled again. Moreover, the action and topological charge density of the spherical vortex have been calculated analytically in the continuum. The spatial and temporal localization of the zeromode(s) for the spherical vortex have been investigated.

Another investigation concerned the lowest nonzero eigenvalues of the Dirac operator for the spherical vortex. It was demonstrated that, for the vortex getting smaller and smaller, these eigenvalues approach the eigenvalues of the free Dirac operator. The zeromode occurs no matter how small the vortex becomes. Moreover, the claim (made in \cite{horvath2001evidence}) that self-dual/anti-self-dual gauge field contributions attract the negative/positive chiral components of the eigenmodes was checked for the lowest non-zero-modes. The results for the localization of the scalar and the chiral density are in agreement with this claim. The same investigation was also done for the lowest non-zero-modes for the instanton. It is interesting to see, that in case of the the instanton (consisting only of self-dual field contributions) only negative chiral components are attracted by the object, in case of the spherical vortex (consisting of both self-dual and anti-self-dual contributions) also positive chiral components are attracted by the object.

Last but not least, the interaction between a spherical vortex and a spherical ``antivortex" (object with $Q_T=-1$) has been investigated. The transformation of the two would-be zeromodes into two near-zero modes was demonstrated.

chiral symmetry breaking, chiral condensate, Dirac operator

http://publik.tuwien.ac.at/files/PubDat_227042.pdf