Topology, symmetry, and phase transitions in lattice gauge ... - tuprints

C O N C L U D I N G R E M A R K S
6
As this thesis draws to an end, we are confronted with the elusive nature of strong
interactions. In spite of the great experimental successes of the Standard Model, a
complete theoretical understanding follows only in baby steps. In particular, the
non-linear, complex nature of the QCD sector hinders its analytical study. We have
searched for an intuitive grasp of the underlying mechanisms, and their connections
with symmetries that emerge under various deformations of the theory.
We have been motivated in particular by the global center symmetry of SU(N)
gauge theories in the limit of infinitely heavy, i.e., static quarks. It allows deconfinement
at finite temperature to be classified as a spontaneous symmetry transition
in close analogy with magnetization in an N-state spin system. As we reviewed in
Chapter 3, global center symmetry allows one to prepare superselection sectors for
the color electric flux of static charges via a combination of boundary conditions
with center vortex interfaces. This places topology at the heart of color confinement
in the pure gauge theory, which results at low temperature from the disorder
generated by vortices and is lost when these interfaces are suppressed at high temperature.
We performed a further deformation, a reduction of spacetime dimensions to
2 + 1 d, to fully exploit the correspondence with spin systems and glean insight
into the universal aspects of the confinement mechanism. This provided us full
access to the powerful tools of universality. The second order transitions of SU(2)
and SU(3) in 2 + 1 d fall into the universality classes of the 2d Ising and 3-state Potts
models, respectively, in which a wealth of exact analytical results are available. By
making use of the universality of interfaces in the gauge and spin systems, we were
able to locate the deconfinement transition on the lattice with extreme precision.
Our subsequent scaling studies unveiled a self-dual relationship between center
vortices and color electric fluxes at criticality, stemming from Kramers-Wannier
self-duality of the 2d N-state Potts models, and led us to precisely determine the
continuum vortex and electric SU(2) string tensions at criticality.
The manifestation of self-duality on finite volumes via ’t Hooft’s Fourier transform
over center sectors emphasizes the relevance of global center symmetry to deconfinement,
at least for static charges. A self-dual relation between center vortices
and electric fluxes in 2 + 1 d is only possible because the center vortex confinement
mechanism pinpoints the pertinent degrees of freedom. As our analysis of SU(4)
revealed, still more may be learnt from these 2 + 1 d models. The effective spin
system and order of the deconfining transition is, in this case, not trivially determined.
A better understanding of the interplay of the larger set of Z 4 center sectors,
via a combination of analytic and numerical methods, would shed further light on
confining gluonic dynamics.
This is a lesson for 3 + 1 d, where the dynamics are similar but universality is not
applicable for N ≥ 3 colors. Our results highlight the benefits of studying simpler
119