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

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40 universal aspects of confinement in 2 + 1 dimensions Figure 3.13: A 2d worldsheet with conformal defects separating different CFT phases. From Ref. [72]. Polyakov loops, and the percolating center flux that disorders it, acquire the same dimensionality as temperature increases. They may then inherit the mapping from the Kramers-Wannier duality of the spin model such that the physics of confined electric fluxes just below T c has the same content as that of confined center fluxes just above T c . This drives home the correctness of the center vortex confinement mechanism. In 3 + 1 d, center vortices become surface like, and their dimensionality no longer matches that of color-electric flux at criticality. Nevertheless, a topological linking remains between the operators that create these fluxes: non-singular gauge transformations for center vortices (’t Hooft loops), and Wilson loops for color-electric fluxes. 3.5.0.1 Duality and conformal theories Self-duality on a 2d torus is a general feature of the N-state Potts models, valid at all temperatures. At the second order phase transitions of Potts models with N ≤ 4, it extends to the 2d conformal field theories (CFT) that describe the universality class. This establishes a link with the keenly studied dualities in conformal and supersymmetric theories. The discovery of dualities between various string theories led to a revolution in the 1990s that spread to gauge theories via Maldacena’s conjectured AdS/CFT correspondence [73]. 2d conformal field theories have a special theoretical significance. For starters, the infinitesimal conformal transformations form an infinite-dimensional algebra in two dimensions. This large symmetry makes 2d CFTs much more tractable than conformal theories in higher dimensions, which have finite dimensional algebras. It allows them to be solved by symmetry considerations alone [74]. In particular, 2d CFTs are classified by the central charge c of their Virasoro algebras, which is specified by a coefficient c = 1/2 for the 2d Ising universality class and c = 4/5 for the 2d 3-state Potts universality class. The peculiarities of 2d CFTs are a blessing not only for condensed matter physicists, but also for string theorists. They emerge naturally from perturbation theory on the 2d worldsheet of strings and bestow string theory with much of its rich mathematical structure [75]. Dualities in 2d CFTs then become important as dualities of the string worldsheet. In the context of 2d CFTs, the defects responsible for Kramers-Wannier duality are a subclass of what have been coined conformal defects [72]. They are obtained by
3.5 finite size scaling and self-duality 41 cutting a 2d surface along a 1d defect line and re-gluing it with boundary conditions for the CFTs on each side. If the boundary conditions conserve the stress-energy tensor, then they are totally transmissive for momentum and the defect is termed topological. This corresponds to translationally invariant boundary conditions (the torus). In the case of the 2d Ising model, periodic and antiperiodic boundary conditions yield a trivial defect (↑↑) and a spin interface (↑↓) respectively. These enact the symmetry group of the conformal theory at criticality, i.e., they flip between Z 2 sectors of the CFT. At criticality, translationally invariant boundary conditions on the torus become boundary states of the conformal field theory. Kramers-Wannier duality is a consequence of the fact that these defects are grouplike. They can be undone by the insertion of another defect [72]. There is a third type of conformal defect in the CFT that has a less straight forward interpretation in the spin model. It can be thought of as enacting Kramers- Wannier duality itself. That is, exchanging the order and disorder fields between the low and high temperature phases. It is associated with free boundary conditions. This conformal defect underlies the duality relation between free and fixed boundary conditions in the 2d Ising model [76, 77]. Note that the framework of conformal defects also allows one to study dualities between different critical theories. For instance, the tetra-critical Ising model and 3-state Potts model both have central charge c = 4/5. One can construct a Kramers- Wannier duality via conformal defects between their in-equivalent CFT phases [72].
Page 1 and 2: T O P O L O G Y, S Y M M E T RY, A
Page 3: To Angela.
Page 6 and 7: Z U S A M M E N FA S S U N G Wir un
Page 8 and 9: viii contents 4.6 Algebraic formula
Page 10 and 11: 2 introduction While waiting on exp
Page 13 and 14: G A U G E T H E O R I E S 2 Before
Page 15 and 16: 2.2 quantization 7 The non-abelian
Page 17 and 18: 2.3 lattice field theory 9 Figure 2
Page 19 and 20: 2.3 lattice field theory 11 the pro
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Page 24 and 25: 16 universal aspects of confinement
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Page 75 and 76: 4.1 magnetic charges in the continu
Page 77 and 78: 4.2 magnetic charge on the lattice
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Page 81 and 82: 4.5 intuitive picture 73 Therefore
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Page 85 and 86: 4.5 intuitive picture 77 This illus
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Page 89 and 90: 4.6 algebraic formulation 81 Figure
Page 91 and 92: 4.6 algebraic formulation 83 Applyi
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90 fractional electric charge and q
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100 fractional electric charge and
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C O N C L U D I N G R E M A R K S 6
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T W I S T AWAY F R O M C R I T I C
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twist away from criticality 123 0.1
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130 more on c-periodic boundary con
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L O W- E N E R G Y R E P R E S E N
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B I B L I O G R A P H Y [1] S. Glas
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ibliography 141 [30] L. von Smekal,
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ibliography 143 [60] J. Engels, E.
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ibliography 145 [89] A. S. Kronfeld
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ibliography 147 [120] Y. Aoki, G. E
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ibliography 149 [151] M. P. Nightin
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ibliography 151 [179] C. Korthals A
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C U R R I C U L U M Name: Samuel Ry
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