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

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82 ’t hooft monopoles in lattice guts 4.6.1 Magnetic flux Since the decomposition of the gauge fixed links (4.30) commutes with charge conjugation, C-periodic boundary conditions imply anti-periodicity of the α a µ(x) in (4.31). The abelian projected fields inherit anti-periodic boundary conditions α a µ(x + Lĵ) = −α a µ(x), (4.86) except for the special cases, in three dimensions corresponding to the links in Eqs. (4.82), where and in Eqs. (4.83), where The fluxes in three dimensions, α a y(L, L − 1, z) = −α a y(0, L − 1, z) − 2π N m 3, (4.87) α a z(L, y, L − 1) = −α a z(0, y, L − 1) + 2π N m 2, α a z(x, L, L − 1) = −α a z(x, 0, L − 1) − 2π N m 1, α a y(L, L − 1, L) = α a y(0, L − 1, 0) + 2π N m 3, (4.88) α a z(L, L, L − 1) = α a z(0, 0, L − 1) − 2π N (m 1 + m 2 ). αij a (⃗x) = αa i (⃗x) + αa j (⃗x + î) − αa i (⃗x + ĵ) − αa j (⃗x) (4.89) are essentially anti-periodic, because the twist angles (2π/N)m i cancel when we compare fluxes on opposite sides of the lattice. There is a single exception, α 23 (L, L − 1, L − 1) = (4.90) −α 23 (0, L − 1, L − 1) − 2π N 2(m 1 + m 2 + m 3 ). Because of the constraint on the possible twists in Eq. (4.52), and because flux is only defined modulo 2π, the additional contribution has no effect, and this is equivalent to anti-periodic boundary conditions also. We therefore have fully antiperiodic abelian field strengths. This means that when we cross the boundary we enter a charge conjugated copy of the same lattice from the opposite side. To determine the magnetic charge we repeat the trick of [100]. The curve shown in Fig. 4.4 divides the boundary into two halves. We denote the magnetic flux through them by Φ + and Φ − choosing the positive direction to be pointing outwards. The two halves are related by the boundary conditions, and in particular, antiperiodicity (4.89) of the field strength implies that they are equal Φ − = Φ + . The magnetic charge inside the lattice is given by the total flux, which is the sum of the two contributions, Q = Φ + + Φ − = 2Φ + . (4.91)
4.6 algebraic formulation 83 Applying Stokes’s theorem, we can write Φ a + = − 1 g L−1 + ∑ z=0 L−1 − ∑ y=0 ( L−1 ∑ α a L−1 x(x, 0, 0) + x=0 α a z(L, L, z) − α a y(0, y, L) − ∑ y=0 α a y(L, y, 0) L−1 ∑ α1 a (x, L, L) x=0 L−1 ∑ α3(0, a 0, z) z=0 ) . (4.92) When we apply the boundary conditions, all terms cancel except those involving the cases, Φ+ a = − 1 ( α a g y(L, L − 1, 0) + αz(L, a L, L − 1) ) −αy(0, a L − 1, L) − αz(0, a 0, L − 1) = 1 g 2π N (m 1 + m 2 + m 3 ) . (4.93) Where we have used the first equation in (4.87) with z = 0 and α a y(0, L − 1, 0) = −α a y(0, L − 1, L), and the second equation in (4.88). As the link angles α a µ are defined modulo 2π, the fluxes Φ ± are only defined modulo (2π/g). Therefore we find Q a = 4π gN (m 1 + m 2 + m 3 ) mod 4π g . (4.94) 4.6.2 Allowed magnetic charges Substituting the constraint on the twists for even N in Eq. (4.52) into Eq. (4.94) gives the charge quantisation condition Q a = 2π g Z 2, (4.95) up to integer multiples of (4π/g). Because all components of the charge vector Q a are the same, modulo (4π/g), the constraint, ∑ Q a = NQ a = 0 a mod 4π g (4.96) is automatically satisfied. This means that we can use twised C-periodic boundary conditions in SU(N) , when N is even, to restrict the ensemble to either of two distinct classes of monopole configurations. If the allowed twists satisfy m 1 + m 2 + m 3 = 0 (modulo N), then their total charges are integer multiples of (4π/g), Q a = 0 mod 4π g for all a. (4.97) If the twists are such that m 1 + m 2 + m 3 = N/2, then every component of the total charge vector is a half-odd integer multiple of (4π/g), Q a = 2π g mod 4π g for all a. (4.98)
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T O P O L O G Y, S Y M M E T RY, A
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To Angela.
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Z U S A M M E N FA S S U N G Wir un
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viii contents 4.6 Algebraic formula
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2 introduction While waiting on exp
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G A U G E T H E O R I E S 2 Before
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2.2 quantization 7 The non-abelian
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2.3 lattice field theory 9 Figure 2
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2.3 lattice field theory 11 the pro
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2.3 lattice field theory 13 picked
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16 universal aspects of confinement
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Page 96 and 97: 88 fractional electric charge and q
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132 more on c-periodic boundary con
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134 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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