SU(N) caloron with non-trivial
holonomy
and its fermionic determinant.
Sergey Slizovskiy,
Nikolay Gromov
Petersburg INP
Outline of the talk
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Finite temperature QCD
Non-trivial holonomy
Caloron with non-trivial holonomy, Nc>2.
Monopole constituents
Why and when to expect calorons with nontrivial holonomy
Partition function, role of zero modes
ADHMN construction
The fermionic determinant.
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Finite temperature QCD
The finite-temperature behavior of any theory is specified by the
partition function
For QCD one finds (Abers and Lee, 1973; Faddeev, 1976)
The functional integral is restricted to fields satisfying the periodicity
conditions
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Non-trivial holonomy
All fields with finite energy can be classified by the behavior of the
Polyakov line = holonomy at spatial infinity
topological charge
and magnetic charges
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Calorons with non-trivial holonomy
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Exact solutions with topological charge 1
Constructed in 1998 by Kraan, van Baal, Lee, Lu.
Composed of Nc monopole constituents
Reduce to the periodical instanton at trivial
holonomy.
Action density of SU(3)
caloron
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BPS Dyon
• Static SU(2) field configuration with unit electric and magnetic charges.
It is embedded in the SU(n) along the simple roots.
• A4 plays the role of the Higgs field and assumed to be Abelian
(lies in the Cartan cubalgebra).
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Dyon embedding
M dyons
Becomes time dependent!
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Field near dyon
The field near the embedded dyon is composed of
Constant part – orthogonal to the root
And the dyon part along t1 , t2 , t3
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Far field – simple formula.
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Dyon sizes are calculated to be
The asymptotic gauge field is
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Why could we expect the nontrivial holonomy?
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1-loop quantum corrections (non-zero modes) prefer
trivial holonomy, but
at non-trivial holonomy the moduli space is richer, which
could overwhelm the non-zero mode determinant.
Some lattice simulations confirm these expectations.
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Partition function with fermions.
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The contribution of one caloron configuration is zero,
since it has an exact zero mode.
Only caloron – anticaloron ensembles with zero
topological charge could contribute.
The would be zero modes are localized on only one dyon
in the caloron. (M.Chernodub, P.van Baal et al. )
It leads to attraction between certain types of
constituent dyons from different calorons and
anticalorons.
Non-zero modes are accounted for by the functional
determinant in the fundamental representation
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Calculation of the determinant
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Calculate the variation of the determinant
in the far-field limit:
In the dyon core
In the rest space
 Find the integration constant.
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Reduce the configuration to the SU(N-1) ! …! SU(2) by
moving the parameters.
Find the variation of determinant during this reduction.
Add the known SU(2) result.
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Periodicity issues
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There are N different calorons with the same
action density. They are related by the nonperiodical gauge transformation e2  i t k/N for
integer k. It is equivalent to twisting the
periodic propagator:
Adding a matrix proportional to the unit matrix
to the gauge field is also equivalent to the
additional twist of the propagator.
We have found a simple exact universal recipe
to account for these twists.
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The variation: general formulas
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Currents
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We separate the current J=Jsingular+Jregular
according to n=0 and n  0 terms in the
propagator and use the standard
regularization for the singular current.
With exponential precision the result is
just J(x) = P’(A4(x))
It suggests the need of unified treatment
without separation, which is still missing.
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Atiyah-Drinfeld-Hitchin-Manin-Nahm
(ADHMN) construction of SU(N) caloron
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ADHM expression for the gauge field
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ADHMN Green’s function
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ADHM expressions for the currents.
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The singular current is
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The regular current is
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We have proved that the regular current is
a total derivative and so can be calculated
exactly from the exponential precision
limit of ADHMN construction
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Results for the currents
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We have calculated the total current far from the
monopoles.
We proved explicitly that the dyon core regions are the
same as in the SU(2) case plus constant field part,
equivalent to the twist. So we easily extract the core
currents from our previous work.*
Since the twist changes only the regular current, only
the P(v) terms are simply changed to P(v+twist)
*
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Finding a constant
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To find a constant in the determinant we
reduce the SU(N) configuration to SU(N-1)
We found explicitly the SU(N-1) ADHMN
construction when i ! i+1 and the i-th
dyon is situated on the line connecting the
(i-1) with (i+1) dyons!
We have proved that the blowed up i-th
dyon does not contribute to the constant.
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The result
=  for anti-periodic boundary conditions
and  is a Pauli-Villars mass.
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Thak
you!
Спасибо!
Merci! Danke!
Tack!
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