A High Temperature Phase Transition in Weakly Coupled Large N Gauge Theories on a Three-sphere Page: 3 of 10
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that all fields in the gauge theory are in the adjoint representation; the generalization to
other cases is straightforward. The effective action of such free gauge theories, as computed
in [5, 6], takes the form, up to an overall additive constant,
S = - > - (zB(n,3) + (-l)n+1zF(n,3)) Tr(Un)Tr(U--), (1)
n
n=1
where zB(/3) (zF(3)) is the generating function for the bosonic (fermionic) modes in this
classical theory on S3, given by the sum of e-3E over all bosonic (fermionic) states of
energy E2, counting each mode in the adjoint representation once. The phase structure of
the matrix model (1) was analyzed in [5, 6]; the action S provides an attraction between
the eigenvalues (at least at short distance), but as in any unitary matrix model there is a
repulsion between the eigenvalues coming from the measure. We can write the action in
terms of the eigenvalues e'0p (p = 1, - - - , N) of U in the form:
S = 7> (1 - zB(n3) - (-1)n+1zF(n)) cos(n(Op - q)), (2)
p n
where the first term comes from the change of measure from the unitary matrix to its
eigenvalues. From now on we will discuss only the large N limit of the gauge theory, in
which we can assume that there is some smooth distribution of the eigenvalues. At low
temperatures the repulsion wins and the eigenvalue distribution is uniform, corresponding
to a confined phase in which the expectation values of the Polyakov-Susskind loops Tr(Un)
vanish. At a temperature Td of order 1/R, given by the solution to zB(/3) + zF(3) = 1,
there is a weakly first order phase transition to a phase where the eigenvalue distribution is
non-uniform and gapped, and this deconfined phase governs the high temperature behavior.
Adding higher loop corrections turns this weakly first order transition either into a first order
transition or into a second order transition followed by a third order transition, depending
on a coefficient which requires a three-loop computation [6, 7].
For T 1/R the eigenvalue distribution becomes highly localized. In this limit the
functions zB and zF go as 2nB(TR)3 and as 2nF(TR)3, respectively, where nB (nF) is the
number of bosonic (fermionic) adjoint degrees of freedom in the theory: two from the vector
field plus one for every additional scalar field. We can then expand the action (2) for small
values of 0. The action includes a quadratic term, which is a one-loop mass term for the zero
mode a; recalling that we are working in a normalization of the fields with a factor of N/A
in front of the classical action (where A is the 't Hooft coupling A g2 MN), this quadratic
term corresponds to a physical mass proportional to AT2(nB - InF). This is precisely the
one-loop "electric mass term" that we expect to find for AO in the large volume limit (see
[8] and references therein); in the large TR limit the diagrams giving this mass term in the
theory on S3 become identical to the same diagrams on R3 (they are not IR-divergent).
The presence of this mass term means that even though classically a is always the lightest
mode, in the theory with finite coupling this is no longer true when AT2 - 1/R2, since then2
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Aharony, Ofer & Hartnoll, Sean A. A High Temperature Phase Transition in Weakly Coupled Large N Gauge Theories on a Three-sphere, report, June 22, 2007; [Menlo Park, California]. (https://digital.library.unt.edu/ark:/67531/metadc883651/m1/3/: accessed April 24, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.