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Exact properties of an integrated correlator in N = 4 SU(N) SYM
Abstract
We present a novel expression for an integrated correlation function of four superconformal primaries in SU(N)
N
= 4 supersymmetric Yang-Mills (
N
= 4 SYM) theory. This integrated correlator, which is based on supersymmetric localisation, has been the subject of several recent developments. In this paper the correlator is re-expressed as a sum over a two dimensional lattice that is valid for all N and all values of the complex Yang-Mills coupling
Ï
=
Ξ
/
2
Ï
+
4
Ïi
/
g
YM
2
. In this form it is manifestly invariant under SL(2, â€) Montonen-Olive duality. Furthermore, it satisfies a remarkable Laplace-difference equation that relates the SU(N) correlator to the SU(N + 1) and SU(N â 1) correlators. For any fixed value of N the correlator can be expressed as an infinite series of non-holomorphic Eisenstein series,
E
s
Ï
Ï
ÂŻ
with s â â€, and rational coefficients that depend on the values of N and s. The perturbative expansion of the integrated correlator is an asymptotic but Borel summable series, in which the n-loop coefficient of order (gYM/Ï)2n is a rational multiple of ζ(2n + 1). The n = 1 and n = 2 terms agree precisely with results determined directly by integrating the expressions in one-loop and two-loop perturbative
N
= 4 SYM field theory. Likewise, the charge-k instanton contributions (|k| = 1, 2, . . .) have an asymptotic, but Borel summable, series of perturbative corrections. The large-N expansion of the correlator with fixed Ï is a series in powers of
N
1
2
â
â
(â â â€) with coefficients that are rational sums of
E
s
Ï
Ï
ÂŻ
with s â †+ 1/2. This gives an all orders derivation of the form of the recently conjectured expansion. We further consider the ât Hooft topological expansion of large-N Yang-Mills theory in which
λ
=
g
YM
2
N
is fixed. The coefficient of each order in the 1/N expansion can be expanded as a series of powers of λ that converges for |λ| < Ï2. For large λ this becomes an asymptotic series when expanded in powers of
1
/
λ
with coefficients that are again rational multiples of odd zeta values, in agreement with earlier results and providing new ones. We demonstrate that the large-λ series is not Borel summable, and determine its resurgent non-perturbative completion, which is
O
exp
â
2
λ
.</jats:p
Volume independence in the large N limit and an emergent fermionic symmetry
Large-N volume independence in circle-compactified QCD with adjoint Weyl fermions implies the absence of any phase transitions as the radius is dialed to arbitrarily small values. This class of theories is believed to possess a Hagedorn density of hadronic states. It turns out that these properties are in apparent tension with each other, because a Hagedorn density of states typically implies a phase transition at some finite radius. This tension is resolved if there are degeneracies between the spectra of bosonic and fermionic states, as happens in the Nf=1 supersymmetric case. Resolution of the tension for Nf>1 then suggests the emergence of a fermionic symmetry at large N, where there is no supersymmetry. We can escape the Coleman-Mandula theorem since the N=â theory is free, with a trivial S matrix. We show an example of such a spectral degeneracy in a nonsupersymmetric toy example which has a Hagedorn spectrum
Large N and Bosonization in Three Dimensions
Bosonization is normally thought of as a purely two-dimensional phenomenon,
and generic field theories with fermions in D>2 are not expected be describable
by local bosonic actions, except in some special cases. We point out that 3D
SU(N) gauge theories on R^{1,1} x S^{1}_{L} with adjoint fermions can be
bosonized in the large N limit. The key feature of such theories is that they
enjoy large N volume independence for arbitrary circle size L. A consequence of
this is a large N equivalence between these 3D gauge theories and certain 2D
gauge theories, which matches a set of correlation functions in the 3D theories
to corresponding observables in the 2D theories. As an example, we focus on a
3D SU(N) gauge theory with one flavor of adjoint Majorana fermions and derive
the large-N equivalent 2D gauge theory. The extra dimension is encoded in the
color degrees of freedom of the 2D theory. We then apply the technique of
non-Abelian bosonization to the 2D theory to obtain an equivalent local theory
written purely in terms of bosonic variables. Hence the bosonized version of
the large N three-dimensional theory turns out to live in two dimensions.Comment: 30 pages, 2 tables. v2 minor revisions, references adde
Group Theory of Non-Abelian Vortices
We investigate the structure of the moduli space of multiple BPS non-Abelian
vortices in U(N) gauge theory with N fundamental Higgs fields, focusing our
attention on the action of the exact global (color-flavor diagonal) SU(N)
symmetry on it. The moduli space of a single non-Abelian vortex, CP(N-1), is
spanned by a vector in the fundamental representation of the global SU(N)
symmetry. The moduli space of winding-number k vortices is instead spanned by
vectors in the direct-product representation: they decompose into the sum of
irreducible representations each of which is associated with a Young tableau
made of k boxes, in a way somewhat similar to the standard group composition
rule of SU(N) multiplets. The K\"ahler potential is exactly determined in each
moduli subspace, corresponding to an irreducible SU(N) orbit of the
highest-weight configuration.Comment: LaTeX 46 pages, 4 figure
The SAGEX Review on Scattering Amplitudes
This is an introduction to, and invitation to read, a series of review
articles on scattering amplitudes in gauge theory, gravity, and superstring
theory. Our aim is to provide an overview of the field, from basic aspects to a
selection of current (2022) research and developments.Comment: 15 pages, overview articl
Vortices and Monopoles in Mass-deformed SO and USp Gauge Theories
Effects of mass deformations on 1/2 Bogomol'nyi-Prasad-Sommerfield (BPS)
non-Abelian vortices are studied in 4d N=2 supersymmetric U(1) \times SO(2n)
and U(1) \times USp(2n) gauge theories, with Nf=2n quark multiplets. The 2d
N=(2,2) effective worldsheet sigma models on the Hermitian symmetric spaces
SO(2n)/U(n) and USp(2n)/U(n) found recently which describe the low-energy
excitations of the orientational moduli of the vortices, are generalized to the
respective massive sigma models. The continuous vortex moduli spaces are
replaced by a finite number (2^{n-1} or 2^{n}) of vortex solutions. The 1/2 BPS
kinks connecting different vortex vacua are magnetic monopoles in the 4d
theory, trapped inside the vortex core, with total configurations being 1/4 BPS
composite states. These configurations are systematically studied within the
semi-classical regime.Comment: 55 pages, 7 figure
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