59 research outputs found

### Scale Invariance + Unitarity => Conformal Invariance?

We revisit the long-standing conjecture that in unitary field theories, scale
invariance implies conformality. We explain why the Zamolodchikov-Polchinski
proof in D=2 does not work in higher dimensions. We speculate which new ideas
might be helpful in a future proof. We also search for possible
counterexamples. We consider a general multi-field scalar-fermion theory with
quartic and Yukawa interactions. We show that there are no counterexamples
among fixed points of such models in 4-epsilon dimensions. We also discuss fake
counterexamples, which exist among theories without a stress tensor.Comment: 17p

### An Introduction to Resurgence, Trans-Series and Alien Calculus

In these notes we give an overview of different topics in resurgence theory
from a physics point of view, but with particular mathematical flavour. After a
short review of the standard Borel method for the resummation of asymptotic
series, we introduce the class of simple resurgent functions, explaining their
importance in physical problems. We define the Stokes automorphism and the
alien derivative and discuss these objects in concrete examples using the
notion of trans-series expansion. With all the tools introduced, we see how
resurgence and alien calculus allow us to extract non-perturbative physics from
perturbation theory. To conclude, we apply Morse theory to a toy model path
integral to understand why physical observables should be resurgent functions.Comment: 48 pages, 7 figures; v2 typos fixed, references added, minor
correction

### Dimensionally reduced SYM$_4$ at large-$N$: an intriguing Coulomb approximation

We consider the light-cone (LC) gauge and LC quantization of the dimensional
reduction of super Yang Mills theory from four to two dimensions. After
integrating out all unphysical degrees of freedom, the non-local LC Hamiltonian
exhibits an explicit ${\cal N}=(2,2)$ supersymmetry. A further SUSY-preserving
compactification of LC-space on a torus of radius $R$, allows for a large-$N$
numerical study where the smooth large-$R$ limit of physical quantities can be
checked. As a first step, we consider a simple, yet quite rich, "Coulomb
approximation" that maintains an ${\cal N}=(1,1)$ subgroup of the original
supersymmetry and leads to a non-trivial generalization of 't Hooft's model
with an arbitrary --but conserved-- number of partons. We compute numerically
the eigenvalues and eigenvectors both in momentum and in position space. Our
results, so far limited to the sectors with 2, 3 and 4 partons, directly and
quantitatively confirm a simple physical picture in terms of a string-like
interaction with the expected tension among pairs of nearest-neighbours along
the single-trace characterizing the large-$N$ limit. Although broken by our
approximation, traces of the full ${\cal N}=(2,2)$ supersymmetry are still
visible in the low-lying spectrum.Comment: 30 pages, 13 figures, Footnote page 3 replaced, Note Added at the
end, 4 References adde

### Two string theory flavours of generalised Eisenstein series

Generalised Eisenstein series are non-holomorphic modular invariant functions of a complex variable, Ï„, subject to a particular inhomogeneous Laplace eigenvalue equation on the hyperbolic upper-half Ï„-plane. Two infinite classes of such functions arise quite naturally within different string theory contexts. A first class can be found by studying the coefficients of the effective action for the low-energy expansion of type IIB superstring theory, and relatedly in the analysis of certain integrated four-point functions of stress tensor multiplet operators in N = 4 supersymmetric Yang-Mills theory. A second class of such objects is known to contain all two-loop modular graph functions, which are fundamental building blocks in the low-energy expansion of closed-string scattering amplitudes at genus one. In this work, we present a PoincarÃ© series approach that unifies both classes of generalised Eisenstein series and manifests certain algebraic and differential relations amongst them. We then combine this technique with spectral methods for automorphic forms to find general and non-perturbative expansions at the cusp Ï„ â†’ iâˆž. Finally, we find intriguing connections between the asymptotic expansion of these modular functions as Ï„ â†’ 0 and the non-trivial zeros of the Riemann zeta function

### Exceptionally simple integrated correlators in N = 4 supersymmetric Yang-Mills theory

Supersymmetric localisation has led to several modern developments in the study of integrated correlators in N = 4 supersymmetric Yang-Mills (SYM) theory. In particular, exact results have been derived for certain integrated four-point functions of superconformal primary operators in the stress tensor multiplet which are valid for all classical gauge groups, SU(N), SO(N), and USp(2N), and for all values of the complex coupling, Ï„ = Î¸/(2Ï€) + 4Ï€i/gYM2. In this work we extend this analysis and provide a unified two-dimensional lattice sum representation valid for all simple gauge groups, in particular for the exceptional series Er (with r = 6, 7, 8), F4 and G2. These expressions are manifestly covariant under Goddard-Nuyts-Olive duality which for the cases of F4 and G2 is given by particular Fuchsian groups. We show that the perturbation expansion of these integrated correlators is universal in the sense that it can be written as a single function of three parameters, called Vogel parameters, and a suitable â€™t Hooft-like coupling. To obtain the perturbative expansion for the integrated correlator with a given gauge group we simply need substituting in this single universal expression specific values for the Vogel parameters. At the non-perturbative level we conjecture a formula for the one-instanton Nekrasov partition function valid for all simple gauge groups and for general Î©-deformation background. We check that our expression reduces in various limits to known results and that it produces, via supersymmetric localisation, the same one-instanton contribution to the integrated correlator as the one derived from the lattice sum representation. Finally, we consider the action of the hyperbolic Laplace operator with respect to Ï„ on the integrated correlators with exceptional gauge groups and derive inhomogeneous Laplace equations very similar to the ones previously obtained for classical gauge groups

### Two string theory flavours of generalised Eisenstein series

Generalised Eisenstein series are non-holomorphic modular invariant functions
of a complex variable, $\tau$, subject to a particular inhomogeneous Laplace
eigenvalue equation on the hyperbolic upper-half $\tau$-plane. Two infinite
classes of such functions arise quite naturally within different string theory
contexts. A first class can be found by studying the coefficients of the
effective action for the low-energy expansion of type IIB superstring theory,
and relatedly in the analysis of certain integrated four-point functions of
stress tensor multiplet operators in $\mathcal{N} = 4$ supersymmetric
Yang-Mills theory. A second class of such objects is known to contain all
two-loop modular graph functions, which are fundamental building blocks in the
low-energy expansion of closed-string scattering amplitudes at genus one. In
this work, we present a Poincar\'e series approach that unifies both classes of
generalised Eisenstein series and manifests certain algebraic and differential
relations amongst them. We then combine this technique with spectral methods
for automorphic forms to find general and non-perturbative expansions at the
cusp $\tau \to i \infty$. Finally, we find intriguing connections between the
asymptotic expansion of these modular functions as $\tau \to 0$ and the
non-trivial zeros of the Riemann zeta function.Comment: 44 pages + 3 figure

### Aspetti Quantistici dei Vortici Non-Abeliani in Teorie di Gauge Supersimmetriche

Studio di vortici non abeliani in teorie di gauge supersimmetriche $N=2$. Problematiche del confinamento come effetto Meissner duale

### To the cusp and back: resurgent analysis for modular graph functions

Modular graph functions arise in the calculation of the low-energy expansion of closed-string scattering amplitudes. For toroidal world-sheets, they are SL(2, â„¤)-invariant functions of the torus complex structure that have to be integrated over the moduli space of inequivalent tori. We use methods from resurgent analysis to construct the non-perturbative corrections arising for two-loop modular graph functions when the argument of the function approaches the cusp on this moduli space. SL(2, â„¤)-invariance will in turn strongly constrain the behaviour of the non-perturbative sector when expanded at the origin of the moduli space

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