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Register a new userFree energy and boundary anomalies on $mathbb{S}^atimes mathbb{H}^b$ spaces
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We compute free energies as well as conformal anomalies associated with boundaries for a conformal free scalar field. To that matter, we introduce the family of spaces of the form $mathbb{S}^atimes mathbb{H}^b$, which are conformally related to $mathbb{S}^{a+b}$. For the case of $a=1$, related to the entanglement entropy across $mathbb{S}^{b-1}$, we provide some new explicit computations of entanglement entropies at weak coupling. We then compute the free energy for spaces $mathbb{S}^atimes mathbb{H}^b$ for different values of $a$ and $b$. For spaces $mathbb{S}^{2n+1}times mathbb{H}^{2k}$ we find an exact match with the free energy on $mathbb{S}^{2n+2k+1}$. For $mathbb{H}^{2k+1}$ and $mathbb{S}^{3}times mathbb{H}^{3}$ we find conformal anomalies originating from boundary terms. We also compute the free energy for strongly coupled theories through holography, obtaining similar results.
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We study confining strings in ${cal{N}}=1$ supersymmetric $SU(N_c)$ Yang-Mills theory in the semiclassical regime on $mathbb{R}^{1,2} times mathbb{S}^1$. Static quarks are expected to be confined by double strings composed of two domain walls - which are lines in $mathbb{R}^2$ - rather than by a single flux tube. Each domain wall carries part of the quarks chromoelectric flux. We numerically study this mechanism and find that double-string confinement holds for strings of all $N$-alities, except for those between fundamental quarks. We show that, for $N_c ge 5$, the two domain walls confining unit $N$-ality quarks attract and form non-BPS bound states, collapsing to a single flux line. We determine the $N$-ality dependence of the string tensions for $2 le N_c le 10$. Compared to known scaling laws, we find a weaker, almost flat $N$-ality dependence, which is qualitatively explained by the properties of BPS domain walls. We also quantitatively study the behavior of confining strings upon increasing the $mathbb{S}^1$ size by including the effect of virtual $W$-bosons and show that the qualitative features of double-string confinement persist.
In this article we complete the classification of the umbilical submanifolds of a Riemannian product of space forms, addressing the case of a conformally flat product $mathbb{H}^ktimes mathbb{S}^{n-k+1}$, which has not been covered in previous works on the subject. We show that there exists precisely a $p$-parameter family of congruence classes of umbilical submanifolds of $mathbb{H}^ktimes mathbb{S}^{n-k+1}$ with substantial codimension~$p$, which we prove to be at most $mbox{min},{k+1, n-k+2}$. We study more carefully the cases of codimensions one and two and exhibit, respectively, a one-parameter family and a two-parameter family (together with three extra one-parameter families) that contain precisely one representative of each congruence class of such submanifolds. In particular, this yields another proof of the classification of all (congruence classes of) umbilical submanifolds of $mathbb{S}^ntimes mathbb{R}$, and provides a similar classification for the case of $mathbb{H}^ntimes mathbb{R}$. We determine all possible topological types, actually, diffeomorphism types, of a complete umbilical submanifold of $mathbb{H}^ktimes mathbb{S}^{n-k+1}$. We also show that umbilical submanifolds of the product model of $mathbb{H}^ktimes mathbb{S}^{n-k+1}$ can be regarded as rotational submanifolds in a suitable sense, and explicitly describe their profile curves when $k=n$. As a consequence of our investigations, we prove that every conformal diffeomorphism of $mathbb{H}^ktimes mathbb{S}^{n-k+1}$ onto itself is an isometry.
We compute conformal anomalies for conformal field theories with free conformal scalars and massless spin $1/2$ fields in hyperbolic space $mathbb{H}^d$ and in the ball $mathbb{B}^d$, for $2leq dleq 7$. These spaces are related by a conformal transformation. In even dimensional spaces, the conformal anomalies on $mathbb{H}^{2n}$ and $mathbb{B}^{2n}$ are shown to be identical. In odd dimensional spaces, the conformal anomaly on $mathbb{B}^{2n+1}$ comes from a boundary contribution, which exactly coincides with that of $mathbb{H}^{2n+1}$ provided one identifies the UV short-distance cutoff on $mathbb{B}^{2n+1}$ with the inverse large distance IR cutoff on $mathbb{H}^{2n+1}$, just as prescribed by the conformal map. As an application, we determine, for the first time, the conformal anomaly coefficients multiplying the Euler characteristic of the boundary for scalars and half-spin fields with various boundary conditions in $d=5$ and $d=7$.
It is known that for $Omega subset mathbb{R}^{2}$ an unbounded convex domain and $H>0$, there exists a graph $Gsubset mathbb{R}^{3}$ of constant mean curvature $H$ over $Omega $ with $partial G=$ $partial Omega $ if and only if $Omega $ is included in a strip of width $1/H$. In this paper we obtain results in $mathbb{H}^{2}times mathbb{R}$ in the same direction: given $Hin left( 0,1/2right) $, if $Omega $ is included in a region of $mathbb{ H}^{2}times left{ 0right} $ bounded by two equidistant hypercycles $ell(H)$ apart, we show that, if the geodesic curvature of $partial Omega $ is bounded from below by $-1,$ then there is an $H$-graph $G$ over $Omega $ with $partial G=partial Omega$. We also present more refined existence results involving the curvature of $partialOmega,$ which can also be less than $-1.$
By employing the $1/N$ expansion, we compute the vacuum energy~$E(deltaepsilon)$ of the two-dimensional supersymmetric (SUSY) $mathbb{C}P^{N-1}$ model on~$mathbb{R}times S^1$ with $mathbb{Z}_N$ twisted boundary conditions to the second order in a SUSY-breaking parameter~$deltaepsilon$. This quantity was vigorously studied recently by Fujimori et al. using a semi-classical approximation based on the bion, motivated by a possible semi-classical picture on the infrared renormalon. In our calculation, we find that the parameter~$deltaepsilon$ receives renormalization and, after this renormalization, the vacuum energy becomes ultraviolet finite. To the next-to-leading order of the $1/N$ expansion, we find that the vacuum energy normalized by the radius of the~$S^1$, $R$, $RE(deltaepsilon)$ behaves as inverse powers of~$Lambda R$ for~$Lambda R$ small, where $Lambda$ is the dynamical scale. Since $Lambda$ is related to the renormalized t~Hooft coupling~$lambda_R$ as~$Lambdasim e^{-2pi/lambda_R}$, to the order of the $1/N$ expansion we work out, the vacuum energy is a purely non-perturbative quantity and has no well-defined weak coupling expansion in~$lambda_R$.
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