More stringy effects in target space from Double Field Theory

In Double Field Theory, the mass-squared of doubled fields associated with bosonic closed string states is proportional to NL + NR− 2. Massless states are therefore not only the graviton, anti-symmetric, and dilaton fields with (NL = 1, NR = 1) such theory is focused on, but also the symmetric traceless tensor and the vector field relative to the states (NL = 2, NR = 0) and (NL = 0, NR = 2) which are massive in the lower-dimensional non-compactified space. While they are not even physical in the absence of compact dimensions, they provide a sample of states for which both momenta and winding numbers are non-vanishing, differently from the states (NL = 1, NR = 1). A quadratic action is therefore here built for the corresponding doubled fields. It results that its gauge invariance under the linearized double diffeomorphisms is based on a generalization of the usual weak constraint, giving rise to an extra mass term for the symmetric traceless tensor field, not otherwise detectable: this can be interpreted as a mere stringy effect in target space due to the simultaneous presence of momenta and windings. Furthermore, in the context of the generalized metric formulation, a non-linear extension of the gauge transformations is defined involving the constraint extended from the weak constraint that can be uniquely defined in triple products of fields. Finally, we show that the above mentioned stringy effect does not appear in the case of only one compact doubled space dimension.

Infrared renormalon in $SU(N)$ QCD(adj.) on $\mathbb{R}^3\times S^1$

We study the infrared renormalon in the gluon condensate in the $SU(N)$ gauge theory with $n_W$-flavor adjoint Weyl fermions (QCD(adj.)) on $\mathbb{R}^3\times S^1$ with the $\mathbb{Z}_N$ twisted boundary conditions. We rely on the so-called large-$\beta_0$ approximation as a conventional tool to analyze the renormalon, in which only Feynman diagrams that dominate in the large-$n_W$ limit are considered, while the coefficient of the vacuum polarization is set by hand to the one-loop beta function $\beta_0=11/3-2n_W/3$. In the large $N$ limit within the large-$\beta_0$ approximation, the W-boson, which acquires the twisted Kaluza–Klein momentum, produces the renormalon ambiguity corresponding to the Borel singularity at $u=2$. This provides an example that the system in the compactified space $\mathbb{R}^3\times S^1$ possesses the renormalon ambiguity identical to that in the uncompactified space $\mathbb{R}^4$. We also discuss the subtle issue that the location of the Borel singularity can change depending on the order of two necessary operations.

Infrared renormalon in the supersymmetric $\mathbb{C}P^{N-1}$ model on $\mathbb{R}\times S^1$

In the leading order of the large-$N$ approximation, we study the renormalon ambiguity in the gluon (or, more appropriately, photon) condensate in the 2D supersymmetric $\mathbb{C}P^{N-1}$ model on $\mathbb{R}\times S^1$ with the $\mathbb{Z}_N$ twisted boundary conditions. In our large-$N$ limit, the combination $\Lambda R$, where $\Lambda$ is the dynamical scale and $R$ is the $S^1$ radius, is kept fixed (we set $\Lambda R\ll1$ so that the perturbative expansion with respect to the coupling constant at the mass scale $1/R$ is meaningful). We extract the perturbative part from the large-$N$ expression of the gluon condensate and obtain the corresponding Borel transform $B(u)$. For $\mathbb{R}\times S^1$, we find that the Borel singularity at $u=2$, which exists in the system on the uncompactified $\mathbb{R}^2$ and corresponds to twice the minimal bion action, disappears. Instead, an unfamiliar renormalon singularity emerges at $u=3/2$ for the compactified space $\mathbb{R}\times S^1$. The semi-classical interpretation of this peculiar singularity is not clear because $u=3/2$ is not dividable by the minimal bion action. It appears that our observation for the system on $\mathbb{R}\times S^1$ prompts reconsideration on the semi-classical bion picture of the infrared renormalon.

Finite-size effects of linear sigma model in compactified space–time

The finite-sized effect caused by compactified space–time is scrutinized by means of the linear sigma model with constituent quarks at finite temperature T and chemical potential μ, where the compactified spatial dimension with length L is taken along the Oz direction. We find several finite-size effects associated with compactified length L: (a) There are two types of Casimir energy corresponding to two types of quarks, untwisted and twisted quarks. (b) For untwisted quarks, a first-order phase transition emerges at intermediate values of L when the Casimir effect is not taken into account and is enhanced by Casimir energy at small L. (c) For twisted quarks, the phase transition is cross-over everywhere when μ≤200 MeV . When μ> 200 MeV there occurs a first-order phase transition at large L and becomes cross-over at smaller L.

CHARGED PION CONDENSATION PHENOMENON OF DENSE BARYONIC MATTER INDUCED BY FINITE VOLUME: THE NJL2 MODEL CONSIDERATION

The properties of two-flavored massless Nambu–Jona-Lasinio (NJL) model in (1+1)-dimensional R1 × S1 space–time with compactified space coordinate are investigated in the presence of isospin and quark number chemical potentials μI, μ. The consideration is performed in the large Nc limit, where Nc is the number of colored quarks. It is shown that at L = ∞ (L is the length of the circumference S1) the charged pion condensation (PC) phase with zero quark number density is realized at arbitrary nonzero μI and for rather small values of μ. However, at arbitrary finite values of L the phase portrait of the model contains the charged PC phase with nonzero quark number density (in the case of periodic boundary conditions for quark fields). Hence, finite sizes of the system can serve as a factor promoting the appearance of the charged PC phase in quark matter with nonzero baryon densities. In contrast, the phase with chiral symmetry breaking may exist only at rather large values of L.

A FULLY PSEUDOSPECTRAL SCHEME FOR SOLVING SINGULAR HYPERBOLIC EQUATIONS ON CONFORMALLY COMPACTIFIED SPACE-TIMES

With the example of the spherically symmetric scalar wave equation on Minkowski space-time we demonstrate that a fully pseudospectral scheme (i.e. spectral with respect to both spatial and time directions) can be applied for solving hyperbolic equations. The calculations are carried out within the framework of conformally compactified space-times. In our formulation, the equation becomes singular at null infinity and yields regular boundary conditions there. In this manner, it becomes possible to avoid "artificial" conditions at some numerical outer boundary at a finite distance. We obtain highly accurate numerical solutions possessing exponential spectral convergence, a feature known from solving elliptic PDEs with spectral methods. Our investigations are meant as a first step towards the goal of treating time evolution problems in General Relativity with spectral methods in space and time.

SHORTCUTS IN PARTICLE PRODUCTION IN A TOROIDALLY COMPACTIFIED SPACE–TIME

We investigate the particle production in a toroidally compactified space–time due to the expansion of a Friedmann cosmological model in ℝ3 × S1 outside a U(1) local cosmic string. The case of a Friedmann space–time is also investigated where torsion is incorporated in the connection. We present a generalization to toroidal compactification of p extra dimensions, where the topology is given by ℝ3 × Tp. Some implications are presented and discussed. Besides the dynamics of space–time, we investigate in detail the physical consequences of the topological transformations.