Narrow Results

We study the effects of having multiple Spin structures on the partition function of the spacetime fields in M-theory. This leads to a potential anomaly which appears in the eta invariants upon variation of the Spin structure. The main sources of such spaces are manifolds with nontrivial fundamental group, which are also important in realistic models. We extend the discussion to the Spin^c case and find the phase of the partition function, and revisit the quantization condition for the C-field in this case. In type IIA string theory in 10 dimensions, the (mod 2) index of the Dirac operator is the obstruction to having a well-defined partition function. We geometrically characterize manifolds with and without such an anomaly and extend to the case of nontrivial fundamental group. The lift to KO-theory gives the α-invariant, which in general depends on the Spin structure. This reveals many interesting connections to positive scalar curvature manifolds and constructions related to the Gromov–Lawson–Rosenberg conjecture. In the 12-dimensional theory bounding M-theory, we study similar geometric questions, including choices of metrics and obtaining elements of K-theory in 10 dimensions by pushforward in K-theory on the disk fiber. We interpret the latter in terms of the families index theorem for Dirac operators on the M-theory circle and disk. This involves superconnections, eta forms, and infinite-dimensional bundles, and gives elements in Deligne cohomology in lower dimensions. We illustrate our discussion with many examples throughout.

Singular sectors ${{𝒵}}_{\mathrm{\text{sing}}}$ (loci of zeros) for real-valued non-positively defined partition functions ${𝒵}$ of $n$ variables are studied. It is shown that ${{𝒵}}_{\mathrm{\text{sing}}}$ have a stratified structure where each stratum is a set of certain hypersurfaces in ${ℝ}^n$. The concept of statistical amoeba is introduced and the properties of a family of statistical amoebas are studied. The relation with algebraic amoebas is discussed. Tropical limits of statistical amoebas are considered too. Applications of the concept of statistical amoeba to the analysis of singular sectors for integrable equations and properties of macroscopic systems with multiple equilibria, including frustrated systems, are discussed.

We report on the exact computation of the S³ partition function of U(N)_k × U(N)_-k ABJM theory for k = 1, N = 1, …, 19. The result is a polynomial in π^-1 with rational coefficients. As an application of our results, we numerically determine the coefficient of the membrane 1-instanton correction to the partition function.

In this study, we work out thermodynamic functions for a quantum gas of mesons described as color-electric charge dipoles. They refer to a particular parametrization of the trigonometric Rosen–Morse potential which allows to transform it to a perturbation of free quantum motion on the three-dimensional hypersphere, $S^3$, a manifold that can host only charge-neutral systems, the charge dipoles being the configuration of the minimal number of constituents. To the amount charge neutrality manifests itself as an important aspect of the color confinement in the theory of strong interaction, the Quantum Chromodynamics, we expect our findings to be of interest to the evaluation of temperature phenomena in the physics of hadrons and in particular in a quantum gas of color charge dipoles as are the mesons. The results are illustrated for $f_0$ and $J/\psi$ mesons.

In this paper, our focus is to discuss galaxy clustering in an accelerated universe through different approaches, viz., the extended structure of galaxies, the halo approximation, in which we evaluate the configurational integral between two finite limits and the most recent technique known as mean-field theory or approximation. Through these approximations, we were able to eliminate the divergence of the involved configurational integral. These approximations have been employed to develop a gravitational partition function, which has been used to evaluate the thermodynamics of galaxy clustering and the distribution function that goes with it. Throughout this paper, we will use the concept of Maxwell–Boltzmann’s statistics to study the galaxy clustering under the modified Yukawa potential. These thermodynamical quantities, such as Helmholtz free energy, entropy internal energy, and so on, were later plotted.

This work examines some aspects related to the existence of negative mass. The requirement for the partition function to converge leads to two distinct approaches. Initially, convergence is achieved by assuming a negative absolute temperature, which results in an imaginary partition function and complex entropy. Subsequently, convergence is maintained by keeping the absolute temperature positive while introducing an imaginary velocity. This modification leads to a positive partition function and real entropy. It seems the utilization of imaginary velocity may yield more plausible physical results compared to the use of negative temperature, at least for the partition function and entropy.

This work attempts to investigate the influence of the linear and quadratic generalized uncertainty principle (LQGUP) on the thermodynamical characteristics of relativistic Fermi gas. The general thermodynamical characteristics of relativistic Fermi gas, such as mean energy, pressure, entropy, and specific heat, are obtained. According to the modified function of temperature for mean energy and specific heat of relativistic Fermi gas, we find that at high temperature limit, the value of deformation parameter in the framework of LQGUP has an important effect on thermodynamic characteristics. It appears that the deformation parameter’s value is rising at higher energy levels of science, resulting in the minimal measurable length reaching close to the Planck length scale.

The modular invariance of the one-loop partition function of the closed bosonic string in four dimensions in the presence of certain homogeneous exact pp-wave backgrounds is studied. In the absence of an axion field, the partition function is found to be modular invariant and equal to the free field partition function. The partition function remains unchanged also in the presence of a fixed axion field. However, in this case, the covariant form of the action suggests summation over all possible twists generated by the axion field. This is shown to modify the partition function. In the light-cone gauge, the axion field generates twists only in the worldsheet σ-direction, so the resulting partition function is not modular invariant, hence wrong. To obtain the correct partition function one needs to sum over twists in the t-direction as well, as suggested by the covariant form of the action away from the light-cone gauge.

We investigate the theory of an open parafermionic string between two parallel Dp-, Dq-branes in Ramond and Neveu–Schwarz sectors. Trilinear commutation relations between the string variables are postulated and the corresponding ones in terms of modes are derived. The analysis of the spectrum shows that one can again have a free tachyon Neveu–Schwarz model for some values of the order of the paraquantization associated to some values of p and q. The consistency of this model requires the calculation of the partition function and its confrontation with the results of the degeneracies. A perfect agreement between the two results is obtained and the closure of the Virasoro superalgebra is confirmed.

Based on the generalized uncertainty principle (GUP) with a deformation of the phase-space, the partition function has recently been modified. In the present work, we analyze the self-consistency of the axiomatic thermodynamics derived from the deformed partition function. First, we set up the thermodynamic quantities such as pressure, energy density, entropy and number density, then we extend the study for testing the approval of consistency. We found that the deformed phase-space satisfies the axiomatic self-consistency of thermodynamics. We would expect the effects of GUP to be more pronounced at high frequencies, but the used deformed phase-space factor would still diverge as $\nu$ approaches ${\nu }_{\alpha }$.

We discuss the Casimir effect in heterotic string theory. This is done by considering a ${\mathbb{Z}}_2$ twist acting on one external compact direction and three internal coordinates. The hyperplanes fixed by the orbifold generator $G$ realize the two infinite parallel plates. For the latter to behave as “conducting material,” we implement in a modular invariant way the projection $(1-G)/2$ on the spectrum running in the vacuum-to-vacuum amplitude at one-loop. Hence, the relevant projector to account for the Casimir effect is orthogonal to that commonly used in string orbifold models which is $(1+G)/2$. We find that this setup yields the same net force acting on the plates in the context of quantum field theory and string theory. However, when supersymmetry is not present from the onset, finiteness of the resultant force in field theory is reached by adding formally infinite forces acting on either side of each plate, while in string theory, both contributions are finite. On the contrary, when supersymmetry is spontaneously broken à la Scherk–Schwarz, finiteness of each contribution is fulfilled in field and string theory.

We compute the partition functions of ${𝒩}=1$ gauge theories on $S^2\times {ℝ}_{𝜀}^2$ using supersymmetric localization. The path integral reduces to a sum over vortices at the poles of $S^2$ and at the origin of ${ℝ}_{𝜀}^2$. The exact partition functions allow us to test Seiberg duality beyond the supersymmetric index. We propose the ${𝒩}=1$ partition functions on the $\mathrm{\Omega }$-background, and show that the Nekrasov partition functions can be recovered from these building blocks.

We consider models of classical statistical mechanics satisfying natural stability conditions: a finite spin space, translation-periodic finite potential of finite range, a finite number of ground states meeting Peierls or Gertzik–Pirogov–Sinai condition. The Pirogov–Sinai theory describes the phase diagrams of these models at low temperature regimes. By using the method of doubling and mixing of partition functions we give an alternative elementary proof of the uniqueness of limiting Gibbs states at low temperatures in ground state uniqueness region.

In this article the equilibrious gas-liquid coexistent system is studied, and a new expression of partition function (PF) corresponding to the two-phase region is derived. Based on this expression, the horizontal line in the isotherm of pressure versus volume is obtained naturally for a finite particle system (i.e., without the necessity of taking the thermodynamic limit). Extending this PF, we can gain a unitive form of the one-component fluid in any system (i.e., one-phase or multi-phase). Then the whole isotherm will have reasonable statistical foundation. The VDW fluid system is discussed as a concrete example.

In this paper, an importance sampling method based on the Generalized Feynman–Kac (GFK) method has been used to calculate the mean values of quantum observables from quantum correlation functions for many-body systems with the Born–Oppenheimer approximation in the nonrelativistic limit both at zero and finite temperature. Specifically, the expectation values $〈r_i^n〉$, $〈r_{ij}^n〉$, $〈r_i^{-n}〉$ and $〈r_{ij}^{-n}〉$ for the ground state of the lithium and beryllium and the density matrix, the partition function, the internal energy and the specific heat of a system of quantum harmonic oscillators are computed, in good agreement with the best nonrelativistic values for these quantities. Although the initial results are encouraging, more experimentation will be needed to improve the other existing numerical results beyond chemical accuracies specially for the last two properties for lithium and beryllium. Also more work needs to be done to improve the trial functions for finite temperature calculations. Although these results look promising, more work needs to be done to achieve the spectroscopic accuracy at zero temperature and to estimate the finite temperature effects from the non-Born–Oppenheimer calculations. Also more experimentation will be needed to study the convergence criteria for the inverse properties for atoms at zero temperature.

In this article, the equation of state for a simple fluid is studied. A new universal equation of state is presented from statistical physics. It gives better properties for a liquid-gas coexisting system. Also, a reduced equation of state correlated with the reduced form of this universal one is obtained. It results in a phase diagram which fits the Guggenheim expressions perfectly well, and further the correct critical exponent β.

By virtue of the technique of integration within an ordered product of operators we derive the normally ordered expansion of time evolution operator for the case of a central oscillator immersed in a heat bath composed of a large number of oscillators. The time evolution of the system and heat bath into coherent states is discussed based on energy conservation. As a by-product, the partition function of two coupled oscillators is also calculated in this way.

The absence of phase transitions in one-dimensional Widom–Rowlinson model with long-range interaction is established in the non-symmetric case when different particles have different activity parameters.

The interaction between a monolayer of fine ferromagnetic particles and a semi-infinite superconductor has been investigated in the mixed state. The frozen and diamagnetic images model was employed to calculate the levitation force as a function of the levitation height as well as the temperature of the monolayer under the zero-field-cooled (ZFC) and the field-cooled (FC) conditions. Results showed the well-known monotonic decrease of the levitation force as a function of the levitation height while it increases rapidly as a function of temperature up to saturation. As a result of the first order approximation used in our calculations in which the interaction was represented by the forces between the magnetic dipoles and their images, the levitation force was dominated by the diamagnetic properties rather than the flux pinning effects of the superconductor for small values of levitation heights compared to the initial field cooling height.

In this paper, we consider a one-dimensional long range Widom–Rowlinson model when particle activity parameters are periodic and biased. We show that if the interaction is sufficiently large versus particle activities then the model does not exhibit a phase transition at low temperatures.