scholarly journals Equivalences between Calabi–Yau manifolds and roofs of projective bundles

2021 ◽  
Author(s):  
Marco Rampazzo
Keyword(s):  
Phase Transitions ◽  
Wide Class ◽  
Sigma Model ◽  
Derived Categories ◽  
Linear Sigma Model ◽  
Derived Equivalence ◽  
Birational Transformations ◽  
Projective Bundles ◽  
Special Case ◽  
Model Phase

It is conjectured that many birational transformations, called K-inequalities, have a categorical counterpart in terms of an embedding of derived categories. In the special case of simple K-equivalence (or more generally K-equivalence), a derived equivalence is expected: we propose a method to prove derived equivalence for a wide class of such cases. This method is related to the construction of roofs of projective bundles introduced by Kanemitsu. Such roofs can be related to candidate pairs of derived equivalent, L-equivalent and non isomorphic Calabi–Yau varieties, we prove such claims in some examples of this construction. In the same framework, we show that a similar approach applies to prove derived equivalence of pairs of Calabi–Yau fibrations, we provide some working examples and we relate them to gauged linear sigma model phase transitions.

PHASE TRANSITION PATTERNS OF NUCLEAR MATTER BASED ON EXTENDED LINEAR SIGMA MODEL

2013 ◽  
Vol 22 (11) ◽  
pp. 1350077 ◽  
Author(s):  
TRAN HUU PHAT ◽  
NGUYEN TUAN ANH ◽  
PHUNG THI THU HA
Keyword(s):  
Phase Transition ◽  
Phase Transitions ◽  
Nuclear Matter ◽  
Degrees Of Freedom ◽  
Sigma Model ◽  
Chemical Potential ◽  
Linear Sigma Model ◽  
Baryon Chemical Potential ◽  
Chiral Phase ◽  
The Relationship

We study systematically various types of phase transitions in nuclear matter at finite temperature T and baryon chemical potential μ based on the extended linear sigma model with nucleon degrees of freedom. It is shown that there are three types of phase transitions in nuclear matter: the chiral symmetry nonrestoration (SNR) at high temperature, the well-known liquid–gas (LG) phase transition at sub-saturation density and the Lifshitz phase transition (LPT) from the fully-gapped state to the state with Fermi surface. Their phase diagrams are established in the (T, μ)-plane and their physical properties are investigated in detail. The relationship between the chiral phase transition and the LG phase transition in nuclear matter is discussed.

Derived Category and Canonical Bundle --II

2007 ◽  
Author(s):  
D. Huybrechts
Keyword(s):  
Hochschild Cohomology ◽  
Derived Categories ◽  
Kodaira Dimension ◽  
Derived Category ◽  
Canonical Bundle ◽  
Derived Equivalence ◽  
One Step ◽  
Canonical Ring ◽  
Special Case

Based on the work of Orlov, Kawamata, and others, this chapter shows that the (numerical) Kodaira dimension and the canonical ring are preserved under derived equivalence. The same techniques can be used to derive the invariance of Hochschild cohomology under derived equivalence. Going one step further, it is shown that the nefness of the canonical bundle is detected by the derived category. The chapter also studies the relation between derived and birational (or rather K-) equivalence. The special case of a central conjecture predicts that two birational Calabi-Yau varieties have equivalent derived categories.

scholarly journals Kinematic numerators from the worldsheet: cubic trees from labelled trees

2021 ◽  
Vol 2021 (8) ◽  
Author(s):  
Song He ◽  
Linghui Hou ◽  
Jintian Tian ◽  
Yong Zhang
Keyword(s):  
Sigma Model ◽  
Building Blocks ◽  
Linear Sigma Model ◽  
Tree Level ◽  
Natural Class ◽  
Yang Mills ◽  
Tree Expansion ◽  
Labelled Trees ◽  
Logarithmic Functions ◽  
Special Case

Abstract In this note we revisit the problem of explicitly computing tree-level scattering amplitudes in various theories in any dimension from worldsheet formulas. The latter are known to produce cubic-tree expansion of tree amplitudes with kinematic numerators automatically satisfying Jacobi-identities, once any half-integrand on the worldsheet is reduced to logarithmic functions. We review a natural class of worldsheet functions called “Cayley functions”, which are in one-to-one correspondence with labelled trees, and natural expansions of known half-integrands onto them with coefficients that are particularly compact building blocks of kinematic numerators. We present a general formula expressing kinematic numerators of all cubic trees as linear combinations of coefficients of labelled trees, which satisfy Jacobi identities by construction and include the usual combinations in terms of master numerators as a special case. Our results provide an efficient algorithm, which is implemented in a Mathematica package, for computing all tree amplitudes in theories including non-linear sigma model, special Galileon, Yang-Mills-scalar, Einstein-Yang-Mills and Dirac-Born-Infeld.

scholarly journals Quark and Nuclear Matter in the Linear Chiral Meson Model

2003 ◽  
Vol 18 (18) ◽  
pp. 3189-3219 ◽  
Author(s):  
J. Berges ◽  
D.-U. Jungnickel ◽  
C. Wetterich
Keyword(s):  
Phase Transitions ◽  
Nuclear Matter ◽  
Sigma Model ◽  
Chemical Potential ◽  
Analytical Description ◽  
Baryon Number ◽  
Baryon Density ◽  
Linear Sigma Model ◽  
Density Enhancement ◽  
Quark Models

We present an analytical description of the phase transitions from a nucleon gas to nuclear matter and from nuclear matter to quark matter within the same model. The equation of state for quark and nuclear matter is encoded in the effective potential of a linear sigma model. We exploit an exact differential equation for its dependence upon the chemical potential μ associated to conserved baryon number. An approximate solution for vanishing temperature is used to discuss possible phase transitions as the baryon density increases. For a nucleon gas and nuclear matter we find a substantial density enhancement as compared to quark models which neglect the confinement to baryons. The results point out that the latter models are not suitable to discuss the phase diagram at low temperature.

scholarly journals Extended DBI and its generalizations from graded soft theorems

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Karol Kampf ◽  
Jiří Novotný ◽  
Petr Vaško
Keyword(s):  
Sigma Model ◽  
Power Counting ◽  
Linear Sigma Model ◽  
Tree Level ◽  
Bottom Up ◽  
Slight Generalization ◽  
S Matrix ◽  
New Type ◽  
Mixed Power ◽  
Special Case

Abstract We analyze a theory known as extended DBI, which interpolates between DBI and the U(N) × U(N)/U(N) non-linear sigma model and represents a nontrivial example of theories with mixed power counting. We discuss symmetries of the action and their geometrical origin; the special case of SU(2) extended DBI theory is treated in great detail. The revealed symmetries lead to a new type of graded soft theorem that allows us to prove on-shell constructibility of the tree-level S-matrix. It turns out that the on-shell constructibility of the full extended DBI remains valid, even if its DBI sub-theory is modified in such a way to preserve its own on-shell constructibility. We thus propose a slight generalization of the DBI sub-theory, which we call 2-scale DBI theory. Gluing it back to the rest of the extended DBI theory gives a new set of on-shell reconstructible theories — the 2-scale extended DBI theory and its descendants. The uniqueness of the parent theory is confirmed by the bottom-up approach that uses on-shell amplitude methods exclusively.

scholarly journals Chiral phase transitions in the linear sigma model in the Tsallis nonextensive statistics

2016 ◽  
Vol 25 (09) ◽  
pp. 1650066 ◽  
Author(s):  
Masamichi Ishihara
Keyword(s):  
Phase Transitions ◽  
Critical Temperature ◽  
Energy Density ◽  
Sigma Model ◽  
Free Particle ◽  
Linear Sigma Model ◽  
Chiral Phase ◽  
Expectation Value ◽  
Nonextensive Statistics ◽  
Tsallis Distribution

In this paper, we studied chiral phase transitions in the Tsallis nonextensive statistics which has two parameters, the temperature [Formula: see text] and entropic parameter [Formula: see text]. The linear sigma model was used in this study. The critical temperature, condensate, masses and energy density were calculated under the massless free particle approximation. The critical temperature decreases as [Formula: see text] increases. The condensate at [Formula: see text] is smaller than that at [Formula: see text]. The sigma mass at [Formula: see text] is heavier than the mass at [Formula: see text] at high temperature, while the sigma mass at [Formula: see text] is lighter than the mass at [Formula: see text] at low temperature. The pion mass at [Formula: see text] is heavier than the mass at [Formula: see text]. The energy density increases remarkably as [Formula: see text] increases. The [Formula: see text] dependence in the case of the [Formula: see text]-expectation value is weaker than that in the case of the conventional expectation value with a Tsallis distribution. The parameter [Formula: see text] should be smaller than [Formula: see text] from energetic point of view. The validity of the Tsallis statistics can be determined by the difference in [Formula: see text] of the restriction for [Formula: see text] when the interaction is weak, because the parameter [Formula: see text] is smaller than [Formula: see text] in the case of the conventional expectation value with a Tsallis distribution.

scholarly journals On the magnetic catalysis and inverse catalysis of phase transitions in the linear sigma model

2015 ◽  
Vol 258-259 ◽  
pp. 209-212
Author(s):  
Alejandro Ayala ◽  
M. Loewe ◽  
C. Villavicencio ◽  
R. Zamora
Keyword(s):  
Phase Transitions ◽  
Sigma Model ◽  
Linear Sigma Model ◽  
Magnetic Catalysis

scholarly journals Long way to Ricci flatness

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Jin Chen ◽  
Chao-Hsiang Sheu ◽  
Mikhail Shifman ◽  
Gianni Tallarita ◽  
Alexei Yung
Keyword(s):  
Sigma Model ◽  
Ultra Violet ◽  
The Other ◽  
Target Space ◽  
Linear Sigma Model ◽  
Close Relative ◽  
Two Dimensional ◽  
World Sheet ◽  
Ricci Flat ◽  
The World

Abstract We study two-dimensional weighted $$ \mathcal{N} $$ N = (2) supersymmetric ℂℙ models with the goal of exploring their infrared (IR) limit. 𝕎ℂℙ(N,$$ \tilde{N} $$ N ˜ ) are simplified versions of world-sheet theories on non-Abelian strings in four-dimensional $$ \mathcal{N} $$ N = 2 QCD. In the gauged linear sigma model (GLSM) formulation, 𝕎ℂℙ(N,$$ \tilde{N} $$ N ˜ ) has N charges +1 and $$ \tilde{N} $$ N ˜ charges −1 fields. As well-known, at $$ \tilde{N} $$ N ˜ = N this GLSM is conformal. Its target space is believed to be a non-compact Calabi-Yau manifold. We mostly focus on the N = 2 case, then the Calabi-Yau space is a conifold. On the other hand, in the non-linear sigma model (NLSM) formulation the model has ultra-violet logarithms and does not look conformal. Moreover, its metric is not Ricci-flat. We address this puzzle by studying the renormalization group (RG) flow of the model. We show that the metric of NLSM becomes Ricci-flat in the IR. Moreover, it tends to the known metric of the resolved conifold. We also study a close relative of the 𝕎ℂℙ model — the so called zn model — which in actuality represents the world sheet theory on a non-Abelian semilocal string and show that this zn model has similar RG properties.

scholarly journals Magnetic field dependence of the neutral pion mass in the linear sigma model with quarks: The strong field case

2021 ◽  
Vol 103 (5) ◽  
Author(s):  
Alejandro Ayala ◽  
José Luis Hernández ◽  
L. A. Hernández ◽  
Ricardo L. S. Farias ◽  
R. Zamora
Keyword(s):  
Magnetic Field ◽  
Field Dependence ◽  
Magnetic Field Dependence ◽  
Sigma Model ◽  
Neutral Pion ◽  
Strong Field ◽  
Pion Mass ◽  
Linear Sigma Model ◽  
Field Case
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