Foliation Structure Research Articles

Higgs branch of 6D (1, 0) SCFTs and little string theories with Dynkin DE-type SUSY enhancement

We detail the Higgs branches of 6D (1, 0) superconformal field theories (SCFTs) and little string theories (LSTs) that exhibit supersymmetry-enhancing Higgs branch renormalization group flows to the 6D (2, 0) SCFTs and LSTs of type DE. Generically, such theories are geometrically engineered in F-theory via a configuration of (−2)-curves, arranged in an (affine) DE-type Dynkin diagram, and supporting special unitary gauge algebras; this describes the effective field theory on the tensor branch of the SCFT. For the Higgsable to D-type (2, 0) SCFTs/LSTs, there generically also exists a type IIA brane description, involving a Neveu-Schwarz orientifold plane, which allows for the derivation of a magnetic quiver for the Higgs branch. These are 3D N=4 unitary-orthosymplectic quivers whose Coulomb branch is isomorphic to the Higgs branch of the 6D theories. From this magnetic quiver, together with an extended quiver subtraction algorithm that we explain, the foliation structure of the Higgs branch as a symplectic singularity is unveiled. For this class of 6D SCFTs, we observe a simple rule, which we refer to as “slice subtraction,” to read off the transverse slice in the foliation from the tensor branch. Based on this slice subtraction observation, we conjecture the transverse slices in the Higgsable to E-type (2, 0) Hasse diagram, where the SCFTs lack any known magnetic quiver for their Higgs branches. Published by the American Physical Society 2024

Preliminary Study of the Graphite Phyllite Rock from the Luk Ulo Melange Complex, Central Java, Indonesia

Abstract Phyllite rocks containing graphite minerals are exposed at the Luk Ulo Melange Complex (LUMC), Central Java, Indonesia. This study aims to determine the occurrence, petrographic, and geochemical characteristics of the LUMC phyllite rocks. The method used in this research consists of field observation and laboratory analysis, including petrographic and geochemical analysis using ICP AES/MS. Field observations were carried out in six locations, namely in the Karangsambung, Totogan, Duren, Kebondalem, Kebutuhjurang, and Pesangkalan villages. The phyllites are present as blocks in LUMC with 2 to >10 m in dimension and are associated with other metamorphic rocks such as greenschist, quartzite, and marble. LUMC phyllite rock has a fresh grey-black color, phyllitic foliated structure, and lepidoblastic texture. The main minerals are graphite, muscovite, quartz, and chlorite. Whole rock geochemical analysis was carried out on a phyllite sample from the Karangsambung area and showed that the major elements that dominated were SiO2 (69 wt%), Al2O3 (13 wt%), and Fe2O3 (5.78 wt%) that typical of metapelite rocks.

Tropological sigma models

With the use of mathematical techniques of tropical geometry, it was shown by Mikhalkin some twenty years ago that certain Gromov-Witten invariants associated with topological quantum field theories of pseudoholomorphic maps can be computed by going to the tropical limit of the geometries in question. Here we examine this phenomenon from the physics perspective of topological quantum field theory in the path integral representation, beginning with the case of the topological sigma model before coupling it to topological gravity. We identify the tropicalization of the localization equations, investigate its geometry and symmetries, and study the theory and its observables using the standard cohomological BRST methods. We find that the worldsheet theory exhibits a nonrelativistic structure, similar to theories of the Lifshitz type. Its path-integral formulation does not require a worldsheet complex structure; instead, it is based on a worldsheet foliation structure.

Leaves of foliated projective structures

The PSL(4,R) Hitchin component of a closed surface group π1(S) consists of holonomies of properly convex foliated projective structures on the unit tangent bundle of S. We prove that the leaves of the codimension-1 foliation of any such projective structure are all projectively equivalent if and only if its holonomy is Fuchsian. This implies constraints on the symmetries and shapes of these leaves.We also give an application to the topology of the non-T0 space C(RPn) of projective classes of properly convex domains in RPn. Namely, Benzécri asked in 1960 if every closed subset of C(RPn) that contains no proper nonempty closed subset is a point. Our results imply a negative resolution for n≥2.

Global flows of foliated gravity-matter systems

Asymptotic safety is a promising mechanism for obtaining a consistent and predictive quantum theory for gravity. The ADM formalism allows to introduce a (Euclidean) time-direction in this framework. It equips spacetime with a foliation structure by encoding the gravitational degrees of freedom in a lapse function, shift vector, and a metric measuring distances on the spatial slices. We use the Wetterich equation to study the renormalization group flow of the graviton 2-point function extracted from the spatial metric. The flow is driven by the 3- and 4-point vertices generated by the foliated Einstein-Hilbert action supplemented by minimally coupled scalar and vector fields. We derive bounds on the number of matter fields cast by asymptotic safety. Moreover, we show that the phase diagram obtained in the pure gravity case is qualitatively stable within these bounds. An intriguing feature is the presence of an IR-fixed point for the graviton mass which prevents the squared mass taking negative values. This feature persists for any number of matter fields and, in particular, also in situations where there is no suitable interacting fixed point rendering the theory asymptotically safe. Our work complements earlier studies of the subject by taking contributions from the matter fields into account.

Anomaly of subsystem symmetries in exotic and foliated BF theories

We study the mixed ’t Hooft anomaly of the subsystem symmetries in the exotic BF theory and the foliated BF theory in 2+1 dimensions, both of which are fractonic quantum field theories describing the equivalent physics. In the anomaly inflow mechanism, the ’t Hooft anomaly of the subsystem symmetries can be canceled by combining a subsystem symmetry-protected topological (SSPT) phase in one dimension higher. In this work, we construct the exotic and foliated BF theories with background gauge fields, and the exotic and foliated forms of the SSPT phases using the foliated-exotic duality. In the foliated form, we see that the non-topological defect that describes a fracton can be viewed as a symmetry-like operator. We also newly construct the foliated and exotic SSPT phases with different foliation structures via the foliated-exotic duality. We can show that the SSPT phases with different foliation structures cancel the same anomaly. This may provide a clue to the characterization of the ’t Hooft anomaly of subsystem symmetries.

Dilatancy and Multi-Scale failure characteristics of a foliated rock under triaxial confinement unloading conditions

Controlled by the foliation structure, foliated rocks exhibit mechanical anisotropy and cause squeezing deformation and subsequent supporting structure failure during tunnel excavation. Here, the deformation and failure characteristics of a phyllite under unloading stress paths were experimentally investigated. The strength and elasticity parameters of phyllite exhibited moderate ‘shoulder’ shaped anisotropy, and specimens failed in three different modes: cross-plane shear-tensile fracture, in-plane shear sliding, and interlayer buckling, depending on the foliation angle and initial confining pressure. Meanwhile, the evolution of dilatancy angle during volume dilatancy also showed significant anisotropy, either increasing or decreasing with the increasing plastic shear strain during post-peak loading. We further discussed the mechanism of the evolution of dilatancy angle based on the multi-scale observation of the failure characteristics. Generally, interlayer buckling improved the dilatancy angle and roughness loss during shear failure resulted in dilatancy angle degradation. These findings provided evidence for a deeper understanding of the failure mechanisms of tunnel rock mass in foliated strata.

GEOCHEMISTRY AND PETROGRAPHY OF METAMORPHIC SOLE AMPHIBOLITES FROM THE SLATINA QUARRY, MT. ZRINSKA GORA, CROATIA

The investigated Mt. Zrinska Gora amphibolites are a part of the metamorphic sole of the Banovina ophiolite complex. Mineral composition and other petrographic characteristics of amphibolites and adjacent peridotite were investigated by polarised microscopy and chemical analyses of rocks were obtained by a combination of inductively coupled plasma mass and emission spectrometry. The main minerals in these garnet-bearing amphibolites are amphibole and plagioclase with accessory garnet, sphene, spinel, opaque mineral ± clinopyroxene ± quartz ± actinolite ± zoisite-clinozoisite ± prehnite ± pumpelyite and ± clay minerals. Kelyphitic corona is developed around garnet. The amphibolites have nematoblastic, nematogranoblastic, porphyroblastic and porphyroclastic textures and homogenous to foliated structures. The presence of clinopyroxene in some of the investigated amphibolites points to their possible formation under P‒T conditions of upper amphibolite facies. The greenschist facies retrograde metamorphism and subsequent surface weathering of amphibolites is also evident. The chemical composition of rocks indicates that protoliths of most amphibolites were tholeiitic basalts, initially formed in island arcs and the adjoining back-arc basins. During the intraoceanic subduction in Jurassic Neotethys Ocean basalts/basaltic tuffs of island arc tholeiite affinity and back-arc basin basalts, positioned on the top of the down-going oceanic slab, were welded to the base of the hot over-riding mantle wedge, metamorphosed to amphibolite rocks and obducted together with ophiolites on the Adria continental margine. The new described amphibolite type having peridotite protolith characteristics originated as the consequence of the hydration of the bottom part of the overlying mantle wedge.

Anisotropic compactification of nonrelativistic M-theory

We study a decoupling limit of M-theory where the three-form gauge potential becomes critical. This limit leads to nonrelativistic M-theory coupled to a non-Lorentzian spacetime geometry. Nonrelativistic M-theory is U-dual to M-theory in the discrete light cone quantization, a non-perturbative approach related to the Matrix theory description of M-theory. We focus on the compactification of nonrelativistic M-theory over a two-torus that exhibits anisotropic behaviors due to the foliation structure of the spacetime geometry. We develop a frame covariant formalism of the toroidal geometry, which provides a geometrical interpretation of the recently discovered polynomial realization of SL(2 , ℤ) duality in nonrelativistic type IIB superstring theory. We will show that the nonrelativistic IIB string background fields transform as polynomials of an effective Galilean “boost velocity” on the two-torus. As an application, we construct an action principle describing a single M5-brane in nonrelativistic M-theory and study its compactification over the anisotropic two-torus. This procedure leads to a D3-brane action in nonrelativistic IIB string theory that makes the SL(2 , ℤ) invariance manifest in the polynomial realization.

Foliated asymptotically safe gravity in the fluctuation approach

The gravitational asymptotic safety program envisions a high-energy completion of gravity based on a non-Gaussian renormalization group fixed point. A key step in this program is the transition from Euclidean to Lorentzian signature spacetimes. One way to address this challenge is to formulate the quantum theory based on the Arnowitt-Deser-Misner decomposition of the metric field. This equips the Euclidean spacetime with a preferred direction which may serve as the time-direction in the Lorentzian setting. In this work we use the Wetterich equation in order to compute the renormalization group flow of the graviton two-point function. The resulting beta functions possess a non-Gaussian renormalization group fixed point suitable for rendering the theory asymptotically safe. The phase diagram underlying the flow of the two-point function is governed by the interplay between this non-Gaussian fixed point, the Gaussian fixed point, and an infrared fixed point. The latter ensures that the renormalized squared graviton mass cannot take negative values. These results are in qualitative agreement with fluctuation computations carried out in the covariant setting. We take this as non-trivial evidence that the asymptotic safety mechanism remains intact when considering quantum gravity on spacetimes carrying a foliation structure. Technically, our work constitutes the first fluctuation computation carried out within the ADM-framework. Therefore, we also provide a detailed discussion of the conceptual framework, highlighting the elements which differ from fluctuation computations in the covariant setting.

Petrology and Mineralogy of Metamorphic Rocks in The Pringsewu District, Lampung Province, Indonesia

The origin of metamorphic rock is greatly influenced by the temperature and pressure changes in every tectonic setting, especially in active subduction margins. One of the wide outcrops of metamorphic rock occurs in the Pringsewu District, Lampung, and it is a part of the Palaeozoic Gunung Kasih Complex. The presence of metamorphic rocks in Pringsewu has raised several questions and debates for some time due to the lack of research and field evidence found in this area, especially, since the origin of metamorphic rock in Lampung has been rarely studied. This research aims to determine the metamorphic rock facies and the tectonic setting underlying the formation of metamorphic rocks in the study area.
 Petrographic analysis on 19 thin-section samples shows that metamorphic rock in the research area can be divided into two regions i.e., western and eastern regions. The Eastern part is characterized by muscovite-epidote schist and greenschist which consist of quartz, muscovite, actinolite, epidote, and garnet as index minerals. Whereas the western part is characterized by quartzite and biotite-epidote schist that consist of quartz, biotite, and muscovite. Based on the mineral index, metamorphic rock’s protoliths are pelitic rock, mafic rock, and quartz-feldspathic sandstone. The metamorphic rock zonation shows the created temperature is from 280-550oC.
 The foliation structure such as schistose and porphyroblastic texture in the whole rocks sample indicated that metamorphic rocks are created in medium-grade metamorphism and are characterized by the greenschist facies to epidote-amphibolite facies. The abundance of quartz, k-feldspar, and labradorite minerals show that metamorphic rocks came from the microcontinent which was caused by a collision between the intra-oceanic Woyla plate and West Sumatra microcontinent in the Cretaceous. Indications of tectonic activity that create the lithology in Lampung Province need more geological study, especially to determine the absolute age of the metamorphic rock.

Characterization of Metapelite and Metasiltstone as Uranium-REE hosted rocks in Rirang Area, West Kalimantan

The Rirang River, Kalan, West Kalimantan has U and REE resources in the form of mineralization found at the intercalation of metapelite and metasiltstone. However, there are only a few studies about the characterization of U and REE-bearing host rocks in Indonesia. Determining the geological influence of the metasedimentary rock on the mineralization of U and REE is crucial. Thus, the characterization of metapelite and metasiltstone has been substantiated in this study to reveal the potency of the host rock in hosting the U and REE. Petrographic, micro-XRF, and AMICS methods were used to conduct mineralogical analysis. These two rocks share a mineral composition that includes clay minerals, hydrothermal alteration products, and several minerals that indicate metamorphism processes. The foliation structure with a spotted shale texture further indicates the metamorphism process in metapelite and metasiltstone. REE and trace elements associated with U are constituents of clay minerals such as wonesite, staurolite, montmorillonite, chlorite, illite, and kaolinite. The geochemical properties of metapelite and metasiltstone suggest that REE mineralization and trace elements associated with U come from hydrothermal alteration processes.

Foliated affine and projective structures

We formalize the concepts of holomorphic affine and projective structures along the leaves of holomorphic foliations by curves on complex manifolds. We show that many foliations admit such structures, we provide local normal forms for them at singular points of the foliation, and we prove some index formulae in the case where the ambient manifold is compact. As a consequence of these, we establish that a regular foliation of general type on a compact algebraic manifold of even dimension does not admit a foliated projective structure. Finally, we classify foliated affine and projective structures along regular foliations on compact complex surfaces.

Hermitian Calabi Functional in Complexified Orbits

Let $(M,\omega)$ be a compact symplectic manifold. We denote by $\ac$ the space of all almost complex structure compatible with $\omega$. $\ac$ has a natural foliation structure with the complexified orbit as leaf. We obtain an explicit formula of the Hessian of Hermitian Calabi functional at an extremal almost K\"ahler metric in $\ac$. We prove that the Hessian of Hermitian Calabi functional is semi-positive definite at critical point when restricted to a complexified orbit, as corollaries we obtain some results analogy to K\"ahler case. We also show weak parabolicity of the Hermitian Calabi flow.

Renormalization of supersymmetric Lifshitz sigma models

We study the renormalization of an mathcal{N} = 1 supersymmetric Lifshitz sigma model in three dimensions. The sigma model exhibits worldvolume anisotropy in space and time around the high-energy z = 2 Lifshitz point, such that the worldvolume is endowed with a foliation structure along a preferred time direction. In curved backgrounds, the target-space geometry is equipped with two distinct metrics, and the interacting sigma model is power-counting renormalizable. At low energies, the theory naturally flows toward the relativistic sigma model where Lorentz symmetry emerges. In the superspace formalism, we develop a heat kernel method that is covariantized with respect to the bimetric target-space geometry, using which we evaluate the one-loop beta-functions of the Lifshitz sigma model. This study forms an essential step toward a thorough understanding of the quantum critical supermembrane as a candidate high-energy completion of the relativistic supermembrane.

A mesostructure-informed cohesion-based numerical method for fracture behavior of slate with foliation structure

The foliation structure and spatial variability are important factors affecting the fracture behavior of slate. According to the statistical analysis of mesostructural information, random slate specimens with the foliation structure are physically reconstructed by coupling a one-dimensional random field with a two-dimensional isotropic random field. A cohesion-based numerical method incorporating the foliation structure is developed to simulate the fracture behavior of slate-like materials. The elastic and fracture parameters are evaluated by direct tensile tests, and the numerical model is validated to be self-consistency. It has been confirmed by numerical simulations and tests that the elastic modulus and tensile fracture increase with the foliation angle, and there is a critical foliation angle from the tensile fracture to the shear fracture for the slate under uniaxial tension. Affected by the spatial variability of the foliation structure, slate fractures in the dense and thick area of the foliation plane, and the mechanical properties follow lognormal distribution.

Symplectic Foliation Structures of Non-Equilibrium Thermodynamics as Dissipation Model: Application to Metriplectic Nonlinear Lindblad Quantum Master Equation.

The idea of a canonical ensemble from Gibbs has been extended by Jean-Marie Souriau for a symplectic manifold where a Lie group has a Hamiltonian action. A novel symplectic thermodynamics and information geometry known as "Lie group thermodynamics" then explains foliation structures of thermodynamics. We then infer a geometric structure for heat equation from this archetypal model, and we have discovered a pure geometric structure of entropy, which characterizes entropy in coadjoint representation as an invariant Casimir function. The coadjoint orbits form the level sets on the entropy. By using the KKS 2-form in the affine case via Souriau's cocycle, the method also enables the Fisher metric from information geometry for Lie groups. The fact that transverse dynamics to these symplectic leaves is dissipative, whilst dynamics along these symplectic leaves characterize non-dissipative phenomenon, can be used to interpret this Lie group thermodynamics within the context of an open system out of thermodynamics equilibrium. In the following section, we will discuss the dissipative symplectic model of heat and information through the Poisson transverse structure to the symplectic leaf of coadjoint orbits, which is based on the metriplectic bracket, which guarantees conservation of energy and non-decrease of entropy. Baptiste Coquinot recently developed a new foundation theory for dissipative brackets by taking a broad perspective from non-equilibrium thermodynamics. He did this by first considering more natural variables for building the bracket used in metriplectic flow and then by presenting a methodical approach to the development of the theory. By deriving a generic dissipative bracket from fundamental thermodynamic first principles, Baptiste Coquinot demonstrates that brackets for the dissipative part are entirely natural, just as Poisson brackets for the non-dissipative part are canonical for Hamiltonian dynamics. We shall investigate how the theory of dissipative brackets introduced by Paul Dirac for limited Hamiltonian systems relates to transverse structure. We shall investigate an alternative method to the metriplectic method based on Michel Saint Germain's PhD research on the transverse Poisson structure. We will examine an alternative method to the metriplectic method based on the transverse Poisson structure, which Michel Saint-Germain studied for his PhD and was motivated by the key works of Fokko du Cloux. In continuation of Saint-Germain's works, Hervé Sabourin highlights the, for transverse Poisson structures, polynomial nature to nilpotent adjoint orbits and demonstrated that the Casimir functions of the transverse Poisson structure that result from restriction to the Lie-Poisson structure transverse slice are Casimir functions independent of the transverse Poisson structure. He also demonstrated that, on the transverse slice, two polynomial Poisson structures to the symplectic leaf appear that have Casimir functions. The dissipative equation introduced by Lindblad, from the Hamiltonian Liouville equation operating on the quantum density matrix, will be applied to illustrate these previous models. For the Lindblad operator, the dissipative component has been described as the relative entropy gradient and the maximum entropy principle by Öttinger. It has been observed then that the Lindblad equation is a linear approximation of the metriplectic equation.

Causal functional calculus

We construct a new topology on the space of stopped paths and introduce a calculus for causal functionals on generic domains of this space. We propose a generic approach to pathwise integration without any assumption on the variation index of a path and obtain functional change of variable formulae which extend the results of Föllmer [Séminaire de probabilités 15 (1981), 143–150] and Cont and Fournié [J. Funct. Anal. 259 (2010), no. 4, 1043–1072] to a larger class of functionals, including Föllmer's pathwise integrals. We show that a class of smooth functionals possess a pathwise analogue of the martingale property. For paths that possess finite quadratic variation, our approach extends the Föllmer–Ito calculus and removes previous restriction on the time partition sequence. We introduce a foliation structure on this path space and show that harmonic functionals may be represented as pathwise integrals of closed 1-forms.

Chaos in Topological Foliations

We call a foliation (M,F) on a manifold M chaotic if it is topologically transitive and the union of closed leaves is dense in M. A foliated manifold M is not assumed to be compact. The chaotic foliations can be considered as multidimensional generalization of chaotic dynamical systems in the sense of Devaney. For foliations covered by fibrations we prove that a foliation is chaotic if and only if its global holonomy group is chaotic. We introduce the concept of the integrable Ehresmann connection for a foliation as a natural generalization of the integrable Ehresmann connection for smooth foliations. A description of the global structure of foliations with integrable Ehresmann connection and a criterion for the chaotic behavior of such foliations are obtained. Applying the method of suspension, a new countable family of pairwise nonisomorphic chaotic foliations of codimension two on 3-dimensional closed and nonclosed manifolds is constructed.

Variscan vs. Alpine structural controls: Karstic proto-conduit development within Palaeozoic marble post-conditioned by Alpine faulting (the Niedźwiedzia Cave, NE Bohemian Massif)

Detailed geological mapping and structural analysis of Niedźwiedzia Cave (in the Orlica-Śnieżnik Dome, NE Bohemian Massif, Central Europe) provides new insights into proto-conduit formation in karstic drainage. Mapping has allowed us to document the structure of Variscan-age foliation and non-outcropping, post-Variscan fault zones at meter-scale resolution. The resulting data has been used to build a detailed 3D model of the marble lens hosting the cave, allowing us to define the primary structural guidance trends. We observe that karst development has been controlled by WSW-ENE trending sub-vertical faults and fractures, with N-S to NNW-SSE steeply dipping foliation playing a subsidiary role. Based on micro-movement sensors in the cave and other neotectonic indicators, and the relationship between structural discontinuities and topography, we propose that structural discontinuities have only become favorable for karst development under Alpine stress conditions, with little or no influence of exhumation relaxation. Furthermore, Variscan foliation in the marble only appears to become relevant in karst guidance in mature stages of karst development, with foliation parting, and in the absence of significant faulting or fracturing. This pattern is only disrupted in the presence of tectonic brecciation or comminution along faults or fractures. Structural guidance in the development of the Niedźwedzia Cave has been further modulated by water-table evolution in the overlying Kleśnica river valley. The result is a cave with three distinct water-table controlled levels and a map pattern of passages that grades from a mature N-S dominated orientation in the south (upstream) to a grid-like arrangement in the younger northern segment of the cave (downstream).