We utilize the deformation theory of algebraic singularities to study charged matter in compactifications of M-theory, F-theory, and type IIa string theory on elliptically fibered Calabi-Yau manifolds. In F-theory, this description is more physical than that of resolution. We describe how two-cycles can be identified and systematically studied after deformation. For ADE singularities, we realize non-trivial ADE representations as sublattices of Z^N, where N is the multiplicity of the codimension one singularity before deformation. We give a method for the determination of Picard-Lefschetz vanishing cycles in this context and utilize this method for one-parameter smooth deformations of ADE singularities. We give a general map from junctions to weights and demonstrate that Freudenthal's recursion formula applied to junctions correctly reproduces the structure of high-dimensional ADE representations, including the 126 of SO(10) and the 43,758 of E_6. We identify the Weyl group action in some examples, and verify its order in others. We describe the codimension two localization of matter in F-theory in the case of heterotic duality or simple normal crossing and demonstrate the branching of adjoint representations. Finally, we demonstrate geometrically that deformations correctly reproduce the appearance of non-simply-laced algebras induced by monodromy around codimension two singularities, showing the reduction of D_4 to G_2 in an example. A companion mathematical paper will follow.
We provide a local geometric description of how charged matter arises in type IIA, M-theory, or F-theory compactifications on Calabi-Yau manifolds. The basic idea is to deform a higher singularity into a lower one through Cartan deformations which vary over space. The results agree with expectations based on string dualities.
We introduce a geometric realization of noncommutative singularity resolutions. To do this, we first present a new conjectural method of obtaining conventional resolutions using coordinate rings of matrix-valued functions. We verify this conjecture for all cyclic quotient surface singularities, the Kleinian D_n and E_6 surface singularities, the conifold singularity, and a non-isolated singularity, using appropriate quiver algebras. This conjecture provides a possible new generalization of the classical McKay correspondence. Then, using symplectic reduction within these rings, we obtain new, n…
We present an algorithm for obtaining the matter content of effective six-dimensional theories resulting from compactification of F-theory on elliptic Calabi-Yau threefolds which are hypersurfaces in toric varieties. The algorithm allows us to read off the matter content of the theory from the polyhedron describing the Calabi-Yau manifold. This is based on the generalized Green-Schwarz anomaly cancellation condition.
The fermion doubling problem has an important impact on quantum gravity, by revealing the tension between fermion and the fundamental discreteness of quantum spacetime. In this work, we discover that in Loop Quantum Gravity, the quantum geometry involving superposition of states associated with lattice refinements provides a resolution to the fermion doubling problem. We construct and analyze the fermion propagator on the quantum geometry, and we show that all fermion doubler modes are suppressed in the propagator. Our result suggests that the superposition nature of quantum geometry should re…
We propose Seiberg-Witten geometry for N=2 gauge theory with gauge group $E_6$ with massive $N_f$ fundamental hypermultiplets. The relevant manifold is described as a fibration of the ALE space of $E_6$ type. It is observed that the fibering data over the base ${\bf CP}^1$ has an intricate dependence on hypermultiplet bare masses.
Motivated by questions related to the landscape of flux compactifications, we combine new and existing techniques into a systematic, streamlined approach for computing vertical fluxes and chiral matter multiplicities in 4D F-theory models. A central feature of our approach is the conjecturally resolution-independent intersection pairing of the vertical part of the integer middle cohomology of smooth elliptic CY fourfolds, relevant for computing chiral indices and related aspects of 4D F-theory flux vacua. We illustrate our approach by analyzing vertical flux backgrounds for F-theory models wit…
We show that in modified $f(R)$ type gravity models with non-minimal coupling between matter and geometry, both the matter Lagrangian, and the energy-momentum tensor, are completely and uniquely determined by the form of the coupling. This result is obtained by using the variational formulation for the derivation of the equations of motion in the modified gravity models with geometry-matter coupling, and the Newtonian limit for a fluid obeying a barotropic equation of state. The corresponding energy-momentum tensor of the matter in modified gravity models with non-minimal coupling is more gene…
We identify the dual geometry of the hadron phase of dense nuclear matter and investigate the confinement/deconfinement phase transition. We suggest that the low temperature phase of the RN black hole with the full backreaction of the bulk gauge field is described by the zero mass limit of the RN black hole with hard wall. We calculated the density dependence of critical temperature and found that the phase diagram closes. We also study the density dependence of the rho meson mass.
In this talk I discuss some features of the entanglement entropy for fuzzy geometry, focusing on its dependence on the background fields and the spin connection of the emergent continuous manifold in a large $N$ limit. Using the Landau-Hall paradigm for fuzzy geometry, this is argued to be given by a generalized Chern-Simons form, making a point of connection with the thermodynamic view of gravity. Matter-gravity couplings are also considered in the same framework; they naturally lead to certain specific nonminimal couplings involving powers of the curvature.
We give a brief exposition of the approach based on the methods of spectral geometry and the spectral action principle to construction and analysis of models on a foliated space-time.
We investigate the possibility that the behavior of the rotational velocities of test particles gravitating around galaxies can be explained in the framework of modified gravity models with non-minimal matter-geometry coupling. Generally, the dynamics of test particles around galaxies, as well as the corresponding mass deficit, is explained by postulating the existence of dark matter. The extra-terms in the gravitational field equations with geometry-matter coupling modify the equations of motion of test particles, and induce a supplementary gravitational interaction. Starting from the variati…
In the present manuscript, I examine an intriguing relation at the classical level between general relativity and a theory where matter couples uniquely multiplicatively to geometry in the Lagrangian density. Interestingly, the gravitational constant $G$ is replaced by a novel fundamental constant, whose value is not tied to any classical phenomenon; while the value of $G$ itself becomes related to the dynamics of the universe. I concentrate on different aspects of the Equivalence Principle, as the theory is expected to violate all of its different formulations.
We present a systematic means to impose Galilean invariance within field theory. We begin by defining the most general background geometries consistent with Galilean invariance and then turn to applications within effective field theory, fluid dynamics, and the quantum Hall effect.
We investigate a strong coupling expansion of N=6 superconformal Chern-Simons theory obtained from the semiclassical analysis of low energy, effective degrees of freedom given by the eigenvalues of a certain matrix model. We show how the orbifolded sphere S^7/Z_k of the dual geometry emerges dynamically from the distribution of the eigenvalues. As a test of this approach we compute the energy of off-diagonal excitations, finding perfect agreement with the dispersion relation of giant magnons.
We study renormalizable nonlinear sigma-models in two dimensions with N=2 supersymmetry described in superspace in terms of chiral and complex linear superfields. The geometrical structure of the underlying manifold is investigated and the one-loop divergent contribution to the effective action is computed. The condition of vanishing beta-function allows to identify a class of models which satisfy this requirement and possess N=4 supersymmetry.
We obtain the Seiberg-Witten geometry for four-dimensional N=2 gauge theory with gauge group SO(2N_c) (N_c \leq 5) with massive spinor and vector hypermultiplets by considering the gauge symmetry breaking in the N=2 $E_6$ theory with massive fundamental hypermultiplets. In a similar way the Seiberg-Witten geometry is determined for N=2 SU(N_c) (N_c \leq 6) gauge theory with massive antisymmetric and fundamental hypermultiplets. Whenever possible we compare our results expressed in the form of ALE fibrations with those obtained by geometric engineering and brane dynamics, and find a remarkable …
This thesis is dedicated to the study of the geometry of six-dimensional superspace, endowed with the minimal amount of supersymmetry. In the first part of it, we unfold the main geometrical features of such superspace by solving completely the Bianchi identities for the constrained superspace torsion, which allow us to determine the full six-dimensional derivate superalgebra. Next, the conformal structure of the supergeometry is considered. Specifically, it is shown that the conventional torsion constraints remain invariant under super-Weyl transformations generated by a real scalar superfiel…
We initiate the study of positive geometry and scattering forms for tree-level amplitudes with matter particles in the (anti-)fundamental representation of the color/flavor group. As a toy example, we study the bi-color scalar theory, which supplements the bi-adjoint theory with scalars in the (anti-)fundamental representations of both groups. Using a recursive construction we obtain a class of unbounded polytopes called open associahedra (or associahedra with certain facets at infinity) whose canonical form computes amplitudes in bi-color theory, for arbitrary number of legs and flavor assign…
The geometrical formulation of gravity is not unique and can be set up in a variety of spacetimes. Even though the gravitational sector enjoys this freedom of different geometrical interpretations, consistent matter couplings have to be assured for a steady foundation of gravity. In generalised geometries, further ambiguities arise in the matter couplings unless the minimal coupling principle (MCP) is adopted that is compatible with the principles of relativity, universality and inertia. In this work, MCP is applied to all Standard Model gauge fields and matter fields in a completely general (…
The field equations of a generalized $f(R)$ type gravity model, in which there is an arbitrary coupling between matter and geometry, are obtained. The equations of motion for test particles are derived from a variational principle in the particular case in which the Lagrange density of the matter is an arbitrary function of the energy-density of the matter only. Generally, the motion is non-geodesic, and takes place in the presence of an extra force orthogonal to the four-velocity. The Newtonian limit of the model is also considered. The perihelion precession of an elliptical planetary orbit i…
We investigate McVittie and generalized McVittie solutions for Horndeski gravity with a spatially homogeneous gravitational scalar field, which is stealth at small scales near the central object but, at large scales, sources the FLRW universe in which the central inhomogeneity is embedded. Unlike previous studies, we include matter and obtain generalized McVittie solutions in the extended cuscuton model. The possible configurations are classified according to the time-dependence of the gravitational coupling, the radial energy flow, the accretion rate onto the central object, and the Hubble ra…
We consider a set of physical degrees of freedom coupled to a finite-dimensional Hilbert space, which may be taken as modeling a fuzzy space or as the lowest Landau level of a Landau-Hall problem. These may be viewed as matter fields on a fuzzy space. Sequentially generalizing to arbitrary backgrounds, we argue that the effective action is given by the Chern-Simons form associated with the Dirac index density (with gauge and gravitational fields), with an abelian gauge field shifted by the Poincaré-Cartan form for matter dynamics. The result is an action for matter fields where the Lagrangian …
We present a study of the evolution of entanglement entropy of matter and geometry in quantum cosmology. For a variety of Gaussian initial states and their linear combinations, and with evolution defined with respect to a relational time, we show numerically that (i) entanglement entropy increases rapidly at very early times, and subsequently saturates to a constant non-zero value, and (ii) that the saturation value of this entropy is a linear function of the energy associated to the quantum state: $S_{\text{ent}}^ψ= γ\langle \hat{H} \rangle_ψ$. These results suggest a remnant of quantum entan…
We study the polymeric nature of quantum matter fields using the example of a Friedmann-Lemaitre-Robertson-Walker universe sourced by a minimally coupled massless scalar field. The model is treated in the symmetry reduced regime via deparametrization techniques, with the scale factor playing the role of time. Subsequently, the remaining dynamic degrees of freedom corresponding to the matter are polymer quantized. The analysis of the resulting genuine quantum dynamic shows that the big bang singularity is resolved, although with the form of the resolution differing significantly from that in th…
We consider a deviation of the physical length from the Riemann geometry toward the Randers'. We construct a consistent second-order relativistic theory of gravity that dynamically reduces to the Einstein-Hilbert theory for the strong and Newtonian gravity while its weak gravitational regime reproduces MOND and the gravitational lensing attributed to the dark matter halo. It also naturally accommodates the observed value of the cosmological constant. We show that it predicts a few percent deviation for the post Newtonian parameter $γ$ in a part of the regime that interpolates the Newtonian reg…
Modified gravity theories with geometry - matter coupling, in which the action is an arbitrary function of the Ricci scalar and the matter Lagrangian ($f\left(R,L_m\right)$ gravity), and of the Ricci scalar and of the trace of the matter energy-momentum tensor ($f(R,T)$ gravity), respectively, have the intriguing property that the divergence of the matter energy - momentum tensor is nonzero. In the present paper, by using the formalism of open thermodynamic systems, we interpret the generalized conservation equations in these gravitational theories from a thermodynamic point of view as describ…
A closed universe containing pressureless dust, more generally perfect fluid matter with pressure-to-density ratio w in the range (1/3, - 1/3), violates holographic principle applied according to the Fischler-Susskind proposal. We show, first for a class of two-fluid solutions and then for the general multifluid case, that the closed universe will obey the holographic principle if it also contains matter with w < - 1/3, and if the present value of its total density is sufficiently close to the critical density. It is possible that such matter can be realised by some form of `quintessence', muc…
General 2d dilaton theories, containing spherically symmetric gravity and hence the Schwarzschild black hole as a special case, are quantized by an exact path integral of their geometric (Cartan-) variables. Matter, represented by minimally coupled massless scalar fields is treated in terms of a systematic perturbation theory. The crucial prerequisite for our approach is the use of a temporal gauge for the spin connection and for light cone components of the zweibeine which amounts to an Eddington Finkelstein gauge for the metric. We derive the generating functional in its most general form wh…
A number of recent proposals for a quantum theory of gravity are based on the idea that spacetime geometry and gravity are derivative concepts and only apply at an approximate level. There are two fundamental challenges to any such approach. At the conceptual level, there is a clash between the "timelessness" of general relativity and emergence. Second, the lack of a fundamental spacetime makes difficult the straightforward application of well-known methods of statistical physics to the problem. We recently initiated a study of such problems using spin systems based on evolution of quantum net…
In the traditional Katz-Vafa method, matter representations are determined by decomposing the adjoint representation of a parent simple Lie algebra $\mathfrak{m}$ as the direct sum of irreducible representations of a semisimple subalgebra $\mathfrak{g}$. The Katz-Vafa method becomes ambiguous as soon as $\mathfrak{m}$ contains several subalgebras isomorphic to $\mathfrak{g}$ but giving different decompositions of the adjoint representation. We propose a selection rule that characterizes the matter representations observed in generic constructions in F-theory and M-theory: the matter representa…
We review and present a few new results of the program of emergent matter as braid excitations of quantum geometry that is represented by braided ribbon networks. These networks are a generalisation of the spin networks proposed by Penrose and those in models of background independent quantum gravity theories, such as Loop Quantum Gravity and Spin Foam models. This program has been developed in two parallel but complimentary schemes, namely the trivalent and tetravalent schemes. The former studies the braids on trivalent braided ribbon networks, while the latter investigates the braids on tetr…
Non commutative geometry is creating new possibilities for physics. Quantum spacetime geometry and post inflationary models of the universe with matter creation have an enormous range of scales of time, distance and energy in between. There is a variety of physics possible till the nucleosynthesis epoch is reached. The use of topology and non commutative geometry in cosmology is a recent approach. This paper considers the possibility of topological solutions of a vortex kind given by non commutative structures. These are interpreted as dark matter, with the grand unified Yang-Mills field theor…
In this paper we demonstrate that Boltzmann's $H$-theorem does not necessarily hold in the context of theories of gravity with nonminimally coupled matter fields. We find sufficient conditions for the violation of Boltzmann's $H$-theorem and derive an expression for the evolution of Boltzmann's $H$ in terms of the nonminimal coupling function, valid in the case of a collisionless gas in a homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker universe. We highlight the implications of this result for the evolution of the entropy of the matter fields, briefly discussing the role played b…
Motivated by the Horava-Lifshitz type theories, we study the physical motion of matter coupled to a foliated geometry in non-diffeomorphism invariant way. We use the concept of a spectral action as a guiding principle in writing down the matter action. Based on the deformed Dirac operator compatible with the reduced symmetry - foliation preserving diffeomorphisms, this approach provides a natural generalization of the minimal coupling. Focusing on the IR version of the Dirac operator, we derive the physical motion of a test particle and discuss in what sense it still can be considered as a geo…
Exactly soluble models can serve as excellent tools to explore conceptual issues in non-perturbative quantum gravity. In perturbative approaches, it is only the two radiative modes of the linearized gravitational field that are quantized. The goal of this investigation is to probe the `Coulombic' aspects of quantum geometry that are governed entirely by matter sources. Since there are no gravitational waves in 3 dimensions, 3-d gravity coupled to matter provides an ideal arena for this task. Our analysis will reveal novel aspects of quantum gravity that bring out limitations of classical and s…
We determine a smooth Euclidean 4-geometry on R^4 from quantum interacting spin matter like in the multichannel Kondo effect. The CFT description of both: the $k$-channel Kondo effect of spin magnetic impurities quantum interacting with spins of conducting electrons and exotic smooth R^4, by the level $k$ WZW model on SU(2), indicates the relation between smooth R^4's and the quantum matter. We propose a model which shows: exotic smooth R^4_k generates fermionic fields via the topological structure of Casson handles and when this handle is attached to some subspace A of R^4 these fermions repr…
We investigate the coupling of matter to geometry in conformal quadratic Weyl gravity, by assuming a coupling term of the form $L_m\tilde{R}^2$, where $L_m$ is the ordinary matter Lagrangian, and $\tilde{R}$ is the Weyl scalar. The coupling explicitly satisfies the conformal invariance of the theory. By expressing $\tilde{R}^2$ with the help of an auxiliary scalar field and of the Weyl scalar, the gravitational action can be linearized, leading in the Riemann space to a conformally invariant $f\left(R,L_m\right)$ type theory, with the matter Lagrangian nonminimally coupled to the Ricci scalar.…
We show that the principal part of the Dirac Hamiltonian in 3+1 dimensions emerges in a semi-classical approximation from a construction which encodes the kinematics of quantum gravity. The construction is a spectral triple over a configuration space of connections. It involves an algebra of holonomy loops represented as bounded operators on a separable Hilbert space and a Dirac type operator. Semi-classical states, which involve an averaging over points at which the product between loops is defined, are constructed and it is shown that the Dirac Hamiltonian emerges as the expectation value of…
