Group-theoretical Structure Research Articles

Towards natural and realistic E7 GUTs in F-theory

We consider phenomenological aspects of a natural class of Standard Model-like supersymmetric F-theory vacua realized through flux breaking of rigid E7 gauge factors. Three generations of Standard Model matter are realized in many of these vacua. We further find that many other Standard Model-like features are naturally compatible with these constructions. For example, dimension-4 and 5 terms associated with proton decay are ubiquitously suppressed. Many of these features are due to the group theoretical structure of E7 and associated F-theory geometry. In particular, a set of approximate global symmetries descends from the E7 group, leading to exponential suppression of undesired couplings.

Identifying the group-theoretic structure of machine-learned symmetries

Deep learning was recently successfully used in deriving symmetry transformations that preserve important physics quantities. Being completely agnostic, these techniques postpone the identification of the discovered symmetries to a later stage. In this letter we propose methods for examining and identifying the group-theoretic structure of such machine-learned symmetries. We design loss functions which probe the subalgebra structure either during the deep learning stage of symmetry discovery or in a subsequent post-processing stage. We illustrate the new methods with examples from the U(n) Lie group family, obtaining the respective subalgebra decompositions. As an application to particle physics, we demonstrate the identification of the residual symmetries after the spontaneous breaking of non-Abelian gauge symmetries like SU(3) and SU(5) which are commonly used in model building.

Chaos from Symmetry: Navier Stokes Equations, Beltrami Fields and the Universal Classifying Crystallographic Group

The core of this paper is the group-theoretical approach, initiated in 2015 by one of the present authors in collaboration with Alexander Sorin that brings into the classical field of mathematical fluid-mechanics a brand new vision, allowing for a more systematic classification and algorithmic construction of Beltrami flows on torii R3/Λ where Λ is a crystallographic lattice. Here this new hydro-theory based on the focal idea of a Universal Classifying Group UGΛ is revised, reorganized, improved and extended. In particular we construct the so far missing UGΛHex for the hexagonal lattice and we advocate that, mastering the cubic and hexagonal instances of this group, we can cover all cases. The relation between the classification of Beltrami Flows with that of contact structures is enlightened. The recent developments about the framework of b-manifolds are considered and it is shown that the choice of the allowed critical surfaces for the b-deformation of a Beltrami field seems to be strongly related to the group-theoretical structure of the latter. This opens new directions of investigations about a group theoretical classification of critical surfaces. Apart from that the most promising research direction opened by the present work streams from the fact that the Fourier series expansion of a generic Navier-Stokes solution can be regrouped into an infinite sum of contributions Wr, each associated with a spherical layer of quantized radius r in the momentum lattice. Each Wr is the superposition of a Beltrami field Wr+ plus an anti-Beltrami field Wr−. These latter have a priori exactly the same decomposition into irreps of the group UGΛ that are variously repeated on higher layers. This crucial property enables the construction of generic Fourier series with prescribed hidden symmetries as candidate solutions of the NS equations. Alternatively the Fourier series representation of known solutions can be analyzed from the point of view of such symmetries. As a further result of this research program a complete and versatile system of MATHEMATICA Codes named AlmafluidaNSPsystem has been constructed and is now available through the site of the Wolfram Community. The exact solutions presented in this paper have to be considered as an illustration of the new conceptions and ideas that have emerged and of what can be further done utilizing the computer codes as an instrument. The main message streaming from our constructions is that the more symmetric is the Beltrami Flow the highest is the probability of the on-set of chaotic trajectories.

Implications for colored HOMFLY polynomials from explicit formulas for group-theoretical structure

We have recently proposed [1] a powerful method for computing group factors of the perturbative series expansion of the Wilson loop in the Chern-Simons theory with SU(N) gauge group. In this paper, we apply the developed method to obtain and study various properties, including nonperturbative ones, of such vacuum expectation values.First, we discuss the computation of Vassiliev invariants. Second, we discuss the Vogel theorem of not distinguishing chord diagrams by weight systems coming from semisimple Lie (super)algebras. Third, we provide a method for constructing linear recursive relations for the colored Jones polynomials considering a special case of torus knots T[2,2k+1]. Fourth, we give a generalization of the one-hook scaling property for the colored Alexander polynomials. And finally, for the group factors we provide a combinatorial description, which has a clear dependence on the rank N and the representation R.

Structural losses, structural realism and the stability of Lie algebras

One of the key assumptions associated with structural realism is the claim that successful scientific theories approximately preserve their structurally based content as they are progressively developed and that this content alone can explain their relevant predictions. The precise way in which these theories are preserved is not trivial but, according to this realist thesis, any kind of structural loss should not occur among theoretical transitions. Although group theory has been proven effective in accounting for preserved structures in the context of physics, structural realists are confronted with the fact that even group-theoretic structures are not immune to these structural discontinuities. Under such circumstances, my contribution consists in a two-fold task. Firstly, I will establish a general condition at the level of the group-theoretic structure to avoid the pessimistic induction argument by appealing to Lie algebra deformation and stability theory; and secondly, I will provide a case study associated with quantum-relativistic kinematics to demonstrate that this condition is actually satisfied. Specifically, through this case study I will support the claim that if the full Lie algebras of our current successful theories are stable, it is possible to disregard any kind of structural loss in the future and explain the relevant successful predictions in a way that we can support structural realism accordingly.

Tensor network approach to two-dimensional Yang–Mills theories

Abstract We propose a novel tensor network representation for two-dimensional Yang–Mills theories with arbitrary compact gauge groups. In this method, tensor indices are given directly by group elements with no direct use of the character expansion. We apply the tensor renormalization group method to this tensor network for SU(2) and SU(3), and find that the free energy density and the energy density are accurately evaluated. We also show that the singular value decomposition of a tensor has a group-theoretic structure and can be associated with the character expansion.

Chern-Simons perturbative series revisited

A group-theoretical structure in a perturbative expansion of the Wilson loops in the 3d Chern-Simons theory with SU(N) gauge group is studied in symmetric approach. A special basis in the center of the universal enveloping algebra ZU(slN) is introduced. This basis allows one to present group factors in an arbitrary irreducible finite-dimensional representation. Developed methods have wide applications, the most straightforward and evident ones are mentioned. Namely, Vassiliev invariants of higher orders are computed, a conjecture about existence of new symmetries of the colored HOMFLY polynomials is stated, and the recently discovered tug-the-hook symmetry of the colored HOMFLY polynomial is proved.

Kleiss-Kuijf relations from momentum amplituhedron geometry

In recent years, it has been understood that color-ordered scattering amplitudes can be encoded as logarithmic differential forms on positive geometries. In particular, amplitudes in maximally supersymmetric Yang-Mills theory in spinor helicity space are governed by the momentum amplituhedron. Due to the group-theoretic structure underlying color decompositions, color-ordered amplitudes enjoy various identities which relate different orderings. In this paper, we show how the Kleiss-Kuijf relations arise from the geometry of the momentum amplituhedron. We also show how similar relations can be realised for the kinematic associahedron, which is the positive geometry of bi-adjoint scalar cubic theory.

A Novel Symmetry of Colored HOMFLY Polynomials Coming from $$\mathfrak {sl}(N|M)$$ Superalgebras

We present a novel symmetry of the colored HOMFLY polynomial. It relates pairs of polynomials colored by different representations at specific values of $N$ and generalizes the previously known "tug-the-hook" symmetry of the colored Alexander polynomial. As we show, the symmetry has a superalgebra origin, which we discuss qualitatively. Our main focus are the constraints that such a property imposes on the general group-theoretical structure, namely the $\mathfrak{sl}(N)$ weight system, arising in the perturbative expansion of the invariant. Finally, we demonstrate its tight relation to the eigenvalue conjecture.

A New Symmetry of the Colored Alexander Polynomial

We present a new conjectural symmetry of the colored Alexander polynomial, that is the specialization of the quantum $\mathfrak{sl}_N$ invariant widely known as the colored HOMFLY-PT polynomial. We provide arguments in support of the existence of the symmetry by studying the loop expansion and the character expansion of the colored HOMFLY-PT polynomial. We study the constraints this symmetry imposes on the group theoretic structure of the loop expansion and provide solutions to those constraints. The symmetry is a powerful tool for research on polynomial knot invariants and in the end we suggest several possible applications of the symmetry.

Groups generated by derangements

We examine the subgroup D(G) of a transitive permutation group G which is generated by the derangements in G. Our main results bound the index of this subgroup: we conjecture that, if G has degree n and is not a Frobenius group, then |G:D(G)|⩽n−1; we prove this except when G is a primitive affine group. For affine groups, we translate our conjecture into an equivalent form regarding |H:R(H)|, where H is a linear group on a finite vector space and R(H) is the subgroup of H generated by elements having eigenvalue 1.If G is a Frobenius group, then D(G) is the Frobenius kernel, and so G/D(G) is isomorphic to a Frobenius complement. We give some examples where D(G)≠G, and examine the group-theoretic structure of G/D(G); in particular, we construct groups G in which G/D(G) is not a Frobenius complement.

Multivariate group entropies, super-exponentially growing complex systems, and functional equations.

We define the class of multivariate group entropies as a novel set of information-theoretical measures, which extends significantly the family of group entropies. We propose new examples related to the "super-exponential" universality class of complex systems; in particular, we introduce a general entropy, representing a suitable information measure for this class. We also show that the group-theoretical structure associated with our multivariate entropies can be used to define a large family of exactly solvable discrete dynamical models. The natural mathematical framework allowing us to formulate this correspondence is offered by the theory of formal groups and rings.

Higher spin partition functions via the quasinormal mode method in de Sitter quantum gravity

In this note we compute the 1-loop partition function of spin-ss fields on Euclidean de Sitter space S^{2n+1}S2n+1 using the quasinormal mode method. Instead of computing the quasinormal mode frequencies from scratch, we use the analytic continuation prescription L_{\text{AdS}}\to iL_{\text{dS}}LAdS→iLdS, appearing in the dS/CFT correspondence, and Wick rotate the normal mode frequencies of fields on thermal \text{AdS}_{2n+1}AdS2n+1 into the quasinormal mode frequencies of fields on de Sitter space. We compare the quasinormal mode and heat kernel methods of calculating 1-loop determinants, finding exact agreement, and furthermore explicitly relate these methods via a sum over the conformal dimension. We discuss how the Wick rotation of normal modes on thermal \text{AdS}_{2n+1}AdS2n+1 can be generalized to calculating 1-loop partition functions on the thermal spherical quotients S^{2n+1}/\mathbb{Z}_{p}S2n+1/ℤp. We further show that the quasinormal mode frequencies encode the group theoretic structure of the spherical spacetimes in question, analogous to the analysis made for thermal AdS in [1-3] .

Quantum mirror map for del Pezzo geometries

Mirror maps play an important role in studying supersymmetric gauge theories. In these theories the dynamics is often encoded in an algebraic curve where two sets of periods enjoy the symplectic structure. The A-periods contribute to redefinitions of chemical potentials known as mirror maps. Using the quantization of the D5 del Pezzo geometry, which enjoys the symmetry of the D5 Weyl group, we are able to identify clearly the group-theoretical structure and the multi-covering structure for the mirror map. With the structures, we can apply the mirror map to superconformal Chern–Simons theories describing the worldvolume of multiple M2-branes on various backgrounds, where we find that the redefinition of the chemical potential is obtained directly from the mirror map. Besides, we have interesting observations for the mirror map: the representations appearing in the quantum mirror map are the same as those appearing in the BPS indices except for the trivial case of degree 1 and the coefficients are all integers.

Profinite extensions of centralizers and the profinite completion of limit groups

We introduce and investigate a class of profinite groups defined via extensions of centralizers analogous to the extensively studied class of finitely generated fully residually free groups, that is, limit groups (in the sense of Z. Sela). From the fact that the profinite completion of limit groups belong to this class, results on their group-theoretical structure and homological properties are obtained.

Imprints of the underlying structure of physical theories

In the context of scientific realism, this paper intends to provide a formal and accurate description of the structural-based ontology posited by classical mechanics, quantum mechanics and special relativity, which is preserved across the empirical domains of these theories and explain their successful predictions. Along the lines of ontic structural realism, such a description is undertaken by a particular ontological commitment: the belief in the existence of a freestanding actual structure, approximately represented by a subgroup of the Inhomogeneous Symplectic Group (up to group homomorphisms), and their corresponding state-space representations. Accordingly, the hierarchy and the complexity of this group-theoretical structure is represented by appropriate philosophical tools, namely, by the language of partial structures. Upon this approach, the lack of knowledge of some relations that hold at the boundary between mathematics and physics, and the presence of surplus structure within the structural edifice are explored and represented. The conclusive issue appeals to an interesting example of a surplus but fruitful structure, where superposition of states with different mass are suggested to be actual relativistic remnants within non-relativistic quantum mechanics, as opposed to the standard interpretation in which they are empirically meaningless.

The construction of the mKdV cyclic symmetric N-soliton solution by the Bäcklund transformation

We study group theoretical structures of the mKdV equation. The Schwarzian-type mKdV equation has the global Möbius group symmetry. The Miura transformation makes a connection between the mKdV equation and the KdV equation. We find the special local Möbius transformation on the mKdV one-soliton solution which can be regarded as the commutative KdV Bäcklund transformation and can generate the mKdV cyclic symmetric N-soliton solution. In this algebraic construction to obtain multi-soliton solutions, we could observe the addition formula.

A new class of entropic information measures, formal group theory and information geometry.

In this work, we study generalized entropies and information geometry in a group-theoretical framework. We explore the conditions that ensure the existence of some natural properties and at the same time of a group-theoretical structure for a large class of entropies. In addition, a method for defining new entropies, using previously known ones with some desired group-theoretical properties is proposed. In the second part of this work, the information geometrical counterpart of the previous construction is examined and a general class of divergences are proposed and studied. Finally, a method of constructing new divergences from known ones is discussed; in particular, some results concerning the Riemannian structure associated with the class of divergences under investigation are formulated.

Torsion-free G2-structures with identical Riemannian metric

Based on a general formula due to Bryant, we work out the topological structure of the space of torsion-free [Formula: see text]-structures generating the same associated Riemannian metric on a compact [Formula: see text]-manifold. We also identify a corresponding Lie group-theoretic structure of the space. These observations are then used to describe the moduli space of torsion-free [Formula: see text]-structures in certain cases — by way of covering spaces.

String dynamics in NS5-F1-P geometries

String theory dynamics on certain fivebrane supertube backgrounds is described by an exactly solvable null-gauged WZW model. We use this description to compute the spectrum of closed string excitations on the three-charge non-supersymmetric solution found by Jejjala, Madden, Ross and Titchener, as well as its supersymmetric limit. The low-lying spectrum matches that of supergravity modes in the effective geometry and exhibits an underlying group-theoretic structure. Winding sectors describe strings carrying the same charges as the background; processes whereby strings turn into flux or vice-versa are mediated by large gauge transformations on the worldsheet. The S-matrix of such wound strings probes microstructure of the fivebrane source that is hidden in the supergravity approximation.