Spinors, algebraic geometry, and the classification of second-order symmetric tensors in general relativity

Recent developments in string theory suggest that there might exist extra spatial dimensions, which are not small nor compact. The framework of most brane cosmological models is that the matter fields are confined on a brane-world embedded in five dimensions (the bulk). Motivated by this we re-examine the classification of the second-order symmetric tensors in 5-D, and prove two theorems which collect together some basic results on the algebraic structure of these tensors in five-dimensional spacetimes. We also briefly indicate how one can obtain, by induction, the classification of symmetric two-tensors (and the corresponding canonical forms) on n-dimensional (n>4) spaces from the classification on four-dimensional spaces. This is important in the context of 11-D supergravity and 10-D superstrings.

Download Full-text

Tensors with eigenvectors in a given subspace

Rendiconti del Circolo Matematico di Palermo (1952 -) ◽

10.1007/s12215-021-00600-2 ◽

2021 ◽

Author(s):

Giorgio Ottaviani ◽

Zahra Shahidi

Keyword(s):

Algebraic Geometry ◽

Linear Subspace ◽

Chern Classes ◽

Symmetric Tensors

AbstractThe first author with B. Sturmfels studied in [16] the variety of matrices with eigenvectors in a given linear subspace, called the Kalman variety. We extend that study from matrices to symmetric tensors, proving in the tensor setting the irreducibility of the Kalman variety and computing its codimension and degree. Furthermore, we consider the Kalman variety of tensors having singular t-tuples with the first component in a given linear subspace and we prove analogous results, which are new even in the case of matrices. Main techniques come from Algebraic Geometry, using Chern classes for enumerative computations.

Download Full-text

Fine-structure classification of multiqubit entanglement by algebraic geometry

Physical Review Research ◽

10.1103/physrevresearch.2.043003 ◽

2020 ◽

Vol 2 (4) ◽

Author(s):

Masoud Gharahi ◽

Stefano Mancini ◽

Giorgio Ottaviani

Keyword(s):

Algebraic Geometry ◽

Fine Structure

Download Full-text

Application of group analysis to classification of systems of three second-order ordinary differential equations

Mathematical Methods in the Applied Sciences ◽

10.1002/mma.3430 ◽

2015 ◽

Vol 38 (18) ◽

pp. 5097-5113 ◽

Cited By ~ 3

Author(s):

S. Suksern ◽

S. Moyo ◽

S. V. Meleshko

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Group Analysis ◽

Second Order

Download Full-text

Limit-point / limit-circle classification of second-order differential operators arising in PT quantum mechanics

PAMM ◽

10.1002/pamm.201610424 ◽

2016 ◽

Vol 16 (1) ◽

pp. 871-872 ◽

Cited By ~ 1

Author(s):

Florian Büttner ◽

Carsten Trunk

Keyword(s):

Quantum Mechanics ◽

Limit Point ◽

Differential Operators ◽

Second Order ◽

Limit Circle ◽

Pt Quantum Mechanics

Download Full-text

SATANIC AND THELEMIC CIRCLES ON KNOTS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s021821651350017x ◽

2013 ◽

Vol 22 (05) ◽

pp. 1350017 ◽

Cited By ~ 1

Author(s):

G. FLOWERS

Keyword(s):

Geometric Interpretation ◽

Second Order ◽

Geometric Properties ◽

Vassiliev Invariants ◽

Vassiliev Invariant ◽

Knot Diagrams

While Vassiliev invariants have proved to be a useful tool in the classification of knots, they are frequently defined through knot diagrams, and fail to illuminate any significant geometric properties the knots themselves may possess. Here, we provide a geometric interpretation of the second-order Vassiliev invariant by examining five-point cocircularities of knots, extending some of the results obtained in [R. Budney, J. Conant, K. P. Scannell and D. Sinha, New perspectives on self-linking, Adv. Math. 191(1) (2005) 78–113]. Additionally, an analysis on the behavior of other cocircularities on knots is given.

Download Full-text