scholarly journals Invariant torsion and G2-metrics

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Diego Conti ◽  
Thomas Bruun Madsen
Keyword(s):  
Local Homogeneity ◽  
Spinor Bundle ◽  
Intrinsic Torsion

AbstractWe introduce and study a notion of invariant intrinsic torsion geometrywhich appears, for instance, in connection with the Bryant-Salamon metric on the spinor bundle over S3. This space is foliated by sixdimensional hypersurfaces, each of which carries a particular type of SO(3)-structure; the intrinsic torsion is invariant under SO(3). The last condition is sufficient to imply local homogeneity of such geometries, and this allows us to give a classification. We close the circle by showing that the Bryant-Salamon metric is the unique complete metric with holonomy G2 that arises from SO(3)-structures with invariant intrinsic torsion.

Gauge Field Theory in Natural Geometric Language

2020 ◽  
Author(s):  
Daniel Canarutto
Keyword(s):  
Particle Physics ◽  
Quantum Physics ◽  
Brst Symmetry ◽  
Natural Extension ◽  
Integrated Approach ◽  
Arbitrary Spin ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Standard View ◽  
Spinor Bundle

This monograph addresses the need to clarify basic mathematical concepts at the crossroad between gravitation and quantum physics. Selected mathematical and theoretical topics are exposed within a not-too-short, integrated approach that exploits standard and non-standard notions in natural geometric language. The role of structure groups can be regarded as secondary even in the treatment of the gauge fields themselves. Two-spinors yield a partly original ‘minimal geometric data’ approach to Einstein-Cartan-Maxwell-Dirac fields. The gravitational field is jointly represented by a spinor connection and by a soldering form (a ‘tetrad’) valued in a vector bundle naturally constructed from the assumed 2-spinor bundle. We give a presentation of electroweak theory that dispenses with group-related notions, and we introduce a non-standard, natural extension of it. Also within the 2-spinor approach we present: a non-standard view of gauge freedom; a first-order Lagrangian theory of fields with arbitrary spin; an original treatment of Lie derivatives of spinors and spinor connections. Furthermore we introduce an original formulation of Lagrangian field theories based on covariant differentials, which works in the classical and quantum field theories alike and simplifies calculations. We offer a precise mathematical approach to quantum bundles and quantum fields, including ghosts, BRST symmetry and anti-fields, treating the geometry of quantum bundles and their jet prolongations in terms Frölicher's notion of smoothness. We propose an approach to quantum particle physics based on the notion of detector, and illustrate the basic scattering computations in that context.

Efficient 3D mesh salient region detection using local homogeneity measure

2018 ◽  
Vol 12 (9) ◽  
pp. 1663-1672 ◽  
Author(s):  
Abdul Rahman El Sayed ◽  
Abdallah El Chakik ◽  
Hassan Alabboud ◽  
Adnan Yassine
Keyword(s):  
Salient Region Detection ◽  
3D Mesh ◽  
Local Homogeneity ◽  
Salient Region ◽  
Region Detection

scholarly journals $$ \mathcal{N} $$ = 2 consistent truncations from wrapped M5-branes

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Davide Cassani ◽  
Grégoire Josse ◽  
Michela Petrini ◽  
Daniel Waldram
Keyword(s):  
Riemann Surface ◽  
Gauge Group ◽  
Tangent Bundle ◽  
Dimensional Manifold ◽  
Five Dimensions ◽  
Intrinsic Torsion ◽  
Consistent Truncations

Abstract We discuss consistent truncations of eleven-dimensional supergravity on a six-dimensional manifold M, preserving minimal $$ \mathcal{N} $$ N = 2 supersymmetry in five dimensions. These are based on GS ⊆ USp(6) structures for the generalised E6(6) tangent bundle on M, such that the intrinsic torsion is a constant GS singlet. We spell out the algorithm defining the full bosonic truncation ansatz and then apply this formalism to consistent truncations that contain warped AdS5×wM solutions arising from M5-branes wrapped on a Riemann surface. The generalised U(1) structure associated with the $$ \mathcal{N} $$ N = 2 solution of Maldacena-Nuñez leads to five-dimensional supergravity with four vector multiplets, one hypermultiplet and SO(3) × U(1) × ℝ gauge group. The generalised structure associated with “BBBW” solutions yields two vector multiplets, one hypermultiplet and an abelian gauging. We argue that these are the most general consistent truncations on such backgrounds.

scholarly journals Killing superalgebras for lorentzian five-manifolds

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Andrew Beckett ◽  
José Figueroa-O’Farrill
Keyword(s):  
Spin Manifold ◽  
Field Equations ◽  
Spinor Bundle ◽  
Killing Spinors ◽  
Spinor Connection ◽  
Spencer Cohomology ◽  
Definition Of ◽  
Supersymmetric Theories ◽  
Filtered Deformations

Abstract We calculate the relevant Spencer cohomology of the minimal Poincaré superalgebra in 5 spacetime dimensions and use it to define Killing spinors via a connection on the spinor bundle of a 5-dimensional lorentzian spin manifold. We give a definition of bosonic backgrounds in terms of this data. By imposing constraints on the curvature of the spinor connection, we recover the field equations of minimal (ungauged) 5-dimensional supergravity, but also find a set of field equations for an $$ \mathfrak{sp} $$ sp (1)-valued one-form which we interpret as the bosonic data of a class of rigid supersymmetric theories on curved backgrounds. We define the Killing superalgebra of bosonic backgrounds and show that their existence is implied by the field equations. The maximally supersymmetric backgrounds are characterised and their Killing superalgebras are explicitly described as filtered deformations of the Poincaré superalgebra.

scholarly journals On certain classes of $$\mathbf{Sp}(4,\mathbb{R})$$ symmetric $$G_2$$ structures

2020 ◽  
Author(s):  
Paweł Nurowski
Keyword(s):  
Homogeneous Space ◽  
Lie Group ◽  
Real Form ◽  
Intrinsic Torsion

AbstractWe find two different families of $$\mathbf{Sp}(4,\mathbb{R})$$ Sp ( 4 , R ) symmetric $$G_2$$ G 2 structures in seven dimensions. These are $$G_2$$ G 2 structures with $$G_2$$ G 2 being the split real form of the simple exceptional complex Lie group $$G_2$$ G 2 . The first family has $$\tau _2\equiv 0$$ τ 2 ≡ 0 , while the second family has $$\tau _1\equiv \tau _2\equiv 0$$ τ 1 ≡ τ 2 ≡ 0 , where $$\tau _1$$ τ 1 , $$\tau _2$$ τ 2 are the celebrated $$G_2$$ G 2 -invariant parts of the intrinsic torsion of the $$G_2$$ G 2 structure. The families are different in the sense that the first one lives on a homogeneous space $$\mathbf{Sp}(4,\mathbb{R})/\mathbf{SL}(2,\mathbb{R})_l$$ Sp ( 4 , R ) / SL ( 2 , R ) l , and the second one lives on a homogeneous space $$\mathbf{Sp}(4,\mathbb{R})/\mathbf{SL}(2,\mathbb{R})_s$$ Sp ( 4 , R ) / SL ( 2 , R ) s . Here $$\mathbf{SL}(2,\mathbb{R})_l$$ SL ( 2 , R ) l is an $$\mathbf{SL}(2,\mathbb{R})$$ SL ( 2 , R ) corresponding to the $$\mathfrak{sl}(2,\mathbb{R})$$ sl ( 2 , R ) related to the long roots in the root diagram of $$\mathfrak{sp}(4,\mathbb{R})$$ sp ( 4 , R ) , and $$\mathbf{SL}(2,\mathbb{R})_s$$ SL ( 2 , R ) s is an $$\mathbf{SL}(2,\mathbb{R})$$ SL ( 2 , R ) corresponding to the $$\mathfrak{sl}(2,\mathbb{R})$$ sl ( 2 , R ) related to the short roots in the root diagram of $$\mathfrak{sp}(4,\mathbb{R})$$ sp ( 4 , R ) .

scholarly journals Superstrings with intrinsic torsion

2004 ◽  
Vol 69 (8) ◽  
Author(s):  
Jerome P. Gauntlett ◽  
Dario Martelli ◽  
Daniel Waldram
Keyword(s):  
Intrinsic Torsion
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