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Forces from Connes' geometry [article]

Thomas Schucker
2002 arXiv   pre-print
We try to give a pedagogical introduction to Connes' derivation of the standard model of electro-magnetic, weak and strong forces from gravity.  ...  The image of the column vector ψ under J is obtained by taking the complex conjugate of ψ and then multiplying it with a unitary n × n matrix U, Jψ = Uψ or J = U • complex conjugation.  ...  An anti-linear operator J is anti-unitary if it is invertible and preserves the scalar product, (Jψ, Jψ) = (ψ, ψ).  ... 
arXiv:hep-th/0111236v2 fatcat:saprakbszvarzguxzij2uhrb5q

Swampland geometry and the gauge couplings [article]

Sergio Cecotti
2021 arXiv   pre-print
First we review in detail the geometric aspects of the swampland program for supersymmetric 4d effective theories using a new and unifying language we dub 'domestic geometry', the generalization of special  ...  As special K\"ahler geometries, domestic geometries carry formal brane amplitudes: when the domestic geometry describes the supersymmetric low-energy limit of a consistent quantum theory of gravity, its  ...  We consider the symmetric Riemannian manifolds of type III, i.e. of the form G/K where G is some real Lie algebra and K its maximal compact subgroup.  ... 
arXiv:2102.03205v3 fatcat:xfgbhks3kbg4xjniyxryyc5gtq

Lectures on Microstate Geometries [article]

Nicholas P. Warner
2019 arXiv   pre-print
The last sections contain brief comments on the larger setting of microstate geometries and describe some of the more recent developments.  ...  The subsequent lectures move on to the construction and properties of bubbled microstate geometries in five dimensions.  ...  I would also like to thank Dominique Hirondel for his careful reading of these notes and for sending me very helpful corrections.  ... 
arXiv:1912.13108v1 fatcat:muxivneidrekjdsqnl27exmqm4

Introduction to Differential Geometry [chapter]

Jean Lévine
2009 Analysis and Control of Nonlinear Systems  
EXISTENCE OF A SURFACE WITH PRESCRIBED FIRST AND SECOND FUNDAMENTAL FORMS Let M 2 , S 2 , and S 2 > denote the sets of all square matrices of order two, of all symmetric matrices of order two, and of all  ...  EXISTENCE OF AN IMMERSION DEFINED ON AN OPEN SET IN R 3 WITH A PRESCRIBED MET-RIC TENSOR Let M 3 , S 3 , and S 3 > denote the sets of all square matrices of order three, of all symmetric matrices of order  ...  components of the first and second fundamental forms of the surface θ(ω) and c αβ = a στ b ασ b βτ .  ... 
doi:10.1007/978-3-642-00839-9_2 fatcat:7joietyxyrbz7n6csv6frh7qh4

Introduction to Differential Geometry [chapter]

Ivan G. Avramidi
2015 Heat Kernel Method and its Applications  
EXISTENCE OF A SURFACE WITH PRESCRIBED FIRST AND SECOND FUNDAMENTAL FORMS Let M 2 , S 2 , and S 2 > denote the sets of all square matrices of order two, of all symmetric matrices of order two, and of all  ...  EXISTENCE OF AN IMMERSION DEFINED ON AN OPEN SET IN R 3 WITH A PRESCRIBED MET-RIC TENSOR Let M 3 , S 3 , and S 3 > denote the sets of all square matrices of order three, of all symmetric matrices of order  ...  components of the first and second fundamental forms of the surface θ(ω) and c αβ = a στ b ασ b βτ .  ... 
doi:10.1007/978-3-319-26266-6_3 fatcat:6vbrqbpacbanraanv4rhtuzmuq

Introduction to Differential Geometry [chapter]

Harry G. Kwatny, Gilmer L. Blankenship
2000 Nonlinear Control and Analytical Mechanics  
EXISTENCE OF A SURFACE WITH PRESCRIBED FIRST AND SECOND FUNDAMENTAL FORMS Let M 2 , S 2 , and S 2 > denote the sets of all square matrices of order two, of all symmetric matrices of order two, and of all  ...  EXISTENCE OF AN IMMERSION DEFINED ON AN OPEN SET IN R 3 WITH A PRESCRIBED MET-RIC TENSOR Let M 3 , S 3 , and S 3 > denote the sets of all square matrices of order three, of all symmetric matrices of order  ...  components of the first and second fundamental forms of the surface θ(ω) and c αβ = a στ b ασ b βτ .  ... 
doi:10.1007/978-1-4612-2136-4_3 fatcat:vb5bv7jh4jd4fjsluqkhbiwjvy

The algebra of entanglement and the geometry of composition [article]

Amar Hadzihasanovic
2017 arXiv   pre-print
We define a category of non-degenerate shapes, and show how to calculate their tensor products.  ...  String diagrams turn algebraic equations into topological moves that have recurring shapes, involving the sliding of one diagram past another.  ...  algebra and geometry.  ... 
arXiv:1709.08086v2 fatcat:ts5oz4iqsnaonfk4m2damc7jyq

"Discrete" vacuum geometry as a tool for Dirac fundamental quantization of Minkowskian Higgs model [article]

Leonid Lantsman
2019 arXiv   pre-print
More precisely, assuming the "discrete" geometry for the appropriate vacuum manifold implies the presence of thread topological defects (side by side with point hedgehog topological defects and walls between  ...  different topological domains) inside this manifold in the shape of specific (rectilinear) threads: gauge and Higgs fields located in the spatial region intimately near the axis z of the chosen (rest)  ...  The topological variablesṄ(t) (respectively, N(t)), may be specified [3, 7, 8, 12] via the relation (1.10) taking account of the natural duality between the tensors F a i0 and F a ij .  ... 
arXiv:hep-th/0701097v16 fatcat:dg6rwgp5rbaknd4daoyixgicum

An Introduction to Differential Geometry

W. E. Byrne, Luther Pfahler Eisenhart
1941 National Mathematics Magazine  
EXISTENCE OF A SURFACE WITH PRESCRIBED FIRST AND SECOND FUNDAMENTAL FORMS Let M 2 , S 2 , and S 2 > denote the sets of all square matrices of order two, of all symmetric matrices of order two, and of all  ...  EXISTENCE OF AN IMMERSION DEFINED ON AN OPEN SET IN R 3 WITH A PRESCRIBED MET-RIC TENSOR Let M 3 , S 3 , and S 3 > denote the sets of all square matrices of order three, of all symmetric matrices of order  ...  components of the first and second fundamental forms of the surface θ(ω) and c αβ = a στ b ασ b βτ .  ... 
doi:10.2307/3028560 fatcat:wetpp3z4evfd7cvxe4lblxunri

An Introduction to Differential Geometry

E. T. Davies, T. J. Willmore
1961 Mathematical Gazette  
EXISTENCE OF A SURFACE WITH PRESCRIBED FIRST AND SECOND FUNDAMENTAL FORMS Let M 2 , S 2 , and S 2 > denote the sets of all square matrices of order two, of all symmetric matrices of order two, and of all  ...  EXISTENCE OF AN IMMERSION DEFINED ON AN OPEN SET IN R 3 WITH A PRESCRIBED MET-RIC TENSOR Let M 3 , S 3 , and S 3 > denote the sets of all square matrices of order three, of all symmetric matrices of order  ...  components of the first and second fundamental forms of the surface θ(ω) and c αβ = a στ b ασ b βτ .  ... 
doi:10.2307/3614797 fatcat:tk7pkjciyfa4vogktdt3ct4apa

An Introduction to Differential Geometry

C. E. Springer, T. J. Wilmore
1960 The American mathematical monthly  
EXISTENCE OF A SURFACE WITH PRESCRIBED FIRST AND SECOND FUNDAMENTAL FORMS Let M 2 , S 2 , and S 2 > denote the sets of all square matrices of order two, of all symmetric matrices of order two, and of all  ...  EXISTENCE OF AN IMMERSION DEFINED ON AN OPEN SET IN R 3 WITH A PRESCRIBED MET-RIC TENSOR Let M 3 , S 3 , and S 3 > denote the sets of all square matrices of order three, of all symmetric matrices of order  ...  components of the first and second fundamental forms of the surface θ(ω) and c αβ = a στ b ασ b βτ .  ... 
doi:10.2307/2309204 fatcat:o2cr7sckobgphoc7tdpcjsazji

On matrix geometry and effective actions

Frank Ferrari
2013 Nuclear Physics B  
We provide an elementary systematic discussion of single-trace matrix actions and of the group of matrix reparameterization that acts on them.  ...  The action of this group yields a generalized notion of gauge invariance which encompasses ordinary diffeomorphism and gauge invariances.  ...  studied in 4.3 and T-duality.  ... 
doi:10.1016/j.nuclphysb.2013.02.005 fatcat:clgmofekbne7zadqdxhpqenwwu

Special Geometry and the Swampland [article]

Sergio Cecotti
2020 arXiv   pre-print
More generally: the Kähler moduli of X do not receive quantum corrections if and only if X has infinite fundamental group.  ...  Seiberg-Witten geometry which holds both in the rigid and local cases.  ...  Then the Ricci curvatures of the two metrics are equal and non-singular.  ... 
arXiv:2004.06929v1 fatcat:uyjvzeglcfcnppnpbfw3nc4q2y

From Special Geometry to Black Hole Partition Functions [article]

Thomas Mohaupt
2008 arXiv   pre-print
Lecture I: special geometry from the superconformal point of view. Lecture II: black hole attractor mechanism, its underlying variational principle, and black hole partition functions.  ...  Lecture III: large and small BPS black holes in N=4 supergravity. Lecture IV: state counting for N=4 string compactifications. Appendix A: special geometry from the mathematical point of view.  ...  The participants of the lecture series in both Frascati and Vienna have helped me to shape these notes. A special thanks goes to Maximilian Kreuzer and his group for their active participation.  ... 
arXiv:0812.4239v1 fatcat:ptfqbhyyovgl3iomqfnz4cqtcy

Special geometry and the swampland

Sergio Cecotti
2020 Journal of High Energy Physics  
More generally: the Kähler moduli of X do not receive quantum corrections if and only if X has infinite fundamental group.  ...  Seiberg-Witten geometry which holds both in the rigid and local cases.  ...  A Pluri-harmonic ⇒ solution to tt * PDEs In this appendix: X is a Kähler manifold which in the standard application of tt * geometry to 2d (2,2) QFTs plays the role of the space of coupling constants associated  ... 
doi:10.1007/jhep09(2020)147 fatcat:kzoru2zqzjhaxkjhxii6ddtuga
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