Pietro Fre
Università degli Studi di Torino, Dipartimento di Fisica, Faculty Member
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This chapter introduces the Cartan approach to differential geometry, the vielbein and the spin connection, discusses Bianchi identities and their relation with gauge invariances and eventually introduces Einstein field equations. The... more
This chapter introduces the Cartan approach to differential geometry, the vielbein and the spin connection, discusses Bianchi identities and their relation with gauge invariances and eventually introduces Einstein field equations. The dynamical equations of gravity and their derivation from an action principle are developed in a parallel way to their analogues for electrodynamics and non-Abelian gauge theories whose structure and features are constantly compared to those of gravity. The linearization of Einstein field equations and the spin of the graviton are then discussed. After that the bottom-up approach to gravity is discussed, namely, following Feynman’s ideas, it is shown how a special relativistic linear theory of the graviton field could be uniquely inferred from the conservation of the stress-energy tensor and its non-linear upgrading follows, once the stress-energy tensor of the gravitational field itself is taken into account. The last section of this chapter contains the derivation of the Schwarzschild metric from Einstein equations.
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In this contribution, emphasizing new developments, we plan to review the group manifold-rheonomic symmetry approach to super- gravity1,2 which has already been presented to other conferences3. In particular, we want to emphasize the... more
In this contribution, emphasizing new developments, we plan to review the group manifold-rheonomic symmetry approach to super- gravity1,2 which has already been presented to other conferences3. In particular, we want to emphasize the central role of the mathematical concept of the graded Lie algebra cohomology class, which gives a constructive criterion for Lagrangians and which was not discussed in previous papers. The cohomological foundations of geometrical theories will be fully explained in a forthcoming paper4.
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Research Interests: Mathematics and Physics
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We show that the superspace formulation of anomaly free supergravity based on the 2-form B(2) and on the 6-form B(6) are asymmetrical. B(6) is the electric potential while B(2) is the magnetic one. As a consequence, the Bianchi-identity... more
We show that the superspace formulation of anomaly free supergravity based on the 2-form B(2) and on the 6-form B(6) are asymmetrical. B(6) is the electric potential while B(2) is the magnetic one. As a consequence, the Bianchi-identity associated with the B(2) formulation is the inhomogeneous Maxwell equation which provides a full account of the dynamics. The Bianchi identity associated with B(6) is the homogeneous Maxwell equation: a mere identity which yields no information on the dynamics. It admits a universal “off-shell” solution.
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Research Interests: Mathematics and Harmonic
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SIGLEITItal
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We consider the class of superstring vacua arising from the free fermion construction based on the SU (3) ⊗ SO (5) or SU (2) ⊗ SU (4) supercurrent. The internal free fermions have complex boundary conditions. We show that these vacua... more
We consider the class of superstring vacua arising from the free fermion construction based on the SU (3) ⊗ SO (5) or SU (2) ⊗ SU (4) supercurrent. The internal free fermions have complex boundary conditions. We show that these vacua admit a geometric interpretation as compactifications on twisted WZW models.
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D3 brane solutions of type IIB supergravity can be obtained by means of a classical Ansatz involving a harmonic warp factor, $$H(\textbf{y},\bar{\textbf{y}})$$ H ( y , y ¯ ) multiplying at power $$-1/2$$ - 1 / 2 the first summand, i.e.,... more
D3 brane solutions of type IIB supergravity can be obtained by means of a classical Ansatz involving a harmonic warp factor, $$H(\textbf{y},\bar{\textbf{y}})$$ H ( y , y ¯ ) multiplying at power $$-1/2$$ - 1 / 2 the first summand, i.e., the Minkowski metric of the D3 brane world-sheet, and at power 1/2 the second summand, i.e., the Ricci-flat metric on a six-dimensional transverse space $$\mathcal {M}_6$$ M 6 , whose complex coordinates y are the arguments of the warp factor. Of particular interest is the case where $$\mathcal {M}_6={\text {tot}}[ K\left[ \left( \mathcal {M}_B\right) \right] $$ M 6 = tot [ K M B is the total space of the canonical bundle over a complex Kähler surface $$\mathcal {M}_B$$ M B . This situation emerges in many cases while considering the resolution à la Kronheimer of singular manifolds of type $$\mathcal {M}_6=\mathbb {C}^3/\Gamma $$ M 6 = C 3 / Γ , where $$\Gamma \subset \mathrm {SU(3)} $$ Γ ⊂ SU ( 3 ) is a discrete subgroup. When $$\Gamma = \mathbb {Z}...
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Research Interests: Physics and Supergravity
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In Chap. 5, before exposing the long and exciting history of Lie group discovery, we remarked that differential geometry is at the basis not only of General Relativity but of all those Gauge Theories by means of which XXth century Physics... more
In Chap. 5, before exposing the long and exciting history of Lie group discovery, we remarked that differential geometry is at the basis not only of General Relativity but of all those Gauge Theories by means of which XXth century Physics obtained a consistent and experimentally verified description of all Fundamental Interactions.
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The word symmetry comes from the Greek \(\sigma \upsilon \mu \mu \varepsilon \tau \rho \acute{\iota }\alpha \) which is composed of two words \(\sigma \acute{\upsilon }\nu \) (with) and \(\mu \acute{\varepsilon }\tau \rho o\nu \)... more
The word symmetry comes from the Greek \(\sigma \upsilon \mu \mu \varepsilon \tau \rho \acute{\iota }\alpha \) which is composed of two words \(\sigma \acute{\upsilon }\nu \) (with) and \(\mu \acute{\varepsilon }\tau \rho o\nu \) (measure). Literally \(\sigma \upsilon \mu \mu \varepsilon \tau \rho \acute{\iota }\alpha \) indicates the adequate proportion of the different parts of something, material or immaterial, and it is well represented in classical Greek literature. For instance from Plato we have \(\eta \) \(\nu \upsilon \kappa \tau \acute{o}\varsigma \) \(\pi \rho o\varsigma \) \(\eta \mu \acute{\varepsilon }\rho \alpha \nu \) \(\sigma \upsilon \mu \mu \varepsilon \tau \rho \acute{\iota }\alpha \), the right proportion of night to day time, or \(\tau \grave{o}\) \(\sigma \acute{\upsilon }\mu \mu \varepsilon \tau \rho o\nu \) \(\kappa \alpha \grave{\iota }\) \(\kappa \alpha \lambda \acute{o}\nu \), namely what is proportionate is also beautiful.
Research Interests: Mathematics and Physics
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In this chapter which is the last we turn to the analysis of important developments in complex geometry which took place in the 1980s–1990s, directly motivated by supersymmetry and supergravity and completely inconceivable outside such a... more
In this chapter which is the last we turn to the analysis of important developments in complex geometry which took place in the 1980s–1990s, directly motivated by supersymmetry and supergravity and completely inconceivable outside such a framework. Notwithstanding their roots in the theoretical physics of the superworld, such developments constitute, by now, the basis of some of the most innovative and alive research directions of contemporary geometry.
Research Interests: Physics and Supergravity
In this paper we address the question how to discriminate whether the gauged isometry group G_Sigma of the Kahler manifold Sigma that produces a D-type inflaton potential in a Minimal Supergravity Model is elliptic, hyperbolic or... more
In this paper we address the question how to discriminate whether the gauged isometry group G_Sigma of the Kahler manifold Sigma that produces a D-type inflaton potential in a Minimal Supergravity Model is elliptic, hyperbolic or parabolic. We show that the classification of isometries of symmetric cosets can be extended to non symmetric Sigma.s if these manifolds satisfy additional mathematical restrictions. The classification criteria established in the mathematical literature are coherent with simple criteria formulated in terms of the asymptotic behavior of the Kahler potential K(C) = 2 J(C) where the real scalar field C encodes the inflaton field. As a by product of our analysis we show that all phenomenologically admissible potentials for the description of inflation and in particular alpha-attractors are mostly obtained from the gauging of a parabolic isometry. The requirement of regularity of the manifold Sigma poses strong constraints on the alpha-attractors and reduces the...
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So far we outlined the conceptual development of group theory paying particular attention to finite groups. It is historically correct to do so, since finite groups were the first to be considered and studied. Indeed the very notion of... more
So far we outlined the conceptual development of group theory paying particular attention to finite groups. It is historically correct to do so, since finite groups were the first to be considered and studied. Indeed the very notion of group is to be credited to Galois and, by definition, Galois groups are finite.