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Mathematical Aspects of Twistor Theory: Null Decomposition of Conformal Algebras and Self-dual Metrics on Intersections of Quadrics

Mihai, Daniela (2007) Mathematical Aspects of Twistor Theory: Null Decomposition of Conformal Algebras and Self-dual Metrics on Intersections of Quadrics. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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<p>The conformal algebra of an n-dimensional affine space with a metric of arbitrary signature (p, q) with p + q = n is considered. The case of broken conformal invariance is studied, by considering the subalgebra of the enveloping algebra of the conformal algebra that commutes with the squared-mass operator. This algebra, denoted R, is generated by the generators of the Poincaré Lie algebra and an additional vector operator R which preserves the relevant information when the conformal invariance is broken. Due to the nonlinearity of the algebra, finding the Casimir invariants becomes extremely difficult. The R-algebra is constructed for arbitrary dimensions, but the Casimir invariants are only determined for n ≤ 5. The second part of this thesis describes the geometric properties of metrics on the twistor space on intersections of quadrics. Consider a generic pencil of quadrics in a complex projective space ℂℙ⁵. The base locus of this pencil is considered as a three-dimensional projective twistor space, such that each point of the associated space-time is a projective two-plane lying inside one quadric of the pencil. The time coordinate can be described as a hyperelliptic curve of genus two, over which the space time is fibered. The metrics arising on the associated twistor space of the completely null two-planes are studied. It emerges that for pencils generated by simultaneously diagonalizable quadrics, these metrics are always self-dual and, in certain cases, conformal to vacuum. </p>


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Item Type: University of Pittsburgh ETD
Status: Unpublished
CreatorsEmailPitt UsernameORCID
Mihai, Danieladam33@pitt.eduDAM33
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairSparling, George A.Jsparling@math.pitt.eduSPARLING
Committee MemberDunajski,
Committee MemberGartside, Paulgartside@math.pitt.eduPMG20
Committee MemberHeath, Robertrwheath@pitt.eduRWHEATH
Date: 31 January 2007
Date Type: Completion
Defense Date: 26 October 2006
Approval Date: 31 January 2007
Submission Date: 4 December 2006
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Institution: University of Pittsburgh
Schools and Programs: Dietrich School of Arts and Sciences > Mathematics
Degree: PhD - Doctor of Philosophy
Thesis Type: Doctoral Dissertation
Refereed: Yes
Uncontrolled Keywords: Casimir invariants; conformal algebra; self-dual metrics; twistors
Other ID:, etd-12042006-210854
Date Deposited: 10 Nov 2011 20:07
Last Modified: 15 Nov 2016 13:53


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