This thesis is divided in two parts, that can be read separately even if both use the possibility of replacing spinors with differential forms in theories with supersymmetry. The ﬁrst part explores some recent results that have been obtained by applying the Gstructure approach to type II supergravities. Using generalized complex geometry it is possible to reformulate the conditions for unbroken supersymmetry in type II supergravity in terms of differential forms. We use this result to ﬁnd a classiﬁcation for AdS7 and AdS6 solutions in type II supergravity. Concerning AdS7 solutions we ﬁnd that in type IIB no solutions can be found, whereas in massive type IIA many new AdS7×M3 solutions are at disposal with the topology of the internal manifold M3 given by a threesphere. We develop a classiﬁcation for such solutions. Concerning AdS6 solutions, very few AdS6×M4 supersymmetric solutions are known in literature: one in massive IIA, and two IIB solutions dual to it. The IIA solution is known to be unique. We obtain a classiﬁcation for IIB supergravity, by reducing the problem to two PDEs on a twodimensional space Σ. The fourdimensional space M4 is then given by a ﬁbration of S2 over Σ. We also explore other two contexts in which the Gstructure approach has revealed its usefulness: ﬁrst of all we derive the conditions for unbroken supersymmetry for a Mink2 (2,0) vacuum, arising from type II supergravity on a compact eightdimensional manifold M8. When M8 enjoys SU(4)×SU(4) structure the resulting system is elegantly rewritten in terms of generalized complex geometry. Finally we rewrite the equations for tendimensional supersymmetry in a way formally identical to an analogous system in N = 2 gauged supergravity; this provides a way to look for lifts of BPS solutions without having to reduce the tendimensional action. The second part is devoted to study some aspects of two different ChernSimons like theories: holomorphic ChernSimons theory on a sixdimensional CalabiYau space and threedimensional supersymmetric theories involving vector multiplets (both with YangMills and ChernSimons terms in the action). Concerning holomorphic ChernSimons theory, we construct an action that couples the gauge ﬁeld to offshell gravitational backgrounds, comprising the complex structure and the (3,0)form of the target space. Gauge invariance of this offshell action is achieved by enlarging the ﬁeld space to include an appropriate system of Lagrange multipliers, ghost and ghostforghost ﬁelds. From this reformulation it is possible to uncover a twisted supersymmetric algebra for this model that strongly constrains the antiholomorphic dependence of physical correlators. Concerning threedimensional theories, we will develop a new way of computing the exact partition function of supersymmetric threedimensional gauge theories, involving vector supermultiplet only. Our approach will reduce the problem of computing the exact partition function to the problem of solving an anomalous Ward identity. To obtain such a result we will describe the coupling of threedimensional topological gauge theories to background topological gravity. The Seifert condition for manifolds supporting global supersymmetry is elegantly deduced from the topological gravity BRST transformations. We will show how the geometrical moduli that affect the partition function can be characterized cohomologically. In the Seifert context ChernSimons topological (framing) anomaly is BRST trivial and we will compute explicitly the corresponding local WessZumino functional. As an application, we obtain the dependence on the Seifert moduli of the partition function of threedimensional supersymmetric gauge theory on the squashed sphere by solving the anomalous topological Ward identities, in a regularization independent way and without the need of evaluating any functional determinant.
(2014). From spinors to forms: results on gstructures in supergravity and on topological field theories. (Tesi di dottorato, Università degli Studi di MilanoBicocca, 2014).
From spinors to forms: results on gstructures in supergravity and on topological field theories
ROSA, DARIO
2014
Abstract
This thesis is divided in two parts, that can be read separately even if both use the possibility of replacing spinors with differential forms in theories with supersymmetry. The ﬁrst part explores some recent results that have been obtained by applying the Gstructure approach to type II supergravities. Using generalized complex geometry it is possible to reformulate the conditions for unbroken supersymmetry in type II supergravity in terms of differential forms. We use this result to ﬁnd a classiﬁcation for AdS7 and AdS6 solutions in type II supergravity. Concerning AdS7 solutions we ﬁnd that in type IIB no solutions can be found, whereas in massive type IIA many new AdS7×M3 solutions are at disposal with the topology of the internal manifold M3 given by a threesphere. We develop a classiﬁcation for such solutions. Concerning AdS6 solutions, very few AdS6×M4 supersymmetric solutions are known in literature: one in massive IIA, and two IIB solutions dual to it. The IIA solution is known to be unique. We obtain a classiﬁcation for IIB supergravity, by reducing the problem to two PDEs on a twodimensional space Σ. The fourdimensional space M4 is then given by a ﬁbration of S2 over Σ. We also explore other two contexts in which the Gstructure approach has revealed its usefulness: ﬁrst of all we derive the conditions for unbroken supersymmetry for a Mink2 (2,0) vacuum, arising from type II supergravity on a compact eightdimensional manifold M8. When M8 enjoys SU(4)×SU(4) structure the resulting system is elegantly rewritten in terms of generalized complex geometry. Finally we rewrite the equations for tendimensional supersymmetry in a way formally identical to an analogous system in N = 2 gauged supergravity; this provides a way to look for lifts of BPS solutions without having to reduce the tendimensional action. The second part is devoted to study some aspects of two different ChernSimons like theories: holomorphic ChernSimons theory on a sixdimensional CalabiYau space and threedimensional supersymmetric theories involving vector multiplets (both with YangMills and ChernSimons terms in the action). Concerning holomorphic ChernSimons theory, we construct an action that couples the gauge ﬁeld to offshell gravitational backgrounds, comprising the complex structure and the (3,0)form of the target space. Gauge invariance of this offshell action is achieved by enlarging the ﬁeld space to include an appropriate system of Lagrange multipliers, ghost and ghostforghost ﬁelds. From this reformulation it is possible to uncover a twisted supersymmetric algebra for this model that strongly constrains the antiholomorphic dependence of physical correlators. Concerning threedimensional theories, we will develop a new way of computing the exact partition function of supersymmetric threedimensional gauge theories, involving vector supermultiplet only. Our approach will reduce the problem of computing the exact partition function to the problem of solving an anomalous Ward identity. To obtain such a result we will describe the coupling of threedimensional topological gauge theories to background topological gravity. The Seifert condition for manifolds supporting global supersymmetry is elegantly deduced from the topological gravity BRST transformations. We will show how the geometrical moduli that affect the partition function can be characterized cohomologically. In the Seifert context ChernSimons topological (framing) anomaly is BRST trivial and we will compute explicitly the corresponding local WessZumino functional. As an application, we obtain the dependence on the Seifert moduli of the partition function of threedimensional supersymmetric gauge theory on the squashed sphere by solving the anomalous topological Ward identities, in a regularization independent way and without the need of evaluating any functional determinant.File  Dimensione  Formato  

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