Papers by Francesco Toppan
Abstract. Generalized superymmetries going beyond the HLS scheme and admitting thepresence of bos... more Abstract. Generalized superymmetries going beyond the HLS scheme and admitting thepresence of bosonic tensorial central charges are constructed and classified in terms ofthe division algebras R , C , H and O . The eleven-dimensional M-algebra falls into thisclass of supersymmetries. Division-algebra compatible constraints can be introduced andfully classified. They can be used to construct and analyze various dynamical systems, thesimplest examples being the superparticles with tensorial central charges which generalizethe Rudychev-Sezgin and the Bandos-Lukierski models. Key words: Supersymmetry, M-Theory, Tensorial Central Charges 1. INTRODUCTIONThegeneralized supersymmetriesgoing beyond the Haag, Lopusza´nskiandSohniusclassi-fication[1]werefirstintroducedbyD’AuriaandFr´e in1982 [2]. Thefermionicsupersymmetrygenerators are, essentially, square roots operators. Their anticommutators produce a r.h.s.which is totally saturated and has to be expanded in terms of higher-rank bosonic tensors...
Conformal Galilei Algebras labeled by d,l (where d is the number of space dimensions and l denote... more Conformal Galilei Algebras labeled by d,l (where d is the number of space dimensions and l denotes a spin-l representation w.r.t. the sl(2) subalgebra) admit two types of central extensions, the ordinary one (for any d and half-integer l) and the exotic central extension which only exists for d = 2 and l ∈ N. For both types of central extensions invariant second-order PDEs with continuous spectrum were constructed in [1]. It was later proved in [2] that the ordinary central extensions also lead to oscillator-like PDEs with discrete spectrum. We close in this paper the existing gap, constructing a new class of second-order invariant PDEs for the exotic centrally extended CGAs; they admit a discrete and bounded spectrum when applied to a lowest weight representation. These PDEs are markedly different with respect to their ordinary counterparts. The l = 1 case (which is the prototype of this class of � � , � �
Proceedings of Fourth International Winter Conference on Mathematical Methods in Physics — PoS(WC2004)
In this talk we present a division-algebra classification of the generalized supersymmetries admi... more In this talk we present a division-algebra classification of the generalized supersymmetries admitting bosonic tensorial central charges. We show that for complex and quaternionic supersymmetries a whole class of compatible division-algebra constraints can be imposed. Possible applications to M-theory related dynamical systems are briefly mentioned.
TheN -extended Supersymmetric Quantum Mechanics is deformed via an abelian twist which preserves ... more TheN -extended Supersymmetric Quantum Mechanics is deformed via an abelian twist which preserves the super-Hopf algebra structure of its Universal Enveloping Superalgebra. Two constructions are possible. For evenN one can identify the 1DN -extended superalgebra with the fermionic Heisenberg algebra. Alternatively, supersymmetry generators can be realized as operators belonging to the Universal Enveloping Superalgebra of one bosonic and several fermionic oscillators. The deformed system is described in terms of twisted operators satisfying twist-deformed (anti)commutators. The main dierences between an abelian twist defined in terms of fermionic operators and an abelian twist defined in terms of bosonic operators are discussed.
arXiv: High Energy Physics - Theory, 1996
It is shown that supersymmetric integrable models in two dimensions, both relativistic (i.e. supe... more It is shown that supersymmetric integrable models in two dimensions, both relativistic (i.e. super-Toda type theories) and non-relativistic (reductions of super-KP hierarchies) can be associated to general Poisson-brackets structures given by superaffinizations of any bosonic Lie or any super-Lie algebra. This result allows enlarging the set of supersymmetric integrable models, which are no longer restricted to the subclass of superaffinizations of purely fermionic super-Lie algebras (that is admitting fermionic simple roots only).
In this talk I will report some results obtained in a joint collaboration with A. Pashnev, concer... more In this talk I will report some results obtained in a joint collaboration with A. Pashnev, concerning the classification of the irreducible representations of the N -extended Supersymmetry in 1 dimension and which find applications to the construction of Supersymmetric Quantum Mechanical Systems [1]. This mathematical problem finds immediate application to the theory of dimensionally (to one temporal dimension) supersymmetric 4d theories, which gets 4 times the number of supersymmetries of the original models (the N = 8 supergravity being e.g. associated with the a N = 32 Supersymmetric Quantum Mechanical theory). Due to a lack of superfield formalism for N > 4, only partial results are known [2] and [3]. More recently, Supersymmetric and Superconformal Quantum Mechanics have been applied in describing e.g. the low-energy effective dynamics of a certain class of black holes, for testing the AdS/CFT correspondence in the case of AdS2, in investigating the light-cone dynamics of su...
There is a growing interest in the logical possibility that exceptional mathematical structures (... more There is a growing interest in the logical possibility that exceptional mathematical structures (exceptional Lie and superLie algebras, the exceptional Jordan algebra, etc.) could be linked to an ultimate “exceptional” formulation for a Theory Of Everything (TOE). The maximal division algebra of the octonions can be held as the mathematical responsible for the existence of the exceptional structures mentioned above. In this context it is quite motivating to systematically investigate the properties of octonionic spinors and the octonionic realizations of supersymmetry. In particular the M-algebra can be consistently defined for two structures only, a real structure, leading to the standard Malgebra, and an octonionic structure. The octonionic version of the M-algebra admits striking properties induced by octonionic p-forms identities.
arXiv: High Energy Physics - Theory, 2004
In this talk we present a division-algebra classification of the generalized supersymmetries admi... more In this talk we present a division-algebra classification of the generalized supersymmetries admitting bosonic tensorial central charges. We show that for complex and quaternionic supersymmetries a whole class of compatible division-algebra constraints can be imposed. Possible applications to M-theory related dynamical systems are briefly mentioned.
Proceedings of Fifth International Conference on Mathematical Methods in Physics — PoS(IC2006)
The complete classification of the irreducible representations of the N-extended one-dimensional ... more The complete classification of the irreducible representations of the N-extended one-dimensional supersymmetry algebra linearly realized on a finite number of fields is presented. Off-shell invariant actions of one-dimensional supersymmetric sigma models are constructed. The role of both Clifford algebras and the Cayley-Dickson's doublings of algebras in association with the N-extended supersymmetries is discussed. We prove in specific examples that the octonionic structure constants enter the N = 8 invariant actions as coupling constants. We further explain how to relate one-dimensional supersymmetric quantum mechanical systems to the dimensional reduction of higher-dimensional supersymmetric theories.
Proceedings of Workshop on Integrable Theories, Solitons and Duality — PoS(unesp2002)
The generalized supersymmetries admitting abelian bosonic tensorial central charges are classifie... more The generalized supersymmetries admitting abelian bosonic tensorial central charges are classified in accordance with their division algebra structure (over R, C, H or O). It is shown in particular that in D = 11 dimensions, the M-superalgebra admits a consistent octonionic formulation, involving 52 real bosonic generators (in place of the 528 of the standard M-superalgebra). The octonionic M 5 (super-5-brane) sector coincides with the octonionic M 1 and M 2 sectors, while in the standard formulation these sectors are all independent. The octonionic conformal and superconformal M-algebras are explicitly constructed. They are respectively given by the Sp(8|O) (OSp(1, 8|O)) (super)algebra of octonionic-valued (super)matrices, whose bosonic subalgebra consists of 232 (and respectively 239) generators. * Speaker. † A large part of the results here reported is a fruit of a collaboration with J. Lukierski. PrHEP unesp2002 Workshop on Integrable Theories, Solitons and Duality Francesco Toppan length in the following. Perhaps the most remarkable and the most unexpected of such features consists in the fact that the different bosonic sectors expressed by the tensorial abelian central charges are no longer independent, as for the standard generalized supersymmetries admitting associative realizations, but they are all interrelated. This phenomenon is a peculiar characteristic of the octonionic construction. It is worth noticing that the Minkowskian 11-dimensional spacetime supports two inequivalent structures, the real structure and the octonionic one. Therefore, besides the standard M-algebra leading to the OSp(1|32) superalgebra [9] (and its OSp(1|64) superconformal algebra), a different M-algebra can be introduced in terms of the octonionic structure and consistently defined as a closed algebra. This is the octonionic M-algebra (it will also sometimes be referred to as M-superalgebra) which will be discussed in this talk. Of course, it is too early to say whether this octonionic M-algebra can be of any relevance for physics. On the other hand, the mere fact that it exists, side by side with the standard M-algebra (not to mention its puzzling features) justifies a thorough investigation of this and its related mathematical structures. The plan of this talk is as follows. In the next section the classification of Clifford algebras and spinors (i.e. the necessary ingredients to introduce supersymmetry) is recalled. Later, in section 3, the connection of division algebras with the classification of Clifford algebras will be elucidated. In particular the octonionic-valued realizations (which are usually disregarded in the literature) of the Clifford algebras and their corresponding spinors will be introduced. This paves the way for the construction, in Section 4, of the generalized supersymmetries based on the division algebras and, in Section 5, of the octonionic M-algebra. A detailed discussion of its properties will also be given. In particular a table, based on the octonionic structure constants, expressing the equivalence of the different brane sectors in the octonionic description, will be furnished. In Section 6 the octonionic superconformal M-algebra will be introduced. Finally, in the Conclusions, the relation of the octonionic M-algebra with other algebraic structures such as Jordan algebras will be elucidated. Some possible geometrical interpretations underlining the octonionic description will be pointed out and the outline for further future investigations will be given.
Nuclear Physics B, 2016
We extend to a possibly infinite chain the conformally invariant mechanical system that was intro... more We extend to a possibly infinite chain the conformally invariant mechanical system that was introduced earlier as a toy model for understanding the topological Yang-Mills theory. It gives a topological quantum model that has interesting and computable zero modes and topological observables. It confirms the recent conjecture by several authors that supersymmetric quantum mechanics may provide useful tools for understanding robotic mechanical systems (Vitelli et al.) and condensed matter properties (Kane et al.), where trajectories of effective models are allowed or not by the conservation of topological indices. The absences of ground state and mass gaps are special features of such systems.
The 1D N-Extended Superalgebra, with N odd generators Q l (I = 1, 2,. .. ,N) and a single even ge... more The 1D N-Extended Superalgebra, with N odd generators Q l (I = 1, 2,. .. ,N) and a single even generator H satisfying the (anti)-commutation relations {Q l , Q J } = 5 lJ H and [H, Q l ] = 0 is the superalgebra underlying the Supersymmetric Quantum Mechanics An important subclass of non-minimal representations is given by the reducible but indecomposable supermultiplets In this work we present a systematical investigation of the inequivalent non-minimal linear supermultiplets carrying a representation of the one-dimensional N = 4-Extended Superalgebra They act on eight bosonic and eight fermionic fields. Inequivalent representations are specified by the mass dimension of the fields and the connectivity of the associated graphs. The oxidation to minimal N = 5 linear representations is given. Two types of N = 4 sigma-models based on non minimal representations are obtained: the resulting off-shell actions are either manifestly invariant or depend on a constrained prepotential. The connectivity properties of the graphs play a decisive role in discriminating inequivalent actions. These results find application in partial breaking of super symmetric theories. Extended Supersymmetries in 1D can be used to constrain, possible higher-dimensional supersymmetric theories (for instance, constraining the number of auxiliary fields in super gravity theories). The inequivalent N = 4 non-minimal supermultiplets and their inequivalent N = 4-invariant off-shell actions can therefore be regarded as building blocks for constructing supersymmetric models obtained from dimensional reduction of partial spontaneous supersymmetry breaking of N = 2, D = 4 supersymmetry.
The connection of (split-)division algebras with Clifford algebras and supersymmetry is investiga... more The connection of (split-)division algebras with Clifford algebras and supersymmetry is investigated. At first we introduce the class of superalgebras constructed from any given (split-)division algebra. We further specify which real Clifford algebras and real fundamental spinors can be reexpressed in terms of split-quaternions. Finally, we construct generalized supersymmetries admitting bosonic tensorial central charges in terms of (split-)division algebras. In particular we prove that split-octonions allow to introduce a split-octonionic M-algebra which extends to the (6, 5) signature the properties of the 11-dimensional octonionic M-algebras (which only exist in the (10, 1) Minkowskian and (2, 9) signatures).
AIP Conference Proceedings, 2001
The division algebras R, C, H, O are used to construct and analyze the N = 1, 2, 4, 8 supersymmet... more The division algebras R, C, H, O are used to construct and analyze the N = 1, 2, 4, 8 supersymmetric extensions of the KdV hamiltonian equation. In particular a global N = 8 super-KdV system is introduced and shown to admit a Poisson bracket structure given by the "Non-Associative N = 8 Superconformal Algebra".
AIP Conference Proceedings, 2005
Quaternionic and octonionic spinors are introduced and their fundamental properties (such as the ... more Quaternionic and octonionic spinors are introduced and their fundamental properties (such as the space-times supporting them) are reviewed. The conditions for the existence of their associated Dirac equations are analyzed. Quaternionic and octonionic supersymmetric algebras defined in terms of such spinors are constructed. Specializing to the D = 11-dimensional case, the relation of both the quaternionic and the octonionic supersymmetries with the ordinary M-algebra are discussed.
In this talk we present a division-algebra classification of the generalized supersymmetries admi... more In this talk we present a division-algebra classification of the generalized supersymmetries admitting bosonic tensorial central charges. We show that for complex and quaternionic supersymmetries a whole class of compatible division-algebra constraints can be imposed. Possible applications to M-theory related dynamical systems are briefly mentioned.
Eprint Arxiv Solv Int 9710001, Oct 1, 1997
We review some basic features of the Lie-algebraic classification of Walgebras and related integr... more We review some basic features of the Lie-algebraic classification of Walgebras and related integrable hierarchies in 1 + 1 dimensions, pointing out the role of affine Lie algebras. We emphasize that the supersymmetric extensions of the above construction possibly lead, though some questions are still opened, to the classification of supersymmetric hierarchies based on "generic" supersymmetric affine Lie algebras. Here the word generic is used to make clear that well-known procedures, as those introduced by Inami and Kanno, are too restricted and do not lead to the full spectrum of supersymmetric integrable hierarchies one can construct. A particular attention is devoted to the large-N supersymmetric extensions (here N = 4). The attention paid by large-N theories being due to the fact that they arise as dimensional reduction of N = 1 models, and moreover that they realize an "unification" of known hierarchies.
Notas De Fisica, May 12, 2014
ABSTRACT Some key features of the symmetries of the Schr\"odinger equation that are comm... more ABSTRACT Some key features of the symmetries of the Schr\"odinger equation that are common to a much broader class of dynamical systems (some under construction) are illustrated. I discuss the algebra/superalgebra duality involving first and second-order differential operators. It provides different viewpoints for the spectrum-generating subalgebras. The representation-dependent notion of on-shell symmetry is introduced. The difference in associating the time-derivative symmetry operator with either a root or a Cartan generator of the $sl(2)$ subalgebra is discussed. In application to one-dimensional Lagrangian superconformal sigma-models it implies superconformal actions which are either supersymmetric or non-supersymmetric.
The complete classification of the irreducible representations of the N-extended one-dimensional ... more The complete classification of the irreducible representations of the N-extended one-dimensional supersymmetry algebra linearly realized on a finite number of fields is presented. Off-shell invariant actions of one-dimensional supersymmetric sigma models are constructed. The role of both Clifford algebras and the Cayley-Dickson's doublings of algebras in association with the N-extended supersymmetries is discussed. We prove in specific examples that the octonionic structure constants enter the N = 8 invariant actions as coupling constants. We further explain how to relate one-dimensional supersymmetric quantum mechanical systems to the dimensional reduction of higher-dimensional supersymmetric theories.
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Papers by Francesco Toppan