Integral representations on supermanifolds: super Hodge duals, PCOs and Liouville forms

Castellani, Leonardo; Catenacci, R.; Grassi, Pietro Antonio

doi:10.1007/s11005-016-0895-x

Cited by 23 publications

(34 citation statements)

References 31 publications

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“…As seen in [3][4][5]15,18,50,51,55], the space of differential forms has to be extended in order to define a meaningful integration theory. We define Ω (•|•) (SM) as the complete complex of forms; they are graded w.r.t.…”

Section: Superforms Integral Forms and Pseudoformsmentioning

confidence: 99%

Pictures from super Chern-Simons theory

Cremonini

Grassi

2020

J. High Energ. Phys.

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We study super-Chern-Simons theory on a generic supermanifold. After a self-contained review of integration on supermanifolds, the complexes of forms (superforms, pseudo-forms and integral forms) and the extended Cartan calculus are discussed. We then introduce Picture Changing Operators. We provide several examples of computation of PCO's acting on different type of forms. We illustrate also the action of the η operator, crucial ingredient to define the interactions of super Chern-Simons theory. Then, we discuss the action for super Chern-Simons theory on any supermanifold, first in the factorized form (3-form × PCO) and then, we consider the most general expression. The latter is written in term of psuedo-forms containing an infinite number of components. We show that the free equations of motion reduce to the usual Chern-Simons equations yielding the proof of the equivalence between the formulations at different pictures of the same theory. Finally, we discuss the interaction terms. They require a suitable definition in order to take into account the picture number. That implies the construction of a 2-product which is not associative that inherits an A ∞ algebra structure. That shares several similarities with a recent construction of a super string field theory action by

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Section: Superforms Integral Forms and Pseudoformsmentioning

confidence: 99%

Pictures from super Chern-Simons theory

Cremonini

Grassi

2020

J. High Energ. Phys.

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“…A similar description in terms of the Voronov integral transform can be found in [26]. The Z operator is in general not invertible but it is possible to find a non unique operator Y such that Z • Y is an isomorphism in the cohomology.…”

Section: Pco's and Their Algebraic Propertiesmentioning

confidence: 83%

“…In this section we recall a few definitions and useful computations about the PCO's in our notations. For more details see [22] and [26]. We start with the Picture Lowering Operators that map cohomology classes in picture q to cohomology classes in picture r < q.…”

Section: Pco's and Their Algebraic Propertiesmentioning

confidence: 99%

“…They are essential to provide a sensible theory of geometric integration for supermanifolds and they are the natural generalizations of differential forms of a conventional manifold. Their properties and their integration theory are briefly described in the text and we refer to the literature [22,23,26,7] and the book by Voronov [5]) for more details.…”

Section: Introductionmentioning

confidence: 99%

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Super-Quantum Mechanics in the Integral Form Formalism

Castellani

Catenacci

Grassi

2018

Ann. Henri Poincaré

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We reformulate Super Quantum Mechanics in the context of integral forms. This framework allows to interpolate between different actions for the same theory, connected by different choices of Picture Changing Operators (PCO). In this way we retrieve component and superspace actions, and prove their equivalence. The PCO are closed integral forms, and can be interpreted as super Poincaré duals of bosonic submanifolds embedded into a supermanifold.. We use them to construct Lagrangians that are top integral forms, and therefore can be integrated on the whole supermanifold. The D = 1, N = 1 and the D = 1, N = 2 cases are studied, in a flat and in a curved supermanifold. In this formalism we also consider coupling with gauge fields, Hilbert space of quantum states and observables.

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“…It is here worth remarking that in [58] the use of integral forms, in the framework of the group manifold geometrical approach [77,78] (intermediate between the superfield and the component approaches) to supergravity, led to the proof of the aforementioned EIT, showing that the origin of that formula can be understood by interpreting the superfield action itself as an integral form. Subsequent further developments dealt with the construction of the super Hodge dual, the integral representation of Picture Changing Operators of string theories, as well as the construction of the super-Liouville form of a symplectic supermanifold [79].…”

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confidence: 99%

One-dimensional super Calabi-Yau manifolds and their mirrors

Noja

Cacciatori

Piazza

et al. 2017

J. High Energ. Phys.

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Abstract:We apply a definition of generalised super Calabi-Yau variety (SCY) to supermanifolds of complex dimension one. One of our results is that there are two SCY's having reduced manifold equal to P 1 , namely the projective super space P 1|2 and the weighted projective super space WP 1|1(2) . Then we compute the corresponding sheaf cohomology of superforms, showing that the cohomology with picture number one is infinite dimensional, while the de Rham cohomology, which is what matters from a physical point of view, remains finite dimensional. Moreover, we provide the complete real and holomorphic de Rham cohomology for generic projective super spaces P n|m . We also determine the automorphism groups: these always match the dimension of the projective super group with the only exception of P 1|2 , whose automorphism group turns out to be larger than the projective super group. By considering the cohomology of the super tangent sheaf, we compute the deformations of P 1|m , discovering that the presence of a fermionic structure allows for deformations even if the reduced manifold is rigid. Finally, we show that P 1|2 is self-mirror, whereas WP 1|1 (2) has a zero dimensional mirror. Also, the mirror map for P 1|2 naturally endows it with a structure of N = 2 super Riemann surface.

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Integral representations on supermanifolds: super Hodge duals, PCOs and Liouville forms

Cited by 23 publications

References 31 publications

Pictures from super Chern-Simons theory

Pictures from super Chern-Simons theory

Super-Quantum Mechanics in the Integral Form Formalism

One-dimensional super Calabi-Yau manifolds and their mirrors

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