Online one-day workshop

26th October 2021, 1:30-5:00pm GMT

Higher Dirac structures have been approached from different perspectives and several definitions are available in the literature. Where do all these definitions come from? How do they relate to each other? Where is the theory going?

This event was held on ZOOM on 26th October 2021, 1:30-5:00pm GMT.

Photo (end of the meeting)

Organizing committee:

Henrique Bursztyn (IMPA)Roberto Rubio (Barcelona)

Aissa Wade

(1:30-2:00pm GMT)

In this talk, we will discuss holomorphic Dirac-Jacobi structures. They naturally appear in the study of holomorphic Jacobi manifolds. Holomorphic Dirac-Jacobi structures are dissimilar to "real" Dirac structures and their study requires different techniques. We will start by reviewing holomorphic gauge algebroids and holomorphic first jet bundles of holomorphic vector bundles. Then we will introduce holomorphic Courant-Jacobi algebroids and holomorphic Dirac-Jacobi structures before explaining their relationship with generalized contact geometry. Part of this talk is based on a joint work with Luca Vitagliano.

In this talk, we will discuss holomorphic Dirac-Jacobi structures. They naturally appear in the study of holomorphic Jacobi manifolds. Holomorphic Dirac-Jacobi structures are dissimilar to "real" Dirac structures and their study requires different techniques. We will start by reviewing holomorphic gauge algebroids and holomorphic first jet bundles of holomorphic vector bundles. Then we will introduce holomorphic Courant-Jacobi algebroids and holomorphic Dirac-Jacobi structures before explaining their relationship with generalized contact geometry. Part of this talk is based on a joint work with Luca Vitagliano.

Yunhe Sheng

(p,k)-Dirac structures for higher analogues of Courant algebroids

Slides here

Slides here

(2:05-2:35pm GMT)

We introduce the notion of a (p,k)-Dirac structurein TM⊕⋀p T*M, where 0≤k≤p-1. The (p,0)-Dirac structures are exactly the higher analogues ofDirac structures of order p introduced by Zambon. The (p,p-1)-Dirac structures are exactly theNambu-Dirac structures introduced by Hagiwara.In the regular case, such a (p,k)-Dirac structure is characterized by a characteristic pair.

We introduce the notion of a (p,k)-Dirac structurein TM⊕⋀p T*M, where 0≤k≤p-1. The (p,0)-Dirac structures are exactly the higher analogues ofDirac structures of order p introduced by Zambon. The (p,p-1)-Dirac structures are exactly theNambu-Dirac structures introduced by Hagiwara.In the regular case, such a (p,k)-Dirac structure is characterized by a characteristic pair.

Marco Zambon

(2:40-3:10pm GMT)

For every positive integer p, consider the “higher Courant algebroid” TM⊕⋀p T*M. We will make some remarks on its isotropic involutive subbundles (including the Lagrangian ones). Then we show that to any isotropic involutive subbundle one can associate an L∞-algebra of observables

For the special case of closed differential forms, we display an embedding of these observables that can be interpreted as a prequantization map.

For every positive integer p, consider the “higher Courant algebroid” TM⊕⋀p T*M. We will make some remarks on its isotropic involutive subbundles (including the Lagrangian ones). Then we show that to any isotropic involutive subbundle one can associate an L∞-algebra of observables

For the special case of closed differential forms, we display an embedding of these observables that can be interpreted as a prequantization map.

Nicolás Martínez-Alba

From mechanics to weakly-Lagrangian condition of higher Dirac structures

Slides here

Slides here

(3:15-3:45pm GMT)

By studying the equation of Classical Field Theory and a suitable geometrical settingfor these equations, we will motivate a weaker notion of the usual Lagrangian condition definingDirac structures. After this motivation some other results associated to this weaker notion willbe shown.

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By studying the equation of Classical Field Theory and a suitable geometrical settingfor these equations, we will motivate a weaker notion of the usual Lagrangian condition definingDirac structures. After this motivation some other results associated to this weaker notion willbe shown.

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Miquel Cueca

Higher Dirac as Lagrangian Q-submanifolds

Slides here

Slides here

(3:50-4:20pm GMT)

The Severa-Roytenberg correspondence states that degree 2 symplectic Q-manifolds are Courant algebroids. Moreover, it sends Lagrangian Q-submanifolds to Dirac structures. I will explain how this result extends to higher Courant algebroids and higher Dirac structures. If time permits, I will comment on applications such as reduction, quantization or field theories for higher Courant.

The Severa-Roytenberg correspondence states that degree 2 symplectic Q-manifolds are Courant algebroids. Moreover, it sends Lagrangian Q-submanifolds to Dirac structures. I will explain how this result extends to higher Courant algebroids and higher Dirac structures. If time permits, I will comment on applications such as reduction, quantization or field theories for higher Courant.

Discussion

Open to everyone

(4:25-5:00pm GMT)

During the discussion time we would like to draw connections between all the approaches and put our understanding to test. Please, join us and share your knowledge and insights.