Higher groupoid bundles, higher spaces, and self‐dual tensor field equations

Jurčo, Branislav; Sämann, Christian; Wolf, Martin

doi:10.1002/prop.201600031

Cited by 28 publications

(55 citation statements)

References 70 publications

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“…The mathematical framework of higher principal bundles with connections, which underlies higher gauge theory, has been developed to full extent, see e.g. [9] or [10] and references therein. There is, however, a severe lack of concrete, interesting examples.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Towards an M5-brane model I: A 6d superconformal field theory

Sämann

Schmidt

2018

Journal of Mathematical Physics

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We present an action for a six-dimensional superconformal field theory containing a non-abelian tensor multiplet. All of the ingredients of this action have been available in the literature. We bring these pieces together by choosing the string Lie 2-algebra as a gauge structure, which we motivated in previous work. The kinematical data contains a connection on a categorified principal bundle, which is the appropriate mathematical description of the parallel transport of self-dual strings. Our action can be written down for each of the simply laced Dynkin diagrams, and each case reduces to a four-dimensional supersymmetric Yang-Mills theory with corresponding gauge Lie algebra. Our action also reduces nicely to an M2-brane model which is a deformation of the ABJM model. While this action is certainly not the desired M5-brane model, we regard it as a key stepping stone towards a potential construction of the (2,0)-theory.

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Section: Introduction and Resultsmentioning

confidence: 99%

Towards an M5-brane model I: A 6d superconformal field theory

Sämann

Schmidt

2018

Journal of Mathematical Physics

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“…Just as a Lie group differentiates to a Lie algebra, a Lie groupoid differentiates to a Lie algebroid and a very general prescription for the Lie differentiation of

L_{\infty}

‐groupoids is found in [], see also [] for all details.…”

Section: Mathematical Toolsmentioning

confidence: 99%

L_∞‐Algebras of Classical Field Theories and the Batalin–Vilkovisky Formalism

Jurčo

Raspollini

Sämann

et al. 2019

Fortschritte der Physik

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221

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We review in detail the Batalin–Vilkovisky formalism for Lagrangian field theories and its mathematical foundations with an emphasis on higher algebraic structures and classical field theories. In particular, we show how a field theory gives rise to an L∞‐algebra and how quasi‐isomorphisms between L∞‐algebras correspond to classical equivalences of field theories. A few experts may be familiar with parts of our discussion, however, the material is presented from the perspective of a very general notion of a gauge theory. We also make a number of new observations and present some new results. Most importantly, we discuss in great detail higher (categorified) Chern–Simons theories and give some useful shortcuts in usually rather involved computations.

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“…It turns out that the 1‐jet of a Lie quasi‐group is an

L_{\infty}

‐algebra []; see also [] for a constructive proof. In particular, letting

G : = \{\dots \begin{matrix} ⟶ \\ ⟶ \\ ⟶ \\ ⟶ \end{matrix} G_{2} \begin{matrix} ⟶ \\ ⟶ \\ ⟶ \end{matrix} G_{1} \begin{matrix} ⟶ \\ ⟶ \end{matrix} *\}

be a Lie quasi‐group with face maps

f_{i}^{p}

and degeneracy maps

d_{i}^{p}

, the 1‐jet of

scriptG

is parametrised as []

L false[1 false] = ⨁_{k \leq 0} {sans-serifL}_{k} false[1 false] with {sans-serifL}_{k} false[1 false] : = \cap_{i = 0}^{- k} ker (f_{i *}^{1 - k}) false[1 - k false],

where

f_{i *}^{p}

denotes the linearisation of

f_{i}^{p}

. Furthermore,

μ_{1} {false|}_{{sans-serifL}_{k} false[1 false]} = {sans-seriff}_{1 - k *}^{1 - k}

and the

μ_{i}

for

i > 1

are given in terms of j ‐th order derivatives of the face maps with

j \leq i

…”

Section: Quasi‐groupsmentioning

confidence: 99%

‐Algebras, the BV Formalism, and Classical Fields

Jurčo

Macrelli

Raspollini

et al. 2019

Fortschritte der Physik

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We summarise some of our recent works on L∞‐algebras and quasi‐groups with regard to higher principal bundles and their applications in twistor theory and gauge theory. In particular, after a lightning review of L∞‐algebras, we discuss their Maurer–Cartan theory and explain that any classical field theory admitting an action can be reformulated in this context with the help of the Batalin–Vilkovisky formalism. As examples, we explore higher Chern–Simons theory and Yang–Mills theory. We also explain how these ideas can be combined with those of twistor theory to formulate maximally superconformal gauge theories in four and six dimensions by means of L∞‐quasi‐isomorphisms, and we propose a twistor space action.

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Higher groupoid bundles, higher spaces, and self‐dual tensor field equations

Cited by 28 publications

References 70 publications

Towards an M5-brane model I: A 6d superconformal field theory

Towards an M5-brane model I: A 6d superconformal field theory

L_∞‐Algebras of Classical Field Theories and the Batalin–Vilkovisky Formalism

‐Algebras, the BV Formalism, and Classical Fields

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Higher groupoid bundles, higher spaces, and self‐dual tensor field equations

Cited by 28 publications

References 70 publications

Towards an M5-brane model I: A 6d superconformal field theory

Towards an M5-brane model I: A 6d superconformal field theory

L∞‐Algebras of Classical Field Theories and the Batalin–Vilkovisky Formalism

‐Algebras, the BV Formalism, and Classical Fields

Contact Info

Product

Resources

About

L_∞‐Algebras of Classical Field Theories and the Batalin–Vilkovisky Formalism