Algebraic Structure of Yang-Mills Theory

Movshev, M. V.; Schwarz, Albert

doi:10.1007/0-8176-4467-9_14

Cited by 37 publications

(96 citation statements)

References 11 publications

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“…We will see that in fact tym(n) is isomorphic (as graded Lie algebras) to the graded free Lie algebra on a graded vector space W (n), i.e., tym f gr (W (n)) (cf. [Mov], [MS06] and §3).…”

Section: Generalities and Finite Dimensional Modulesmentioning

confidence: 99%

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Representations of Yang-Mills algebras

Herscovich

Solotar

2011

Ann. Math.

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The aim of this article is to describe families of representations of the Yang-Mills algebras YM(n) (n ∈ N ≥2 ) defined by A. Connes and M. DuboisViolette. We first describe some irreducible finite dimensional representations. Next, we provide families of infinite dimensional representations of YM, big enough to separate points of the algebra. In order to prove this result, we prove and use that all Weyl algebras Ar(k) are epimorphic images of YM(n).

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“…We will see that in fact tym(n) is isomorphic (as graded Lie algebras) to the graded free Lie algebra on a graded vector space W (n), i.e., tym f gr (W (n)) (cf. [Mov], [MS06] and §3).…”

Section: Generalities and Finite Dimensional Modulesmentioning

confidence: 99%

“…We suggest [LH], [Kel] as a reference. Although some of the results of this section are mentioned in [MS06], our aim here is to give detailed proofs of the results that we will need later.…”

Section: The Ideal Tym(n)mentioning

confidence: 99%

Representations of Yang-Mills algebras

Herscovich

Solotar

2011

Ann. Math.

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“…The

L_{\infty}

‐algebra

L_{{YM}_{2}}

was first given in [] in its dual formulation as a differential graded algebra. The same

L_{\infty}

‐algebra was then rederived from string field theory considerations and further discussed in [].…”

Section: Classical L∞‐structure Of Field Theoriesmentioning

confidence: 99%

“…In particular, the Batalin–Vilkovisky (BV) formalism associates to each classical field theory an

L_{\infty}

‐algebra and for interacting field theories, this

L_{\infty}

‐algebra is not merely a differential graded Lie algebra. This fact is well‐known to experts on BV quantisation, see for example [], in particular [], which is based on the earlier work [], or the later works [], but it seems to be much less known in general. The recent paper [] revived interest in the

L_{\infty}

‐algebras of classical field theories, but only a very partial picture of the categorified structures and their origin was given.…”

Section: Introductionmentioning

confidence: 99%

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

Jurčo

Raspollini

Sämann

et al. 2019

Fortschritte der Physik

102

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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“…We introduce the second‐order Yang–Mills complex by setting []

\begin{matrix} true \\ left \underset{= : L_{0}}{\underset{︸}{{normalΩ}^{0} false(X, frakturg false)}} \overset{μ_{1} : = normald}{\to} \underset{= : L_{1}}{\underset{︸}{{normalΩ}^{1} false(X, frakturg false)}} \\ left \overset{μ_{1} : = normald ★ normald}{\to} \underset{= : L_{2}}{\underset{︸}{{normalΩ}^{3} false(X, frakturg false)}} \overset{μ_{1} : = normald}{\to} \underset{= : L_{3}}{\underset{︸}{{normalΩ}^{4} false(X, frakturg false)}}, \end{matrix}

where ⋆ is the Hodge operator on X . This complex can be given an

L_{\infty}

‐structure by defining the non‐vanishing products by []

\begin{matrix} true \\ right μ_{1} (c_{1}) & left : = normald c_{1}, \\ right μ_{1} (A_{1}) & left : = normald ★ d A_{1}, \\ right μ_{1} (A_{1}^{+}) & left : = normald A_{1}^{+}, \\ right μ_{2} (c_{1} \end{matrix}

…”

Section: Homotopy Maurer–cartan Theorymentioning

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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Algebraic Structure of Yang-Mills Theory

Cited by 37 publications

References 11 publications

Representations of Yang-Mills algebras

Representations of Yang-Mills algebras

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

‐Algebras, the BV Formalism, and Classical Fields

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Algebraic Structure of Yang-Mills Theory

Cited by 37 publications

References 11 publications

Representations of Yang-Mills algebras

Representations of Yang-Mills algebras

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

‐Algebras, the BV Formalism, and Classical Fields

Contact Info

Product

Resources

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