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

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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.

“…If, in addition, both

sans-serifA

and

sans-serifL

come with inner products, then

\hat{L}

admits a natural inner product defined by

false⟨ a_{1} \otimes ℓ_{1}, a_{2} \otimes ℓ_{2} false⟩ : = {(- 1)}^{false| a_{2} false| false| ℓ_{1} false|} false⟨ a_{1}, a_{2} false⟩ false⟨ ℓ_{1}, ℓ_{2} false⟩

for homogeneous

a_{1}, a_{2} \in A

and

ℓ_{1}, ℓ_{2} \in L

and again extended to general elements by linearity. Detailed proofs on checking the higher Jacobi identities for the products

{\hat{μ}}_{i}

and the cyclicity of this inner product can be found in [].…”

Section: L∞‐algebrasmentioning

confidence: 99%

‐Algebras, the BV Formalism, and Classical Fields

Jurčo

Macrelli

Raspollini

et al. 2019

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“…The result is a differential graded vector space which is equivalent to an

L_{\infty}

‐algebra. See [] for a recent discussion of the

L_{\infty}

‐perspective of the BV formalism.…”

Section: Contributions To This Volumementioning

confidence: 99%

Higher Structures in M‐Theory

Jurčo

Sämann

Schreiber

et al. 2019

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The key open problem of string theory remains its non‐perturbative completion to M‐theory. A decisive hint to its inner workings comes from numerous appearances of higher structures in the limits of M‐theory that are already understood, such as higher degree flux fields and their dualities, or the higher algebraic structures governing closed string field theory. These are all controlled by the higher homotopy theory of derived categories, generalised cohomology theories, and L∞‐algebras. This is the introductory chapter to the proceedings of the LMS/EPSRC Durham Symposium on ‘Higher Structures in M‐Theory’: We first review higher structures as well as their motivation in string theory and beyond. Then we list the contributions in this volume, putting them into context.

“…These, however, are just the strict morphisms. More general morphisms are readily derived from the dual description of

L_{\infty}

‐algebras in terms of differential graded commutative algebras, see [] or also [] for more details. The end result is that a morphism of

L_{\infty}

‐algebras

ϕ : L \to {sans-serifL}^{'}

corresponds to a set of maps

ϕ_{i} : {sans-serifL}^{\land i} \to {sans-serifL}^{'}

of degree

1 - i

such that

\begin{matrix} true \\ left \sum_{j + k = i} \sum_{σ \in Sh false(j; i false)} {false(- 1 false)}^{i j} χ (σ; ℓ_{1}, \dots, ℓ_{i}) \\ left \times ϕ_{k + 1} (μ_{j} false(ℓ_{σ false(1 false)}, \dots, ℓ_{σ false(j false)} false), ℓ_{σ (j + 1)}, \dots, ℓ_{σ (i)}) \\ left = \sum_{j = 1}^{i} \frac{1}{j!} \sum_{k_{1} + \dots + k_{j} = i} \sum_{σ \in Sh false(k_{1}, \dots, k_{j - 1}; i false)} \\ left \times χ (σ; ℓ_{1}, \dots, ℓ_{i}) ζ (σ; ℓ_{1}, \dots, ℓ_{i}) \\ left \times μ_{j}^{'} true(ϕ_{k_{1}} true(ℓ_{σ false(1 false)} \end{matrix}

…”

Section: Higher Gauge Theoriesmentioning

confidence: 99%

“…For the latter, we have the Maurer–Cartan equation, which naturally generalizes to

L_{\infty}

‐algebras, see e.g. [] for a detailed review. A gauge potential is here an element

a \in {sans-serifL}_{1}

, and we call it a homotopy Maurer–Cartan element if it satisfies the homotopy Maurer–Cartan equation

\begin{matrix} true \\ right f & left : = μ_{1} (a) + 0false \frac{1}{2} μ_{2} (a, a) + 0false \frac{1}{3!} μ_{3} (a, a, a) + \dots \\ left = \sum_{i = 1}^{\infty} 0false \frac{1}{i!} μ_{i} (a, \dots, a) = 0 . \end{matrix}

The element

f \in {sans-serifL}_{2}

is the curvature of a and it satisfies the Bianchi identity

\sum_{i = 0}^{\infty} \frac{{false(- 1 false)}^{i}}{i!} μ_{i + 1} false(f, a, \dots, a false) = 0 .

The gauge transformations are given by

δ a = \sum_{i = 0}^{\infty} \frac{1}{i!} μ_{i + 1} false(a, \dots, a, α false)

for α an element of

L_{0}

.…”

Section: Higher Gauge Theoriesmentioning

confidence: 99%

Higher Structures, Self‐Dual Strings and 6d Superconformal Field Theories

Sämann

2019

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I summarize and discuss some recent results on formulating actions of six‐dimensional superconformal field theories using the language of higher gauge theory. The latter guarantees mathematical consistency of our constructions and we review crucial aspects of this framework, such as L∞‐algebras and corresponding kinematical data given by higher connections. We then show that there is a mathematically consistent non‐Abelian extension of the self‐dual string equation which satisfies many physical expectations. Our construction favors a particular higher gauge group leading us to higher principal bundles known as string structures. Using these, we manage to formulate a six‐dimensional action which shares many properties with the famous (2,0)‐theory but also still differs from it in some key points.