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Algebras over not too little discs

Damien Calaque, Victor Carmona

TL;DR

The paper develops a scale-aware formalism for observables in topological field theories by introducing $R$-truncated little discs operads $\mathsf{D}_n^R$ and proving that locally constant $\mathsf{D}_n^R$-algebras are equivalent to $\mathbb{E}_n$-algebras, thereby allowing renormalization at a fixed scale to propagate to all scales. It extends the framework to theories with defects via operads for fattened, nested, and constructible prefactorization algebras controlled by linear and corner stratifications, culminating in a generalization to defect setups and a cubical version. The paper then applies these results to quantization of constant Poisson structures, constructing Weyl-type algebras as global sections of scale-$R$ algebras (with $R=\tfrac{1}{2}$) and exploring two concrete quantization routes via Costello–Gwilliam's discretization and a discrete model, followed by a deformation-quantization program using Beilinson–Drinfeld operads. In particular, constant ($-1$)-shifted Poisson structures are quantized using BD operads, yielding BD-algebras whose cohomology recovers Weyl-type algebras, thereby connecting scale propagation with explicit quantum observables. The results provide a robust, scale-local perspective on renormalization for TFTs and offer discrete models that illustrate the interaction between topology, defects, and quantization.

Abstract

By the introduction of locally constant prefactorization algebras at a fixed scale, we show a mathematical incarnation of the fact that observables at a given scale of a topological field theory propagate to every scale over euclidean spaces. The key is that these prefactorization algebras over $\mathbb{R}^n$ are equivalent to algebras over the little $n$-disc operad. For topological field theories with defects, we get analogous results by replacing $\mathbb{R}^n$ with the spaces modelling corners $\mathbb{R}^p\times\mathbb{R}^{q}_{\geq 0}$. As a toy example in $1d$, we quantize, once more, constant Poisson structures.

Algebras over not too little discs

TL;DR

The paper develops a scale-aware formalism for observables in topological field theories by introducing -truncated little discs operads and proving that locally constant -algebras are equivalent to -algebras, thereby allowing renormalization at a fixed scale to propagate to all scales. It extends the framework to theories with defects via operads for fattened, nested, and constructible prefactorization algebras controlled by linear and corner stratifications, culminating in a generalization to defect setups and a cubical version. The paper then applies these results to quantization of constant Poisson structures, constructing Weyl-type algebras as global sections of scale- algebras (with ) and exploring two concrete quantization routes via Costello–Gwilliam's discretization and a discrete model, followed by a deformation-quantization program using Beilinson–Drinfeld operads. In particular, constant ()-shifted Poisson structures are quantized using BD operads, yielding BD-algebras whose cohomology recovers Weyl-type algebras, thereby connecting scale propagation with explicit quantum observables. The results provide a robust, scale-local perspective on renormalization for TFTs and offer discrete models that illustrate the interaction between topology, defects, and quantization.

Abstract

By the introduction of locally constant prefactorization algebras at a fixed scale, we show a mathematical incarnation of the fact that observables at a given scale of a topological field theory propagate to every scale over euclidean spaces. The key is that these prefactorization algebras over are equivalent to algebras over the little -disc operad. For topological field theories with defects, we get analogous results by replacing with the spaces modelling corners . As a toy example in , we quantize, once more, constant Poisson structures.
Paper Structure (23 sections, 29 theorems, 110 equations, 4 figures)

This paper contains 23 sections, 29 theorems, 110 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Little discs at large scale
  3. Fattened configurations in euclidean space
  4. Fattened configurations.
  5. Nested configurations.
  6. Operads of closed discs
  7. Proof of Theorem \ref{['thm_LocalizationForTruncatedDisj']}
  8. A cubical version
  9. Defects
  10. Linear defects
  11. Fattened configurations in the presence of linear defects.
  12. Nested configurations with linear defects.
  13. Corner defects
  14. Application: quantization of constant Poisson structures
  15. Poisson additivity for constant Poisson structures
  16. ...and 8 more sections

Key Result

Theorem 2.3

For any symmmetric monoidal $\infty$-category $\EuScript{V}$, the functor is an equivalence of $\infty$-categories, between $\mathbbst{E}_n$-algebras and locally constant prefactorization algebras at scale $R$ over $\mathbbst{R}^n$.

Figures (4)

  • Figure 1: On the right hand side, a $k=3$ chain $\upalpha\colon [3]\to \mathop{\mathrm{\mathsf{Fin}}}\nolimits_*$ of active maps. On the left hand side, a point in $\mathop{\mathrm{\mathsf{D}}}\nolimits^R\mathop{\mathrm{\mathsf{Conf}}}\nolimits_{\upalpha}(\mathbbst{R}^n)$ for the previous $\upalpha$.
  • Figure 2: On the left, components of two elements $\overline{D}\!\,^{*}_{\bullet},\, \overline{D}\!\,^{\prime,*}_{\bullet}\in \mathsf{weq}\left[[k],\overline{\mathdutchcal{D}}\right]_{/\upsigma}$ which admit no upper nor lower bound in that poset, i.e. the slice is not filtered nor cofiltered. On the right, components of an element $\upsigma\in \mathsf{weq}\left[[k],\mathdutchcal{D}\right]$ such that $\mathsf{weq}\left[[k],\overline{\mathdutchcal{D}}\right]_{\upsigma/}$ is empty.
  • Figure 3: A point $(f_{i_r})_{i_r}\in \mathop{\mathrm{\mathsf{D}}}\nolimits^R\mathop{\mathrm{\mathsf{Conf}}}\nolimits_{\upalpha} (\mathbbst{R}^n)$ and $\underline{U}^*\in \mathop{\mathrm{\mathsf{P}}}\nolimits$ such that $(f_{i_r})_{i_r}\in \upzeta(\underline{U}^*)$.
  • Figure 4: An operation in $\mathbbst{E}_{\mathop{\mathrm{\mathbin{\ThisStyle{\ensurestackMath{ \stackinset{c}{}{c}{-.2\LMpt} {\SavedStyle\scaleobj{.5}{\bullet}}{\SavedStyle\boxminus}}}}}}\nolimits}\genfrac{[}{]}{0pt}{}{\{\mathop{\mathrm{\mathbin{\ThisStyle{\ensurestackMath{ \stackinset{c}{}{c}{-.2\LMpt} {\SavedStyle\scaleobj{.5}{\bullet}}{\SavedStyle\boxminus}}}}}}\nolimits,\square_{u},\boxminus_{l}\}}{\mathop{\mathrm{\mathbin{\ThisStyle{\ensurestackMath{ \stackinset{c}{}{c}{-.2\LMpt} {\SavedStyle\scaleobj{.5}{\bullet}}{\SavedStyle\boxminus}}}}}}\nolimits}$, where $\mathbbst{E}_{\mathop{\mathrm{\mathbin{\ThisStyle{\ensurestackMath{ \stackinset{c}{}{c}{-.2\LMpt} {\SavedStyle\scaleobj{.5}{\bullet}}{\SavedStyle\boxminus}}}}}}\nolimits}$ represents the little $(\upchi,2)$-discs operad associated to the linear stratification $\mathbbst{R}^2_{\mathop{\mathrm{\mathbin{\ThisStyle{\ensurestackMath{ \stackinset{c}{}{c}{-.2\LMpt} {\SavedStyle\scaleobj{.5}{\bullet}}{\SavedStyle\boxminus}}}}}}\nolimits}$.

Theorems & Definitions (90)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • Lemma 2.8
  • proof
  • Definition 2.9
  • ...and 80 more