pith. sign in

arxiv: 2606.03368 · v2 · pith:4XK6V2IJnew · submitted 2026-06-02 · 🧮 math-ph · math.CT· math.MP· math.QA· math.SG

Classical Symmetry TFTs for Continuous Symmetries via Higher Symplectic Geometry

Hao Xu This is my paper

Pith reviewed 2026-06-28 08:08 UTC · model grok-4.3

classification 🧮 math-ph math.CTmath.MPmath.QAmath.SG
keywords symmetry TFTshifted symplectic geometryAKSZ constructionBF theorycontinuous symmetriesgaugingLagrangian boundarieshigher gerbes
0
0 comments X

The pith

The bulk theory for a continuous G-symmetry acting on an n-dimensional topological sigma model is the AKSZ construction with target the shifted cotangent stack T^*[n](BG), or equivalently (n+1)-dimensional BF theory for G.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a classical continuous version of symmetry TFTs by embedding them in shifted symplectic geometry. When a Lie group G acts by topological defects on a sigma model whose target is an (n-1)-shifted symplectic derived stack, the associated bulk theory in one higher dimension is identified with the AKSZ sigma model whose target is the shifted cotangent stack T^*[n](BG). Boundary conditions are then described as shifted Lagrangians inside this bulk, and gauging the original symmetry is realized by inserting a topological domain wall between appropriate boundary conditions. The same framework introduces shifted versions of Hamiltonian, symplectic, and Lagrangian reduction and incorporates prequantum data through higher gerbes on BG that encode classical anomaly information.

Core claim

We argue that the corresponding (n+1)-dimensional bulk theory should be the AKSZ theory with target the shifted cotangent stack T^*[n](BG), equivalently the (n+1)-dimensional BF theory for G. We characterize the Dirichlet and Neumann boundary conditions, and more general topological boundaries, in terms of shifted Lagrangians in T^*[n](BG). We realize the gauging of the G-symmetry in the original theory as inserting a topological domain wall between the corresponding topological boundaries in the BF bulk, and introduce the notion of Hamiltonian reduction, symplectic reduction, and Lagrangian reduction in the shifted symplectic setting. We also discuss prequantum refinements of continuous Sym

What carries the argument

The shifted cotangent stack T^*[n](BG) with its canonical shifted symplectic form, used as the target space for the AKSZ sigma model that defines the (n+1)-dimensional bulk theory.

If this is right

  • Gauging a continuous symmetry is equivalent to inserting a domain wall between Dirichlet and Neumann boundaries in the BF bulk.
  • All topological boundary conditions are classified by shifted Lagrangian sub-stacks of T^*[n](BG).
  • Shifted versions of Hamiltonian, symplectic, and Lagrangian reduction become available for constructing reduced theories.
  • In three dimensions the infinitesimal model B(g ⋉ g^∨) is compared with the factorizable double B(g ⊕ g), relating boundaries to Lagrangian Lie subalgebras and r-matrices.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same Lagrangian-reduction language may supply a uniform way to construct reduced phase spaces when the original theory is already a higher-dimensional sigma model.
  • Quantizing the shifted symplectic structure on T^*[n](BG) would give a direct route from the classical construction to a quantum symmetry TFT.
  • The gerbe decoration of the bulk may generalize to other higher categorical structures that capture mixed anomalies beyond the 't Hooft case.

Load-bearing premise

The bulk theory that encodes the G-symmetry is exactly the AKSZ sigma model whose target is the shifted cotangent stack T^*[n](BG) (or the equivalent BF theory).

What would settle it

An explicit continuous symmetry action on a concrete shifted symplectic target whose topological defects cannot be reproduced by any collection of shifted Lagrangian boundary conditions inside the BF theory on BG.

read the original abstract

We propose a shifted-symplectic formulation of a classical continuous analogue of the symmetry TFT paradigm. Let $G$ be an algebraic or Lie group acting by topological defects on an $n$-dimensional classical topological sigma model with target an $(n-1)$-shifted symplectic derived stack $(X,\omega)$ via the AKSZ construction. We argue that the corresponding $(n+1)$-dimensional bulk theory should be the AKSZ theory with target the shifted cotangent stack $T^*[n] (\mathrm B G)$, equivalently the $(n+1)$-dimensional BF theory for $G$. We characterize the Dirichlet and Neumann boundary conditions, and more general topological boundaries, in terms of shifted Lagrangians in $T^*[n] (\mathrm B G)$. We realize the gauging of the $G$-symmetry in the original theory as inserting a topological domain wall between the corresponding topological boundaries in the BF bulk, and introduce the notion of Hamiltonian reduction, syplectic reduction, and Lagrangian reduction in the shifted symplectic setting. We also discuss prequantum refinements of continuous SymTFTs. In this refinement, higher gerbes on $\mathrm B G$ encode classical analogues of 't Hooft anomaly data by decorating the shifted cotangent bulk and its Lagrangian boundary conditions. Finally, in dimension three we compare the infinitesimal BF model $\mathrm B(\mathfrak g\ltimes\mathfrak g^\vee)$ with the factorizable double $\mathrm B(\mathfrak g\oplus \mathfrak g)$. The resulting topological boundaries are described by Lagrangian Lie subalgebras, and the factorizable case relates the SymTFT dictionary to $r$-matrices and Belavin--Drinfeld data.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper proposes a shifted-symplectic formulation of classical continuous symmetry TFTs. For a Lie or algebraic group G acting by topological defects on an n-dimensional AKSZ sigma-model with target an (n-1)-shifted symplectic derived stack (X,ω), it argues that the corresponding (n+1)-dimensional bulk theory is the AKSZ sigma-model with target the shifted cotangent stack T^*[n](BG), equivalently (n+1)-dimensional BF theory for G. It characterizes Dirichlet/Neumann and general topological boundaries as shifted Lagrangians in this bulk, realizes gauging via domain walls between such boundaries, introduces notions of Hamiltonian/syplectic/Lagrangian reduction in the shifted setting, and discusses prequantum refinements in which higher gerbes on BG encode classical 't Hooft anomaly data. In dimension three it compares the infinitesimal BF model B(g ⋉ g^∨) with the factorizable double B(g ⊕ g), relating boundaries to Lagrangian Lie subalgebras and r-matrices/Belavin-Drinfeld data.

Significance. If the central identification of the bulk holds, the work supplies a geometric language for continuous SymTFTs that unifies boundary conditions, gauging, and anomaly data inside shifted symplectic geometry and AKSZ constructions. The explicit comparison in dimension three with factorizable doubles and r-matrix data offers a concrete bridge to existing structures in Poisson-Lie theory.

major comments (2)
  1. [Abstract] Abstract (and the opening paragraphs of the introduction): the claim that the (n+1)-dimensional bulk 'should be' the AKSZ theory with target T^*[n](BG) (equivalently BF theory) is presented as an argument, yet no explicit functoriality statement, derivation from the given G-action on the original AKSZ model, or reference establishing why this particular shifted symplectic stack (rather than another extension encoding the same defects) is canonically forced appears. All subsequent constructions—shifted Lagrangian boundaries, domain-wall gauging, Hamiltonian reduction, and prequantum gerbe refinements—rest on this identification being the correct one.
  2. [Abstract] The three-dimensional comparison (final paragraph of the abstract): the statement that the infinitesimal BF model B(g ⋉ g^∨) is compared with the factorizable double B(g ⊕ g) and that the resulting boundaries are described by Lagrangian Lie subalgebras is asserted without an explicit statement of the equivalence or the precise relation between the two models that would justify identifying their Lagrangian subalgebras with the SymTFT dictionary.
minor comments (1)
  1. [Abstract] The abstract contains the typographical error 'syplectic reduction' (should be 'symplectic reduction').

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the two major comments point by point below, acknowledging where additional clarification is warranted and outlining the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and the opening paragraphs of the introduction): the claim that the (n+1)-dimensional bulk 'should be' the AKSZ theory with target T^*[n](BG) (equivalently BF theory) is presented as an argument, yet no explicit functoriality statement, derivation from the given G-action on the original AKSZ model, or reference establishing why this particular shifted symplectic stack (rather than another extension encoding the same defects) is canonically forced appears. All subsequent constructions—shifted Lagrangian boundaries, domain-wall gauging, Hamiltonian reduction, and prequantum gerbe refinements—rest on this identification being the correct one.

    Authors: We agree that the identification of the bulk theory is presented as a natural proposal motivated by the AKSZ formalism rather than derived from a fully explicit functoriality statement in the current text. The choice of T^*[n](BG) follows from the requirement that the bulk encode the G-action on the original model via topological defects in a manner compatible with shifted symplectic structures; however, we acknowledge that a dedicated discussion of why this stack is canonically selected (as opposed to other possible extensions) would strengthen the argument. In the revised manuscript we will add a short subsection in the introduction that sketches the functorial reasons, referencing the universal property of shifted cotangent stacks for Hamiltonian G-actions in derived geometry, and we will cite supporting literature on AKSZ constructions for symmetries. revision: yes

  2. Referee: [Abstract] The three-dimensional comparison (final paragraph of the abstract): the statement that the infinitesimal BF model B(g ⋉ g^∨) is compared with the factorizable double B(g ⊕ g) and that the resulting boundaries are described by Lagrangian Lie subalgebras is asserted without an explicit statement of the equivalence or the precise relation between the two models that would justify identifying their Lagrangian subalgebras with the SymTFT dictionary.

    Authors: The three-dimensional comparison is offered as an illustrative link between the infinitesimal BF model and structures from Poisson-Lie theory, with boundaries corresponding to Lagrangian Lie subalgebras and r-matrix data. We accept that the current text does not state the equivalence between B(g ⋉ g^∨) and B(g ⊕ g) with sufficient precision. In the revision we will insert an explicit remark (or short proposition) in the relevant section that records the precise relation at the level of the underlying dg-Lie algebras and clarifies how the sets of Lagrangian subalgebras are identified, thereby making the connection to the SymTFT dictionary rigorous. revision: yes

Circularity Check

0 steps flagged

Bulk theory identification draws from standard AKSZ/shifted symplectic constructions; no self-referential reduction.

full rationale

The paper states that the (n+1)-dimensional bulk 'should be the AKSZ theory with target the shifted cotangent stack T^*[n](BG), equivalently the (n+1)-dimensional BF theory for G' and proceeds to characterize boundaries, gauging, and reductions in that setting. This choice is presented as an argument grounded in the existing AKSZ construction and shifted symplectic geometry literature rather than a quantity fitted or defined by the present work itself. No equation reduces a claimed prediction to an input parameter by construction, no load-bearing uniqueness theorem is imported solely via self-citation, and no ansatz is smuggled through prior author work. The derivation chain therefore remains independent of the paper's own outputs and receives the default low-circularity assessment.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction relies on the validity of the AKSZ sigma-model construction for shifted symplectic targets and on the existence of shifted Lagrangian submanifolds in T^*[n](BG) that encode boundary conditions; no free parameters are introduced. The main axioms are standard results in derived symplectic geometry.

axioms (2)
  • domain assumption The AKSZ construction produces a valid (n+1)-dimensional topological field theory when the target is the shifted cotangent stack T^*[n](BG).
    Invoked when the bulk theory is identified with the AKSZ theory on T^*[n](BG).
  • domain assumption Shifted Lagrangian submanifolds in T^*[n](BG) classify topological boundary conditions for the symmetry TFT.
    Used to characterize Dirichlet, Neumann, and general topological boundaries.

pith-pipeline@v0.9.1-grok · 5843 in / 1578 out tokens · 17907 ms · 2026-06-28T08:08:32.274889+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

72 extracted references

  1. [1]

    Anomalies and gauging of U(1) symmetries

    Andrea Antinucci and Francesco Benini. Anomalies and gauging of U(1) symmetries. Physical Review B , 111:024110, 2025

    2025

  2. [2]

    SymTh for non-finite symmetries, 2024

    Fabio Apruzzi, Francesco Bedogna, and Nicola Dondi. SymTh for non-finite symmetries, 2024

    2024

  3. [3]

    Shifted symplectic reduction of derived critical loci

    Mathieu Anel and Damien Calaque. Shifted symplectic reduction of derived critical loci. Advances in Theoretical and Mathematical Physics , 26(6):1543--1583, 2022

    2022

  4. [4]

    Arvanitakis, Leron T

    Alex S. Arvanitakis, Leron T. Cole, Saskia Demulder, and Daniel C. Thompson. On topological defects in Chern--Simons theory. Physical Review D , 111:105014, 2025

    2025

  5. [5]

    Non-invertible T -duality at any radius via non-compact SymTFT , 2024

    Riccardo Argurio, Andr \'e s Collinucci, Giovanni Galati, Ondrej Hulik, and Elise Paznokas. Non-invertible T -duality at any radius via non-compact SymTFT , 2024

    2024

  6. [6]

    Symmetry TFTs for continuous spacetime symmetries, 2025

    Fabio Apruzzi, Nicola Dondi, I \ n aki Garc \'i a Etxebarria , Ho Tat Lam, and Sakura Sch \"a fer-Nameki. Symmetry TFTs for continuous spacetime symmetries, 2025

    2025

  7. [7]

    SymTFTs from string theory

    Fabio Apruzzi, Federico De Luca, Alessandra Gnecchi, Gabriele Lo Monaco, and Sakura Sch \"a fer-Nameki. SymTFTs from string theory. Communications in Mathematical Physics , 402:895--949, 2023

    2023

  8. [8]

    Alexandrov, M

    M. Alexandrov, M. Kontsevich, A. Schwarz, and O. Zaboronsky. The geometry of the master equation and topological quantum field theory. International Journal of Modern Physics A , 12(7):1405--1429, 1997

    1997

  9. [9]

    Topological quantum field theory

    Michael Atiyah. Topological quantum field theory. Publications Math \'e matiques de l'IH \'E S , 68:175--186, 1988

    1988

  10. [10]

    Multiplicative forms at the infinitesimal level

    Henrique Bursztyn, Alejandro Cabrera, and Cristian Ortiz. Multiplicative forms at the infinitesimal level. Mathematische Annalen , 353(3):663--705, 2012

    2012

  11. [11]

    A. A. Belavin and V. G. Drinfeld. Solutions of the classical Yang--Baxter equation for simple Lie algebras. Functional Analysis and Its Applications , 16(3):159--180, 1982

    1982

  12. [12]

    Baez and James Dolan

    John C. Baez and James Dolan. Higher-dimensional algebra and topological quantum field theory. Journal of Mathematical Physics , 36(11):6073--6105, 1995

    1995

  13. [13]

    SymTFTs for continuous non- Abelian symmetries, 2024

    Federico Bonetti, Michele Del Zotto, and Ruben Minasian. SymTFTs for continuous non- Abelian symmetries, 2024

    2024

  14. [14]

    SymTFT for continuous symmetries: non-linear realizations and spontaneous breaking, 2025

    Federico Bonetti, Michele Del Zotto, and Ruben Minasian. SymTFT for continuous symmetries: non-linear realizations and spontaneous breaking, 2025

    2025

  15. [15]

    Sandwich construction of symmetry TFTs for the centre symmetries of Chern--Simons , Yang--Mills , and Einstein gravity, 2025

    Leron Borsten, Dimitri Kanakaris, and Hyungrok Kim. Sandwich construction of symmetry TFTs for the centre symmetries of Chern--Simons , Yang--Mills , and Einstein gravity, 2025

    2025

  16. [16]

    Higher generalized morphisms and Morita equivalence of geometric \( \)-groupoids

    Christian Blohmann, Kalin Krishna, and Chenchang Zhu. Higher generalized morphisms and Morita equivalence of geometric \( \)-groupoids. In preparation

  17. [17]

    Daniel Brennan and Zhengdi Sun

    T. Daniel Brennan and Zhengdi Sun. A SymTFT for continuous symmetries, 2024

    2024

  18. [18]

    Lagrangian structures on mapping stacks and semi-classical TFTs

    Damien Calaque. Lagrangian structures on mapping stacks and semi-classical TFTs . In Stacks and Categories in Geometry, Topology, and Algebra , volume 643 of Contemporary Mathematics . American Mathematical Society, 2015

    2015

  19. [19]

    Shifted cotangent stacks are shifted symplectic

    Damien Calaque. Shifted cotangent stacks are shifted symplectic. Annales de la Facult \'e des Sciences de Toulouse , 28(1):67--90, 2019

    2019

  20. [20]

    Coste, P

    A. Coste, P. Dazord, and A. Weinstein. Groupo \"i des symplectiques. Publications du D \'e partement de math \'e matiques (Lyon) , pages 1--62, 1987

    1987

  21. [21]

    The AKSZ Construction in Derived Algebraic Geometry as an Extended Topological Field Theory , volume 308 of Memoirs of the American Mathematical Society

    Damien Calaque, Rune Haugseng, and Claudia Scheimbauer. The AKSZ Construction in Derived Algebraic Geometry as an Extended Topological Field Theory , volume 308 of Memoirs of the American Mathematical Society . American Mathematical Society, 2025. No. 1555

    2025

  22. [22]

    Dirac manifolds

    Theodore James Courant. Dirac manifolds. Transactions of the American Mathematical Society , 319(2):631--661, 1990

    1990

  23. [23]

    Cattaneo and Carlo A

    Alberto S. Cattaneo and Carlo A. Rossi. Wilson surfaces and higher dimensional knot invariants. Communications in Mathematical Physics , 256(3):513--537, 2005

    2005

  24. [24]

    Shifted symplectic higher Lie groupoids and classifying spaces

    Miquel Cueca and Chenchang Zhu. Shifted symplectic higher Lie groupoids and classifying spaces. Advances in Mathematics , 413:Paper No.\ 108829, 2023

    2023

  25. [25]

    Centre of an algebra

    Alexei Davydov. Centre of an algebra. Advances in Mathematics , 225(1):319--348, 2010

    2010

  26. [26]

    Classification des triples de Manin pour les alg\`ebres de Lie r\'eductives complexes

    Patrick Delorme. Classification des triples de Manin pour les alg\`ebres de Lie r\'eductives complexes. Journal of Algebra , 246(1):97--174, 2001

    2001

  27. [27]

    Drinfeld

    Vladimir G. Drinfeld. On Poisson homogeneous spaces of Poisson--Lie groups. Theoretical and Mathematical Physics , 95(2):524--525, 1993

    1993

  28. [28]

    On the variety of Lagrangian subalgebras, I

    Sam Evens and Jiang-Hua Lu. On the variety of Lagrangian subalgebras, I . Annales Scientifiques de l'\'Ecole Normale Sup\'erieure , 34(5):631--668, 2001

    2001

  29. [29]

    On the variety of Lagrangian subalgebras, II

    Sam Evens and Jiang-Hua Lu. On the variety of Lagrangian subalgebras, II . Annales Scientifiques de l'\'Ecole Normale Sup\'erieure , 39(2):347--379, 2006

    2006

  30. [30]

    Abelian ideals and the variety of Lagrangian subalgebras

    Sam Evens and Yanpeng Li. Abelian ideals and the variety of Lagrangian subalgebras. Journal of Algebra , 590:1--32, 2022

    2022

  31. [31]

    Moore, Adam Schwimmer, and Nathan Seiberg

    Shmuel Elitzur, Gregory W. Moore, Adam Schwimmer, and Nathan Seiberg. Remarks on the canonical quantization of the Chern--Simons--Witten theory. Nuclear Physics B , 326(1):108--134, 1989

    1989

  32. [32]

    Fusion categories and homotopy theory

    Pavel Etingof, Dmitri Nikshych, and Victor Ostrik. Fusion categories and homotopy theory. Quantum Topology , 1(3):209--273, 2010

    2010

  33. [33]

    Freed and Michael J

    Daniel S. Freed and Michael J. Hopkins. Reflection positivity and invertible topological phases. Geometry & Topology , 25(3):1165--1330, 2021

    2021

  34. [34]

    Freed, Gregory W

    Daniel S. Freed, Gregory W. Moore, and Constantin Teleman. Topological symmetry in quantum field theory. Quantum Topology , 2024

    2024

  35. [35]

    Daniel S. Freed. Classical Chern--Simons theory, Part 1 . Advances in Mathematics , 113(2):237--303, 1995

    1995

  36. [36]

    Freed and Constantin Teleman

    Daniel S. Freed and Constantin Teleman. Relative quantum field theory. Communications in Mathematical Physics , 326(2):459--476, 2014

    2014

  37. [37]

    SymTFTs for U(1) symmetries from descent, 2024

    Finn Gagliano and I \ n aki Garc \'i a Etxebarria . SymTFTs for U(1) symmetries from descent, 2024

    2024

  38. [38]

    Generalized global symmetries

    Davide Gaiotto, Anton Kapustin, Nathan Seiberg, and Brian Willett. Generalized global symmetries. Journal of High Energy Physics , 2015(2):172, 2015

    2015

  39. [39]

    Symplectic Techniques in Physics

    Victor Guillemin and Shlomo Sternberg. Symplectic Techniques in Physics . Cambridge University Press, 1984

    1984

  40. [40]

    Generalized complex geometry

    Marco Gualtieri. Generalized complex geometry . PhD thesis, University of Oxford, 2004. Published as Ann. of Math. (2) 174 (2011), no. 1, 75--123

    2004

  41. [41]

    Shifted coisotropic correspondences

    Rune Haugseng, Valerio Melani, and Pavel Safronov. Shifted coisotropic correspondences. Journal of the Institute of Mathematics of Jussieu , 21(3):785--849, 2022

    2022

  42. [42]

    Categorical continuous symmetry, 2025

    Qiang Jia, Ran Luo, Jiahua Tian, Yi-Nan Wang, and Yi Zhang. Categorical continuous symmetry, 2025

    2025

  43. [43]

    SymTFT approach for flavor symmetry, 2025

    Qiang Jia, Ran Luo, Jiahua Tian, Yi-Nan Wang, and Yi Zhang. SymTFT approach for flavor symmetry, 2025

    2025

  44. [44]

    Categorical symmetries via operator algebras, 2026

    Qiang Jia, Ran Luo, Jiahua Tian, Yi-Nan Wang, and Yi Zhang. Categorical symmetries via operator algebras, 2026

    2026

  45. [45]

    Hilbert space and defect Hilbert spaces associated with categorical symmetries, 2026

    Qiang Jia and Jiahua Tian. Hilbert space and defect Hilbert spaces associated with categorical symmetries, 2026

    2026

  46. [46]

    Karolinsky

    Eugene A. Karolinsky. A classification of Poisson homogeneous spaces of complex reductive Poisson--Lie groups. Banach Center Publications , 51:103--108, 2000

    2000

  47. [47]

    Algebraic higher symmetry and categorical symmetry -- a holographic and entanglement view of symmetry

    Liang Kong, Tian Lan, Xiao-Gang Wen, Zhi-Hao Zhang, and Hao Zheng. Algebraic higher symmetry and categorical symmetry -- a holographic and entanglement view of symmetry. Physical Review Research , 2(4):043086, 2020

    2020

  48. [48]

    Symmetry TFTs for non-invertible defects

    Justin Kaidi, Kantaro Ohmori, and Yunqin Zheng. Symmetry TFTs for non-invertible defects. Communications in Mathematical Physics , 404:1021--1124, 2023

    2023

  49. [49]

    Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions, 2014

    Liang Kong and Xiao-Gang Wen. Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions, 2014

    2014

  50. [50]

    Boundary-bulk relation for topological orders as the functor mapping higher categories to their centers, 2015

    Liang Kong, Xiao-Gang Wen, and Hao Zheng. Boundary-bulk relation for topological orders as the functor mapping higher categories to their centers, 2015

    2015

  51. [51]

    On the classification of topological field theories

    Jacob Lurie. On the classification of topological field theories. Current Developments in Mathematics , 2008:129--280, 2009

    2008

  52. [52]

    Group orbits and regular partitions of Poisson manifolds

    Jiang-Hua Lu and Milen Yakimov. Group orbits and regular partitions of Poisson manifolds. Communications in Mathematical Physics , 283(3):729--748, 2008

    2008

  53. [53]

    Kenneth R. Meyer. Symmetries and integrals in mechanics. In Maur \'i cio M. Peixoto, editor, Dynamical Systems , pages 259--272. Academic Press, New York, 1973

    1973

  54. [54]

    Marsden and Tudor S

    Jerrold E. Marsden and Tudor S. Ratiu. Introduction to Mechanics and Symmetry . Springer, 2nd edition, 1999

    1999

  55. [55]

    Marsden and Alan Weinstein

    Jerrold E. Marsden and Alan Weinstein. Reduction of symplectic manifolds with symmetry. Reports on Mathematical Physics , 5(1):121--130, 1974

    1974

  56. [56]

    Principal -bundles: general theory

    Thomas Nikolaus, Urs Schreiber, and Danny Stevenson. Principal -bundles: general theory. Journal of Homotopy and Related Structures , 10(4):749--801, 2015

    2015

  57. [57]

    A non-abelian duality for (higher) gauge theories

    J \'a n Pulmann, Pavol S evera, and Fridrich Valach. A non-abelian duality for (higher) gauge theories. Advances in Theoretical and Mathematical Physics , 25(1):241--274, 2021

    2021

  58. [58]

    Shifted symplectic structures

    Tony Pantev, Bertrand To \"e n, Michel Vaqui \'e , and Gabriele Vezzosi. Shifted symplectic structures. Publications Math \'e matiques de l'IH \'E S , 117:271--328, 2013

    2013

  59. [59]

    Duals of higher vector bundles and cotangents of Lie 2-groupoids

    Stefano Ronchi. Duals of higher vector bundles and cotangents of Lie 2-groupoids . Phd thesis, Georg-August-Universit \"a t G \"o ttingen, 2025

    2025

  60. [60]

    Duals of higher vector spaces, 2024

    Stefano Ronchi and Chenchang Zhu. Duals of higher vector spaces, 2024

    2024

  61. [61]

    Poisson-- Lie structures as shifted Poisson structures

    Pavel Safronov. Poisson-- Lie structures as shifted Poisson structures. Advances in Mathematics , 381:107633, 2021

    2021

  62. [62]

    Shifted geometric quantization

    Pavel Safronov. Shifted geometric quantization. Journal of Geometry and Physics , 194:104992, 2023

    2023

  63. [63]

    Differential cohomology in a cohesive -topos, 2013

    Urs Schreiber. Differential cohomology in a cohesive -topos, 2013

    2013

  64. [64]

    Poisson-- Lie T -duality as a boundary phenomenon of Chern--Simons theory

    Pavol S evera. Poisson-- Lie T -duality as a boundary phenomenon of Chern--Simons theory. Journal of High Energy Physics , 2016(5):044, 2016

    2016

  65. [65]

    Structure des syst \`e mes dynamiques

    Jean-Marie Souriau. Structure des syst \`e mes dynamiques . Dunod, Paris, 1969

    1969

  66. [66]

    Schommer-Pries

    Christopher J. Schommer-Pries. The Classification of Two-Dimensional Extended Topological Field Theories . PhD thesis, University of California, Berkeley, 2009

    2009

  67. [67]

    Homotopy Quantum Field Theory , volume 10 of EMS Tracts in Mathematics

    Vladimir Turaev. Homotopy Quantum Field Theory , volume 10 of EMS Tracts in Mathematics . European Mathematical Society, 2010

    2010

  68. [68]

    Homotopical algebraic geometry II : geometric stacks and applications

    Bertrand To \"e n and Gabriele Vezzosi. Homotopical algebraic geometry II : geometric stacks and applications. Memoirs of the American Mathematical Society , 193(902), 2008

    2008

  69. [69]

    Quantum field theory and the Jones polynomial

    Edward Witten. Quantum field theory and the Jones polynomial. Communications in Mathematical Physics , 121(3):351--399, 1989

    1989

  70. [70]

    On holomorphic factorization of WZW and coset models

    Edward Witten. On holomorphic factorization of WZW and coset models. Communications in Mathematical Physics , 144(1):189--212, 1992

    1992

  71. [71]

    Momentum maps and Morita equivalence

    Ping Xu. Momentum maps and Morita equivalence. Journal of Differential Geometry , 67(2):289--333, 2004

    2004

  72. [72]

    n -groupoids and stacky groupoids

    Chenchang Zhu. n -groupoids and stacky groupoids. International Mathematics Research Notices , 2009(21):4087--4141, 2009

    2009