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A Morse gauge and gradient-flow analysis construct the continuum Yang–Mills measure on any compact surface as a random distributional connection, recovering Witten’s partition function and Migdal–Lévy holonomy laws without lattice limits.

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A Morse-gauge continuum construction of the 2D Yang–Mills measure on any compact surface yields Witten’s partition function and Migdal–Lévy holonomy laws for admissible loops.

T0 review reviewed 2026-07-31 challenge →

load-bearing objection Solid continuum Morse-gauge construction of YM₂; one clear sign typo in the Thm 2.3 disk law that does not match their own Z_YM, almost certainly fixable.

arxiv 2607.24640 v1 pith:DZXZ4DNE submitted 2026-07-27 math.PR math-phmath.APmath.DGmath.DSmath.MP

The Yang-Mills measure on surfaces via Morse theory

Reda Chhaibi , Nguyen Viet Dang , Yannick Guedes Bonthonneau , Gabriel Rivi\`ere , Tat Dat T\^o This is my paper
classification math.PR math-phmath.APmath.DGmath.DSmath.MP MSC 81T1358E1560H1537D2053C07
keywords Yang-Mills measureMorse gaugeMorse-Smale flowsrandom connectionsholonomy processwhite noisetwo-dimensional gauge theoryWitten partition function
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

The paper builds the two-dimensional Yang–Mills measure directly on the space of rough connections of a compact oriented Riemannian surface. The construction begins by putting connections into a new Morse gauge: a Morse–Smale gradient flow is used to rewrite any connection in terms of its curvature plus a finite sum of currents supported on unstable manifolds of the saddles. For a white-noise curvature the same flow solves a random cohomological equation, producing a free random connection. Holonomies of this free field along a large class of curves are defined by stochastic differential equations and shown to be reparametrized Brownian motions on the structure group. Conditioning the free measure so that the holonomy around a small loop about the Morse maximum equals the identity yields a non-Gaussian measure whose total mass is Witten’s formula and whose law on boundaries of admissible disks recovers the classical Migdal–Lévy heat-kernel expressions. The result gives a continuum, lattice-independent construction that works for every genus and every compact structure group.

Core claim

On any compact oriented Riemannian surface there exists a finitely additive functional MYM, upgraded to a genuine probability measure μ_YM on a weighted negative Sobolev space of g-valued 1-forms, obtained by conditioning a free Morse-gauge Gaussian-plus-Morse-complex measure so that holonomy near the Morse maximum is the identity; its partition function equals ∑_ρ e^{-c₂(ρ)υ(Σ)/2}(dim V_ρ)^{2-2g} and the law of Hol(∂D) for admissible disks D is the product of heat kernels predicted by Migdal, Witten and Lévy.

What carries the argument

The Morse gauge: any connection is rewritten, after parallel transport along a Morse–Smale gradient flow, as a sum of the resolvent of the Lie derivative applied to the curvature plus a linear combination of integration currents on the unstable manifolds of the saddles; the same resolvent applied to white noise defines the free random connection that is later conditioned.

Load-bearing premise

The argument needs a sharp exponential contraction estimate for the Morse–Smale gradient flow on weighted L^p spaces; if that spectral-gap bound fails at the stated exponents, the free random connection cannot be constructed in the claimed Sobolev spaces.

What would settle it

Compute the law of holonomy around a small admissible disk that contains no critical points and check whether it equals the Migdal heat-kernel formula p_{υ(D)}(g) μ_G(dg); any systematic deviation would falsify the conditioning construction.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • The continuum Yang–Mills measure on every closed surface is now available as a random distributional connection without passage through a lattice limit.
  • Partition functions and Wilson-loop expectations for small loops are given by the classical Migdal–Witten–Lévy formulae and are independent of the auxiliary Morse function.
  • The same Morse-gauge free field can be conditioned onto fixed Chern classes in the abelian case, producing measures supported on nontrivial line bundles.
  • Random holonomies are defined pathwise via SDEs for every admissible curve, opening the way to Driver–Sengupta-type formulae for arbitrary embedded graphs.

Where Pith is reading between the lines

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

  • Because the Morse gauge is global yet singular along unstable manifolds, it may supply a concrete analytic setting in which Gribov copies are absent for two-dimensional gauge theories.
  • The same weighted contraction estimates could be tried on Morse–Smale flows in dimension three to construct continuum Maxwell or abelian Chern–Simons measures.
  • Comparing the present measure with the Coulomb-gauge measure on the torus would give a direct continuum proof that the two gauge-fixed theories are equivalent.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Circularity Check

0 steps flagged

No significant circularity: continuum Morse-gauge construction derives Witten/Migdal outputs from white noise, Haar data, and Hol≈Id conditioning; self-citations supply analytic tools, not the target formulas.

full rationale

The derivation chain is constructive and self-contained. Random connections are built as A(ξ,b)=L_V^{-1}(ξι_V(υ))+∑ log(b_j)[W^u(a_j)] from an independent white-noise measure and Haar measure on G^{2g} (Def. 2.5, §6–9). The free boundary measure is the product law of these inputs. The Yang–Mills measure is obtained by conditioning Hol near the Morse maximum to Id_G (Thm 9.16, informal (9.11)); the partition function Z_YM=∑_ρ e^{-c_2(ρ)υ(Σ)/2}(dim V_ρ)^{2-2g} is the heat-kernel density p_Hol,0(Id) evaluated after that conditioning (Lemma 9.12, Thm 2.3), not a fitted parameter. Disk holonomy laws are computed from the same SDEs and Markov/abelianization properties (§8, §10), recovering external Migdal–Lévy formulas as outputs. Self-citations (Dang–Rivière Ruelle/Morse decay; Nohra–Dang lattice [24]; Jia–Stewart–Sverak weighted contraction ideas) supply or motivate analytic ingredients (esp. Thm 4.1, proved in §4) and a parallel lattice comparison; they do not define Z_YM or force the holonomy laws by renaming. No parameter is tuned to data; no uniqueness theorem is imported to forbid alternatives; no ansatz is smuggled as a prediction. A possible sign error in the dim V_ρ power in Thm 2.3 (correctness, not circularity) does not make the construction reduce to its inputs by definition.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 2 invented entities

The construction rests on standard compact Lie group harmonic analysis, white noise on surfaces, Itô/Stratonovich SDEs, Morse–Smale gradient dynamics, and classical YM₂ target formulas used only as checks. The main paper-specific inputs are the Morse gauge ansatz, the weighted contraction Theorem 4.1, and the Hol≈Id conditioning scheme.

axioms (6)
  • domain assumption Existence of a perfect Morse function f with Morse–Smale gradient for a locally flat metric near Crit(f), with distinct critical values and |Crit(f)|=2g+2.
    Fixed once and for all in §3; used to define unstable currents, filtration near a_{2g+2}, and the Morse complex correction.
  • standard math G compact connected linear Lie group; Ad-invariant inner product on g; heat kernel/Casimir spectral expansion on G.
    §8.1 and throughout holonomy laws; standard representation theory.
  • standard math g-valued white noise ξ exists in H^{-1-κ} and generates the stated Gaussian cylinder measures; independent restrictions to disjoint sets.
    §6.1; classical Gaussian space construction.
  • standard math Itô/Stratonovich and Marcus canonical SDEs for reparametrized Brownian motion on G (including finitely many deterministic jumps) have unique strong solutions in G.
    §8 and Appendix A; used to define Hol(γ) and Hol_r.
  • ad hoc to paper Weighted L^p contraction / exponential decay for Morse–Smale gradient pullbacks on Y_{p,γ} (Theorem 4.1), refining Dang–Rivière and Jia–Stewart–Sverak.
    Proved in §4; indispensable for rough RHS white noise and continuum gauge limit.
  • ad hoc to paper Yang–Mills measure is obtained by conditioning free boundary measure on Hol_0 = Id_G (regularized via Hol_r and heat-kernel abelianization).
    Definitional construction in §9; justified by matching known holonomy/partition formulas, not by an external uniqueness theorem alone.
invented entities (2)
  • Morse gauge (connections of the form ∑ b_j [W^u(a_j)] + β with ι_V(β)=0) independent evidence
    purpose: Global singular gauge fixing that slices A/G, separates curvature (white noise) from topological zero modes (Morse complex), and avoids continuous Gribov copies.
    Introduced in Thm 2.1 / §5; central organizing device of the paper.
  • Free boundary Yang–Mills measure μ_free_YM / P_free_YM on Ω × G^{2g} no independent evidence
    purpose: Gaussian random connection in Morse gauge plus independent Haar data on unstable curves, before topology-coupling conditioning.
    Def. 2.5 and §9.1; intermediate object used to define μ_YM by conditioning.

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Pith. "Pith review of The Yang-Mills measure on surfaces via Morse theory." pith.science (2026). https://pith.science/paper/DZXZ4DNE

@misc{pith2026260724640,
  author       = {Pith},
  title        = {Pith review of: The Yang-Mills measure on surfaces via Morse theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DZXZ4DNE}},
  note         = {Machine review of arXiv:2607.24640}
}
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read the original abstract

We introduce a Morse theoretical approach to the construction of the Yang--Mills measure on the space of connections of a compact Riemannian surface. This provides a direct continuous version of this measure which was previously obtained through lattice approximations by Chevyrev in the case of the flat torus and by one of the authors and Nohra for general compact Riemannian surfaces. The starting point is the new notion of a Morse gauge together with the resolution of random cohomological equations associated to Morse--Smale vector fields. This is achieved by improving exponential convergence to equilibrium results for Morse--Smale gradient flows that were obtained by two of the authors in the context of the study of Ruelle spectra and by Jia, Stewart and Sverak in the context of simplified models from fluid mechanics. Combining these random solutions with the data given by the Morse complex, we introduce a free Yang-Mills measure on space of connections and, using classical tools from stochastic differential equations, we show how to make sense of holonomies for random connections along a large class of curves. Finally, by setting a proper conditioning of this free measure through these random holonomies, we define the Yang--Mills measure and we compute its partition function together with the law of random holonomies with respect to this measure, recovering the formulas from the works of Migdal, Witten and L\'evy.

Figures

Figures reproduced from arXiv: 2607.24640 by Gabriel Rivi\`ere, Nguyen Viet Dang, Reda Chhaibi, Tat Dat T\^o, Yannick Guedes Bonthonneau.

Figure 1
Figure 1. Figure 1: An admissible disk. With these conventions, we can define a notion of random holonomy on the probability space Ω × G2g and a simplified statement reads as follows: Theorem 2.3 (Random holonomies). There exists a finitely additive functional MYM : B G ∞ → R+ such that the following holds 1. for all n ⩾ 1, MYM is a measure on (Ω × G2g , B G n ); 2. for every continuous and piecewise C 1 curve γ : [0, 1] → Σ … view at source ↗
Figure 2
Figure 2. Figure 2: Morse flow on Σ and Blow-up at max(f) 5.3 Slicing the space of connections by the Morse gauge In this paragraph, we examine in which precise sense the Morse gauge realizes a slicing of the space of connections, meaning that we prove the last item of Theorem 2.1. Assume that A2 = g−1dg + g−1A1g for some g ∈ C∞(Σ, G) and observe that L A2 V = g−1L A1 V g, where L A V = LV + A(V ) = ∇AιV + ιV ∇A (with ∇A = d … view at source ↗
Figure 3
Figure 3. Figure 3: An example of curve types. • If γ is of type I, we set [△(γ)] := [γ] + [Lγ(1)] − [Lγ(0)], which is a current of degree 1. • If γ is of type II± and if γ(1) ∈ Σ \ Wu (a1) (resp. γ(0)), we set [∆(γ)] := [γ] + [L ± γ(1)] − [Lγ(0)],  resp. := [γ] + [Lγ(1)] − [L ± γ(0)]  which is a current of degree 1. In both cases, if we denote by ▲(γ) = {φ −t f (γ(s)) : t ≥ 0, s ∈ (0, 1)}, we orient this triangular domain … view at source ↗
Figure 4
Figure 4. Figure 4: An example of rectangle of γ [PITH_FULL_IMAGE:figures/full_fig_p043_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: A triangle △(γ). Recall that type I (or type II) implies by definition that γ is elementary, V -oriented and primitive [PITH_FULL_IMAGE:figures/full_fig_p043_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Index for various Lipschitz curves. Hopf index of γ. See [39, Prop 4.16 p. 42] and we refer to [PITH_FULL_IMAGE:figures/full_fig_p046_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Tetris curves. where the constant in the remainder depends on ψ, γ, υ and f. Finally, if γ is V -oriented and has no type III elementary curve, then there exists a constant cγ > 0 such that ∀t ∈ [0, 1] \ {t1, . . . , tM}, [PITH_FULL_IMAGE:figures/full_fig_p047_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Example for Ribbon graph Lemma 9.3. With the above conventions, the permutation ϱf consists of one cycle of length 4g. Proof. This topological argument was indicated to us by Baptiste Chantraine and Stéphane Guillermou. In the following, we suppose that g ⩾ 1. We consider the Morse chart around a2g+2 and the small disk Dr0 := {x 2 1 +x 2 2 ≤ r 2 0 } centered at a2g+2. It follows from [64, Th. 3.1, Th. 3.2]… view at source ↗
Figure 9
Figure 9. Figure 9: Example for an order of 4g intersection points that the case r = 0 corresponds formally to the case where the curve is reduced to a point as in §7.4. Equivalently, if we denote by S the bordered surface obtained by blowing up the initial closed surface Σ at a2g+2, the surface S has one boundary component ∂S corresponding to the case r = 0 introduced in §7.4. In the following, we will sometimes make the sma… view at source ↗
Figure 10
Figure 10. Figure 10: Picture of the holonomy process Holr We give in [PITH_FULL_IMAGE:figures/full_fig_p068_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Example for γ, γ0 r and γer. Recall that, by construction, Hol(γ) can be viewed as the holonomy along the closed curve composed by γ([0, 1]) and by the flow lines joining from γ(0) and γ(1) to the critical point a1. Recall that these curves were denoted by Lγ(0) and Lγ(1). In the case of type I curves, the area delimited by these three curves is given by Aγ(1). The assumption that γ(t) ∈ Ws (a2g+2) makes … view at source ↗
Figure 12
Figure 12. Figure 12: Example for (H1). • γ 1 , γ2 : [0, 1] → Σ are C 1 curves such that γ j ([0, 1]) does not contain any critical point; • γ 1 and γ 2 are primitive and verify that t ∈ [0, 1] 7→ Aj (t) := Aγ j (t) is increasing for each j ∈ {1, 2}; • γ 1 (0) = γ 2 (0) and γ 1 (1) = γ 2 (1). Our goal is to compute the random holonomy along the piecewise C 1 curve γ = γ 1 ⋆ γ 2. (10.2) We also suppose that γ is small enough so… view at source ↗
Figure 13
Figure 13. Figure 13: Example for (H2) [PITH_FULL_IMAGE:figures/full_fig_p090_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Example for (H3). (H3) Both γ j are the concatenation of two type II curves γ j 1 ⋆ γ j 2 such that γ j 1 (1) = γ j 2 (0) belongs to some Wu (a(γ)) with a(γ) a critical point of index 1 (that does not depend on j). In that case, one has still γ 1 1 ≼ γ 2 1 and γ 1 2 ≼ γ 2 2 . Moreover, there exists some 1 ≤ k ≤ 4g such that a(tk) = a(γ) and γr(tk) lies on the forward orbit of γ j 1 (1) under the gradient … view at source ↗

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Reference graph

Works this paper leans on

81 extracted references · 4 linked inside Pith

  1. [1]

    and Gourevitch, D.Schwartz functions on Nash manifolds, International Mathematics Research Notices 2008: rnm155 (2008)

    Aizenbud, A. and Gourevitch, D.Schwartz functions on Nash manifolds, International Mathematics Research Notices 2008: rnm155 (2008)

    2008

  2. [2]

    C., Morando, P

    Albeverio, S., De Vecchi, F. C., Morando, P. and Ugolini, S.Weak symmetries of stochastic differential equations driven by semimartingales with jumps, Electron. J. Probab, 25(44), 1-34 (2020)

    2020

  3. [3]

    Applebaum, D.Lévy processes and stochastic calculus, Cambridge university press (2009)

    2009

  4. [4]

    L., and Fickler, S.I.Quantization of the Yang-Mills field, Physical Review 127.5: 1821 (1962)

    Arnowitt, R. L., and Fickler, S.I.Quantization of the Yang-Mills field, Physical Review 127.5: 1821 (1962)

    1962

  5. [5]

    Atiyah, M.F.andBott, R.The Yang-Mills equations over Riemann surfaces, Philosophical Transactionsof the Royal Societyof London.Series A, Mathematical and Physical Sciences 308.1505: 523-615 (1983)

    1983

  6. [6]

    Baladi, V.Positive transfer operators and decay of correlations, Singapore: World Scien- tific (2000)

    2000

  7. [7]

    A functional approach, Springer (2018)

    Baladi, V.Dynamical zeta functions and dynamical determinants for hyperbolic maps. A functional approach, Springer (2018)

    2018

  8. [8]

    Becherer, D., and Sun, Y.Rough backward SDEs with discontinuous Young drivers, arXiv preprint arXiv:2505.20437 (2025)

    Pith/arXiv arXiv 2025

  9. [9]

    Bonatti, C., Lorenzo J.D., and Viana, M.Dynamics beyond uniform hyperbolicity: A global geometric and probabilistic perspective, Berlin, Heidelberg: Springer Berlin Heidelberg (2005)

    2005

  10. [10]

    Canadian Mathematical Bulletin

    Bott R.On Some Recent Interactions Between Mathematics and Physics. Canadian Mathematical Bulletin. 1985;28(2):129-164

    1985

  11. [11]

    Springer (2008)

    Bowen, R.Equilibrium states and the ergodic theory of Anosov diffeomorphisms, 2nd revised ed. Springer (2008)

    2008

  12. [12]

    Math., Vol

    Bowen, R., and Ruelle, D.The ergodic theory of Axiom A flows, Inv. Math., Vol. 29, 181-202 (1975)

    1975

  13. [13]

    Bringmann, B., and Cao, S.A para-controlled approach to the stochastic Yang-Mills equation in two dimensions, arXiv preprint arXiv:2305.07197 (2023)

    Pith/arXiv arXiv 2023

  14. [14]

    and Cao, S.Global well-posedness of the stochastic Abelian-Higgs equa- tions in two dimensions, arXiv preprint arXiv:2403.16878 (2024)

    Bringmann, B. and Cao, S.Global well-posedness of the stochastic Abelian-Higgs equa- tions in two dimensions, arXiv preprint arXiv:2403.16878 (2024)

    Pith/arXiv arXiv 2024

  15. [15]

    and Chatterjee, S.The Yang-Mills heat flow with random distributional initial data, Communications in Partial Differential Equations 48.2: 209-251 (2023)

    Cao, S. and Chatterjee, S.The Yang-Mills heat flow with random distributional initial data, Communications in Partial Differential Equations 48.2: 209-251 (2023)

    2023

  16. [16]

    and Chatterjee, S.A state space for 3D Euclidean Yang–Mills theories, Commu- nications in Mathematical Physics 405.1: 3 (2024)

    Cao, S. and Chatterjee, S.A state space for 3D Euclidean Yang–Mills theories, Commu- nications in Mathematical Physics 405.1: 3 (2024). 108

    2024

  17. [17]

    Bruhat filtrations and Whit- taker vectors for real groups

    Casselman, William, Henryk Hecht, and Dragan Milicic. "Bruhat filtrations and Whit- taker vectors for real groups." Proceedings of symposia in pure mathematics. Vol. 68. Providence, RI; American Mathematical Society; 1998, 2000

    1998

  18. [18]

    and Lefeuvre, T.The holonomy inverse problem, Journal of the European Mathematical Society (EMS Publishing) 27.6 (2025)

    Cekic, M. and Lefeuvre, T.The holonomy inverse problem, Journal of the European Mathematical Society (EMS Publishing) 27.6 (2025)

    2025

  19. [19]

    Chevyrev, I.Yang–Mills Measure on the Two-Dimensional Torus as a Random Distri- bution, Commun. Math. Phys. 372, 1027–1058 (2019)

    2019

  20. [20]

    and Shen, H.Invariant measure and universality of the 2D Yang-Mills Langevin dynamic, arXiv preprint arXiv:2302.12160 (2023)

    Chevyrev, I. and Shen, H.Invariant measure and universality of the 2D Yang-Mills Langevin dynamic, arXiv preprint arXiv:2302.12160 (2023)

    Pith/arXiv arXiv 2023

  21. [21]

    Chandra, A., Chevyrev, I., Hairer, M., and Shen, H.Langevin dynamic for the 2D Yang–Mills measure, Publications mathématiques de l’IHÉS, 136(1), 1-147 (2022)

    2022

  22. [22]

    Chandra, A., Chevyrev, I., Hairer, M., and Shen, H.Stochastic quantisation of Yang–Mills–Higgs in 3d, Inventiones mathematicae, 237(2), 541-696 (2024)

    2024

  23. [23]

    and Herscovich, E.Renormalization of quantum field theory on Riemannian manifolds, Reviews in Mathematical Physics 31.06: 1950017 (2019)

    Dang, N.V. and Herscovich, E.Renormalization of quantum field theory on Riemannian manifolds, Reviews in Mathematical Physics 31.06: 1950017 (2019)

    2019

  24. [24]

    and Nohra, E.The Yang–Mills measure on compact surfaces as a universal scaling limit of lattice gauge models, arXiv preprint arXiv:2602.08591 (2026)

    Dang, N.V. and Nohra, E.The Yang–Mills measure on compact surfaces as a universal scaling limit of lattice gauge models, arXiv preprint arXiv:2602.08591 (2026)

    arXiv 2026

  25. [25]

    and Nohra, E.Semiclassical analysis for Yang–Mills random connections on compact surfaces, in preparation

    Dang, N.V. and Nohra, E.Semiclassical analysis for Yang–Mills random connections on compact surfaces, in preparation

  26. [26]

    and Rivière, G.Equidistribution of the conormal cycle of random nodal sets, Journal of the European Mathematical Society 20.12: 3017-3071 (2018)

    Dang, N.V. and Rivière, G.Equidistribution of the conormal cycle of random nodal sets, Journal of the European Mathematical Society 20.12: 3017-3071 (2018)

    2018

  27. [27]

    and Rivière, G.Spectral analysis of Morse-Smale gradient flows, Ann

    Dang, N.V. and Rivière, G.Spectral analysis of Morse-Smale gradient flows, Ann. Sci. ENS, Vol. 52, 1403-1458 (2019)

    2019

  28. [28]

    and Rivière, G.Pollicott–Ruelle spectrum and Witten Laplacians, Journal of the European Mathematical Society 23.6: 1797-1857 (2021)

    Dang, N.V. and Rivière, G.Pollicott–Ruelle spectrum and Witten Laplacians, Journal of the European Mathematical Society 23.6: 1797-1857 (2021)

    2021

  29. [29]

    and Rivière, G.Poincaré series and linking of Legendrian knots, Duke Math- ematical Journal 173.1: 1-74 (2024)

    Dang, N.V. and Rivière, G.Poincaré series and linking of Legendrian knots, Duke Math- ematical Journal 173.1: 1-74 (2024)

    2024

  30. [30]

    uniroma2.it/~liverani/Lavori/33CBM16-eBook.pdf(2021)

    Demers, M.F., Kiamari, N., and Liverani, C.Transfer operators in Hyperbolic Dy- namics An introduction, Publisher: IMPAISBN: 978-65-89124-26-9,https://www.mat. uniroma2.it/~liverani/Lavori/33CBM16-eBook.pdf(2021)

    2021

  31. [31]

    Donaldson, S.K., and Kronheimer, P.B.The geometry of four-manifolds, Oxford univer- sity press (1997)

    1997

  32. [32]

    Driver, B.K.YM2: continuum expectations, lattice convergence, and lassos, Comm. Math. Phys., 123(4):575–616 (1989)

    1989

  33. [33]

    K.A functional integral approaches to the Makeenko-Migdal equations

    Driver, B. K.A functional integral approaches to the Makeenko-Migdal equations. Comm. Math. Phys.,370(1):49–116 (2019). 109

    2019

  34. [34]

    J.The Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac Operator, Modern Birkhäuser Classics, Birkhäuser Boston, MA, 2197-1803, VIII, 247 (2011)

    Duistermaat, J. J.The Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac Operator, Modern Birkhäuser Classics, Birkhäuser Boston, MA, 2197-1803, VIII, 247 (2011)

    2011

  35. [35]

    Eguchi, T., Gilkey P. B. and Hanson, A. J.Gravitation, gauge theories and differential geometry, Physics Reports, Volume 66, Issue 6, Pages 213-393, (1980)

    1980

  36. [36]

    Evans, L.C.An introduction to stochastic differential equations, AMS (2013)

    2013

  37. [37]

    Fine, D.S.Quantum Yang-Mills on a Riemann surface, Communications in mathematical physics 140: 321-338 (1991)

    1991

  38. [38]

    and Le Jan, Y.Hyperbolic dynamics and Brownian motion, an introduction, Oxford Science Publication (2012)

    Franchi, J. and Le Jan, Y.Hyperbolic dynamics and Brownian motion, an introduction, Oxford Science Publication (2012)

    2012

  39. [39]

    Fulton, W.Algebraic Topology: A First Course, Springer (1995)

    1995

  40. [40]

    Gel’fand, I. M. and Vilenkin, N. Y.Generalized functions: Applications of harmonic analysis, Vol. 4. Academic press (2014)

    2014

  41. [41]

    Griffiths, P., andHarris, J.Principles of algebraic geometry.JohnWileyandSons, (1994)

    1994

  42. [42]

    Gross, L., King, C., and Sengupta, A.Two dimensional Yang-Mills theory via stochastic differential equations, Annals of Physics 194.1: 65-112 (1989)

    1989

  43. [43]

    and Rhodes, R.Compactified imaginary Liouville theory, Commun

    Guillarmou, C., Kupiainen, A. and Rhodes, R.Compactified imaginary Liouville theory, Commun. Am. Math. Soc. 5, 12, 571-694 (2025)

    2025

  44. [44]

    Hadfield, C., Kandel, S., andSchiavina, M.Ruelle zeta function from field theory, Annales Henri Poincaré. Vol. 21. No. 12 (2020)

    2020

  45. [45]

    Harvey, F.R., and Lawson, H.B.Morse theory and Stokes theorem, Surveys in differential geometry: 259-311 (2000)

    2000

  46. [46]

    153(1), 1-25 (2001)

    Harvey, F.R., and Lawson, H.B.Finite Volume Flows and Morse Theory, Annals of Math. 153(1), 1-25 (2001)

    2001

  47. [47]

    213, 272 p

    Harvey, F.R., and Lawson, H.B.A theory of characteristic currents associated with a singular connection, Astérisque, no. 213, 272 p. (1993)

    1993

  48. [48]

    Hatcher, A.Algebraic topology, Cambridge University Press (2002)

    2002

  49. [49]

    99-118 (1989)

    Hitchin, N.J.Gauge theory on Riemann surfaces, Lectures on Riemann surfaces, World Scientific, p. 99-118 (1989)

    1989

  50. [50]

    Jia, H., Stewart, S., and Sverak, V..On the De Gregorio Modification of the Con- stantin–Lax–Majda Model, ARMA 231 1269-1304 (2019)

    2019

  51. [51]

    Knapp, A.W.Lie groups beyond an introduction(2023)

    2023

  52. [52]

    Koralov, L., and Sinai, Y.G.Theory of probability and random processes, Springer (2007)

    2007

  53. [53]

    Austriaca 14, 149 (1961) 110

    Kummer, W., Acta Phys. Austriaca 14, 149 (1961) 110

    1961

  54. [54]

    Labourie, F.Lectures on Representations of Surface Groups, EMS publishing house, pp.145, Zurich Lectures in Advanced Mathematics (2013)

    2013

  55. [55]

    Laudenbach, F.On the Thom-Smale complex, Astérisque 205: 219-233 (1992)

    1992

  56. [56]

    With a contributed chapter by Yann Chaubet.Société Mathématique de France (2025)

    Lefeuvre, T.Microlocal analysis in hyperbolic dynamics and geometry. With a contributed chapter by Yann Chaubet.Société Mathématique de France (2025)

    2025

  57. [57]

    Lévy, T.Yang-Mills measure on compact surfaces, American Mathematical Soc., 2003

    2003

  58. [58]

    Lévy, T.Two-dimensional Markovian holonomy fields, Astérisque 329 (2010)

    2010

  59. [59]

    Cham: Springer International Publishing (2020)

    Lévy, T.Two-dimensional quantum Yang–Mills theory and the Makeenko–Migdal equa- tions, Frontiers in Analysis and Probability: In the Spirit of the Strasbourg-Zürich Meet- ings. Cham: Springer International Publishing (2020)

    2020

  60. [60]

    Lévy, T.The Master Field on the Plane, Astérisque 388 (2017)

    2017

  61. [61]

    Lévy, T.Discrete and continuous Yang-Mills measure for non-trivial bundles over com- pact surfaces, Probability theory and related fields 136.2: 171-202 (2006)

    2006

  62. [62]

    Malgrange, B.Ideals of differentiable functions, Oxford University Press (1966)

    1966

  63. [63]

    Migdal, A.A.Phase transitions in gauge and spin-lattice systems, Zh. Eksp. Teor. Fiz 69: 1457 (1975)

    1975

  64. [64]

    Milnor, J.Morse theory, Princeton University Press (1963)

    1963

  65. [65]

    Morgan, F.Geometric Measure Theory: A Beginner’s Guide, Academic Press, 272 pages (2009)

    2009

  66. [66]

    Polyakov, A.M.Gauge fields and strings, 2018

    2018

  67. [67]

    Berlin, Heidelberg: Springer Berlin Heidelberg, 249-361 (2012)

    Protter, P.E.Stochastic differential equations, Stochastic integration and differential equations. Berlin, Heidelberg: Springer Berlin Heidelberg, 249-361 (2012)

    2012

  68. [68]

    Reed, M., and Simon, B.Modern mathematical physics 1: Functional analysis, Academic press (1981)

    1981

  69. [69]

    293, Springer Science & Business Media, (2013)

    Revuz, D., and Yor, M.Continuous martingales and Brownian motion, Vol. 293, Springer Science & Business Media, (2013)

    2013

  70. [70]

    Y.Loop averages and partition functions inU(N)gauge theory on two- dimensional manifolds, Modern Phys

    Rusakov, B. Y.Loop averages and partition functions inU(N)gauge theory on two- dimensional manifolds, Modern Phys. Lett. A 5, no. 9, 693–703 (1990)

    1990

  71. [71]

    and Stucker, T.Perturbative BF theory in axial, Anosov gauge, Annales Henri Poincaré

    Schiavina, M. and Stucker, T.Perturbative BF theory in axial, Anosov gauge, Annales Henri Poincaré. Vol. 25. No. 10. (2024)

    2024

  72. [72]

    Schwartz, L.Théorie des distributions, Hermann (1966)

    1966

  73. [73]

    Sengupta, A.Gauge theory on compact surfaces, Vol. 600. American Mathematical Soc. (1997). 111

    1997

  74. [74]

    Sepanski, M.R.Compact Lie groups, New York, NY: Springer New York, (2007)

    2007

  75. [75]

    Shen, H., Smith, S.A., and Zhu, R.A new derivation of the finiteNmaster loop equation for lattice Yang–Mills, Electronic Journal of Probability 29:1-8 (2024)

    2024

  76. [76]

    Shen, H., Zhu, R., and Zhu, X.A stochastic analysis approach to lattice Yang–Mills at strong coupling, Communications in Mathematical Physics, 400(2), 805-851 (2023)

    2023

  77. [77]

    Smale, S.Differentiable dynamical systems, Bull. Am. Math. Soc. 73, No. 6, 747-817 (1967)

    1967

  78. [78]

    151, (2016)

    Viana, M., and Oliveira, K.Foundations of ergodic theoryCambridge University Press, Vol. 151, (2016)

    2016

  79. [79]

    145, (2014)

    Viana, M.Lectures on Lyapunov exponents, Cambridge University Press Vol. 145, (2014)

    2014

  80. [80]

    Witten, E.On quantum gauge theories in two dimensions, Communications in Mathe- matical Physics 141.1: 153-209 (1991)

    1991

Showing first 80 references.

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