Langevin dynamic for the 2D Yang–Mills measure

Ajay Chandra; Ilya Chevyrev; Martin Hairer; Hao Shen

doi:10.1007/s10240-022-00132-0

Article

Langevin dynamic for the 2D Yang–Mills measure

Ajay Chandra ¹; Ilya Chevyrev ¹; Martin Hairer ¹; Hao Shen ¹

¹

Publications Mathématiques de l'IHÉS, Volume 136 (2022), pp. 1-147

Abstract

We define a natural state space and Markov process associated to the stochastic Yang–Mills heat flow in two dimensions.

To accomplish this we first introduce a space of distributional connections for which holonomies along sufficiently regular curves (Wilson loop observables) and the action of an associated group of gauge transformations are both well-defined and satisfy good continuity properties. The desired state space is obtained as the corresponding space of orbits under this group action and is shown to be a Polish space when equipped with a natural Hausdorff metric.

To construct the Markov process we show that the stochastic Yang–Mills heat flow takes values in our space of connections and use the “DeTurck trick” of introducing a time dependent gauge transformation to show invariance, in law, of the solution under gauge transformations.

Our main tool for solving for the Yang–Mills heat flow is the theory of regularity structures and along the way we also develop a “basis-free” framework for applying the theory of regularity structures in the context of vector-valued noise – this provides a conceptual framework for interpreting several previous constructions and we expect this framework to be of independent interest.

Received: 2020-06-10
Accepted: 2022-05-04
Online First: 2022-06-07
Published online: 2022-12-07

Zbl

DOI: 10.1007/s10240-022-00132-0

Author's affiliations:

Ajay Chandra ¹; Ilya Chevyrev ¹; Martin Hairer ¹; Hao Shen ¹

¹

@article{PMIHES_2022__136__1_0,
     author = {Ajay Chandra and Ilya Chevyrev and Martin Hairer and Hao Shen},
     title = {Langevin dynamic for the {2D} {Yang{\textendash}Mills} measure},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {1--147},
     year = {2022},
     publisher = {Springer International Publishing},
     address = {Cham},
     volume = {136},
     doi = {10.1007/s10240-022-00132-0},
     zbl = {1518.70029},
     language = {en},
     url = {https://pmihes.centre-mersenne.org/articles/10.1007/s10240-022-00132-0/}
}

TY  - JOUR
AU  - Ajay Chandra
AU  - Ilya Chevyrev
AU  - Martin Hairer
AU  - Hao Shen
TI  - Langevin dynamic for the 2D Yang–Mills measure
JO  - Publications Mathématiques de l'IHÉS
PY  - 2022
SP  - 1
EP  - 147
VL  - 136
PB  - Springer International Publishing
PP  - Cham
UR  - https://pmihes.centre-mersenne.org/articles/10.1007/s10240-022-00132-0/
DO  - 10.1007/s10240-022-00132-0
LA  - en
ID  - PMIHES_2022__136__1_0
ER  -

%0 Journal Article
%A Ajay Chandra
%A Ilya Chevyrev
%A Martin Hairer
%A Hao Shen
%T Langevin dynamic for the 2D Yang–Mills measure
%J Publications Mathématiques de l'IHÉS
%D 2022
%P 1-147
%V 136
%I Springer International Publishing
%C Cham
%U https://pmihes.centre-mersenne.org/articles/10.1007/s10240-022-00132-0/
%R 10.1007/s10240-022-00132-0
%G en
%F PMIHES_2022__136__1_0

Ajay Chandra; Ilya Chevyrev; Martin Hairer; Hao Shen. Langevin dynamic for the 2D Yang–Mills measure. Publications Mathématiques de l'IHÉS, Volume 136 (2022), pp. 1-147. doi: 10.1007/s10240-022-00132-0

References
Cited by

[AB83.] M. F. Atiyah; R. Bott The Yang-Mills equations over Riemann surfaces, Philos. Trans. R. Soc. Lond. Ser. A, Volume 308 (1983), pp. 523-615 | MR | DOI | Zbl

[AK20.] S. Albeverio; S. Kusuoka The invariant measure and the flow associated to the $Φ_{3}^{4}$ -quantum field model, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), Volume 20 (2020), pp. 1359-1427 | MR | DOI | Zbl

[BCCH21.] Y. Bruned; A. Chandra; I. Chevyrev; M. Hairer Renormalising SPDEs in regularity structures, J. Eur. Math. Soc., Volume 23 (2021), pp. 869-947 | MR | DOI | Zbl

[BCFP19.] Y. Bruned; I. Chevyrev; P. K. Friz; R. Preiß A rough path perspective on renormalization, J. Funct. Anal., Volume 277 (2019) | MR | DOI | Zbl

[BG20.] N. Barashkov; M. Gubinelli A variational method for $Φ_{3}^{4}$ , Duke Math. J., Volume 169 (2020), pp. 3339-3415 | MR | DOI | Zbl

[BHST87.] Z. Bern; M. B. Halpern; L. Sadun; C. Taubes Continuum regularization of quantum field theory. II. Gauge theory, Nucl. Phys. B, Volume 284 (1987), pp. 35-91 | MR | DOI

[BHZ19.] Y. Bruned; M. Hairer; L. Zambotti Algebraic renormalisation of regularity structures, Invent. Math., Volume 215 (2019), pp. 1039-1156 | MR | DOI | Zbl

[Bog07.] V. I. Bogachev Measure Theory. Vols. I, II, Springer, Berlin, 2007 (Vol. I: xviii+500 pp., Vol. II: xiv+575.) | DOI | Zbl

[Bou94.] J. Bourgain Periodic nonlinear Schrödinger equation and invariant measures, Commun. Math. Phys., Volume 166 (1994), pp. 1-26 | DOI | Zbl

[CCHS22.] A. Chandra, I. Chevyrev, M. Hairer and H. Shen, Stochastic quantisation of Yang-Mills-Higgs in 3D, ArXiv e-prints (2022). | arXiv

[CG13.] N. Charalambous; L. Gross The Yang-Mills heat semigroup on three-manifolds with boundary, Commun. Math. Phys., Volume 317 (2013), pp. 727-785 | MR | DOI | Zbl

[CH16.] A. Chandra and M. Hairer, An analytic BPHZ theorem for regularity structures, ArXiv e-prints (2016). | arXiv

[Cha19.] S. Chatterjee Yang-Mills for probabilists, Probability and Analysis in Interacting Physical Systems, 283, Springer, Cham, 2019, pp. 1-16 | MR | Zbl | DOI

[Che19.] I. Chevyrev Yang-Mills measure on the two-dimensional torus as a random distribution, Commun. Math. Phys., Volume 372 (2019), pp. 1027-1058 | MR | DOI | Zbl

[CW16.] T. Cass and M. P. Weidner, Tree algebras over topological vector spaces in rough path theory, ArXiv e-prints (2016). | arXiv

[DeT83.] D. M. DeTurck Deforming metrics in the direction of their Ricci tensors, J. Differ. Geom., Volume 18 (1983), pp. 157-162 | MR | DOI | Zbl

[DH87.] P. H. Damgaard; H. Hüffel Stochastic quantization, Phys. Rep., Volume 152 (1987), pp. 227-398 | MR | DOI

[DK90.] S. K. Donaldson; P. B. Kronheimer The Geometry of Four-Manifolds, The Clarendon Press, Oxford University Press, New York, 1990 (x+440. Oxford Science Publications) | Zbl | MR | DOI

[Don85.] S. K. Donaldson Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. Lond. Math. Soc. (3), Volume 50 (1985), pp. 1-26 | MR | DOI | Zbl

[Dri89.] B. K. Driver YM₂: continuum expectations, lattice convergence, and lassos, Commun. Math. Phys., Volume 123 (1989), pp. 575-616 | DOI | Zbl | MR

[FH20.] P. K. Friz; M. Hairer A Course on Rough Paths, Springer, Cham, 2020 (xvi+346. With an introduction to regularity structures) | DOI | Zbl | MR

[FV10.] P. K. Friz; N. B. Victoir Multidimensional Stochastic Processes as Rough Paths, 120, Cambridge University Press, Cambridge, 2010 (xiv+656. Theory and applications) | DOI | Zbl | MR

[GH19a.] M. Gerencsér; M. Hairer Singular SPDEs in domains with boundaries, Probab. Theory Relat. Fields, Volume 173 (2019), pp. 697-758 | MR | DOI | Zbl

[GH19b.] M. Gerencsér; M. Hairer A solution theory for quasilinear singular SPDEs, Commun. Pure Appl. Math., Volume 72 (2019), pp. 1983-2005 | MR | DOI | Zbl

[GH21.] M. Gubinelli; M. Hofmanová A PDE construction of the Euclidean $φ_{3}^{4}$ quantum field theory, Commun. Math. Phys., Volume 384 (2021), pp. 1-75 | DOI | Zbl | MR

[GHM22.] A. Gerasimovics, M. Hairer and K. Matetski, Directed mean curvature flow in noisy environment, ArXiv e-prints (2022). | arXiv | MR

[GIP15.] M. Gubinelli; P. Imkeller; N. Perkowski Paracontrolled distributions and singular PDEs, Forum Math. Pi, Volume 3 (2015) | MR | DOI | Zbl

[Gro85.] L. Gross A Poincaré lemma for connection forms, J. Funct. Anal., Volume 63 (1985), pp. 1-46 | MR | DOI | Zbl

[Gro17.] L. Gross, Stability of the Yang-Mills heat equation for finite action, ArXiv e-prints (2017). | arXiv

[Gro22.] L. Gross The Yang-Mills heat equation with finite action in three dimensions, Mem. Am. Math. Soc., Volume 275 (2022), p. v+111 | MR | DOI | Zbl

[Hai14.] M. Hairer A theory of regularity structures, Invent. Math., Volume 198 (2014), pp. 269-504 | MR | DOI | Zbl

[HM18a.] M. Hairer; K. Matetski Discretisations of rough stochastic PDEs, Ann. Probab., Volume 46 (2018), pp. 1651-1709 | MR | DOI | Zbl

[HM18b.] M. Hairer; J. Mattingly The strong Feller property for singular stochastic PDEs, Ann. Inst. Henri Poincaré Probab. Stat., Volume 54 (2018), pp. 1314-1340 | MR | DOI | Zbl

[HP21.] M. Hairer; E. Pardoux Fluctuations around a homogenised semilinear random PDE, Arch. Ration. Mech. Anal., Volume 239 (2021), pp. 151-217 | MR | DOI | Zbl

[HS22.] M. Hairer; P. Schönbauer The support of singular stochastic partial differential equations, Forum Math. Pi, Volume 10 (2022), p. 127 | MR | DOI | Zbl

[JW06.] A. Jaffe; E. Witten Quantum Yang-Mills theory, The Millennium Prize Problems, Clay Math. Inst., Cambridge, 2006, pp. 129-152 | MR | Zbl

[Kec95.] A. S. Kechris Classical Descriptive Set Theory, 156, Springer, New York, 1995 (xviii+402) | Zbl | MR | DOI

[Kna02.] A. W. Knapp Lie Groups Beyond an Introduction, 140, Birkhäuser Boston, Inc., Boston, 2002 (xviii+812) | DOI | Zbl | MR

[Lév03.] T. Lévy Yang-Mills measure on compact surfaces, Mem. Am. Math. Soc., Volume 166 (2003), p. xiv+122 | MR | DOI | Zbl

[Lév06.] T. Lévy Discrete and continuous Yang-Mills measure for non-trivial bundles over compact surfaces, Probab. Theory Relat. Fields, Volume 136 (2006), pp. 171-202 | MR | DOI | Zbl

[LN06.] T. Lévy; J. R. Norris Large deviations for the Yang-Mills measure on a compact surface, Commun. Math. Phys., Volume 261 (2006), pp. 405-450 | MR | DOI | Zbl

[Lyo94.] T. Lyons Differential equations driven by rough signals. I. An extension of an inequality of L. C. Young, Math. Res. Lett., Volume 1 (1994), pp. 451-464 | MR | DOI | Zbl

[Man89.] M. Mandelkern Metrization of the one-point compactification, Proc. Am. Math. Soc., Volume 107 (1989), pp. 1111-1115 | MR | Zbl | DOI

[Mei75.] G. H. Meisters Polygons have ears, Am. Math. Mon., Volume 82 (1975), pp. 648-651 | MR | DOI | Zbl

[MV81.] P. K. Mitter; C.-M. Viallet On the bundle of connections and the gauge orbit manifold in Yang-Mills theory, Commun. Math. Phys., Volume 79 (1981), pp. 457-472 | MR | DOI | Zbl

[MW17a.] J.-C. Mourrat; H. Weber The dynamic $Φ_{3}^{4}$ model comes down from infinity, Commun. Math. Phys., Volume 356 (2017), pp. 673-753 | DOI | Zbl | MR

[MW17b.] J.-C. Mourrat; H. Weber Global well-posedness of the dynamic $Φ^{4}$ model in the plane, Ann. Probab., Volume 45 (2017), pp. 2398-2476 | MR | DOI | Zbl

[MW20.] A. Moinat; H. Weber Space-time localisation for the dynamic $Φ_{3}^{4}$ model, Commun. Pure Appl. Math., Volume 73 (2020), pp. 2519-2555 | DOI | Zbl | MR

[PW81.] G. Parisi; Y. S. Wu Perturbation theory without gauge fixing, Sci. Sin., Volume 24 (1981), pp. 483-496 | MR | DOI | Zbl

[Rad92.] J. Rade On the Yang-Mills heat equation in two and three dimensions, J. Reine Angew. Math., Volume 431 (1992), pp. 123-163 | MR | DOI | Zbl

[Sch18.] P. Schönbauer, Malliavin calculus and density for singular stochastic partial differential equations, ArXiv e-prints (2018). | arXiv | MR

[Sen92.] A. Sengupta The Yang-Mills measure for $S^{2}$ , J. Funct. Anal., Volume 108 (1992), pp. 231-273 | MR | DOI | Zbl

[Sen97.] A. Sengupta Gauge theory on compact surfaces, Mem. Am. Math. Soc., Volume 126 (1997), p. viii+85 | MR | DOI | Zbl

[She21.] H. Shen Stochastic quantization of an Abelian gauge theory, Commun. Math. Phys., Volume 384 (2021), pp. 1445-1512 | MR | Zbl | DOI

[ST12.] S. Stolz; P. Teichner Traces in monoidal categories, Trans. Am. Math. Soc., Volume 364 (2012), pp. 4425-4464 | MR | DOI | Zbl

[Whi34.] H. Whitney Analytic extensions of differentiable functions defined in closed sets, Trans. Am. Math. Soc., Volume 36 (1934), pp. 63-89 | MR | Zbl | DOI

[You36.] L. C. Young An inequality of the Hölder type, connected with Stieltjes integration, Acta Math., Volume 67 (1936), pp. 251-282 | MR | DOI | Zbl

[Zwa81.] D. Zwanziger Covariant quantization of gauge fields without Gribov ambiguity, Nucl. Phys. B, Volume 192 (1981), pp. 259-269 | MR | DOI

Cited by Sources: