Geometric Obstructions to the Proof of the Yang–Mills Mass Gap Localization No-Go, Spectral Flow, and Three Independent Analytic Barriers
Authors/Creators
Description
Abstract
We study structural limitations of a class of geometric approaches to the Yang–Mills mass gap problem, based on the Fundamental Modular Region (FMR) with the Gribov–Zwanziger measure dµGZ = det(M[A]) e-SYM DA. This work constitutes a rigorous theory of limitations for the class under consideration, and does not claim to prove the mass gap.
Rigorous results. (1) No-Go Theorem 2: under polynomial vanishing λ1(M[A]) ∼ sk, all exponential localization mechanisms based on a scalar function h(- log λ1) with h = O(√u) are forbidden. (2) Dominance
Theorem 3: when multiple mode classes are present, convergence of
the functional D is governed by the class with the smallest exponent βmin. (3) Explicit computation in the multiscaling regime (Section 4):
with logarithmic corrections, the convergence criterion depends on the correction exponent and the spectral density.
Conditional results. Under six hypotheses (Section 3): classification of spectral flow regimes and the universal survival condition 2β+p > 1; the mass gap problem reduces to three independent barriers.
Files
Yang_Mills_EN.pdf
Files
(128.2 kB)
| Name | Size | Download all |
|---|---|---|
|
md5:eeacbbbc91a22ff0e0d1d5050e64b902
|
128.2 kB | Preview Download |
Additional details
References
- [1] A. Jaffe and E. Witten, Yang–Mills Existence and Mass Gap, Clay Mathematics Institute, 2000. [2] G. Dell'Antonio and D. Zwanziger, Every gauge orbit passes inside the Gribov horizon, Commun. Math. Phys. 138 (1991), 291–299. [3] K. Osterwalder and R. Schrader, Axioms for Euclidean Green's functions, Commun. Math. Phys. 31 (1973), 83–112. [4] V. N. Gribov, Quantization of non-Abelian gauge theories, Nucl. Phys. B 139 (1978), 1–19. [5] I. M. Singer, Some remarks on the Gribov ambiguity, Commun. Math. Phys. 60 (1978), 7–12. [6] O. Babelon and C.-M. Viallet, The Riemannian geometry of the configuration space of gauge theories, Commun. Math. Phys. 81 (1981), 515–525. [7] D. Zwanziger, Local and renormalizable action from the Gribov horizon, Nucl. Phys. B 323 (1989), 513–544. [8] H. Neuberger, Non-perturbative BRS invariance, Phys. Lett. B 183 (1987), 337–340. [9] D. Dudal, J. A. Gracey, S. P. Sorella, N. Vandersickel, and H. Verschelde, A refinement of the Gribov–Zwanziger approach, Phys. Rev. D 78 (2008), 065047. [10] D. Bakry and M. Émery, Diffusions hypercontractives, Séminaire de Probabilités XIX, Springer, 1985, pp. 177–206. [11] J. Cheeger and T. Colding, On the structure of spaces with Ricci curvature bounded below, J. Differential Geom. 46 (1997), 406–480. [12] J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. 169 (2009), 903–991. [13] M. Röckner and F.-Y. Wang, Weak Poincaré inequalities and L2- convergence rates, J. Funct. Anal. 185 (2001), 564–603. [14] S. Agmon, Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations, Princeton University Press, 1982. [15] L. Gross, Convergence of U(1)3 lattice gauge theory, Commun. Math. Phys. 92 (1983), 137–162. [16] B. Driver and B. Hall, Yang–Mills theory and the Segal–Bargmann transform, Commun. Math. Phys. 201 (1999), 249–290. [17] M. Erbar, K. Kuwada, and K.-T. Sturm, On the equivalence of the entropic curvature-dimension condition and Bochner's inequality, Invent. Math. 201 (2015), 993–1071. [18] L. Ambrosio, N. Gigli, and G. Savaré, Metric measure spaces with Riemannian Ricci curvature bounded from below, Duke Math. J. 163 (2014), 1405–1490. [19] P. Malliavin, Stochastic Analysis, Springer, 1997. [20] T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer, 1976. [21] M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69. [22] K.-T. Sturm, On the geometry of metric measure spaces, Acta Math. 196 (2006), 65–177. [23] G. 't Hooft, Topology of the gauge condition and new confinement phases, Nucl. Phys. B 190 (1981), 455–478. [24] G. Parisi and Y.-S. Wu, Perturbation theory without gauge fixing, Sci. Sinica 24 (1981), 483–496. [25] M. Hairer, A theory of regularity structures, Invent. Math. 198 (2014), 269–504. [26] A. Connes, Noncommutative Geometry, Academic Press, 1994.