Uniqueness of Gauge Covariant Renormalisation of Stochastic 3D Yang–Mills–Higgs

We introduce class A spacetimes, i.e. compact vicious spacetimes (M, g) such that the Abelian cover \({(\overline{M}, \overline{g})}\) is globally hyperbolic. We study the main properties of class A spacetimes using methods similar to those introduced in Sullivan (Invent Math 36:225–255, 1976) and Burago (Adv Sov Math 9:205–210, 1992). As a consequence we are able to characterize manifolds admitting class A metrics completely as mapping tori. Further we show that the notion of class A spacetime is equivalent to that of SCTP (spacially compact time-periodic) spacetimes as introduced in Galloway (Comm Math Phys 96:423–429, 1984). The set of class A spacetimes is shown to be open in the C 0-topology on the set of Lorentzian metrics. As an application we prove a coarse Lipschitz property for the time separation of the Abelian cover. This coarse Lipschitz property is an essential part in the study of Aubry-Mather theory in Lorentzian geometry.

Let \(G\) be a semi-simple simply connected group over \(\mathbb {C}\). Following Gerasimov et al. (Comm Math Phys 294:97–119, 2010) we use the \(q\)-Toda integrable system obtained by quantum group version of the Kostant–Whittaker reduction (cf. Etingof in Am Math Soc Trans Ser 2:9–25, 1999, Sevostyanov in Commun Math Phys 204:1–16, 1999) to define the notion of \(q\)-Whittaker functions \(\varPsi _{\check{\lambda }}(q,z)\). This is a family of invariant polynomials on the maximal torus \(T\subset G\) (here \(z\in T\)) depending on a dominant weight \(\check{\lambda }\) of \(G\) whose coefficients are rational functions in a variable \(q\in \mathbb {C}^*\). For a conjecturally the same (but a priori different) definition of the \(q\)-Toda system these functions were studied by Ion (Duke Math J 116:1–16, 2003) and by Cherednik (Int Math Res Notices 20:3793–3842, 2009) [we shall denote the \(q\)-Whittaker functions from Cherednik (Int Math Res Notices 20:3793–3842, 2009) by \(\varPsi '_{\check{\lambda }}(q,z)\)]. For \(G=SL(N)\) these functions were extensively studied in Gerasimov et al. (Comm Math Phys 294:97–119, 2010; Comm Math Phys 294:121–143, 2010; Lett Math Phys 97:1–24, 2011). We show that when \(G\) is simply laced, the function \(\hat{\varPsi }_{\check{\lambda }}(q,z)=\varPsi _{\check{\lambda }}(q,z)\cdot {\prod \nolimits _{i\in I}\prod \nolimits _{r=1}^{\langle \alpha _i,\check{\uplambda }\rangle }(1-q^r)}\) (here \(I\) denotes the set of vertices of the Dynkin diagram of \(G\)) is equal to the character of a certain finite-dimensional \(G[[{\mathsf {t}}]]\rtimes \mathbb {C}^*\)-module \(D(\check{\lambda })\) (the Demazure module). When \(G\) is not simply laced a twisted version of the above statement holds. This result is known for \(\varPsi _{\check{\lambda }}\) replaced by \(\varPsi '_{\check{\lambda }}\) (cf. Sanderson in J Algebraic Combin 11:269–275, 2000 and Ion in Duke Math J 116:1–16, 2003); however our proofs are algebro-geometric [and rely on our previous work (Braverman, Finkelberg in Semi-infinite Schubert varieties and quantum \(K\)-theory of flag manifolds, arXiv/1111.2266, 2011)] and thus they are completely different from Sanderson (J Algebraic Combin 11:269–275, 2000) and Ion (Duke Math J 116:1–16, 2003) [in particular, we give an apparently new algebro-geometric interpretation of the modules \(D(\check{\lambda })]\).

A bstract We apply the positivity bootstrap approach to SU(3) lattice Yang-Mills (YM) theory, extending previous studies of large N and SU(2) theories by incorporating multiple-trace Wilson loop operators. By utilizing Hermitian and reflection positivity conditions, alongside Schwinger-Dyson (SD) loop equations, we compute rigorous bounds for the expectation values of plaquette Wilson loops in 2D, 3D, and 4D YM theories. Our results exhibit clear convergence and are consistent with known analytic or numerical results. To enhance the approach, we introduce a novel twist-reflection positivity condition, which we prove to be exact in 2D YM theory. Additionally, we propose a dimensional-reduction truncation, where Wilson loop operators are effectively restricted to a lower-dimensional subplane, significantly simplifying computations. SD equations for double-trace Wilson loops are also derived in detail. Our findings suggest that the positivity bootstrap method is broadly applicable to higher-rank gauge theories beyond single-trace cases, providing a solid foundation for further non-perturbative investigations of gauge theories using positivity-based methods.

Spin, statistics, and geometry of random walks: T. Jaroszewicz and P. S. Kurzepa. Department of Physics, University of California Los Angeles, Los Angeles, California 90024

Quark confinement: Dual superconductor picture based on a non-Abelian Stokes theorem and reformulations of Yang–Mills theory

We study joint eigenvector distributions for large symmetric matrices in the presence of weak noise. Our main result asserts that every submatrix in the orthogonal matrix of eigenvectors converges to a multidimensional Gaussian distribution. The proof involves analyzing the stochastic eigenstate equation (SEE) (Bourgade and Yau in Comm Math Phys, 2013) which describes the Lie group valued flow of eigenvectors induced by matrix valued Brownian motion. We consider the associated colored eigenvector moment flow defining an SDE on a particle configuration space. This flow extends the eigenvector moment flow first introduced in Bourgade and Yau (Comm Math Phys, 2013) to the multicolor setting. However, it is no longer driven by an underlying Markov process on configuration space due to the lack of positivity in the semigroup kernel. Nevertheless, we prove the dynamics admit sufficient averaged decay and contractive properties. This allows us to establish optimal time of relaxation to equilibrium for the colored eigenvector moment flow and prove joint asymptotic normality for eigenvectors. Applications in random matrix theory include the explicit computations of joint eigenvector distributions for general Wigner type matrices and sparse graph models when corresponding eigenvalues lie in the bulk of the spectrum, as well as joint eigenvector distributions for Lévy matrices when the eigenvectors correspond to small energy levels.

Hedgehogs in Wilson loops and phase transition in SU(2) Yang–Mills theory

The Dyson equation proposed for planar temporal Wilson loops in the context\nof supersymmetric gauge theories is critically analysed thereby exhibiting its\ningredients and approximations involved. We reveal its limitations and identify\nits range of applicability in non-supersymmetric gauge theories. In particular,\nwe show that this equation is applicable only to strongly asymmetric planar\nWilson loops (consisting of a long and a short pair of loop segments) and as a\nconsequence the Wilsonian potential can be extracted only up to intermediate\ndistances. By this equation the Wilson loop is exclusively determined by the\ngluon propagator. We solve the Dyson equation in Coulomb gauge for the temporal\nWilson loop with the instantaneous part of the gluon propagator and for the\nspatial Wilson loop with the static gluon propagator obtained in the\nHamiltonian approach to continuum Yang-Mills theory and on the lattice. In both\ncases we find a linearly rising color potential.\n

In this study, we compute the correlation functions of Wilson(-'t Hooft) loops with chiral primary operators in the supersymmetric Yang-Mills theory with gauge symmetry, which has a holographic dual description of the Type IIB superstring theory on the background. Specifically, we compute the coefficients of the chiral primary operators in the operator product expansion of Wilson loops in the fundamental representation, Wilson-'t Hooft loops in the symmetric representation, Wilson loops in the anti-fundamental representation, and Wilson loops in the spinor representation. We also compare these results to those of the super Yang-Mills theory.

In this paper we consider a Wilson loop in a 2+1 dimensional pure Yang-Mills theory with an SU(2) gauge group. The initial goal is to test a conjecture of A. M. Polyakov's which proposes that if one considers the field-strength, Fa„, and the gauge field, Aa, as independent, random variables, then a sum over surfaces spanning the Wilson loop will re-introduce the Bianchi Identity. We do this by introducing an additional functional integral over a sigma model variable which unravels the path-ordering of the loop variables. Then, via a non-Abelian Stokes' theorem, we express the Wilson loop as a surface integral with separate functional integrals over both Fa and Aa. At the semi-classical level, characterized by a large spin parameter, we find that the conjecture holds true - the Bianchi Identity arises as a natural constraint. Secondly, we find that this reformulation of the Wilson loop naturally allows for an arbitrary distribution of monopoles. We treat both the cases of a single monopole and a monopole gas. In the latter case case we demonstrate the confinement of quarks for states of half-odd-integer spin.

We derive a new version of the non-Abelian Stokes theorem for the Wilson loop in the SU(N) case by making use of the coherent state representation on the coset space SU(N)/U(1)N−1 = FN−1, the flag space. We consider the SU(N) Yang-Mills theory in the maximal Abelian gauge in which SU(N) is broken down to U(1)N−1. First, we show that the Abelian dominance in the string tension follows from this theorem and the Abelian-projected effective gauge theory that was derived by one of the authors. Next (but independently), combining the non-Abelian Stokes theorem with a novel reformulation of the Yang-Mills theory recently proposed by one of the authors, we proceed to derive the area law of the Wilson loop in four-dimensional SU(N) Yang-Mills theory in the maximal Abelian gauge. Owing to dimensional reduction, the planar Wilson loop at least for the fundamental representation in four-dimensional SU(N) Yang-Mills theory can be estimated by the diagonal (Abelian) Wilson loop defined in the two-dimensional CPN−1 model. This derivation shows that the fundamental quarks are confined by a single species of magnetic monopole. The origin of the area law is related to the geometric phase of the Wilczek-Zee holonomy for U(N−1). The calculations are performed using the instanton calculus (in the dilute instanton-gas approximation) and using the large N expansion (in the leading order).

We consider double-winding, triple-winding and multiple-winding Wilson loops in the $SU(N)$ Yang-Mills gauge theory. We examine how the area law falloff of the vacuum expectation value of a multiple-winding Wilson loop depends on the number of color $N$. In sharp contrast to the difference-of-areas law recently found for a double-winding $SU(2)$ Wilson loop average, we show irrespective of the spacetime dimensionality that a double-winding $SU(3)$ Wilson loop follows a novel area law which is neither difference-of-areas nor sum-of-areas law for the area law falloff and that the difference-of-areas law is excluded and the sum-of-areas law is allowed for $SU(N)$ ($N \ge 4$), provided that the string tension obeys the Casimir scaling for the higher representations. Moreover, we extend these results to arbitrary multi-winding Wilson loops. Finally, we argue that the area law follows a novel law, which is neither sum-of-areas nor difference-of-areas law when $N\ge 3$. In fact, such a behavior is exactly derived in the $SU(N)$ Yang-Mills theory in the two-dimensional spacetime.

In this paper we present recent progress on the Dirichlet forms associated with stochastic quantization problems obtained in Rockner et al. (J Funct Anal, 272(10):4263–4303, 2017, [23]), Rockner et al. (Commun Math Phys, 352(3):1061–1090, 2017, [24]), Zhu and Zhu (Dirichlet form associated with the \(\varPhi _3^4\) model, 2017, [27]). In the two dimensional case we have obtained the equivalence of the two notions of solutions, the restricted Markov uniqueness and the uniqueness of martingale problem. In the three dimensional case we construct the Dirichlet form associated with the dynamical \(\varPhi ^4_3\) model obtained in Catellier and Chouk (Paracontrolled distributions and the 3-dimensional stochastic quantization equation, [6]), Hairer (Invent Math, 198:269–504, 2014, [14]), Mourrat and Weber (Global well-posedness of the dynamic \(\varPhi ^4_3\) model on the torus, [20].

There are many Markov chains on infinite dimensional spaces whose one-step transition kernels are mutually singular when starting from different initial conditions. We give results which prove unique ergodicity under minimal assumptions on one hand and the existence of a spectral gap under conditions reminiscent of Harris’ theorem. The first uses the existence of couplings which draw the solutions together as time goes to infinity. Such “asymptotic couplings” were central to (Mattingly and Sinai in Comm Math Phys 219(3):523–565, 2001; Mattingly in Comm Math Phys 230(3):461–462, 2002; Hairer in Prob Theory Relat Field 124:345–380, 2002; Bakhtin and Mattingly in Commun Contemp Math 7:553–582, 2005) on which this work builds. As in Bakhtin and Mattingly (2005) the emphasis here is on stochastic differential delay equations. Harris’ celebrated theorem states that if a Markov chain admits a Lyapunov function whose level sets are “small” (in the sense that transition probabilities are uniformly bounded from below), then it admits a unique invariant measure and transition probabilities converge towards it at exponential speed. This convergence takes place in a total variation norm, weighted by the Lyapunov function. A second aim of this article is to replace the notion of a “small set” by the much weaker notion of a “d-small set,” which takes the topology of the underlying space into account via a distance-like function d. With this notion at hand, we prove an analogue to Harris’ theorem, where the convergence takes place in a Wasserstein-like distance weighted again by the Lyapunov function. This abstract result is then applied to the framework of stochastic delay equations. In this framework, the usual theory of Harris chains does not apply, since there are natural examples for which there exist no small sets (except for sets consisting of only one point). This gives a solution to the long-standing open problem of finding natural conditions under which a stochastic delay equation admits at most one invariant measure and transition probabilities converge to it.

In this thesis we apply the technique of localization to three-dimensional N=2 superconformal field theories. We consider both theories which are exactly superconformal, and those which are believed to flow to nontrivial superconformal fixed points, for which we consider implicitly these fixed points. We find that in such theories, the partition function and certain supersymmetric observables, such as Wilson loops, can be computed exactly by a matrix model. This matrix model consists of an integral over g, the Lie algebra of the gauge group of the theory, of a certain product of 1-loop factors and classical contributions. One can also consider a space of supersymmetric deformations of the partition function corresponding to the set of abelian global symmetries. In the second part of the thesis we apply these results to test dualities. We start with the case of ABJM theory, which is dual to M-theory on an asymptotically AdS_4xS^7 background. We extract strong coupling results in the field theory, which can be compared to semiclassical, weak coupling results in the gravity theory, and a nontrivial agreement is found. We also consider several classes of dualities between two three-dimensional field theories, namely, 3D mirror symmetry, Aharony duality, and Giveon-Kutasov duality. Here the dualities are typically between the IR limits of two Yang-Mills theories, which are strongly coupled in three dimensions since Yang-Mills theory is asymptotically free here. Thus the comparison is again very nontrivial, and relies on the exactness of the localization computation. We also compare the deformed partition functions, which tests the mapping of global symmetries of the dual theories. Finally, we discuss some recent progress in the understanding of general three-dimensional theories in the form of the F-theorem, a conjectured analogy to the a-theorem in four dimensions and c-theorem in two dimensions, which is closely related to the localization computation.