Yang-Mills mass gap at large-N, non-commutative YM theory, topological quantum field theory and hyperfiniteness

We review a number of old and new concepts in quantum gauge theories, some of which are well established but not widely appreciated, some are most recent. Such concepts involve non-commutative gauge theories and their relation to the large-N limit, loop equations and the change to the anti-selfdual variables also known as Nicolai map, topological field theory (TFT) and its relation to localization and Morse-Smale-Floer homology, with an emphasis both on the mathematical aspects and the physical meaning. These concepts, assembled in a new way, enter a line of attack to the problem of the mass gap in large-N SU(N) YM, that is reviewed as well. In the large-N limit of pure SU(N) YM the ambient algebra of Wilson loops is known to be a type II_1 non-hyperfinite factor. Nevertheless, for the mass gap problem at the leading 1/N order, only the subalgebra of local gauge-invariant single-trace operators matters. The connected two-point correlators in this subalgebra must be an infinite sum of propagators of free massive fields, a vast simplification. It is an open problem, determined by the grow of the degeneracy of the spectrum, whether the aforementioned local subalgebra is in fact hyperfinite. For the mass-gap problem, in the search of a hyperfinite subalgebra containing the scalar sector of large-N YM, a major role is played by the existence of a TFT underlying the large-N limit of YM, with twisted boundary conditions on a torus or, what is the same by Morita duality, on a non-commutative torus.Comment: 19 pages, latex; the paper, originally a byproduct of the workshop: Mathematical Foundations of Quantum Field Theory, Jan 16-20 (2012), has been expanded and rewritten as a short review in order to include most recent developments. To appear in IJMP

Stability and Hermitian-Einstein metrics for vector bundles on framed manifolds

We adapt the notions of stability of holomorphic vector bundles in the sense of Mumford-Takemoto and Hermitian-Einstein metrics in holomorphic vector bundles for canonically polarized framed manifolds, i.e. compact complex manifolds X together with a smooth divisor D such that K_X \otimes [D] is ample. It turns out that the degree of a torsion-free coherent sheaf on X with respect to the polarization K_X \otimes [D] coincides with the degree with respect to the complete K\"ahler-Einstein metric g_{X \setminus D} on X \setminus D. For stable holomorphic vector bundles, we prove the existence of a Hermitian-Einstein metric with respect to g_{X \setminus D} and also the uniqueness in an adapted sense.Comment: 21 pages, International Journal of Mathematics (to appear

Remarks on evolution of space-times in 3+1 and 4+1 dimensions

A large class of vacuum space-times is constructed in dimension 4+1 from hyperboloidal initial data sets which are not small perturbations of empty space data. These space-times are future geodesically complete, smooth up to their future null infinity, and extend as vacuum space-times through their Cauchy horizon. Dimensional reduction gives non-vacuum space-times with the same properties in 3+1 dimensions.Comment: 10pp, exposition improved; final versio

A Kummer construction for gravitational instantons

We give a simple and uniform construction of essentially all known deformation classes of gravitational instantons with ALF, ALG or ALH asymptotics and nonzero injectivity radius. We also construct new ALH Ricci flat metrics asymptotic to the product of a real line with a flat 3-manifold.Comment: The construction of locally hyperkahler ALH spaces is corrected and complete

Chiral formulation for hyperkaehler sigma-models on cotangent bundles of symmetric spaces

Starting with the projective-superspace off-shell formulation for four-dimensional N = 2 supersymmetric sigma-models on cotangent bundles of arbitrary Hermitian symmetric spaces, their on-shell description in terms of N = 1 chiral superfields is developed. In particular, we derive a universal representation for the hyperkaehler potential in terms of the curvature of the symmetric base space. Within the tangent-bundle formulation for such sigma-models, completed recently in arXiv:0709.2633 and realized in terms of N = 1 chiral and complex linear superfields, we give a new universal formula for the superspace Lagrangian. A closed form expression is also derived for the Kaehler potential of an arbitrary Hermitian symmetric space in Kaehler normal coordinates.Comment: 11 pages, LaTeX, no figue

Exact beta function from the holographic loop equation of large-N QCD_4

We construct and study a previously defined quantum holographic effective action whose critical equation implies the holographic loop equation of large-N QCD_4 for planar self-avoiding loops in a certain regularization scheme. We extract from the effective action the exact beta function in the given scheme. For the Wilsonean coupling constant the beta function is exacly one loop and the first coefficient agrees with its value in perturbation theory. For the canonical coupling constant the exact beta function has a NSVZ form and the first two coefficients agree with their value in perturbation theory.Comment: 42 pages, latex. The exponent of the Vandermonde determinant in the quantum effective action has been changed, because it has been employed a holomorphic rather than a hermitean resolution of identity in the functional integral. Beta function unchanged. New explanations and references added, typos correcte

Fourier-Laplace transform of a variation of polarized complex Hodge structure, II

We show that the limit, by rescaling, of the `new supersymmetric index' attached to the Fourier-Laplace transform of a polarized variation of Hodge structure on a punctured affine line is equal to the spectral polynomial attached to the same object. We also extend the definition by Deligne of a Hodge filtration on the de Rham cohomology of a exponentially twisted polarized variation of complex Hodge structure and prove a E_1 degeneration property for it.Comment: 51 pages, revised version, to appear in the proceedings volume of the conference `` New developments in Algebraic Geometry, Integrable Systems and Mirror symmetry", RIMS, Kyoto, Jan. 7-11, 200

The Cauchy problems for Einstein metrics and parallel spinors

We show that in the analytic category, given a Riemannian metric $g$ on a hypersurface $M\subset \Z$ and a symmetric tensor $W$ on $M$, the metric $g$ can be locally extended to a Riemannian Einstein metric on $Z$ with second fundamental form $W$, provided that $g$ and $W$ satisfy the constraints on $M$ imposed by the contracted Codazzi equations. We use this fact to study the Cauchy problem for metrics with parallel spinors in the real analytic category and give an affirmative answer to a question raised in B\"ar, Gauduchon, Moroianu (2005). We also answer negatively the corresponding questions in the smooth category.Comment: 28 pages; final versio