BRST invariant PV regularization of SUSY Yang–Mills and SUGRA

Pauli?CVillars regularization of Yang?CMills theories and of supergravity theories is outlined, with an emphasis on BRST invariance. Applications to phenomenology and the anomaly structure of supergravity are discussed.     

Non-perturbative QEG corrections to the Yang–Mills beta function

We discuss the non-perturbative renormalization group evolution of the gauge coupling constant by using a truncated form of the functional flow equation for the effective average action of the Yang–Mills-gravity system. Our result is consistent with the conjecture that quantum Einstein gravity (QEG) is asymptotically safe and has a vanishing gauge coupling constant at the non-trivial fixed point.     

A BRST gauge-fixing procedure for Yang–Mills theory on sphere

A gauge-fixing procedure for the Yang–Mills theory on an n  -dimensional sphere (or a hypersphere) is discussed in a systematic manner. We claim that Adler's gauge-fixing condition used in massless Euclidean QED on a hypersphere is not conventional because of the presence of an extra free index, and hence is unfavorable for the gauge-fixing procedure based on the BRST invariance principle (or simply BRST gauge-fixing procedure). Choosing a suitable gauge condition, which is proved to be equivalent to a generalization of Adler's condition, we apply the BRST gauge-fixing procedure to the Yang–Mills theory on a hypersphere to obtain consistent results. Field equations for the Yang–Mills field and associated fields are derived in manifestly O(n+1)$\mathrm{O}(n+1)$ covariant or invariant forms. In the large radius limit, these equations reproduce the corresponding field equations defined on the n-dimensional flat space.     

Holography and drag force in thermal plasma of non-commutative Yang–Mills theories in diverse dimensions

We use holography and a string probe approach to compute the drag force on a quark moving in a thermal plasma of non-commutative Yang–Mills (NCYM) theories in various dimensions. The gravity background in these cases are described by a particular decoupling limit of non-extremal (D(p−2),Dp)$(\mathrm{D}(p-2),\mathrm{D}p)$-brane bound state system. We show how the drag force on an external quark moving in the dual NCYM theories gets corrected due to non-commutativity and as a result the effective viscosity of the plasma gets reduced. We have obtained the drag force for both small and large non-commutativity. This was known earlier for (3+1)$(3+1)$-dimensional NCYM theory, however, we find that the corrections for the general case typically depend on the dimensionality of the NCYM theories, indicating that the structure of the drag force is non-universal.     

A note on the UV behaviour of maximally supersymmetric Yang–Mills theories

The question of whether BPS invariants are protected in maximally supersymmetric Yang–Mills theories is investigated from the point of view of algebraic renormalisation theory. The protected invariants are those whose cohomology type differs from that of the action. It is confirmed that one-half BPS invariants (F⁴$F^4$) are indeed protected while the double-trace one-quarter BPS invariant (d²F⁴$d^2{F}^4$) is not protected at two loops in D=7$D=7$, but is protected at three loops in D=6$D=6$ in agreement with recent calculations. Non-BPS invariants, i.e. full superspace integrals, are also shown to be unprotected.     

Yang–Mills Algebra

Some unexpected properties of the cubic algebra generated by the covariant derivatives of a generic Yang–Mills connection over the (s+1)-dimensional pseudo Euclidean space are pointed out. This algebra is Koszul of global dimension 3 and Gorenstein but except for s=1 (i.e. in the two-dimensional case) where it is the universal enveloping algebra of the Heisenberg Lie algebra and is a cubic Artin–Schelter regular algebra, it fails to be regular in that it has exponential growth. We give an explicit formula for the Poincaré series of this algebra and for the dimension in degree n of the graded Lie algebra of which is the universal enveloping algebra. In the four-dimensional (i.e. s=3) Euclidean case, a quotient of this algebra is the quadratic algebra generated by the covariant derivatives of a generic (anti) self-dual connection. This latter algebra is Koszul of global dimension 2 but is not Gorenstein and has exponential growth. It is the universal enveloping algebra of the graded Lie algebra which is the semi-direct product of the free Lie algebra with three generators of degree one by a derivation of degree one.     

Cohomological Yang–Mills theories on Kähler 3-folds

We study topological gauge theories with N_c=(2,0) supersymmetry based on stable bundles on general Kähler 3-folds. In order to have a theory that is well defined and well behaved, we consider a model based on an extension of the usual holomorphic bundle by including a holomorphic 3-form. The correlation functions of the model describe complex 3-dimensional generalizations of Donaldson–Witten type invariants. We show that the path integral can be written as a sum of contributions from stable bundles and a complex 3-dimensional version of Seiberg–Witten monopoles. We study certain deformations of the theory, which allow us to consider the situation of reducible connections. We shortly discuss situations of reduced holonomy. After dimensional reduction to a Kähler 2-fold, the theory reduces to Vafa–Witten theory. On a Calabi–Yau 3-fold, the supersymmetry is enhanced to N_c=(2,2). This model may be used to describe classical limits of certain compactifications of (matrix) string theory.     

Modified Faddeev–Jackiw quantization of massive non-Abelian Yang–Mills fields and Lagrange multiplier fields

Massive Yang–Mills fields and Lagrange multiplier fields are quantized by the modified Faddeev–Jackiw quantization method, and the method's comparisons with Dirac method and the usual Faddeev–Jackiw method are also given. We show that this method not only is equivalent to Dirac method, but also remains all the virtues of the usual Faddeev–Jackiw method. Moreover, the modified Faddeev–Jackiw quantization method is simpler than the usual one when obtaining new secondary constraints. Therefore, the modified Faddeev–Jackiw method is more economical and effective than Dirac method and the usual Faddeev–Jackiw method. Meanwhile, we find the new meanings of the Lagrange multipliers, and discover that the Lagrange multipliers and the zeroth components of gauge field are just a pair of canonical field variables except a constant factor in this system.     

Loop Expansion in Yang–Mills Thermodynamics

We argue that a self-consistent spatial coarse graining, which involves interacting (anti)calorons of unit topological charge modulus, implies that real-time loop expansions of thermodynamical quantities in the deconfining phase of SU(2) and SU(3) Yang–Mills thermodynamics are, modulo one-particle irreducible resummations, determined by a finite number of connected bubble diagrams.     

Inclusion of Yang–Mills fields in string corrected supergravity

We consistently incorporate Yang–Mills matter fields into string corrected (deformed) D=10$D=10$, N=1$N=1$ supergravity. We solve the Bianchi identities within the framework of the modified beta function favored constraints to second order in the string slope parameter γ also including the Yang–Mills fields. In the torsion, curvature and H   sectors we find that a consistent solution is readily obtained with a Yang–Mills modified supercurrent A_abc$A_{abc}$. We find a solution in the F sector following our previously developed method.     

Confinement in a three-dimensional Yang–Mills theory

We show that, starting from known exact classical solutions of the Yang–Mills theory in three dimensions, the string tension is obtained and the potential is consistent with a marginally confining theory. The potential we obtain agrees fairly well with preceding findings in the literature but here we derive it analytically from the theory without further assumptions. The string tension is in strict agreement with lattice results and the well-known theoretical result by Karabali–Kim–Nair analysis. Classical solutions depend on a dimensionless numerical factor arising from integration. This factor enters into the determination of the spectrum and has been arbitrarily introduced in some theoretical models. We derive it directly from the solutions of the theory and is now fully justified. The agreement obtained with the lattice results for the ground state of the theory is well below 1% at any value of the degree of the group.