Stability and isolation phenomena for Yang-Mills fields

In this article a series of results concerning Yang-Mills fields over the euclidean sphere and other locally homogeneous spaces are proved using differential geometric methods. One of our main results is to prove that any weakly stable Yang-Mills field overS ⁴ with groupG=SU₂, SU₃ orU ₂ is either self-dual or anti-self-dual. The analogous statement for SO₄-bundles is also proved. The other main series of results concerns gap-phenomena for Yang-Mills fields. As a consequence of the non-linearity of the Yang-Mills equations, we can give explicitC ⁰-neighbourhoods of the minimal Yang-Mills fields which contain no other Yang-Mills fields. In this part of the study the nature of the groupG does not matter, neither is the dimension of the base manifold constrained to be four.Laboratoire Associé au C.N.R.S. No. 169Research partially supported by Volkswagen Grant and NSF Grant MCS-77-23579     

Link Homologies and the Refined Topological Vertex

We establish a direct map between refined topological vertex and sl(N) homological invariants of the of Hopf link, which include Khovanov-Rozansky homology as a special case. This relation provides an exact answer for homological invariants of the Hopf link, whose components are colored by arbitrary representations of sl(N). At present, the mathematical formulation of such homological invariants is available only for the fundamental representation (the Khovanov-Rozansky theory) and the relation with the refined topological vertex should be useful for categorizing quantum group invariants associated with other representations (R ₁, R ₂). Our result is a first direct verification of a series of conjectures which identifies link homologies with the Hilbert space of BPS states in the presence of branes, where the physical interpretation of gradings is in terms of charges of the branes ending on Lagrangian branes.     

Interaction of moving branes with background massless and tachyon fields in superstring theory

Using the boundary state formalism, we study a moving Dp-brane in a partially compact space-time in the presence of background fields: the Kalb-Ramond field B _μν, a U(1) gauge field A _α, and the tachyon field. The boundary state enables us to obtain the interaction amplitude of two branes with the above back-ground fields. The branes are parallel or perpendicular to each other. Because of the presence of background fields, compactification of some space-time directions, motion of the branes, and the arbitrariness of the dimensions of the branes, the system is rather general. Due to the tachyon fields and velocities of the branes, the behavior of the interaction amplitude reveals obvious differences from the conventional behavior.     

The universe dynamics from topological considerations

We explore the possibility that the dynamics of the universe can be reproduced choosing appropriately the initial global topology of the Universe. In this work we start with two concentric spherical three-dimensional branes S ³, with radius a ₁ < a ₂ immersed in a five-dimensional space-time. The novel feature of this model is that in the interior brane there exist only spin-zero fundamental fields (scalar fields), while in the exterior one there exist only spin-one fundamental interactions. As usual, the bulk of the universe is dominated by gravitational interactions. In this model, like in the Ekpyrotic one, the Big Bang is consequence of the collision of the branes and causes the existence of the particles predicted by the standard model in the exterior brane (our universe). The scalar fields on the interior brane interact with the spin-one fields on the exterior one only through gravitation, they induce the effect of Scalar Field Dark Matter with an ultra-light mass on the exterior one. We discuss two different regimes where the energy density and the brane tension are compared, with the aim to obtain the observed dynamics of the universe after the collision of the branes.     

Elliptic-type solutions to the scale invariant Yang-Mills and sigma-model hierarchies

We outline the construction of non-self-dual elliptic solutions by relating the spherically symmetric subsystem of the (scale invariant) Yang-Mills and sigmamodel hierarchies to the hierarchies of ⁴ and Sine-Gordon models in one dimension respectively. The construction is carried out explicitly for the usual Yang-Mills model on ₄, and the first two sigma-models on ₂ and ₄. The solution to the first member of the Yang-Mills hierarchy is related to elliptic solutions found previously.     

Ward identity for gauge field in stochastic quantization with nonlocal form factors

In the framework of the stochastic quantization method with nonlocal form factors, two-, three- and four-point correlation functions for the Yang-Mills field and function are calculated. A Schwinger-Dyson renormalization program is formulated for regularized QCD₄. It is shown that the gauge fixing due to Zwanziger does not break gauge invariance and that the Ward identity is also fulfilled.     

Kähler metrics and gauge kinetic functions for intersecting D6‐branes on toroidal orbifolds – The complete perturbative story

We systematically derive the perturbatively exact holomorphic gauge kinetic function, the open string Kähler metrics and closed string Kähler potential on intersecting D6‐branes by matching open string one‐loop computations of gauge thresholds with field theoretical gauge couplings in 𝒩 = 1 supergravity. We consider all cases of bulk, fractional and rigid D6‐branes on T⁶/Ω ℛ and the orbifolds T⁶/(ℤ_N × Ω ℛ) and T⁶/(ℤ₂ × ℤ_2M × Ω ℛ) without and with discrete torsion, which differ in the number of bulk complex structures and in the bulk Kähler potential. Our analysis includes all supersymmetric configurations of vanishing and non‐vanishing angles among D6‐branes and O6‐planes, and all possible Wilson line and displacement moduli are taken into account. The shape of the Kähler moduli turns out to be orbifold independent but angle dependent, whereas the holomorphic gauge kinetic functions obtain three different kinds of one‐loop corrections: a Kähler moduli dependent one for some vanishing angle independently of the orbifold background, another one depending on complex structure moduli only for fractional and rigid D6‐branes, and finally a constant term from intersections with O6‐planes. These results are of essential importance for the construction of the related effective field theory of phenomenologically appealing D‐brane models. As first examples, we compute the complete perturbative gauge kinetic functions and Kähler metrics for some T⁶/ℤ₂ × ℤ₂ examples with rigid D‐branes of [1]. As a second class of examples, the Kähler metrics and gauge kinetic functions for the fractional QCD and leptonic D6‐brane stacks of the Standard Model on T⁶/ℤ₆^′T⁶/ℤ₆^′ from [2] are given.     

Identities for noncovariant-gauge integrals in then μ * -formalism

Four one-loop identities are derived for axial-type gauges in the unifyingn _μ ^* -formalism, and their usefulness for multi-loop calculations in Yang-Mills theory discussed. Each identity may be decomposed into two distinct identities, one relating the pole terms, the other the finite parts, of various integrals.     

D3 instantons in Calabi–Yau orientifolds with(out) fluxes

We investigate the instanton effects due to D3 branes wrapping a four-cycle in a Calabi–Yau orientifold with D7 branes. We study the condition for the nonzero superpotentials from the D3 instantons. To this aim we work out the zero mode structures of D3 branes wrapping a four-cycle both in the presence of the fluxes and in the absence of the fluxes. In the presence of the fluxes, the condition for the nonzero superpotential could be different from that without the fluxes. We explicitly work out a simple example of the orientifold of K3×T ²/Z ₂ with a suitable flux to show such behavior. The effects of the D3–D7 sectors are interesting and give further constraints for the nonzero superpotential. In a special configuration where D3 branes and D7 branes wrap the same four-cycle, multi-instanton calculus of D3 branes could be reduced to that of a suitable field theory. The structure of D5 instantons in Type I theory is briefly discussed.     

A Hierarchical Model of Quantum Anharmonic Oscillators: Critical Point Convergence

We study the two-dimensional gauge theory of the symmetric group S_n describing the statistics of branched n-coverings of Riemann surfaces. We consider the theory defined on the disc and on the sphere in the large-n limit. A non trivial phase structure emerges, with various phases corresponding to different connectivity properties of the covering surface. We show that any gauge theory on a two-dimensional surface of genus zero is equivalent to a random walk on the gauge group manifold: in the case of S_n, one of the phase transitions we find can be interpreted as a cutoff phenomenon in the corresponding random walk. A connection with the theory of phase transitions in random graphs is also pointed out. Finally we discuss how our results may be related to the known phase transitions in Yang-Mills theory. We discover that a cutoff transition occurs also in two dimensional Yang-Mills theory on a sphere, in a large N limit where the coupling constant is scaled with N with an extra logN compared to the standard t Hooft scaling.     

Minkowski space Yang-Mills fields from Euclidean self-dual fields

It is examined, if it is possible, to obtain solutions of the SU(2) Yang-Mills field equations in Minkowski space from Euclidean self-dual Yang-Mills fields by method proposed by Bernreuther. It is shown that the conditions, which are imposed on the Euclidean self-dual fields by this method, make every Euclidean self-dual field and the corresponding Minkowski space field obtained from it, equivalent to a pure gauge field, F _ab 0.     

Gravity and gauge: A new perspective

Realization of the Poincaré groupP ₁₀ as a subgroup ofGL(5,R) that maps a 4-dimensional affine set into itself has been shown to lead to a direct Yang-Mills gauging process. This paper discusses the differences between direct gauge theory forP ₁₀ and previously published works. These differences are fundamental, both physically and mathematically, and lead to marked departures from previous concepts and interpretations. The translation subgroup is correctly gauged; the metric structure and metric compatibility are derived from the gauging process rather than assumed; spin structures are automatically incorporated in a consistent manner; the local holonomy group is shown to be the component of the Lorentz group connected to the identity; the geometric analog of Yang-Mills minimal coupling precludes dependence of the free gauge field Lagranian on torsion; and the theory reduces exactly to general relativity when the momentumenergy complex is symmetric and all matter fields are spin-free. Gravitational effects on neutral test particles are shown to arise from the compensating 1-forms for local action of Lorentz boosts. The compensating 1-forms for local action of the translation subgroup may be interpreted as space-time dislocations, while the compensating 1-forms for the rotation subgroup can be viewed as space-time disclinations. Unfortunately, there are no clear physical meanings that can be ascribed to space-time dislocations or disclinations.     

Part 1 : Topological aspects of Yang-Mills fields in curved spaces. (Exact solutions)

To explore in its full richness the topological possibilities of gauge fields one should allow for simultaneous presence of gravitational and Yang-Mills ones. Thus if the integral topological indices of the Yang-Mills field for a flat Euclidean base space is associated with the structure of the vacuum, one may ask among other questions of interest, how this spectrum might be modified when the base space itself has non trivial indices. Exact solutions of SU(2) Yang-Mills fields are presented for metrics corresponding to well-known gravitational instantons. Such selfdual solutions, with vanishing energy monien-tunl tensor T_μv for Euclidean signature of the base space, do not perturb the metric. Thus they provide solutions of the combined gravitational-Y.M. system. New topological possibilities, such as finite action SU(2) fields with fractional indices for many centre inetrics are displayed explicitly. As another type of possibility non selfdual, finite action solutions are constructed explicitly on Schwarzschild and de Sitter metrics, the solution being real in the first and complex in second case respectively. It is also shown how various meron type solutions in flat space can be derived systematically from a very simple static solution in de Sitter.     

A Random Surface Wave Equation for Bosonic QCD(<Emphasis Type="Italic">SU</Emphasis>(∞))

We propose a quantum surface wave functional describing the interaction between a colored SU(N _c ) membrane and a quantized Yang-Mills field. Additionally, we deduce its associated wave equation in the t’Hooft N _c →∞ limit. We show that its reproduces the Yang-Mills Field Theory at a large rigid random surface scale.     

Two-Brane Randall-Sundrum Model in AdS 5 and dS 5

Two flat Randall-Sundrum three-branes are analyzed, at fixed mutual distance, in the case where each brane contains an ideal isotropic fluid. Both fluids are to begin with assumed to obey the equation of state p = ( – 1), where is a constant. Thereafter, we impose the condition that there is zero energy flux from the branes into the bulk, and assume that the tension on either brane is zero. It then follows that constant values of the fluid energies at the branes are obtained only if the value of is equal to zero (i.e., a vacuum fluid). The fluids on the branes are related: if one brane is a dS ₄ brane (the effective four-dimensional constant being positive), then the other brane is dS ₄ also, and if the fluid energy density on one brane is positive, the energy density on the other brane is larger in magnitude but negative. This is a non-acceptable result, which sheds some light on how far it is possible to give a physical interpretation of the two-brane scenario. Also, we discuss the graviton localization problem in the two-brane setting, generalizing prior works.     

Quantum Yang-Mills on the two-sphere

We obtain the quantum expectations of gauge-invariant functions of the connection on a principalG=SU(N) bundle overS ². We show that the spaceA/g _m of connections modulo gauge transformations which are the identity at one point is itself a principal bundle over G, based loops in the symmetry group. The fiber inA/g _m is an affine linear space. Quantum expectations are iterated path integrals first over this fiber then over G, each with respect to the push-forward toA/g _m of the measure s^-S(A) DA.S(A) denotes the Yang-Mills action onA. There is a global section ofA/g _m on which the first integral is a Gaussian. The resulting measure on G is the conditional Wiener measure. We explicitly compute the expectations of a special class of Wilson loops.     

Non-transversality of the Yang-Mills self-energy in the planar gauge

It is shown that the Yang-Mills self-energy in the planar gauge is non-transverse, even though this self-energy satisfies the appropriate Ward identity. The non-transversality implies that the one-loop counterterm is no longer proportional to (F_μν^a)².     

Self-dual Yang-Mills fields in eight dimensions

Strongly self-dual Yang-Mills fields in even-dimensional spaces are characterised by a set of constraints on the eigenvalues of the Yang-Mills fieldsF . We derive a topological bound on R⁸, , wherep1 is the first Pontryagin class of the SO(n) Yang-Mills bundle, andk is a constant. Strongly self-dual Yang-Mills fields realise the lower bound.     

The one-loop Green’s functions of dimensionally reduced gauge theories

We apply the dimensional regularization technique as well as that by dimensional reduction to the calculation of the regularized one-loop Green’s functions ind ₀-dimensional Yang-Mills theory with real massless scalars and spinors in arbitrary (real) representations of a gauge groupG. As a particular example, the super-symmetrically regularized one-loop Green’s functions of theN=4 supersymmetric Yang-Mills model are derived.