Quantum ’t Hooft Loops of SYM $${{\mathcal N}=4}$$ as Instantons of YM2 in Dual Groups SU(N) and SU(N)/ZN

A relation between circular 1/2 BPS ’t Hooft operators in 4d N=4{{\mathcal N}=4} SYM and instantonic solutions in 2D Yang-Mills theory (YM₂) has recently been conjectured. Localization indeed predicts that those ’t Hooft operators in a theory with gauge group G are captured by instanton contributions to the partition function of YM₂, belonging to representations of the dual group ^L G. This conjecture has been tested in the case G = U(N) =  ^L G and for fundamental representations. In this paper, we examine this conjecture for the case of the groups G = SU(N) and ^L G = SU(N)/Z _N and loops in different representations. Peculiarities when groups are not self-dual and representations not “minimal” are pointed out.     

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.     

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.     

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.     

Spontaneous Symmetry Breaking in Presence of Electric and Magnetic Charges

Starting with the definition of quaternion gauge theory, we have undertaken the study of SU(2) _e ×SU(2) _m ×U(1) _e ×U(1) _m in terms of the simultaneous existence of electric and magnetic charges along with their Yang-Mills counterparts. As such, we have developed the gauge theory in terms of four coupling constants associated with four-gauge symmetry SU(2) _e ×SU(2) _m ×U(1) _e ×U(1) _m . Accordingly, we have made an attempt to obtain the abelian and non-Abelian gauge structures for the particles carrying simultaneously the electric and magnetic charges (namely dyons). Starting from the Lagrangian density of two SU(2)×U(1) gauge theories responsible for the existence of electric and magnetic charges, we have discussed the consistent theory of spontaneous symmetry breaking and Higgs mechanism in order to generate the masses. From the symmetry breaking, we have generated the two electromagnetic fields, the two massive vector W ^± and Z ⁰ bosons fields and the Higgs scalar fields.     

Stability of proton and maximally symmetric minimal unification model for basic forces and building blocks of matter

With the hypothesis that all independent degrees of freedom of basic building blocks should be treated equally on the same footing and correlated by a possible maximal symmetry, we arrive at a 4-dimensional space-time unification model. In this model the basic building blocks are Majorana fermions in the spinor repre- sentation of 14-dimensional quantum space-time with a gauge symmetry GM4D = SO(1,3)×SU(32)×U(1)A×SU(3)F. The model leads to new physics including mirror particles of the standard model. It enables us to issue some fundamental questions that include: why our living space-time is 4-dimensional, why parity is not con- served in our world, how the stability of proton is, what the origin of CP violation is and what the dark matter can be.     

Quaternionic-valued Gravitation in 8D, Grand Unification and Finsler Geometry

A unification model of 4D gravity and SU(3)×SU(2)×U(1) Yang-Mills theory is presented. It is obtained from a Kaluza-Klein compactification of 8D quaternionic gravity on an internal CP ²=SU(3)/U(2) symmetric space. We proceed to explore the nonlinear connection formalism used in Finsler geometry to show how ordinary gravity in D=4+2 dimensions has enough degrees of freedom to encode a 4D gravitational and SU(5) Yang-Mills theory. This occurs when the internal two-dim space is a sphere S ². This is an appealing result because SU(5) is one of the candidate GUT groups. We conclude by discussing how the nonlinear connection formalism of Finsler geometry provides an infinite hierarchical extension of the Standard Model within a six dimensional gravitational theory due to the embedding of SU(3)×SU(2)×U(1)⊂SU(5)⊂SU(∞).     

On the Prasad-Sommerfield dyon (monopole) solution

We prove that the Prasad-Sommerfield dyon (monopole) solution for an SU(2) Yang-Mills field coupled with an SU(2) Higgs multiplet can be associated to a certain minimal immersion in S ³ (SU(2)) i.e. it has a differential-geometric content similar to that of self-dual solutions for the pure SU(2) Yang-Mills field. Implications of this result as well as possibilities to extend it to higher gauge groups are discussed.     

Geometry of multidimensional universes

LetG be a compact group of transformation (global symmetry group) of a manifoldE (multidimensional universe) with all orbits of the same type (one stratum). We studyG invariant metrics onE and show that there is one-to-one correspondence between those metrics and triples (g _μv,A _μ ^ä ,h _αβ), whereg _μv is a (pseudo-) Riemannian metric on the space of orbits (space-time),A _μ ^ä is a Yang-Mills field for the gauge groupN|H, whereN is the normalizer of the isotropy groupH inG, andh _αβ are certain scalar fields characterizing geometry of the orbits (internal spaces). The scalar curvature ofE is expressed in terms of the component fields onM. Examples and model building recipes are also given. The results generalize those of non-abelian Kaluza-Klein theories to the case where internal spaces are not necessarily group manifolds.     

A note about the t’Hooft ansatz for SU(N) real-time gauge theories

The t’Hooft ansatz, which reduces the classical Yang-Mills theory to the λϕ⁴ one, is under consideration. It is shown that, in the framework of this ansatz, the real-time classical solutions for the arbitrary SU(N) gauge group are obtained by embedding SU(2) × SU(2) into SU(N). It is argued that this group structure is the only possibility in the framework of the considered ansatz. New explicit solutions for SU(3) and SU(5) gauge groups are shown. The text was submitted by the authors in English.     

Applications of a unique group embedding of internal symmetry groups andSU(2): spin-extensions for G2 and for the Sp(6)-Trion model

A recently proposed method of embeddingSU (2) and an internal symmetry groupG into a bigger groupG is applied to construct a spin extension ofG ₂ andSp (6). BecauseG ₂ andSp (6) possess a generalized quark model the embedding group can be proved to be unique and to be given bySp (14) resp.O (12) forG ₂ resp.Sp (6). For a particle classification splittings are calculated and tabulated forSp (14) ↓G ₂ xSU (2) andO (12) ↓Sp (6) xSU (2). The identification of low dimensional irreducible representations ofO (12) is quite satisfactory whereas an unreasonable number of unobserved particles are needed to fill up the supermultiplets of the spin extensionSp (14) of G₂.     

Instantons and algebraic geometry

Minimum action solutions for SU(2) Yang-Mills fields in Euclidean 4-space correspond, via the Penrose twistor transform, to algebraic bundles on the complex projective 3-space. These bundles in turn correspond to algebraic curves. The implication of these results for the Yang-Mills fields is described. In particular all solutions are rational and can be constructed from a series of AnsätzeA _l forl1.     

Spectra of Wess-Zumino-Witten models with arbitrary simple groups

We consider the Wess-Zumino-Witten two-dimensional sigma models with fields taking values in an arbitrary connected (but not necessarily simply connected) simple Lie groupG. The quantum states of the theory are realized geometrically as sections of a line bundle over the loop groupLG. The action of the current algebra of the theory is decomposed into highest weight representations by explicit construction of the highest weight states. This solves for the spectrum of the models. As a by-product, we obtain modular invariant partition functions of the theory on tori. The present paper extends the results of [7] where the casesG=SU(2) andSO(3) were treated.Ce rapport a été publié en partie grâce à une subvention du Fonds FCAR pour l'aide et le soutien à la rercherche     

Non-self-dual Yang-Mills connections with quadrupole symmetry

We prove the existence of non-self-dual Yang-Mills connections onSU(2) bundles over the four-sphere, specifically on all bundles with second Chern number not equal±1. We study connections equivariant under anSU(2) symmetry group to reduce the effective dimensionality from four to one, and then use variational techniques. The existence of non-self-dualSU(2) YM connections on the trivial bundle (second Chern number equals zero) has already been established by Sibner, Sibner, and Uhlenbeck via different methods.Research partially supported by NSF Grant DMS-8806731Most of this research was done while the author was a Bantrell Fellow at the California Institute of Technology, and was partially supported by NSF Grant DMS-8801918     

A basis-free approach to time-reversal for symmetry groups

We develop a basis-free approach to time-reversal for the quantal angular momentum group,SU2, and apply these methods to the physical symmetrySU2_isospin,SU3_flavor,SU3_nuclear and the nuclear collective symmetry groupSL(3,R) of Gell-Mann and Tomonaga.     

Baryon masses in the skyrme Yang-Mills model

The predictions concerning solitonic baryon masses in the general nonlinear chiral SU₂×SU₂ invariant Yang-Mills lagrangian for pions and ϱ-mesons are obtained. It is found that the predictions of this Skyrme Yang-Mills models are almost identical to the corresponding predictions of the original Skyrme model.     

Topological gauge theories and group cohomology

We show that three dimensional Chern-Simons gauge theories with a compact gauge groupG (not necessarily connected or simply connected) can be classified by the integer cohomology groupH ⁴(BG,Z). In a similar way, possible Wess-Zumino interactions of such a groupG are classified byH ³(G,Z). The relation between three dimensional Chern-Simons gauge theory and two dimensional sigma models involves a certain natural map fromH ⁴(BG,Z) toH ³(G,Z). We generalize this correspondence to topological spin theories, which are defined on three manifolds with spin structure, and are related to what might be calledZ ₂ graded chiral algebras (or chiral superalgebras) in two dimensions. Finally we discuss in some detail the formulation of these topological gauge theories for the special case of a finite group, establishing links with two dimensional (holomorphic) orbifold models.     

Strange and singlet form factors of the nucleon: Predictions for G0, A4, and HAPPEX II experiments

We investigate the strange and flavor-singlet electric and magnetic form factors of the nucleon within the framework of the SU(3) chiral quark-soliton model. Isospin symmetry is assumed and the symmetry-conserving SU(3) quantization is employed, rotational and strange-quark mass corrections being included. For the experiments G0, A4, and HAPPEX II we predict the quantities G⁰_E + G⁰_M and G^s_E + G^s_M. The dependence of the results on the parameters of the model and the treatment of the Yukawa asymptotic behavior of the soliton are investigated.     

Mass operator in theSU(3)-symmetric strong coupling theory with pseudoscalar mesons

In this paper theSU(3)-symmetric model of a static baryon octet source interacting with pseudoscalar meson octet fields by the coupling of Yukawa type is considered in the strong coupling limit. Using the result derived earlier that the isobaric states form the basis of the unitary irreducible representation of the dynamical group G=T ₂₄ [SU(3) SU(2)], the mass operator is specified by the kinetic part of the Hamiltonian as a particular element of the universal enveloping algebra of the symmetry groupSU(3) SU(2) acting in the space of isobaric states.     

Self dual solutions of the temperature SU(2) Yang-Mills theory

Continuing previous work we elaborate on the method of “heating” the self-dual axially symmetric fields of the SU(2) Yang-Mills theory to finite temperature. Heating consists of performing—in certain Ansatz functions which are two-dimensional (2D) conformally invariant—a 2D conformal transformation x = x₀ + i ∥x∥ → y(x), where the analytic function y(x) is periodic in the Euclidean time variable x₀. Solutions are preserved by this manipulation, which automatically changes zero-temperature fields into finite temperature ones. One can exploit this simple fact in various ways. The Harrington-Shepard caloron solution of the temperature Yang-Mills theory can be gotten from the T = 0 instanton by the transformation y(x) = (πT)^?1 tan πTx. One can generate a multicaloron solution from the T = 0 one instanton solution by a conformal transformation. Generally, self-dual axially symmetric Yang-Mills fields can be heated without spoiling self duality. The caloron and three other temperature solutions are studied in some detail. One of the new solutions is a generalized caloron with interesting properties. Our study reveals a remarkable property of the self-dual sector of the temperature Yang-Mills theory: it is full of Wu-Yang (color) monopoles at high temperature. At low temperature these monopoles disappear.