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The standard model (SM) is viewed as a variation on the Yang-Mills theory with gauge symmetry $u(1) \times su(2) \times su(3)$, in which the flavour symmetry is framed and to which 3 generations of quarks and leptons are appended as inputs from experiment. The framed standard model (FSM) is then a further variation on the SM in which the colour symmetry is also framed, where the 3 generations now follow as consequences together with their characteristic mass and mixing patterns. In addition, the FSM yields as a bonus a solution to the strong CP problem leading to a unified treatment seemingly of all known CP physics for both quarks and leptons. It predicts, however, also a "hidden" sector populated by particles yet unknown, hence full both of promises (e.g.\ dark matter?) and of threats, which is just beginning to be explored (i) with a modified Weinberg mixing and (ii) with the $g-2$ and some other anomalies., Comment: Invited contribution to the "Festschrift for the Yang Centenary". edited FC Chen et al

The extraordinary success of the Standard Model asks for a more rigorous control beyond the perturbative approach, which is affected by mathematical problems (interaction picture and canonical quantization, non-convergence of the perturbative series, trivialy results for the $\ph^4$ model and related models etc.). We shall briefly discuss sone non-perturbative results concerning the crucial role and realization of symmetry breakings in the Standard Model., Comment: 8

The standard model (SM) is viewed as a variation on the Yang-Mills theory with gauge symmetry $u(1) \times su(2) \times su(3)$, in which the flavour symmetry is framed and to which 3 generations of quarks and leptons are appended as inputs from experiment. The framed standard model (FSM) is then a further variation on the SM in which the colour symmetry is also framed, where the 3 generations now follow as consequences together with their characteristic mass and mixing patterns. In addition, the FSM yields as a bonus a solution to the strong CP problem leading to a unified treatment seemingly of all known CP physics for both quarks and leptons. It predicts, however, also a "hidden" sector populated by particles yet unknown, hence full both of promises (e.g.\ dark matter?) and of threats, which is just beginning to be explored (i) with a modified Weinberg mixing and (ii) with the $g-2$ and some other anomalies., Comment: Invited contribution to the "Festschrift for the Yang Centenary". edited FC Chen et al

The standard model (SM) is viewed as a variation on the Yang-Mills theory with gauge symmetry $u(1) \times su(2) \times su(3)$, in which the flavour symmetry is framed and to which 3 generations of quarks and leptons are appended as inputs from experiment. The framed standard model (FSM) is then a further variation on the SM in which the colour symmetry is also framed, where the 3 generations now follow as consequences together with their characteristic mass and mixing patterns. In addition, the FSM yields as a bonus a solution to the strong CP problem leading to a unified treatment seemingly of all known CP physics for both quarks and leptons. It predicts, however, also a "hidden" sector populated by particles yet unknown, hence full both of promises (e.g.\ dark matter?) and of threats, which is just beginning to be explored (i) with a modified Weinberg mixing and (ii) with the $g-2$ and some other anomalies., Comment: Invited contribution to the "Festschrift for the Yang Centenary". edited FC Chen et al

The extraordinary success of the Standard Model asks for a more rigorous control beyond the perturbative approach, which is affected by mathematical problems (interaction picture and canonical quantization, non-convergence of the perturbative series, trivialy results for the $\ph^4$ model and related models etc.). We shall briefly discuss sone non-perturbative results concerning the crucial role and realization of symmetry breakings in the Standard Model., Comment: 8

We consider (effective) Quantum General Relativity coupled to the Standard Model (QGR-SM) and study the corresponding BRST double complex. This double complex is generated by infinitesimal diffeomorphisms and infinitesimal gauge transformations. To this end, we define the respective BRST differentials $P$ and $Q$ and their corresponding anti-BRST differentials $\overline{P}$ and $\overline{Q}$. Then we show that all differentials mutually anticommute, which allows us to introduce the total BRST differential $D := P + Q$ and the total anti-BRST differential $\overline{D} := \overline{P} + \overline{Q}$. Furthermore, we characterize all non-constant Lagrange densities that are essentially closed with respect to the diffeomorphism differentials $P$ and $\overline{P}$ as scalar tensor densities of weight $w = 1$. As a consequence thereof, we obtain that graviton-ghosts decouple from matter if the Yang--Mills theory gauge fixing and ghost Lagrange densities are tensor densities of weight $w = 1$. In particular, we show that every such Lagrange density can be modified uniquely to satisfy said condition. Finally, we introduce a total gauge fixing fermion $\Upsilon$ and show that we can generate the complete gauge fixing and ghost Lagrange densities of QGR-SM via $D \Upsilon$., Comment: 18 pages, article

The homotopy types of gauge groups of principal SO(4)-bundles over S⁴ are classified p-locally for every prime p, and partial results are obtained integrally. The method generalizes to deal with any quotient of the form (S³)n/Z where Z is a subgroup generated by (−1, …, −1).

The N=4 superconformal Yang-Mills theory on flat four-dimensional Minkowski space is a de-confined gauge theory in the sense that the string tension for fundamental representation coloured quarks vanishes. In fact, static fundamental representation quarks which lie in certain half-BPS super-multiplets do not interact at all. An interesting question asks whether such quarks would carry a well-defined global colour charge which, when the gauge is fixed, should have the status of an internal symmetry. We shall present a simple paradigmatic model which suggests that the answer to this question lies in the way in which infrared divergences are dealt with., Comment: 28 pages, 2 figures, typos fixed and some clarification added

The N=4 superconformal Yang-Mills theory on flat four-dimensional Minkowski space is a de-confined gauge theory in the sense that the string tension for fundamental representation coloured quarks vanishes. In fact, static fundamental representation quarks which lie in certain half-BPS super-multiplets do not interact at all. An interesting question asks whether such quarks would carry a well-defined global colour charge which, when the gauge is fixed, should have the status of an internal symmetry. We shall present a simple paradigmatic model which suggests that the answer to this question lies in the way in which infrared divergences are dealt with., Comment: 28 pages, 2 figures, typos fixed and some clarification added

We study the perturbative renormalization of Quantum Gauge Theories in the Hopf algebra setup of Connes and Kreimer. It was shown by van Suijlekom (2007) that the quantum counterpart of gauge symmetries -- the so-called Ward--Takahashi and Slavnov--Taylor identities -- correspond to Hopf ideals in the respective renormalization Hopf algebra. We generalize this correspondence to super- and non-renormalizable Quantum Field Theories and extend it to theories with multiple coupling constants. In particular, this allows us to apply our results to gravity, possibly coupled to matter, as was suggested by Kreimer (2007). To this end, we introduce different gradings on the renormalization Hopf algebra and study combinatorial properties of the superficial degree of divergence. Then we generalize known coproduct and antipode identities to the super- and non-renormalizable case. Building upon our main result, we provide criteria for the compatibility of these Hopf ideals with the corresponding renormalized Feynman rules. A direct consequence of our results is the well-definedness of the Corolla polynomial for Quantum Yang--Mills Theories without reference to a particular renormalization scheme., Comment: 36 pages, 3 figues, article; minor revisions; version to appear in Mathematical Physics, Analysis and Geometry

The classical matter fields are sections of a vector bundle E with base manifold M. The space L^2(E) of square integrable matter fields w.r.t. a locally Lebesgue measure on M, has an important module action of C_b^\infty(M) on it. This module action defines restriction maps and encodes the local structure of the classical fields. For the quantum context, we show that this module action defines an automorphism group on the algebra A, of the canonical anticommutation relations on L^2(E), with which we can perform the analogous localization. That is, the net structure of the CAR, A, w.r.t. appropriate subsets of M can be obtained simply from the invariance algebras of appropriate subgroups. We also identify the quantum analogues of restriction maps. As a corollary, we prove a well-known "folk theorem," that the algebra A contains only trivial gauge invariant observables w.r.t. a local gauge group acting on E., Comment: 15 pages

We make an extensive use of BLM approach to study details of predicting higher order corrections based on the approach. This way we are able to test two procedures to improve the prediction process. Beside the main line of the two procedures it is found out that overall normalization could change BLM patterns effectively. Finally we try to find out whether a BLM pattern is sufficient for a prediction or not, and how one should use such a pattern., Comment: 8 pages

A heatbath algorithm is proposed for pure SU(N) lattice gauge theory based on the Manton action of the plaquette element for general gauge group N. Comparison is made to the Metropolis thermalization algorithm using both the Wilson and Manton actions. The heatbath algorithm is found to outperform the Metropolis algorithm in both execution speed and decorrelation rate. Results, mostly in D=3, for N=2 through 5 at several values for the inverse coupling are presented., Comment: 9 pages, 10 figures, 1 table, major revision, final version, to appear in PRD

We make an extensive use of BLM approach to study details of predicting higher order corrections based on the approach. This way we are able to test two procedures to improve the prediction process. Beside the main line of the two procedures it is found out that overall normalization could change BLM patterns effectively. Finally we try to find out whether a BLM pattern is sufficient for a prediction or not, and how one should use such a pattern., Comment: 8 pages

A heatbath algorithm is proposed for pure SU(N) lattice gauge theory based on the Manton action of the plaquette element for general gauge group N. Comparison is made to the Metropolis thermalization algorithm using both the Wilson and Manton actions. The heatbath algorithm is found to outperform the Metropolis algorithm in both execution speed and decorrelation rate. Results, mostly in D=3, for N=2 through 5 at several values for the inverse coupling are presented., Comment: 9 pages, 10 figures, 1 table, major revision, final version, to appear in PRD

A Q-manifold is a graded manifold endowed with a vector field of degree one squaring to zero. We consider the notion of a Q-bundle, that is, a fiber bundle in the category of Q-manifolds. To each homotopy class of ``gauge fields'' (sections in the category of graded manifolds) and each cohomology class of a certain subcomplex of forms on the fiber we associate a cohomology class on the base. Any principal bundle yielding canonically a Q-bundle, this construction generalizes Chern-Weil classes. Novel examples include cohomology classes that are locally the de Rham differential of the integrands of topological sigma models obtained by the AKSZ-formalism in arbitrary dimensions. For Hamiltonian Poisson fibrations one obtains a characteristic 3-class in this manner. We also relate to equivariant cohomology and Lecomte's characteristic classes of exact sequences of Lie algebras., Comment: 23 pages, LaTeX, uses diagrams.sty

We present an account of negative differential forms within a natural algebraic framework of differential graded algebras, and explain their relationship with forms on path spaces., Comment: 12 pp.; the Introduction has been rewritten and mention of cohomology dropped in Proposition 3.2; material slightly reorganized

In this paper we study the holomorphic Euler characteristics of determinant line bundles on moduli spaces of rank 2 semistable sheaves on an algebraic surface X, which can be viewed as $K$-theoretic versions of the Donaldson invariants. In particular, if X is a smooth projective toric surface, we determine these invariants and their wallcrossing in terms of the K-theoretic version of the Nekrasov partition function (called 5-dimensional supersymmetric Yang-Mills theory compactified on a circle in the physics literature). Using the results of math.AG/0606180 we give an explicit generating function for the wallcrossing of these invariants in terms of elliptic functions and modular forms., Comment: 72 pages, 2 figures

For a smooth projective toric surface we determine the Donaldson invariants and their wallcrossing in terms of the Nekrasov partition function. Using the solution of the Nekrasov conjecture math.AG/0306198, hep-th/0306238, math.AG/0409441 and its refinement math.AG/0311058, we apply this result to give a generating function for the wallcrossing of Donaldson invariants of good walls of simply connected projective surfaces with $b_+=1$ in terms of modular forms. This formula was proved earlier in alg-geom/9506018 more generally for simply connected 4-manifolds with $b_+=1$, assuming the Kotschick-Morgan conjecture and it was also derived by physical arguments in hep-th/9709193., Comment: 45pages, typos corrected, update the reference to a new version of Mochizuki's paper math.AG/0210211

The violation of the Noether relation between symmetries and charges is reduced to the time dependence of the charge associated to a conserved current. For the U(1) gauge symmetry a non-perturbative control of the charge commutators is obtained by an analysis of the Coulomb charged fields. From this, in the unbroken case we obtain a correct expression for the electric charge on the Coulomb states, its superselection and the presence of massless vector bosons; in the broken case, we obtain a general non-perturbative version of the Higgs phenomenon, i.e. the absence of massless Goldstone bosons and of massless vector bosons. The conservation of the (gauge dependent) current associated to the U(1) axial symmetry in QCD is shown to be compatible with the time dependence of the corresponding charge commutators and a non-vanishing eta' mass, as a consequence of the non locality of the (conserved) current., Comment: Invited contribution to ``The Quantum Universe'', dedicated to G. Ghirardi for his 70th birthday

We consider a Lagrangian system on a fiber bundle and its gauge transformations depending on derivatives of dynamic variables and gauge parameters of arbitrary order. We say that gauge transformations form an algebra if they generate a nilpotent BRST operator., Comment: 12 pages

Let G be one of the local gauge groups C(X,U(n)), C^\infty(X,U(n)), C(X,SU(n)) or C^\infty(X,SU(n)) where X is a compact Riemannian manifold. We observe that G has a nontrivial group topology, coarser than its natural topology, w.r.t. which it is amenable, viz the relative weak topology of C(X,M(n)). This topology seems more useful than other known amenable topologies for G. We construct a simple fermionic model containing an action of G, continuous w.r.t. this amenable topology., Comment: 8 pages, Latex

We present Noether's second theorem for graded Lagrangian systems of even and odd variables on an arbitrary body manifold X in a general case of BRST symmetries depending on derivatives of dynamic variables and ghosts of any finite order. As a preliminary step, Noether's second theorem for Lagrangian systems on fiber bundles over X possessing gauge symmetries depending on derivatives of dynamic variables and parameters of arbitrary order is proved., Comment: 31 pages, to be published in J. Math. Phys

We consider bosons on Euclidean R^4 that are minimally coupled to an external Yang-Mills field. We compute the logarithmically divergent part of the cut-off regularized quantum effective action of this system. We confirm the known result that this term is proportional to the Yang-Mills action. We use pseudodifferential operator methods throughout to prepare the ground for a generalization of our calculation to the noncommutative four-dimensional Moyal plane (also known as noncommutative flat space). We also include a detailed comparison of our cut-off regularization to heat kernel techniques. In the case of the noncommutative space, we complement the usual technique of asymptotic expansion in the momentum variable with operator theoretic arguments in order to keep separated quantum from noncommutativity effects. We show that the result from the commutative space R^4 still holds if one replaces all pointwise products by the noncommutative Moyal product., Comment: 37 pages, v2 contains an improved treatment of the theta function in Appendix A.3

By using supermanifold techniques we prove a generalization of the localization formula in equivariant cohomology which is suitable for studying supersymmetric Yang-Mills theories in terms of ADHM data. With these techniques one can compute the reduced partition functions of topological super Yang-Mills theory with 4, 8 or 16 supercharges. More generally, the superlocalization formula can be applied to any topological field theory in any number of dimensions., Comment: 22 pages, Latex2e

The so-called covariant Poincare lemma on the induced cohomology of the spacetime exterior derivative in the cohomology of the gauge part of the BRST differential is extended to cover the case of arbitrary, non reductive Lie algebras. As a consequence, the general solution of the Wess-Zumino consistency condition with a non trivial descent can, for arbitrary (super) Lie algebras, be computed in the small algebra of the 1 form potentials, the ghosts and their exterior derivatives. For particular Lie algebras that are the semidirect sum of a semisimple Lie subalgebra with an ideal, a theorem by Hochschild and Serre is used to characterize more precisely the cohomology of the gauge part of the BRST differential in the small algebra. In the case of an abelian ideal, this leads to a complete solution of the Wess-Zumino consistency condition in this space. As an application, the consistent deformations of 2+1 dimensional Chern-Simons theory based on iso(2,1) are rediscussed., Comment: 39 pages Latex file, 1 eps figure, typos and proof of lemma 5 corrected

The geometry of minimal surfaces generated by charge 2 Bogomolny monopoles on 3-dimensional Euclidean space is described in terms of the moduli parameter k. We find that the distribution of Gaussian curvature on the surface reflects the monopole structure. This is elucidated by the behaviour of the Gauss maps of the minimal surfaces., Comment: 23 pages, 2 figures

It is well-known that the perturbative prepotentials of four-dimensional N=2 supersymmetric Yang-Mills theories satisfy the generalized WDVV equations, regardless of the gauge group. In this paper we study perturbative prepotentials of the five-dimensional theories for some classical gauge groups and determine whether or not they satisfy the WDVV system., Comment: 15 pages, no figures, web address of paper added in bibliography. Misprints corrected

We prove the conjecture that a monopole in three-dimensional anti-de Sitter space can be completely determined by its ``holographic'' image on the conformal boundary two-sphere., Comment: 18 pages. Complete revision of the paper, focusing on clearer exposition. Although sometimes stated differently, the results remain the same as in the original version

In this paper, using the Atiyah-Ward equivalence and a theorem of Hitchin, one makes to correspond to certain bundles on the projective space, which are extensions of instanton bundles (in particular, these new bundles may have the first Chern class not congruent to 0 modulo the rank), bundles on the grassmanian Gr(2,4) or on the sphere S4, equiped with certain connections constructed from data associated to the bundle. Monads for these extensions are described., Comment: latex file, 12 pages, one uses xypic

The construction of appropriate jet space coordinates for calculating local BRST cohomology groups is discussed. The relation to tensor calculus is briefly reviewed too., Comment: 11 pages, correction in eq. (29), to appear in Lett.Math.Phys

It is shown that the definition of physical integration measures via ``exponential of minus the action times kinematical integration measure'' typically contradicts properties of physical models. In particular, theories with uncountably many non-vanishing Wilson-loop expectation values cannot be gained this way. The results are rigorous within the Ashtekar approach to gauge field theories., Comment: 4 pages, LaTeX

It is shown that the definition of physical integration measures via ``exponential of minus the action times kinematical integration measure'' typically contradicts properties of physical models. In particular, theories with uncountably many non-vanishing Wilson-loop expectation values cannot be gained this way. The results are rigorous within the Ashtekar approach to gauge field theories., Comment: 4 pages, LaTeX

Sengupta's lower bound for the Yang-Mills action on smooth connections on a bundle over a Riemann surface generalizes to the space of connections whose action is finite. In this larger space the inequality can always be saturated. The Yang-Mills critical sets correspond to critical sets of the energy action on a space of paths. This may shed light on Atiyah and Bott's conjecture concerning Morse theory for the space of connections modulo gauge transformations., Comment: 7 pages, 2 figures, Latex2e with epsfig, submitted to Journal of Mathematical Physics

Yang-Mills gauge theory models on a cylinder coupled to external matter charges provide powerful means to find and solve certain non-linear integrable systems. We show that, depending on the choice of gauge group and matter charges, such a Yang-Mills model is equivalent to trigonometric Calogero-Moser systems and certain known spin generalizations thereof. Choosing a more general ansatz for the matter charges allows us to obtain and solve novel integrable systems. The key property we use to prove integrability and to solve these systems is gauge invariance of the corresponding Yang-Mills model., Comment: 33 pages, no figures, to appear in Nucl. Phys. B