The Donaldson–Witten Function for Gauge Groups of Rank Larger Than One

We study correlation functions in topologically twisted , d=4 supersymmetric Yang–Mills theory for gauge groups of rank larger than one on compact four-manifolds X. We find that the topological invariance of the generator of correlation functions of BRST invariant observables is not spoiled by noncompactness of field space. We show how to express the correlators on simply connected manifolds of b _2,+(X)>0 in terms of Seiberg–Witten invariants and the classical cohomology ring of X. For manifolds X of simple type and gauge group SU(N) we give explicit expressions of the correlators as a sum over =1 vacua. We describe two applications of our expressions, one to superconformal field theory and one to large N expansions of SU(N) , d=4 supersymmetric Yang–Mills theory. Received: 30 March 1998 / Accepted: 17 April 1998     

Electrodynamics on matrix space: non-Abelian by coordinates

Faddeev and Niemi have proposed a decomposition of SU(N) Yang–Mills theory in terms of new variables, appropriate for describing the theory in the infrared limit. We extend this method to SO(2N) Yang–Mills theory. We find that the SO(2N) connection decomposes according to irreducible representations of SO(N). The low-energy limit of the decomposed theory is expected to describe soliton-like configurations with nontrivial topological numbers. How the method of decomposition generalizes for SO(2N+1) Yang–Mills theory is also discussed. Received: 22 November 2000 / Published online: 8 June 2001     

On Yang–Mills Instantons over Multi-Centered Gravitational Instantons

 In this paper we explicitly calculate the analogue of the 't Hooft SU(2) Yang–Mills instantons on Gibbons–Hawking multi-centered gravitational instantons, which come in two parallel families: the multi-Eguchi–Hanson, or A _k ALE gravitational instantons and the multi-Taub–NUT spaces, or A _k ALF gravitational instantons. We calculate their energy and find the reducible ones. Following Kronheimer we also exploit the U(1) invariance of our solutions and study the corresponding explicit singular SU(2) magnetic monopole solutions of the Bogomolny equations on flat ℝ³. Received: 16 September 2002 / Accepted: 22 October 2002 Published online: 21 February 2003 Communicated by A. Connes     

Complexified Gravity in Noncommutative Spaces

The presence of a constant background antisymmetric tensor for open strings or D-branes forces the space-time coordinates to be noncommutative. This effect is equivalent to replacing ordinary products in the effective theory by the deformed star product. An immediate consequence of this is that all fields get complexified. The only possible noncommutative Yang–Mills theory is the one with U(N) gauge symmetry. By applying this idea to gravity one discovers that the metric becomes complex. We show in this article that this procedure is completely consistent and one can obtain complexified gravity by gauging the symmetry U(1,D−1) instead of the usual SO(1,D−1). The final theory depends on a Hermitian tensor containing both the symmetric metric and antisymmetric tensor. In contrast to other theories of nonsymmetric gravity the action is both unique and gauge invariant. The results are then generalized to noncommutative spaces. Received: 1 June 2000 / Accepted: 27 November 2000     

Multiple Instantons Representing Higher-Order Chern–Pontryagin Classes

It has been shown in the work of Chakrabarti, Sherry and Tchrakian that the chiral SO _±(4 p) Yang–Mills theory in the Euclidean 4 p (p≥ 2) dimensions allows an axially symmetric self-dual system of equations similar to Witten's instanton equations in the classical 4-dimensional SU(2)∼SO _±(4) theory and the solutions represent a new class of instantons. However the rigorous existence of these higher-dimensional instanton solutions has remained open except for the solution of unit charge representing a single instanton. In this paper we establish an existence and uniqueness theorem for multi-instantons of arbitrary charges in the case p≥ 2. These solutions are the first known instantons, with the Chern–Pontryagin index greater than one, of the Yang–Mills model in higher dimensions. Our approach is a study of a nonlinear variational equation defined on the Poincaré half plane. Received: 20 May 1996 / Accepted: 30 April 1997     

Early and late-time cosmic acceleration in non-minimal Yang–Mills-<Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">G</Emphasis>) gravity

In this paper we show that power-law inflation can be realized in non-minimal gravitational coupling of Yang–Mills field with a general function of the Gauss–Bonnet invariant in the framework of Einstein gravity. Such a non-minimal coupling may appear due to quantum corrections. We also discuss the non-minimal Yang–Mills-f(G) gravity in the framework of modified Gauss–Bonnet action which is widely studied recently. It is shown that both inflation and late-time cosmic acceleration are possible in such a theory.     

A purely algebraic construction of a gauge and renormalization group invariant scalar glueball operator

This paper presents a complete algebraic proof of the renormalizability of the gauge invariant d=4 operator F _{μ ν} ²(x) to all orders of perturbation theory in pure Yang–Mills gauge theory, whereby working in the Landau gauge. This renormalization is far from being trivial as mixing occurs with other d=4 gauge variant operators, which we identify explicitly. We determine the mixing matrix Z to all orders in perturbation theory by using only algebraic arguments and consequently we can uncover a renormalization group invariant by using the anomalous dimension matrix Γ derived from Z. We also present a future plan for calculating the mass of the lightest scalar glueball with the help of the framework we have set up.     

Nahm Transform for Periodic Monopoles¶and ?=2 Super Yang–Mills Theory

We study Bogomolny equations on ℝ²×?¹. Although they do not admit nontrivial finite-energy solutions, we show that there are interesting infinite-energy solutions with Higgs field growing logarithmically at infinity. We call these solutions periodic monopoles. Using the Nahm transform, we show that periodic monopoles are in one-to-one correspondence with solutions of Hitchin equations on a cylinder with Higgs field growing exponentially at infinity. The moduli spaces of periodic monopoles belong to a novel class of hyperk?hler manifolds and have applications to quantum gauge theory and string theory. For example, we show that the moduli space of k periodic monopoles provides the exact solution of ?=2 super Yang–Mills theory with gauge group SU(k) compactified on a circle of arbitrary radius. Received: 20 July 2000 / Accepted: 29 November 2000     

On the nature of the phase transition in <Emphasis Type="Italic">SU</Emphasis>(<Emphasis Type="Italic">N</Emphasis>), <Emphasis Type="Italic">Sp</Emphasis>(2) and <Emphasis Type="Italic">E</Emphasis>(7) Yang–Mills theory

We study the nature of the confinement phase transition in d=3+1 dimensions in various non-abelian gauge theories with the approach put forward in Phys. Lett. B 684, 262 (2010). We compute an order-parameter potential associated with the Polyakov loop from the knowledge of full 2-point correlation functions. For SU(N) with N=3,…,12 and Sp(2) we find a first-order phase transition in agreement with general expectations. Moreover our study suggests that the phase transition in E(7) Yang–Mills theory also is of first order. We find that it is weaker than for SU(N). We show that this can be understood in terms of the eigenvalue distribution of the order parameter potential close to the phase transition.     

Geometric Quantization¶and the Generalized Segal--Bargmann Transform¶for Lie Groups of Compact Type

Let K be a connected Lie group of compact type and let T ^*(K) be its cotangent bundle. This paper considers geometric quantization of T ^*(K), first using the vertical polarization and then using a natural K?hler polarization obtained by identifying T ^*(K) with the complexified group K _ℂ. The first main result is that the Hilbert space obtained by using the K?hler polarization is naturally identifiable with the generalized Segal–Bargmann space introduced by the author from a different point of view, namely that of heat kernels. The second main result is that the pairing map of geometric quantization coincides with the generalized Segal–Bargmann transform introduced by the author. This means that the pairing map, in this case, is a constant multiple of a unitary map. For both results it is essential that the half-form correction be included when using the K?hler polarization. These results should be understood in the context of results of K. Wren and of the author with B. Driver concerning the quantization of (1+1)-dimensional Yang–Mills theory. Together with those results the present paper may be seen as an instance of “quantization commuting with reduction”. Received: 28 June 2001 / Accepted: 17 September 2001     

N-dimensional non-abelian dilatonic,stable black holes and their Born–Infeld extension

We find large classes of non-asymptotically flat Einstein–Yang–Mills–Dilaton and Einstein–Yang–Mills–Born–Infeld–Dilaton black holes in N-dimensional spherically symmetric spacetime expressed in terms of the quasilocal mass. Extension of the dilatonic YM solution to N-dimensions has been possible by employing the generalized Wu-Yang ansatz. Another metric ansatz, which aided in finding exact solutions is the functional dependence of the radius function on the dilaton field. These classes of black holes are stable against linear radial perturbations. In the limit of vanishing dilaton we obtain Bertotti–Robinson type metrics with the topology of AdS ₂×S ^N–2. Since connection can be established between dilaton and a scalar field of Brans–Dicke type we obtain black hole solutions also in the Brans–Dicke–Yang–Mills theory as well.     

The nonstandardq-deformation of enveloping algebraU(so n ): results and problems

We describe properties of the nonstandardq-deformationU _q ^/′ (so _n ) of the universal enveloping algebraU(so _n ) of the Lie algebra so _n which does not coincide with the Drinfeld-Jimbo quantum algebraU _q(so _n ) and is important for quantum gravity. Many unsolved problems are formulated. Some of these problems are solved in special cases. The research of this paper was made possible in part by Award UP1-2115 of U.S. Civilian Research and Development Foundation. Presented at DI-CRM Workshop held in Prague, 18–21 June 2000.     

New Representation of the Self-Duality and Exact Solutions for Yang–Mills Equations

In this paper, we found a new representation for self-duality . In addition, exact solution class of the classical SU(2) Yang–Mills field in four-dimensional Euclidean space and two exact solution classes for SU(2) Yang–Mills when ρ is a complex analytic function are also obtained. PACS numbers: 11.15.-q Gauge field theories, 11.15.Kc Semiclassical theories in gauge fields, 12.10.-g, 12.15.-y Yang–Mills fields     

Sound propagation anomalies and phase diagram of chromium alloys

Calculations of the complex elastic moduli Ĉ(T) in dilute Cr alloys are compared to measurements of the velocity and damping of sound near T _N and at high temperatures T>T _N (T _N — Néel temperature). The thermodynamic calculation is based on the covalent bond model of 3d ions in a state with different numbers n of covalent electrons. The parameters A _ij ⁽ⁿ⁾ of indirect exchange between the ions of the i and j sublattices are expressed in terms of the covalent bond energy Γ _ij ⁽ⁿ⁾ . The stability of the charge and spin density waves (CDWs and SDWs) is found by a variational method and is determined by the dispersion of Γ _ij ⁽ⁿ⁾ and by the Coulomb parameters U _n. For a small structural vector Q the phase diagram contains a superantiferromagnetic phase (SAFM) at temperatures T _N<T≲2T _N. The peak of the defect |ΔE(T)| of the modulus and of the sound damping Δ_h(T _N) near the first-order SDW-SAFM transition is determined by the structure of the transitional domain. Measurements of the anomalous growth of E(T) at temperatures T>T _N make it possible to determine the magnetostriction constants λ(T) of Ce alloys in the SAFM phase on the basis of the SAFM theory. Fiz. Tverd. Tela (St. Petersburg) 41, 1467–1472 (August 1999)     

The structure of the Yang–Mills spectrum for arbitrary simple gauge algebras

The mass spectrum of pure Yang–Mills theory in 3+1 dimensions is discussed for an arbitrary simple gauge algebra within a quasigluon picture. The general structure of the low-lying gluelump and two-quasigluon glueball spectrum is shown to be common to all algebras, while the lightest C=− three-quasigluon glueballs only exist when the gauge algebra is A _r≥2, that is, in particular, \mathfraksu(N 3 3)\mathfrak{su}(N\geq3). Higher-lying C=− glueballs are shown to exist only for the A _r≥2, D_odd−r≥4 and E₆ gauge algebras. The shape of the static energy between adjoint sources is also discussed assuming the Casimir scaling hypothesis and a funnel form; it appears to be gauge-algebra dependent when at least three sources are considered. As a main result, the present framework’s predictions are shown to be consistent with available lattice data in the particular case of an \mathfraksu(N)\mathfrak{su}(N) gauge algebra within ’t Hooft’s large-N limit.     

Geometrical Tools for Quantum Euclidean Spaces

We apply one of the formalisms of noncommutative geometry to ℝ ^N _q , the quantum space covariant under the quantum group SO _q (N). Over ℝ ^N _q there are two SO _q (N)-covariant differential calculi. For each we find a frame, a metric and two torsion-free covariant derivatives which are metric compatible up to a conformal factor and which have a vanishing linear curvature. This generalizes results found in a previous article for the case of ℝ³ _q . As in the case N=3, one has to slightly enlarge the algebra ℝ ^N _q ; for N odd one needs only one new generator whereas for N even one needs two. As in the particular case N=3 there is a conformal ambiguity in the natural metrics on the differential calculi over ℝ ^N _q . While in our previous article the frame was found “by hand”, here we disclose the crucial role of the quantum group covariance and exploit it in the construction. As an intermediate step, we find a homomorphism from the cross product of ℝ ^N _q with U _q so(N) into ℝ ^N _q , an interesting result in itself. Received: 4 March 2000 / Accepted: 11 October 2000     

Instantons and Yang–Mills Flows on Coset Spaces

We consider the Yang–Mills flow equations on a reductive coset space G/H and the Yang–Mills equations on the manifold \mathbbR×G/H{\mathbb{R}\times G/H}. On non-symmetric coset spaces G/H one can introduce geometric fluxes identified with the torsion of the spin connection. The condition of G-equivariance imposed on the gauge fields reduces the Yang–Mills equations to f⁴{\phi^4}-kink equations on \mathbbR{\mathbb{R}}. Depending on the boundary conditions and torsion, we obtain solutions to the Yang–Mills equations describing instantons, chains of instanton–anti-instanton pairs or modifications of gauge bundles. For Lorentzian signature on \mathbbR×G/H{\mathbb{R}\times G/H}, dyon-type configurations are constructed as well. We also present explicit solutions to the Yang–Mills flow equations and compare them with the Yang–Mills solutions on \mathbbR×G/H{\mathbb{R}\times G/H}.     

String Geometry and the Noncommutative Torus

We construct a new gauge theory on a pair of d-dimensional noncommutative tori. The latter comes from an intimate relationship between the noncommutative geometry associated with a lattice vertex operator algebra ? and the noncommutative torus. We show that the tachyon algebra of ? is naturally isomorphic to a class of twisted modules representing quantum deformations of the algebra of functions on the torus. We construct the corresponding real spectral triples and determine their Morita equivalence classes using string duality arguments. These constructions yield simple proofs of the O(d,d;ℤ) Morita equivalences between d-dimensional noncommutative tori and give a natural physical interpretation of them in terms of the target space duality group of toroidally compactified string theory. We classify the automorphisms of the twisted modules and construct the most general gauge theory which is invariant under the automorphism group. We compute bosonic and fermionic actions associated with these gauge theories and show that they are explicitly duality-symmetric. The duality-invariant gauge theory is manifestly covariant but contains highly non-local interactions. We show that it also admits a new sort of particle-antiparticle duality which enables the construction of instanton field configurations in any dimension. The duality non-symmetric on-shell projection of the field theory is shown to coincide with the standard non-abelian Yang–Mills gauge theory minimally coupled to massive Dirac fermion fields. Received: 26 October 1998/ Accepted: 9 April 1999     

The Geometry of¶(Super) Conformal Quantum Mechanics

N-particle quantum mechanics described by a sigma model with an N-dimensional target space with torsion is considered. It is shown that an SL(2,ℝ) conformal symmetry exists if and only if the geometry admits a homothetic Killing vector D ^a δ _a whose associated one-form D _a dX ^a is closed. Further, the SL(2,ℝ) can always be extended to Osp(1|2) superconformal symmetry, with a suitable choice of torsion, by the addition of N real fermions. Extension to SU(1,1|1) requires a complex structure I and a holomorphic U(1) isometry D ^a I _a ^b δ _b . Conditions for extension to the superconformal group D(2,1;α), which involve a triplet of complex structures and SU(2)×SU(2) isometries, are derived. Examples are given. Received: 3 September 1999 / Accepted: 30 January 2000