A. G. Sergeev

Complex Analysis in the Future Tube

The future tube in ℂ n +1 is the unbounded domain τ+ = z = (z o,...,z n ) ∈ ℂ n +1 : (Im z o)2 > (Imz 1)2 +...+ (Im z n )2 Im z o > 0. In other words, τ+ is a tube domain over the future cone V + = y ∈ ℝ n +1 : y 2 0 > y 2 1 +...+ y 2 n , y 0 > 0. The domain τ+ is biholomorphically equivalent to a classical Cartan domain of the IVth type, hence to a bounded symmetric domain in ℂ n +1. The future tube τ+ in ℂ4 (n = 3) is important in mathematical physics, especially in axiomatic quantum field theory, being the natural domain of definition of holomorphic relativistic fields. These specific features of the future tube motivated its investigation by mathematicians and physicists. Beginning with Elie Cartan’s classification of bounded symmetric domains, these domains were examined in many papers where the complex structure of their boundaries, integral representations, boundary values of holomorphic functions and so on were considered. The proof of the “edge-of-the-wedge” theorem by N.N. Bogolubov generated the rapid development of applications of the theory of several complex variables to axiomatic quantum field theory. During this period the “C-convex hull” and “finite covariance” theorems were proved, the Jost-Lehmann-Dyson representation was found et cetera. Recently R. Penrose has proposed a transformation connecting holomorphic solutions of the basic equations of field theory with analytic sheaf cohomologies of domains in ℝℙ3. These two directions developed, to a large extent, independently from each other, and some important results obtained in axiomatic quantum field theory still remain unknown to specialists in several complex variables and differential geometry. One of the goals of this paper is to give a unified presentation of advances in complex analysis in the future tube and related domains achieved in both of these directions.

Seiberg–Witten monopole equations on noncommutative R4

It is well known that, due to vanishing theorems, there are no nontrivial finite action solutions to the Abelian Seiberg–Witten (SW) monopole equations on Euclidean four-dimensional space R4. We show that this is no longer true for the noncommutative version of these equations, i.e., on a noncommutative deformation Rθ4 of R4 there exist smooth solutions to the SW equations having nonzero topological charge. We introduce action functionals for the noncommutative SW equations and construct explicit regular solutions. All our solutions have finite energy. We also suggest a possible interpretation of the obtained solutions as codimension four vortex-like solitons representing D(p−4)- and D(p−4)¯-branes in a Dp-Dp¯ brane system in type II superstring theory.

The Group of Quasisymmetric Homeomorphisms of the Circle and Quantization of the Universal Teichmuller Space

In the first part of the paper we describe the complex geometry of the universal Teichmuller space T , which may be realized as an open subset in the complex Banach space of holomorphic quadratic differentials in the unit disc. The quotient S of the diffeomorphism group of the circle modulo Mobius transformations may be treated as a smooth part of T. In the second part we consider the quantization of universal Teichmuller space T. We explain first how to quantize the smooth part S by embedding it into a Hilbert-Schmidt Siegel disc. This quantization method, however, does not apply to the whole universal Teichmuller space T , for its quantization we use an approach, due to Connes.

Symplectic Twistors and Geometric Quantization of Strings

We present the geometric quantization scheme for the bosonic string theory in twistor terms. Starting from the loop space of a Lie group we define a symplectic twistor bundle over the loop space and reformulate the geometric quantization problem in terms of this bundle. For the standard bosonic string we recover in this way the well known critical dimension condition.

TWISTORS AND GAUGE FIELDS

We describe briefly the basic ideas and results of the twistor theory. The main points: twistor representation of Minkowsky space, Penrose correspondence and its geometrical properties, twistor interpretation of linear massless fields, Yang-Mills fields (including instantons and monopoles) and Einstein-Hilbert equations.

Seiberg-Witten theory as a complex version of Abelian Higgs model

The adiabatic limit procedure associates with every solution of Abelian Higgs model in (2 + 1) dimensions a geodesic in the moduli space of static solutions. We show that the same procedure for Seiberg-Witten equations on 4-dimensional symplectic manifolds introduced by Taubes may be considered as a complex (2+2)-dimensional version of the (2 + 1)-dimensional picture. More precisely, the adiabatic limit procedure in the 4-dimensional case associates with a solution of Seiberg-Witten equations a pseudoholomorphic divisor which may be treated as a complex version of a geodesic in (2+1)-dimensional case.

Adiabatic limit in Abelian Higgs model with application to Seiberg–Witten equations

In this paper we deal with the (2 + 1)-dimensional Higgs model governed by the Ginzburg–Landau Lagrangian. The static solutions of this model, called otherwise vortices, are described by the theorem of Taubes. This theorem gives, in particular, an explicit description of the moduli space of vortices (with respect to gauge transforms). However, much less is known about the moduli space of dynamical solutions. A description of slowly moving solutions may be given in terms of the adiabatic limit. In this limit the dynamical Ginzburg–Landau equations reduce to the adiabatic equation coinciding with the Euler equation for geodesics on the moduli space of vortices with respect to the Riemannian metric (called T-metric) determined by the kinetic energy of the model. A similar adiabatic limit procedure can be used to describe approximately solutions of the Seiberg–Witten equations on 4-dimensional symplectic manifolds. In this case the geodesics of T-metric are replaced by the pseudoholomorphic curves while the solutions of Seiberg–Witten equations reduce to the families of vortices defined in the normal planes to the limiting pseudoholomorphic curve. Such families should satisfy a nonlinear ∂-equation which can be considered as a complex analogue of the adiabatic equation. Respectively, the arising pseudoholomorphic curves may be considered as complex analogues of adiabatic geodesics in (2 + 1)-dimensional case. In this sense the Seiberg–Witten model may be treated as a (2 + 1)-dimensional analogue of the (2 + 1)-dimensional Abelian Higgs model2.

On the Moduli Space of Yang–Mills Fields on \( \mathbb{R}^4 \)

We consider the problem of description of the structure of the moduli space of Yang–Mills fields on ( mathbb{R}^4 ) with gauge group G. According to harmonic spheres conjecture, this moduli space should be closely related to the space of harmonic spheres in the loop space ΩG. Since the structure of the latter space is much better understood, the proof of conjecture will help to clarify the structure of the moduli space of Yang–Mills fields. We propose an idea how to prove the harmonic spheres conjecture using the twistor methods.

Quantization of universal Teichmüller space

Universal Teichmuller space Open image in new window is the quotient of the group QS(S 1) of quasisymmetric homeomorphisms of S 1 modulo Mobius transformations. The quantization problem for Open image in new window arises in the theory of non-smooth closed bosonic strings. Because of non-smoothness of strings the natural QS(S 1)-action on Open image in new window is also not smooth so there is no classical Lie algebra, associated to QS(S 1). However, using methods of non-commutative geometry, we can define a quantum Lie algebra of observables Der q (QS), yielding the quantization of Open image in new window .

Quantization of the universal Teichmüller space

In the first part of the paper, we describe the Kähler geometry of the universal Teichmüller space, which can be realized as an open subset in the complex Banach space of holomorphic quadratic differentials in the unit disc. The universal Teichmüller space contains classical Teichmüller spaces T(G), where G is a Fuchsian group, as complex submanifolds. The quotient Diff+(S1)/Möb(S1) of the diffeomorphism group of the unit circle modulo Möbius transformations can be considered as a “smooth” part of the universal Teichmüller space. In the second part we describe how to quantize Diff+(S1)/Möb(S1) by embedding it in an infinite-dimensional Siegel disc. This quantization method does not apply to the whole universal Teichmüller space. However, this space can be quantized using the “quantized calculus” of A. Connes and D. Sullivan.