Target Manifold Research Articles

Special geometry of euclidean supersymmetry II. Hypermultiplets and thec-map

We construct two new versions of the c-map which allow us to obtain the target manifolds of hypermultiplets in Euclidean theories with rigid N =2 supersymmetry. While the Minkowskian para-c-map is obtained by dimensional reduction of the Minkowskian vector multiplet lagrangian over time, the Euclidean para-c-map corresponds to the dimensional reduction of the Euclidean vector multiplet lagrangian. In both cases the resulting hypermultiplet target spaces are para-hyper-Kahler manifolds. We review and prove the relevant results of para-complex and para-hypercomplex geometry. In particular, we give a second, purely geometrical construction of both c-maps, by proving that the cotangent bundle N=T^*M of any affine special (para-)Kahler manifold M is para-hyper-Kahler.

Extension of CR mappings between generic algebraic submanifolds

We prove a holomorphic extension theorem for CR mappings between real algebraic submanifolds of C N . The source and the target manifolds are assumed to be generic submanifolds of equal dimension with source being connected, holomorphically nondegenerate, and having at least one point at which it is of finite type. The mapping H is assumed to be a smooth mapping which is a local diffeomorphism near at least one point of the source. In proving our main result, we construct a CR function which is nonzero at a given point of the source if and only if H is a local diffeomorphism near that same point.

Higher-spin conserved currents in supersymmetric sigma models on symmetric spaces

Local higher-spin conserved currents are constructed in the supersymmetric sigma models with target manifolds symmetric spaces G / H . One class of currents is based on generators of the de Rham cohomology ring of G / H ; a second class of currents are higher-spin generalizations of the (super)energy–momentum tensor. A comprehensive analysis of the invariant tensors required to construct these currents is given from two complimentary points of view, and sets of primitive currents are identified from which all others can be constructed as differential polynomials. The Poisson bracket algebra of the top component charges of the primitive currents is calculated. It is shown that one can choose the primitive currents so that the bosonic charges all Poisson-commute, while the fermionic charges obey an algebra which is a form of higher-spin generalization of supersymmetry. Brief comments are made on some implications for the quantized theories.

An approach to the regularity for stable-stationary harmonic maps

In this paper we investigate the regularity of stable-stationary harmonic maps. By assuming that the target manifolds do not carry any stable harmonic S 2 , we obtain some compactness results and regularity theorems. In particular, we prove that the Hausdorff dimension of the singular set of these maps cannot exceed n - 3, and the dimension estimate is optimal.

CONFORMAL SIGMA MODELS WITH ANOMALOUS DIMENSIONS AND RICCI SOLITONS

We present new non-Ricci-flat Kähler metrics with U(N) and O(N) isometries as target manifolds of superconformally invariant sigma models with an anomalous dimension. They are so-called Ricci solitons, special solutions to a Ricci-flow equation. These metrics explicitly contain the anomalous dimension and reduce to Ricci-flat Kähler metrics on the canonical line bundles over certain coset spaces in the limit of vanishing anomalous dimension.

P-Harmonic morphisms, biharmonic morphisms, and nonharmonic biharmonic maps

We study the transformation of a p-harmonic morphism into a q-harmonic morphism via biconformal change of the domain metric and/or conformal change of the codomain metric. As an application of p-harmonic morphisms, we characterize a twisted product among doubly twisted products and a warped product among twisted products using p-harmonicity of their projection maps. We describe those p-harmonic morphisms which are also biharmonic morphisms and give a complete classification of polynomial biharmonic morphisms between Euclidean spaces. Finally, we show that a horizontally homothetic harmonic morphism with harmonic energy density pulls back a nonharmonic biharmonic map to a nonharmonic biharmonic map and that totally geodesic immersing the target manifold of a nonharmonic biharmonic map into an ambient manifold produces a new nonharmonic biharmonic map. These methods are used to construct many examples of nontrivial biharmonic maps.

On Hidden Symmetries of a Super Gauge Theory and Twistor String Theory

We discuss infinite-dimensional hidden symmetry algebras (and hence an infinite number of conserved nonlocal charges) of the N-extented self-dual super Yang-Mills equations for general N\leq4 by using the supertwistor correspondence. Furthermore, by enhancing the supertwistor space, we construct the N-extended self-dual super Yang-Mills hierarchies, which describe infinite sets of graded Abelian symmetries. We also show that the open topological B-model with the enhanced supertwistor space as target manifold will describe the hierarchies. Furthermore, these hierarchies will in turn -- by a supersymmetric extension of Ward's conjecture -- reduce to the super hierarchies of integrable models in D<4 dimensions.

Irreversibility of world-sheet renormalization group flow

We demonstrate the irreversibility of a wide class of world-sheet renormalization group (RG) flows to first order in α′ in string theory. Our techniques draw on the mathematics of Ricci flows, adapted to asymptotically flat target manifolds. In the case of somewhere-negative scalar curvature (of the target space), we give a proof by constructing an entropy that increases monotonically along the flow, based on Perelman's Ricci flow entropy. One consequence is the absence of periodic solutions, and we are able to give a second, direct proof of this. If the scalar curvature is everywhere positive, we instead construct a regularized volume to provide an entropy for the flow. Our results are, in a sense, the analogue of Zamolodchikov's c-theorem for world-sheet RG flows on noncompact spacetimes (though our entropy is not the Zamolodchikov C-function).

η-INVARIANT AND CHERN-SIMONS CURRENT

The author presents an alternate proof of the Bismut-Zhang localization formula of η invariants, when the target manifold is a sphere, by using ideas of mod k index theory instead of the difficult analytic localization techniques of Bismut-Lebeau. As a consequence, it is shown that the R/Z part of the analytically defined η invariant of Atiyah-Patodi-Singer for a Dirac operator on an odd dimensional closed spin manifold can be expressed purely geometrically through a stable Chern-Simons current on a higher dimensional sphere. As a preliminary application, the author discusses the relation with the Atiyah-Patodi-Singer R/Z index theorem for unitary flat vector bundles, and proves an R refinement in the case where the Dirac operator is replaced by the Signature operator.

Poincaré Inequalities for Maps with Target Manifold of Negative Curvature

We prove that for any given homotopic C1-maps u, v: G → M in a nontrivial homotopy class from a metric graph into a closed manifold of negative sectional curvature, the distance between u and v can be bounded by 3(length(u) + length(v)) + C(κ, ρ/20), where ρ>0 is a lower bound of the injectivity radius and −κ<0 an upper bound for the sectional curvature of M. The constant C(κ, e) is given by C(κ, e) = 8 sh−1κ(1) + 8 sh−1κ (1/shκ(e)) with shκ(t) = sinh(√{κ} t). Various applications are given.

On holomorphic mappings of complex manifolds with ball model

We consider holomorphic mappings of complex manifolds with ball model into complex manifolds which are quotients of bounded domains and estimate the dimension of the moduli space of holomorphic mappings in terms of the essential boundary dimension of target manifolds. For this purpose, we generalize a classical uniqueness theorem of Fatou-Riesz for bounded holomorphic functions on the unit disk to one for bounded holomorphic mappings on a bounded C2 domain. This generalization enables us to establish rigidity and finiteness theorems for holomorphic mappings. We also discuss the rigidity for holomorphic mappings into quotients of some symmetric bounded domains. In the final section, we construct examples related to our results.

The Dirichlet energy of mappings from ${\bf B^3}$ into a manifold: density results and gap phenomenon

Weak limits of graphs of smooth maps \(u_k: B^n\to \mathcal{Y}\) with equibounded Dirichlet integral give rise to elements of the space \(\mathrm{cart}^{2,1}(B^n\times \mathcal{Y})\). We assume that the 2-homology group of \(\mathcal{Y}\) has no torsion and that the Hurewicz homomorphism \(\pi_2(\mathcal{Y})\to H_2(\mathcal{Y},{\mathbb{Q}})\) is injective. Then, in dimension n = 3, we prove that every element T in \( \mathrm{cart} ^{2,1}(B^3\times \mathcal{Y})\), which has no singular vertical part, can be approximated weakly in the sense of currents by a sequence of smooth graphs {uk} with Dirichlet energies converging to the energy of T. We also show that the injectivity hypothesis on the Hurewicz map cannot be removed. We finally show that a similar topological obstruction on the target manifold holds for the approximation problem of the area functional.

Non-BPS walls and their stability in 5D supersymmetric theory

An exact solution of non-BPS multi-walls is found in supersymmetric massive T ⋆ ( CP 1 ) model in five dimensions. The non-BPS multi-wall solution is found to have no tachyon. Although it is only metastable under large fluctuations, we can give topological stability by considering a model with a double covering of the T ⋆ ( CP 1 ) target manifold. The N = 1 supersymmetry preserved on the four-dimensional world volume of one wall is broken by the coexistence of the other wall. The supersymmetry breaking is exponentially suppressed as the distance between the walls increases.

Low regularity solutions for the wave map equation into the 2-D sphere

A class of weak wave map solutions with initial data in Sobolev space of order s<1 is studied. A non uniqueness result is proved for the case, when the target manifold is a two dimensional sphere. Using an equivariant wave map ansatz a family of self - similar solutions is constructed. This construction enables one to show ill - posedness of the inhomogeneous Cauchy problem for wave maps.

Construction of harmonic map flows through the method of discrete Morse flows

By the method of discrete Morse flows, we construct the Morse flow of harmonic map from a Riemannian manifold with measurable and bounded metric into that with Alexander non-positive curvature. The construction is directly derived without isometrically embedding of the target manifold into Euclidean space. Our method will be in force in the case, as treated in [11], where a target manifold has the bounded positive curvature.

Special Geometry of Euclidean Supersymmetry I: Vector Multiplets

We construct the general action for Abelian vector multiplets in rigid 4-dimensional Euclidean (instead of Minkowskian) N=2 supersymmetry, i.e., over space-times with a positive definite instead of a Lorentzian metric. The target manifolds for the scalar fields turn out to be para-complex manifolds endowed with a particular kind of special geometry, which we call affine special para-Kahler geometry. We give a precise definition and develop the mathematical theory of such manifolds. The relation to the affine special Kahler manifolds appearing in Minkowskian N=2 supersymmetry is discussed. Starting from the general 5-dimensional vector multiplet action we consider dimensional reduction over time and space in parallel, providing a dictionary between the resulting Euclidean and Minkowskian theories. Then we reanalyze supersymmetry in four dimensions and find that any (para-)holomorphic prepotential defines a supersymmetric Lagrangian, provided that we add a specific four-fermion term, which cannot be obtained by dimensional reduction. We show that the Euclidean action and supersymmetry transformations, when written in terms of para-holomorphic coordinates, take exactly the same form as their Minkowskian counterparts. The appearance of a para-complex and complex structure in the Euclidean and Minkowskian theory, respectively, is traced back to properties of the underlying R-symmetry groups. Finally, we indicate how our work will be extended to other types of multiplets and to supergravity in the future and explain the relevance of this project for the study of instantons, solitons and cosmological solutions in supergravity and M-theory.

Nonlinear Riemann-Hilbert Problems and Boundary Interpolation

We study properties of solutions to non-linear Riemann-Hilbert problems with smooth compact regularly traceable target manifold. The investigations focus on the dependence of solutions with positive winding numbers on additional parameters. While previous results investigated the behavior of the solutions inside the unit disk, we also pay attention to the boundary functions.

Winding behaviour of finite-time singularities of the harmonic map heat flow *

We settle a number of questions about the possible behaviour of the harmonic map heat flow at finite-time singularities. In particular, we show that a type of nonuniqueness of bubbles can occur at finite time, we show that the weak limit of the flow at the singular time can be discontinuous, we determine exactly the (polynomial) rate of blow-up in one particular example, and we show that ‘winding’ behaviour of the flow can lead to an unexpected failure of convergence when the flow is (locally) lifted to the universal cover of the target manifold.

Solving variational problems and partial differential equations mapping into general target manifolds

A framework for solving variational problems and partial differential equations that define maps onto a given generic manifold is introduced in this paper. We discuss the framework for arbitrary target manifolds, while the domain manifold problem was addressed in [J. Comput. Phys. 174(2) (2001) 759]. The key idea is to implicitly represent the target manifold as the level-set of a higher dimensional function, and then implement the equations in the Cartesian coordinate system where this embedding function is defined. In the case of variational problems, we restrict the search of the minimizing map to the class of maps whose target is the level-set of interest. In the case of partial differential equations, we re-write all the equation’s geometric characteristics with respect to the embedding function. We then obtain a set of equations that, while defined on the whole Euclidean space, are intrinsic to the implicitly defined target manifold and map into it. This permits the use of classical numerical techniques in Cartesian grids, regardless of the geometry of the target manifold. The extension to open surfaces and submanifolds is addressed in this paper as well. In the latter case, the submanifold is defined as the intersection of two higher dimensional hypersurfaces, and all the computations are restricted to this intersection. Examples of the applications of the framework here described include harmonic maps in liquid crystals, where the target manifold is a hypersphere; probability maps, where the target manifold is a hyperplane; chroma enhancement; texture mapping; and general geometric mapping between high dimensional manifolds.

A tentative theory of large distance physics

A theoretical mechanism is devised to determine the large distance physics of spacetime. It is a two dimensional nonlinear model, the lambda model, set to govern the string worldsurface to remedy the failure of string theory. The lambda model is formulated to cancel the infrared divergent effects of handles at short distance on the worldsurface. The target manifold is the manifold of background spacetimes. The coupling strength is the spacetime coupling constant. The lambda model operates at 2d distance $\Lambda^{-1}$, very much shorter than the 2d distance $\mu^{-1}$ where the worldsurface is seen. A large characteristic spacetime distance $L$ is given by $L^2=\ln(\Lambda/\mu)$. Spacetime fields of wave number up to 1/L are the local coordinates for the manifold of spacetimes. The distribution of fluctuations at 2d distances shorter than $\Lambda^{-1}$ gives the {\it a priori} measure on the target manifold, the manifold of spacetimes. If this measure concentrates at a macroscopic spacetime, then, nearby, it is a measure on the spacetime fields. The lambda model thereby constructs a spacetime quantum field theory, cutoff at ultraviolet distance $L$, describing physics at distances larger than $L$. The lambda model also constructs an effective string theory with infrared cutoff $L$, describing physics at distances smaller than $L$. The lambda model evolves outward from zero 2d distance, $\Lambda^{-1} = 0$, building spacetime physics starting from $L=\infty$ and proceeding downward in $L$. $L$ can be taken smaller than any distance practical for experiments, so the lambda model, if right, gives all actually observable physics. The harmonic surfaces in the manifold of spacetimes are expected to have novel nonperturbative effects at large distances.