Target Manifold Research Articles

Target-local Gromov compactness

We prove a version of Gromov's compactness theorem for pseudo-holomorphic curves which holds locally in the target symplectic manifold. This result applies to sequences of curves with an unbounded number of free boundary components, and in families of degenerating target manifolds which have unbounded geometry (e.g. no uniform energy threshold). Core elements of the proof regard curves as submanifolds (rather than maps) and then adapt methods from the theory of minimal surfaces.

The Refined Analysis on the Convergence Behavior of Harmonic Map Sequence from Cylinders

In this paper, we are concerned with the convergence behavior of a sequence of harmonic maps from long cylinders with uniformly bounded energy. If the bubbling phenomenon does not occur, we give the length formula of the limit map (i.e., geodesic in the target manifold). Furthermore, we provide a geometric explanation of the energy identity for a sequence of harmonic maps from degenerating Riemann surfaces with uniformly bounded energy, proved by M. Zhu.

On higher order geometric and renormalization group flows

Renormalization group (RG) flows of the bosonic nonlinear σ -model are governed, perturbatively, at different orders of α ′ , by perturbatively evaluated β -functions. In regions where α ′ R c 2 ≪ 1 ( 1 R c 2 represents the curvature scale), the flow equations at various orders in α ′ can be thought of as approximating the full, non-perturbative RG flow. On the other hand, taking a different viewpoint, we may consider the above-mentioned RG flow equations as viable geometric flows in their own right, without any reference to the RG aspect. Looked at as purely geometric flows where higher order terms appear, we no longer have the perturbative restrictions (small curvatures). In this paper, we perform our analysis from both these perspectives using specific target manifolds such as S 2 , H 2 , unwarped S 2 × H 2 and simple warped products. We analyse and solve the higher order RG flow equations within the appropriate perturbative domains and find the corrections arising due to the inclusion of higher order terms. Such corrections, within the perturbative regime, are shown to be small and they provide an estimate of the error that arises when higher orders are ignored. We also investigate higher order geometric flows on the same manifolds and figure out generic features of geometric evolution, the appearance of singularities and solitons. The aim, in this context, is to demonstrate the role of higher order terms in modifying the flow. One interesting aspect of our analysis is that, separable solutions of the higher order flow equations for simple warped spacetimes (of the kind used in bulk-brane models with a single extra dimension), correspond to constant curvature anti de Sitter (AdS) spacetimes, modulo an overall flow parameter dependent scale factor. The functional form of this scale factor (that we obtain) changes on the inclusion of successive higher order terms in the flow.

Non-holomorphic modular forms and SL(2,R)/U(1) superconformal field theory

We study the torus partition function of the SL(2,R)/U(1) SUSY gauged WZW model coupled to N=2 U(1) current. Starting from the path-integral formulation of the theory, we introduce an infra-red regularization which preserves good modular properties and discuss the decomposition of the partition function in terms of the N=2 characters of discrete (BPS) and continuous (non-BPS) representations. Contrary to our naive expectation, we find a non-holomorphic dependence (dependence on \bar{\tau}) in the expansion coefficients of continuous representations. This non-holomorphicity appears in such a way that the anomalous modular behaviors of the discrete (BPS) characters are compensated by the transformation law of the non-holomorphic coefficients of the continuous (non-BPS) characters. Discrete characters together with the non-holomorphic continuous characters combine into real analytic Jacobi forms and these combinations exactly agree with the "modular completion" of discrete characters known in the theory of Mock theta functions \cite{Zwegers}. We consider this to be a general phenomenon: we expect to encounter "holomorphic anomaly" (\bar{\tau}-dependence) in string partition function on non-compact target manifolds. The anomaly occurs due to the incompatibility of holomorphy and modular invariance of the theory. Appearance of non-holomorphicity in SL(2,R)/U(1) elliptic genus has recently been observed by Troost \cite{Troost}.

D =7/D =6 heterotic supergravity with gauged R-symmetry

We construct a family of chiral anomaly-free supergravity theories in D =6 starting from D = 7 supergravity with a gauged noncompact R-symmetry, employing a Hořava-Witten bulk-plus-boundary construction. The gauged noncompact R-symmetry yields a positive (de Sitter sign) D = 6 scalar field potential. Classical anomaly inflow which is needed to cancel boundary-field loop anomalies requires careful consideration of the gravitational, gauge, mixed and local supersymmetry anomalies. Coupling of boundary hypermultiplets requires care with the Sp(1) gauge connection required to obtain quaternionic Kähler target manifolds in D = 6. This class of gauged R-symmetry models may be of use as starting points for further compactifications to D = 4 that take advantage of the positive scalar potential, such as those proposed in the scenario of supersymmetry in large extra dimensions.

NON-CANONICAL PERTURBATION THEORY OF NONLINEAR SIGMA MODELS

We explore the O(N)-invariant Non-Linear Sigma Model (NLSM) in a different perturbative regime from the usual relativistic-free-field one, by using non-canonical basic commutation relations adapted to the underlying O(N) symmetry of the system, which also account for the nontrivial (nonflat) geometry and topology of the target manifold.

A Remark on the Global Existence of a Third Order Dispersive Flow into Locally Hermitian Symmetric Spaces

We prove time-global existence of solutions to the initial value problem for a third order dispersive flow into compact locally Hermitian symmetric spaces. The equation under consideration generalizes two-sphere-valued completely integrable systems modeling the motion of vortex filament. Unlike one-dimensional Schrödinger maps, our third order equation is not completely integrable under the curvature condition on the target manifold in general. The idea of our proof is to exploit two conservation laws and an “almost conserved quantity” which prevents the formation of a singularity in finite time.

Energy identity for approximations of harmonic maps from surfaces

We prove the energy identity for min-max sequences of the Sacks-Uhlenbeck and the biharmonic approximation of harmonic maps from surfaces into general target manifolds. The proof relies on Hopf-differential type estimates for the two approximations and on estimates for the concentration radius of bubbles.

Aligning Coupled Manifolds for Face Hallucination

Many learning-based super-resolution methods are based on the manifold assumption, which claims that point-pairs from the low-resolution representation manifold (LRM) and the corresponding high-resolution representation manifold (HRM) possess similar local geometry. However, the manifold assumption does not hold well on the original coupled manifolds (i.e., LRM and HRM) due to the nonisometric one-to-multiple mappings from low-resolution (LR) image patches to high-resolution (HR) ones. To overcome this limitation, we propose a solution from the perspective of manifold alignment. In this context, we perform alignment by learning two explicit mappings which project the point-pairs from the original coupled manifolds into the embeddings of the common manifold (CM). For the task of SR reconstruction, we treat HRM as target manifold and employ the manifold regularization to guarantee that the local geometry of CM is more consistent with that of HRM than LRM is. After alignment, we carry out the SR reconstruction based on neighbor embedding between the new couple of the CM and the target HRM. Besides, we extend our method by aligning the multiple coupled subsets instead of the whole coupled manifolds to address the issue of the global nonlinearity. Experimental results on face image super-resolution verify the effectiveness of our method.

Biharmonic superspace forN=4mechanics

We develop a new superfield approach to $\mathcal{N}=4$ supersymmetric mechanics based on the concept of biharmonic superspace (bi-HSS). It is an extension of the $\mathcal{N}=4$, $d=1$ superspace by two sets of harmonic variables associated with the two SU(2) factors of the $R$-symmetry group SO(4) of the $\mathcal{N}=4$, $d=1$ super Poincar\'e algebra. There are three analytic subspaces in it: two of the Grassmann dimension 2 and one of the dimension 3. They are closed under the infinite-dimensional ``large'' $\mathcal{N}=4$ superconformal group, as well as under the finite-dimensional superconformal group $D(2,1;\ensuremath{\alpha})$. The main advantage of the bi-HSS approach is that it gives an opportunity to treat $\mathcal{N}=4$ supermultiplets with finite numbers of off-shell components on equal footing with their ``mirror'' counterparts. We show how such multiplets and their superconformal properties are described in this approach. We also define nonpropagating gauge multiplets which can be used to gauge various isometries of the bi-HSS actions. We present an example of a nontrivial $\mathcal{N}=4$ mechanics model with a seven-dimensional target manifold obtained by gauging a U(1) isometry in a sum of the free actions of the multiplet $(\mathbf{4},\mathbf{4},\mathbf{0})$ and its mirror counterpart.

Special geometry of Euclidean supersymmetry III: the local r-map, instantons and black holes

We define and study projective special para-Kahler manifolds and show that they appear as target manifolds when reducing five-dimensional vector multiplets coupled to supergravity with respect to time. The dimensional reductions with respect to time and space are carried out in a uniform way using an -complex notation. We explain the relation of our formalism to other formalisms of special geometry used in the literature. In the second part of the paper we investigate instanton solutions and their dimensional lifting to black holes. We show that the instanton action, which can be defined after dualising axions into tensor fields, agrees with the ADM mass of the corresponding black hole. The relation between actions via Wick rotation, Hodge dualisation and analytic continuation of axions is discussed.

One loop renormalizability of the Poisson–Lie sigma models

We present the proof of the one loop renormalizability in the strict field theoretic sense of the Poisson–Lie σ-models. The result is valid for any Drinfeld double and it relies solely on the Poisson–Lie structure encoded in the target manifold.

A third order dispersive flow for closed curves into almost Hermitian manifolds

We discuss a short-time existence theorem of solutions to the initial value problem for a third order dispersive flow for closed curves into a compact almost Hermitian manifold. Our equations geometrically generalize a physical model describing the motion of vortex filament. The classical energy method cannot work for this problem since the almost complex structure of the target manifold is not supposed to be parallel with respect to the Levi-Civita connection. In other words, a loss of one derivative arises from the covariant derivative of the almost complex structure. To overcome this difficulty, we introduce a bounded pseudodifferential operator acting on sections of the pullback bundle, and eliminate the loss of one derivative from the partial differential equation of the dispersive flow.

The quantum equivariant cohomology of toric manifolds through mirror symmetry

Using mirror symmetry as described by Hori and Vafa, we compute the quantum equivariant cohomology ring of toric manifolds. This ring arises naturally in topological gauged sigma-models and is related to the Hamiltonian Gromov-Witten invariants of the target manifold.

Poisson–Lie T-plurality with spectators

Using a suitable extension of Drinfel’d double we derive formulas for the Poisson–Lie T-plurality transformation with spectators, i.e., for σ-models whose group of generalized isometries does not act transitively on the target manifold. These formulas are further applied to transformations of the gluing matrices that define the boundary conditions of the σ-models.

Semi-Implicit Approximation of Wave Maps into Smooth or Convex Surfaces

A semi-implicit scheme using linear finite elements for the approximation of wave maps into smooth or convex surfaces is devised, and its stability is analyzed. Convergence is established for the case of the unit sphere as a target manifold, which is unconditional in the case of $(2+1)$ Minkowski space. Numerical experiments illustrate the theoretical results.

Linear Biharmonic Maps into Sol, Nil and Heisenberg Spaces

In this paper, we study linear biharmonic maps from Euclidean space into Sol, Nil, and Heisenberg spaces using the linear structure of the target manifolds. We show that a linear map from a Euclidean space into Sol, Nil, or Heisenberg space is biharmonic if and only if it is a harmonic map, and give a complete description of such maps.

The heat flow of harmonic maps from noncompact manifolds

We consider the existence, uniqueness and convergence for the long time solution to the harmonic map heat equation between two complete noncompact Riemannian manifolds, where the target manifold is assumed to have nonpositive curvature. As a corollary, we show that if the tension field is in L p , the heat flow exists globally and uniquely which converges at infinity under an additional condition.

Topological sigma B model in 4-dimensions

We propose a 4-dimensional version of topological sigma B-model, governing maps from a smooth compact 4-manifold M to a Calabi-Yau target manifold X. The theory depends on complex structure of X, while is independent of Kahler metric of X. The theory is also a 4-dimensional topological field theory in the sense that the theory is independent of variation of Riemannian metric of the source 4-manifold M, potentially leading to new smooth invariant of 4-manifolds. We argue that the theory also comes with a topological family parametrized by the extended moduli space of complex structures.

Kinks in a Nonlinear Massive Sigma Model

We describe the kink solitary waves of a massive nonlinear sigma model with an S2 sphere as the target manifold. Our solutions form a moduli space of nonrelativistic solitary waves in the long wavelength limit of ferromagnetic linear spin chains.