Pierre Bénard
email: pierreg.benard@inrialpes.fr

Adrien Bousseau
email: adrien.bousseau@imag.fr

Joëlle Thollot
email: joelle.thollot@imag.fr

Dynamic Solid Textures for Real-Time Coherent Stylization

Keywords: CR Categories: I.3.7 [Computing Methodologies]: Computer Graphics-Three-Dimensional Graphics and Realism; Non-photorealistic rendering, temporal coherence, real-time rendering, stylization, infinite zoom

Figure 1: Dynamic solid textures (on the left) serve as a basis to create a large range of temporally coherent styles. From left to right: a complex scene (538k tris) is rendered in real-time in a watercolor style, a collage style and a binary style.

Introduction

Many non-photorealistic rendering approaches aim at depicting 3D scenes with styles that are traditionally produced on 2D media like paper. The main difficulty suffered by these methods is temporal coherence when stylizing dynamic scenes. This problem arises from the contrary goals of depicting a 3D motion (induced by the camera or the 3D objects of the scene) while preserving the 2D characteristics inherent to any style marks (pigments, strokes, etc). Achieving these goals without introducing visual artifacts implies the concurrent fulfilment of three constraints. First, the style marks should have a constant size and density in the image in order to preserve the 2D appearance of the medium. Second, the style marks should follow the motion of the 3D objects they depict to avoid the sliding of the style features over the 3D scene (shower door effect) [START_REF] Meier | Painterly rendering for animation[END_REF]]. Finally, a sufficient temporal continuity between adjacent frames is required to avoid popping and flickering. Many solutions have been proposed to provide a trade-off between these three constraints [START_REF] Gooch | Non-Photorealistic Rendering[END_REF][START_REF] Strothotte | Non-Photorealistic Computer Graphics: Modeling, Rendering and Animation[END_REF]. Unfortunately, the integration of these methods in existing rendering pipelines (such as game engines) remains challenging due to their computation cost and specialized data structures.

In this paper we describe dynamic textures, a method that facilitates the integration of temporally coherent stylization in real-time rendering pipelines. Dynamic textures address the specific problem of temporal coherence for color region stylization, leaving contour stylization for future work. Our method uses textures as simple data structures to represent style marks. This makes our approach especially well suited to media with characteristic textures (eg. watercolor, charcoal, stippling), while ensuring real-time performances due to the optimized texture management of modern graphic cards.

Central to our technique is an object space infinite zoom mechanism that guarantees a quasi-constant size and density of the texture elements in screen space for any distance from the camera. This simple mechanism preserves most of the 2D appearance of the medium supported by the texture while maintaining a strong temporal coherence during animation. The infinite zoom illusion can be applied to both 2D and 3D textures. However, we present our approach for 3D textures (the dynamic solid textures) which alleviate the need for complex computation or manual definition of 2D parameterizations over the 3D surfaces. These solid textures can be produced either procedurally or by synthesis from a 2D exemplar. In order to demonstrate the effectiveness of our approach, we integrated it into the OGRE1 game rendering engine. The performances measured in this environment indicate the low impact of the proposed method on framerate, even for complex 3D scenes. The different styles implemented in this prototype illustrate the simplicity and efficiency of this approach for real-time non photorealistic rendering.

Previous Work

In order to solve the contradiction between a 2D style and a 3D motion, [START_REF] Meier | Painterly rendering for animation[END_REF] proposes in her seminal work on painterly rendering to decorrelate the appearance of style marks from their motion. In her approach, style marks (paint strokes in her case) are drawn with constant size billboards which preserve their 2D appearance. Each mark is then attached to a 3D anchor point on the 3D object that it depicts. Although this approach has been extended to numerous styles (painterly [START_REF] Daniels | Deep canvas in disney's tarzan[END_REF][START_REF] Vanderhaeghe | Dynamic point distribution for stroke-based rendering[END_REF], stippling [START_REF] Pastor | Real-time animated stippling[END_REF], watercolor [START_REF] Bousseau | Interactive watercolor rendering with temporal coherence and abstraction[END_REF]), the data structure required to manage the anchor points and the expensive rendering of each individual style element makes this family of methods not well-suited for real-time rendering engines.

The individual management of style marks can be avoided by grouping the style marks in textures. The Dynamic Canvas method [START_REF] Cunzi | Dynamic canvas for immersive non-photorealistic walkthroughs[END_REF]] applies a 2D paper texture over the screen in order to stylize a 3D environment during a real-time walkthrough. This approach reproduces 3D transformations of the camera with 2D transformations of the texture. Translations in depth of the camera are mimicked with an infinite zoom mechanism that preserves a quasi-constant size of the texture in screen space. Although this approach provides a convincing trade-off between the 3D motion of the camera and the 2D appearance of the paper, it is limited to navigation of static scenes with a restricted set of camera motions. Moreover, the image space mechanisms proposed by Cunzi et al. models the scene as a single plane which leads to sliding artifacts for strong parallax. [START_REF] Coconu | Real-time pen-and-ink illustration of landscapes[END_REF] and [START_REF] Breslav | Dynamic 2d patterns for shading 3d scenes[END_REF] adopt an approach similar to Dynamic Canvas by applying a stylization texture in screen space on each object of the scene. In these methods, the projected 3D transformations of the objects are approximated by their closest 2D transformations. Because the textures are only transformed in screen space, these methods wellpreserve the 2D appearance of style marks. On the other hand, the approximation of a 3D transformation by a 2D transformation can lead to sliding effects for extreme 3D motions. A novel approach proposed recently by [START_REF] Han | Multiscale texture synthesis[END_REF] introduces an infinite zoom mechanism. This 2D multiscale texture synthesis algorithm generates texture elements on the fly during the zoom. The zoom illusion produced by this method is more accurate than the one of Cunzi et al. but its computing cost limits its integration in real-time rendering pipelines. In this paper we extend the Dynamic Canvas infinite zoom mechanism to object space textures, which allows the realtime stylization of dynamic objects without any sliding effects.

The above approaches tend to sacrifice accuracy in 3D motion to preserve the 2D appearance of the style. However, the converse can also lead to reasonable solutions. This is the approach adopted by art maps [START_REF] Klein | Nonphotorealistic virtual environments[END_REF]] and tonal art maps [START_REF] Praun | Real-time hatching[END_REF]] that map a stylization texture directly on the 3D objects of the scene. Using object space textures, the style marks are perfectly attached to the object but are severely distorted by the perspective projection. The art maps solution relies on the mipmapping mechanism to adapt the scale of the texture according to the distance to the camera. This approach corrects the texture compression induced by depth and can be extended to the correction of perspective deformation using the more complex ripmaps mechanism [START_REF] Klein | Nonphotorealistic virtual environments[END_REF]].

As noted by [START_REF] Praun | Real-time hatching[END_REF], higher temporal coherence can be obtained for binary styles by including the binary marks of one tonal art map level into the next level. [START_REF] Freudenberg | Walk-Through Illustrations: Frame-Coherent Pen-and-Ink Style in a Game Engine[END_REF] prove the performance of these methods by integrating an art map based non photorealistic rendering in the Fly3D game engine.

The infinite zoom mechanism described in this paper extends the art maps approaches in several ways. First, while the mipmap mechanism addresses the texture compression due to minification (zooming out), it produces simple linear blending for texture magnification (zooming in). Our mechanism in opposition produces quasi-constant size texture elements for any distance to the camera. Then, by combining several scales of the texture at a time, our approach is less specialized to binary styles than tonal art maps. Finally, mapping art maps on 3D objects requires the definition of a 2D parameterization of the 3D surfaces. The automatic definition of parameterization that avoid texture distortions or visible seams is still an active research topic. In the case of tonal art maps, [START_REF] Praun | Real-time hatching[END_REF] propose to automatically compute the parameterization using lapped textures but this approach requires additional data structures. Because our approach is based on solid textures, it does not require this definition of additional surface parameterization.

Dynamic Solid Textures

We present in this section the object-space infinite zoom mechanism central to the dynamic solid textures. Various applications of our approach for the real-time coherent stylization of 3D animations will be described in section 4.

Object Space Infinite Zoom Mechanism

The contradictory goals of the infinite zoom illusion are to maintain the size of the texture elements as constant as possible in screen space while preserving the enlarging or shrinking of the texture elements required for a convincing feeling of zooming in or out. More generally, the infinite zoom on a signal can be seen as the infinite growth of its frequency. For an 1d signal (i.e. a sound) it corresponds to an endlessly increasing pitch, as demonstrated by [START_REF] Shepard | Circularity in judgments of relative pitch[END_REF]. For 2D images, [START_REF] Cunzi | Dynamic canvas for immersive non-photorealistic walkthroughs[END_REF] and [START_REF] Han | Multiscale texture synthesis[END_REF] consider this process as the infinite generation of new visible details.

Drawing inspiration from the procedural noise function of [START_REF] Perlin | An image synthesizer[END_REF] and the infinite zoom mechanism of Dynamic Canvas [START_REF] Cunzi | Dynamic canvas for immersive non-photorealistic walkthroughs[END_REF]], our method relies on the fractalization of a texture. In Dynamic Canvas, a self-similar image is obtained by linearly blending n octaves (doubled frequency) of a texture. When the observer moves forward or backward, the frequencies of the octaves are continuously shifted to produce an illusion of zoom.

We similarly define a dynamic solid texture as the weighted sum of n octaves Ωi of the original solid texture. Each 3D object is then carved in such a solid texture, which naturally ensures a convincing feeling of zooming in and out: the texture will appear twice bigger when the object is twice closer to the camera. But care must be taken to keep the texture elements at a quasi-constant size in screen space. To this purpose, we introduce the notion of zoom cycle that occurs every time the appearant size of the texture doubles. In that case, each octave is replaced by the following one and a new high frequency octave is created, as illustrated in Fig. 2. Note that the fractalization process introduces new frequencies in the texture, along with a loss of contrast, as discussed in section 5. While reducing the number of octaves limits these artifacts, it also makes the apparition and disparition of texture elements more visible. Empirically we observed that n = 4 octaves is enough to deceive human perception.

Proposed Algorithm

In practice, an object is embedded in its dynamic solid texture by deriving texture coordinates from vertex coordinates in the local mesh frame. This object space texturing solves the Dynamic Canvas limitation of approximating an entire scene with a plane. For the sake of simplicity, we use only one cube of solid texture that we sample at different rates to retrieve all n octaves. For a vertex p(x, y, z) at distance zcam from the camera, the 3D texture coordinates (u, v, w) for the octave i are given by:

(u, v, w)i = 2 i-1 (x, y, z)/2 log 2 (zcam)
In this equation, the 2 i-1 term scales the sampling rate so that the texture retrieved for one octave is twice smaller than the texture for the previous octave. The log 2 (zcam) term accounts for the refreshing of the octaves at each zoom cycle: the distance zcam can be decomposed in log2(zcam) zoom cycles, making log 2 (zcam) the indicator of how many times each octave has been doubled during the zoom.

During a zoom cycle, we modulate each octave with a weight αi=1...n(s). These weights are dependent on s, the interpolation factor between the beginning and the end of the cycle:

s = log 2 (zcam) -log 2 (zcam) ∈ [0, 1]
We must impose three constraints on the weights in order to ensure a smooth transition during the frequency shift (Fig. 3). First, to avoid the sharp appearance/disappearance of texture elements, the first octave should appear at the beginning of the cycle while the last should disappear at its end: α1(0) = 0 and αn(1) = 0

Second, the weight of the intermediate octaves at the end of the cycle should be equal to the weight of the following octaves at the beginning of the next cycle:

αi(1) = αi+1(0) ∀i ∈ {1, . . . , n -1}
Finally, the weights should sum to 1 to preserve a constant intensity.

In our implementation, we use a linear blending which is fast to compute and coherent with the linearity of the zoom. We choose the following weights:

α1(s) = s/2 α2(s) = 1/2 -s/6 α3(s) = 1/3 -s/6 α4(s) = 1/6 -s/6
The full process is illustrated in Fig. 2. The red line highlights the frequency shift. It corresponds to the distance after which the zoom cycle restarts. Note the frequency similarity between the octaves i and i + 1 for two consecutive zoom cycles. This correspondence ensures the texture continuity after the blending (last column).

Implementation details

We have implemented this infinite zoom algorithm in GLSL2 and integrated it in the rendering engine OGRE (see shaders source code in appendix A). The vertex shader assigns the 3D texture coordinates of the vertex which can be the position of the vertices in the local object coordinate system (for deformable objects, this position in an undeformed pose of the object). The user can also specify an additional scaling factor to define the global size of the texture with regards to the depicted object. We compute the distance between a vertex and the camera as its z coordinate in the camera frame.

Then the fragment shader blends linearly the four octaves of the solid texture for each pixel, according to the formula detailed in the previous section.

We produced the solid textures presented in this paper following two different approaches. We generated some of them procedurally (Perlin noise [START_REF] Perlin | An image synthesizer[END_REF][START_REF] Olano | Modified noise for evaluation on graphics hardware[END_REF], for example) using shaders. The more natural and complex textures have been synthesised from 2D exemplars using the algorithm proposed by [START_REF] Kopf | Solid texture synthesis from 2d exemplars[END_REF]. In this case, the texture cube is stored in the graphic card memory (typically 128 × 128 × 128 RGB pixels, i.e. 6 MB without compression in DirectDraw Surface format).

Results

We compare in Fig. 4 our dynamic solid textures with traditional texture mapping. With a standard (meaning fixed scale) mapping, the size of texture elements varies with depth due to perspective projection. On the contrary, with our infinite zoom approach, texture elements keep a globally constant size in image space independent of the zoom factor. Compared with Dynamic Canvas in Fig. 5, our approach suffers from perspective deformations when the surface is almost tangential to the viewing direction and from discontinuities at occlusion boundaries. However, these artifacts are limited by the infinite zoom mechanism that effectively reduces the apparent depth of the scene. On the other hand, as highlighted by the red dot on Fig. 5, sliding effects occur with the Dynamic Canvas approach. In our case, the texture elements follow perfectly the 3D motion of the scene. We refer the reader to the accompanying video for a better illustration of these sliding effects compared to the accurate motion and convincing infinite zoom produced by our approach. Note that the approaches of [START_REF] Coconu | Real-time pen-and-ink illustration of landscapes[END_REF] and [START_REF] Breslav | Dynamic 2d patterns for shading 3d scenes[END_REF] would suffer from the same sliding problem.

The main advantages of our method are its simplicity of integration in existing rendering engines and its real-time performance.

Our implementation in OGRE induces a small additional cost on the order of 10% in comparison with a traditional gouraud shading (computed in a shader). For the complex scene of Fig. 6 (135k tris) rendered at a resolution of 1280 × 1024 pixels, the framerate decrease from 70 fps to 65 fps with a 2.4GHz Core 2 Duo 6600, 4Go memory and a Geforce 8800 GT. This makes dynamic solid textures perfectly suited for real-time applications such as video games.

Application to coherent stylization

The technique of dynamic solid textures introduced in this paper can be used in a variety of rendering methods simply by replacing standard textures. We illustrate this principle with three real-time stylization algorithms (see Fig. 6 and7). Two are inspired by existing styles (watercolor and binary style) and one is an original style that benefits from the variety of textures that our method can combine in one image (collage style).

Watercolor

We propose to extend the static watercolor pipeline of [START_REF] Bousseau | Interactive watercolor rendering with temporal coherence and abstraction[END_REF] to dynamic scenes. This approach makes use of 2D gray level textures to mimic the variation of watercolor pigments density on paper. We simply replace these textures by our dynamic solid textures. As a result, each of the watercolor effects related to the texture (pigmentation pattern, wobbling) remains totally coherent during the animation. Fig. 6b and the accompanying video illustrate the richness of the medium and the temporal coherence obtained with this approach.

We compared our method with the texture advection approach proposed by [START_REF] Bousseau | Video watercolorization using bidirectional texture advection[END_REF] for video watercolorization (see the accompanying video for the visual comparison). Note that the bidirectional advection requires the knowledge of the entire anima- tion sequence, which makes it unsuitable for real-time applications. However it produces animated watercolor with almost no noticeable sliding or deformations and thus gives a very strong 2D appearance on still frames. As can be seen on the video, the motion of the texture elements -perceived and interpreted as tridimensional by the observer -tends to conceal their 2D characteristics during an animation. Consequently we believe that the additional perspective deformation produced by our approach,although noticeable on still frames, is a small degradation compared to the vivid perception of perspective induced by the motion cues, particularly considering the performance gain.

Binary style

We obtain black-and-white shading styles (pen-and-ink, stippling, charcoal. . . ) using a rendering pipeline very similar to the one of [START_REF] Durand | Decoupling strokes and high-level attributes for interactive traditional drawing[END_REF]. In this method, strokes are simulated by truncating a threshold structure -a grayscale texture seen as a heightfield -at different height with respect to the target tone. We apply a similar threshold on our dynamic solid textures. Fig. 6c illustrates the diversity of binary styles we can combine in one image by simply using a different dynamic solid texture for each object. During the animation, binary strokes appear and disappear progressively thanks to the infinite zoom mechanism.

Note that contrary to existing methods for pen-and-ink stylization [START_REF] Hertzmann | Illustrating smooth surfaces[END_REF][START_REF] Praun | Real-time hatching[END_REF], our solid texture based approach does not orient the binary strokes along the principal directions of the surface. This limitation of the solid textures is discussed in section 5.

Collage

We finally propose a new stylization process that we call collage. This new style takes advantage of the diversity of the texture gallery that one can synthesize. Traditional collage consists in creating an image as a composition of several small strips of paper with various colors and textures. We mimic this style by assigning different dynamic solid textures to each tone of the image (obtained by a dis-cretized shading model similar to toon shading [START_REF] Lake | Stylized rendering techniques for scalable real-time 3d animation[END_REF]).

In order to reinforce the paper aspect of the collage, a white border and a wobbling effect is added between the tone strips (Fig. 6d).

The accompanying video illustrates the accurate temporal coherence of the paper strips, which would be difficult to obtain without our dynamic solid textures.

Discussion and Future Work

Infinite zoom mechanism The main limitation of our method, shared with Dynamic Canvas [START_REF] Cunzi | Dynamic canvas for immersive non-photorealistic walkthroughs[END_REF]], texture advection [START_REF] Bousseau | Video watercolorization using bidirectional texture advection[END_REF]] and to some extent with mipmaps-based approaches [START_REF] Klein | Nonphotorealistic virtual environments[END_REF][START_REF] Praun | Real-time hatching[END_REF][START_REF] Freudenberg | Walk-Through Illustrations: Frame-Coherent Pen-and-Ink Style in a Game Engine[END_REF], is the linear blending of multiple octaves that creates new frequencies and induces a global contrast loss compared with the original texture. Self-similar textures, such as Perlin fractal noise, paper or watercolor textures, are not severly affected by this blending. However, more structured textures can be visually altered, as individual features tend to overlap (the checkerboard texture being an extreme case). We plan to address this limitation in future work.

One possible solution would be to replace the linear blending by a salience aware blending in the spirit of the work of [START_REF] Grundland | Cross dissolve without cross fade: preserving contrast, color and salience in image compositing[END_REF]. This blending should better preserve the relevant features of the texture. The required salience data could be easily encoded in solid feature maps [START_REF] Wu | Feature matching and deformation for texture synthesis[END_REF].

Texture mapping

We chose to develop the infinite zoom mechanism for solid textures because it avoids the need for an adequate 2D parameterization. However, the counterpart of this choice is that textures are decorrelated from the 3D surfaces. This can be seen as a limitation for styles that benefit from the orientation of texture elements along the surface. An example of such styles is pen-andink, for which it has been shown that orienting strokes along the principal directions of the surface emphasizes the shape of the objects [START_REF] Hertzmann | Illustrating smooth surfaces[END_REF][START_REF] Praun | Real-time hatching[END_REF]]. Nevertheless, the infinite zoom mechanism described in this paper is independent of the texture dimension and can also be applied on 2D textures if the required parameterization is available. A limitation shared by all texture-based approaches concerns the texturing of the highly deformable objects. In this case, defining the texture coordinates as the vertices positions of the undeformed object creates additional dilatation/shrinking of the simulated medium, decreasing the perception of a two-dimensional look.

Conclusion

We introduced in this paper the dynamic solid textures, that maintain a quasi-constant size of textures in screen space independently of depth. In the context of non-photorealistic rendering, dynamic solid textures preserve the 2D appearance of style marks while ensuring temporal coherence. In addition, the simplicity and efficiency of this approach facilitates its use in real-time applications such as video games.

In the future we plan to extend the range of applications of our method. As an example, any type of feature line that stays fixed on the object surface (ridges [START_REF] Ohtake | Ridgevalley lines on meshes via implicit surface fitting[END_REF]], demarcating curves [START_REF] Kolomenkin | Demarcating curves for shape illustration[END_REF]) could be stylized according to a dynamic texture.

Figure 2 :

 2 Figure2: Tridimensional infinite zoom mechanism (using a solid checkerboard texture for illustration purpose). Observe the globally invariant frequency of the solid texture produced by our algorithm (last column). The red line delineates the boundary between two consecutive zoom cycles. Note the frequency similarity between two adjacent octaves for two consecutive zoom cycles (blue circles).

Figure 3 :

 3 Figure 3: Evolution of the blending weights during a zoom cycle. Note the continuity at the beginning and the end of the zoom cycle, which ensures an infinite number of cycles.

Figure 4 :

 4 Figure 4: Comparison of our dynamic solid textures (d) with standard texture mapping (c). Regardless of the distance from the camera (top and bottom row), our method provides quasi-constant size of texture elements.

Figure 5 :

 5 Figure 5: Comparison with Dynamic Canvas: note the sliding of the texture with the Dynamic Canvas method, highlighted by the red dot.

Figure 6 :

 6 Figure 6: Complex scene (135k tris) rendered with various styles.

Figure 7 :

 7 Figure 7: Simple scene rendered with various styles.

http://www.ogre3d.org

OpenGL Shading Language : http://www.opengl.org/documentation/glsl/

Acknowledgements

Most 3D models used in this research were kindly provided by Laurence Boissieux. The authors would like to thank Kartic Subr, Alexandrina Orzan, Frédo Durand, Franc ¸ois Sillion and the anonymous reviewers for their valuable comments and suggestions. Many thanks to Tilke Judd for the voice-over.

A Infinite Zoom Shaders

Vertex Shader § uniform f l o a t o b j s c a l e ; v a r y i n g f l o a t d i s t ;

/ / o c t a v e w e i g h t a c c o r d i n g t o t h e i n t e r p o l a t i o n

f a c t o r f l o a t a l p h a 1 = s / 2 . 0 ; f l o a t a l p h a 2 = 1 . 0 / 2 . 0s / 6 . 0 ; f l o a t a l p h a 3 = 1 . 0 / 3 . 0s / 6 . 0 ; f l o a t a l p h a 4 = 1 . 0 / 6 . 0s / 6 . 0 ;


Julia Monnin
email: monninjulia@yahoo.fr

Hélène Loevenbruck
email: helene.loevenbruck@gipsa-lab.inpg.fr

Mary E Beckman
email: mbeckman@ling.ohio-state.edu

Jan Edwards
email: jedwards2@wisc.edu

Vowel context and frequency effects in dorsal and coronal acquisition in Drehu and French

Background and aim

The age-typical misarticulations that often occur in phonological development can be described in terms of the interaction of two types of production constraints: (1) -markedness‖ constraints that reflect universal age-related differences in vocal tract morphology and the child's immature motor system (e.g., [START_REF] Locke | Phonological Acquisition and Change[END_REF][START_REF] Davis | Acquisition of serial complexity in speech production: A comparison of phonetic and phonological approaches to first word production[END_REF], and (2) -phonotactic‖ constraints that reflect the potentially language-specific distribution in the child's growing lexicon of elements such as consonants, vowels, and coarticulated consonant-vowel sequences (e.g., [START_REF] Pye | A comparison of initial consonant acquisition in English and Quiché[END_REF][START_REF] Edwards | Some cross-linguistic evidence for modulation of implicational universals by language-specific frequency effects in phonological development[END_REF]. One misarticulation that is very typical for young children who are acquiring English or German is -velar fronting‖the substitution of coronal sounds such as [t] for dorsal sounds such as [k]. Based on his review of diary studies and a few larger norming studies of children acquiring English and several other languages, [START_REF] Locke | Phonological Acquisition and Change[END_REF] followed [START_REF] Jakobson | Kindersprache, aphasie und allgemeine lautgestze[END_REF] in describing more accurate production of [t] relative to [k] as a markedness universal. More recent cross-sectional studies of children acquiring Greek [START_REF] Nicolaidis | Acquisition of lingual obstruents in Greek[END_REF], Japanese [START_REF] Beckman | Language-specific and language-universal aspects of lingual obstruent productions in Japanese-acquiring children[END_REF], and Drehu or French [START_REF] Monnin | Language-specific influence on phoneme development: French and Drehu data[END_REF], by contrast, offer evidence suggesting that any tendency to produce [t] more accurately than [k] is modulated by language-specific frequencies. What is not clear is whether the modulation is due to consonant frequency per se. There may instead be phonotactic constraints related to the language-specific frequencies of -fronted frames‖ versus -backed frames‖ [START_REF] Davis | Acquisition of serial complexity in speech production: A comparison of phonetic and phonological approaches to first word production[END_REF]. To begin to differentiate the two hypotheses, the present study examines Drehu (an Austronesian language spoken in New Caledonia) and French data on [k] and [t] in different vowel contexts.

Evaluation of the vowel context effect

Word-initial consonant-vowel sequences chosen for cross-linguistic comparison were elicited from 4 groups of about 40 French-acquiring children aged 2 through 5 years, and from 3 groups of about 16 Drehu-acquiring children, aged 3 through 5. Words in the child's mother tongue were elicited in a picture-prompted word-repetition task, and target sounds were transcribed by a native speaker. This general trend confirms the results reported for this dataset by [START_REF] Monnin | Language-specific influence on phoneme development: French and Drehu data[END_REF]. What these new figures make clear, however, is that the difference in accuracy is modulated by vowel context. In both languages, the difference in accuracy is substantial only in one of the three vowel contexts, namely [u] for Drehu and [i] for French. The greater accuracy of [k] before [u] in Drehu is in keeping with the intrinsic difficulty of producing coronal constrictions before back vowels described in the frame/content account of markedness universals based on the mandibular cycle. The greater accuracy of [k] before [i] in French, on the other hand, is unexpected.

Evaluation of the frequency effect

In addition to recording productions by Drehu-and French-speaking children, we also had recorded and transcribed two small corpora of child-directed speech. We use these two transcribed corpora to estimate the relative type frequencies of all of the word-initial consonant-vowel sequences that we elicited. Figure 2

Conclusion and future research

The relationships plotted in Figures 1 and2 support the idea that the relative accuracy of [t] and [k] in development reflects the markedness of particular combinations of lingual stop and coarticulated vowel, as modulated by language-specific phonotactics. Future analyses will examine stop burst spectra and vowel formants to evaluate whether there are cross-language phonetic differences in the consonant and vowel targets which might also contribute to the accuracy differences.

Figure 1 .

 1 Figure 1. Growth in accuracy of [k] and [t] productions in the context of [a, i, u]. Curves are summed coefficients of fixed effects in a mixed-effects logistic regression model with age, consonant, vowel, and the consonant-by-vowel interaction as fixed effects and individual-level intercepts to capture differences in overall accuracy across the children.

Figure 1

 1 Figure 1 shows growth in accuracy by age for word-initial [k] and [t] in the 3 vowel contexts where both consonants were elicited for Drehu. (Here French -[a]‖ includes all 3 low vowels: /a, ,  /.) As the figure makes clear, both stops are mastered early in both languages, but [k] is somewhat more accurate for the youngest children.

  shows the relationship between mean accuracy of [k] or [t] and these CV frequencies for the youngest group of Drehu-speaking children and for the two youngest groups of French-speaking children. The otherwise puzzling effects of the front-vowel context on accuracy of [t] seem to reflect the very low frequency of the [ti] sequence in both languages.

Figure 2 .

 2 Figure 2. Accuracy of [k] and [t] by CV frequency, in the context of [a, i, u] for Drehu-speaking 3-year-olds (left), and in the 6 contexts in which both consonants were elicited for French-speaking 2-and 3-year-olds (middle and right).


Osvaldo P Santillan

New non compact Calabi-Yau metrics in D=6

come

Introduction

The development of the subject of Calabi-Yau (CY) manifolds is an illustrative example of the interplay between algebraic geometry and string theory. On one hand, CY spaces are interpreted as internal spaces of string and M-theory giving supersymmetric field theories after compactification. In fact, CY 3-folds may provide compactifications which are more realistic than the ones corresponding to other Ricci flat manifolds such as G 2 holonomy spaces, for which the generation of chiral matter and non abelian gauge symmetries seems harder (but not impossible) to achieve. On the other hand, string theory compactifications stimulated several new trends in the algebro-geometrical aspects of CY spaces, an example is the subject of mirror symmetry.

By definition a CY manifold is a compact Kahler n-dimensional manifold with vanishing first Chern class. The Yau proof of the Calabi conjecture implies that these manifolds admit a Ricci flat metric and their holonomy is reduced from SO(2n) to SU(n) [1]. Although these compact Ricci flat metrics do exist, no explicit expression has been found. For the non compact case, the definition usually adopted is that a CY manifold is a Ricci flat Kahler manifold, which also implies that the holonomy is reduced to SU(n) or to a smaller subgroup. In this case, several explicit metrics are known. One of the oldest examples are the asymptotically conical metrics presented in [2]. Another interesting examples have been found in [3] and [4]. The last ones posses curvature singularities, but for some of them the contribution to the gravitational action is finite and this makes plausible that they may be extended to a six dimensional gravitational instanton. In fact, some of these solutions were identified as the asymptotic form of the so called generalized Bando-Kobayashi-Tian-Yau (BKTY) metrics [START_REF] Bando | Proc. 21st Int. Taniguchi Symp, Notes in Pure Math[END_REF] and [START_REF] Tian | [END_REF], which are by construction Calabi-Yau. More examples were found recently in [START_REF] Gauntlett | [END_REF]- [START_REF] Chen | [END_REF] and higher dimensional ones in [19]- [20], some of these metrics posses conical singularities but in some cases these singularities have been resolved to give complete metrics.

A relatively new achievement in the subject is the characterization of the supergravity backgrounds corresponding to D6 branes wrapping a complex submanifold inside a 4-dimensional hyperkahler space. It was shown by Fayyazuddin in [9] that these D6 backgrounds are described in terms of a linear system of equations and the uplift to eleven dimensions results in a purely geometrical background of the form M 1,4 × Y 6 where Y 6 is a Calabi-Yau space. The Ricci flat Kahler metric on Y 6 is therefore determined in terms of this linear system. The present work generalize this idea and is organized as follows. In section 2 the geometry of certain Calabi-Yau spaces in six dimensions with an isometry preserving the metric and the full SU(3) structure is characterized. In addition, a class of these Calabi-Yau 3-folds constructed in terms of an initial hyperkahler structure are presented and the non linear system describing these geometries is written explicitly. It is also shown that this non linear system reduce to the linear one found in [9] when the objects describing the geometry depends on a single complex coordinate and its complex conjugated and also that, in accordance with [9], when the initial hyperkahler metric is the flat one on R 4 the resulting CY is the direct sum of R 2 and the multi-centered hyperkahler 4-metrics [21]. These metrics are of holonomy SU (2) which is a subgroup of SU (3). Section 3 and 4 improve all these situations by dealing with the full non linear problem associated to the flat hyperkahler structure. Particular solutions are constructed and it is shown that the corresponding metrics are of holonomy exactly SU (3). In addition, an infinite family of non compact Calabi-Yau metrics is presented.

Calabi-Yau metrics with an isometry preserving the SU(3) structure

The general form of the SU(3) structure

In this subsection a large family of Calabi-Yau (CY) manifolds with an isometry group of codimension one will be characterized. It will be assumed that the Killing vector V corresponding to this isometry preserve not only the metric, but the full SU(3) structure. It will be convenient to give an operative definition of CY manifolds in six dimensions first, for more details see for instance [START_REF] Gross | Calabi-Yau Manifolds and Related Geometries[END_REF]- [START_REF] Voisin | Hodge theory and complex algebraic geometry[END_REF]. Roughly speaking, a Calabi-Yau manifold M 6 is Kahler manifold, thus complex sympletic, which in addition admits a Ricci-flat metric g 6 . This definition means in particular that there exist an endomorphism of the tangent space J : T M 6 → T M 6 such that J 2 = -I d and for which g 6 (X, JY ) = g 6 (JX, Y ) being X and Y arbitrary vector fields. It is commonly said that the metric g 6 is hermitian with respect to J and the tensor (g 6 ) µα J α ν is skew symmetric, therefore locally it defines a 2-form

ω 6 = 1 2 (g 6 ) µα J α ν dx µ ∧ dx ν . (2.1)
Here x µ is a local choice of coordinates for M 6 . The endomorphism J it is called an almost complex structure. If the Nijenhuis tensor

N (X, Y ) = [X, Y ] + J [X, J Y ] + J [J X, Y ] -[J X, J Y ],
vanish identically for any choice of X and Y then the tensor J will be called a complex structure and M 6 a complex manifold. This is the case for a CY manifold. The Newlander-Niremberg theorem states that there is an atlas of charts for M 6 which are open subsets in C n , in such a way that the transition maps are holomorphic functions. These local charts are parameterized by complex coordinates (z i , z i ) with i = 1, 2, 3 for which the complex structure looks like

J j i = -J j i = iδ j i , J j i = J i j = 0, (2.2)
and for which the metric and the 2-form (2.1) are expressed as follows

g 6 = (g 6 ) ij dz i ⊗ dz j , ( 2.3
)

ω 6 = i 2 (g 6 ) ij dz i ∧ dz j . (2.4)
The form (2.4) is called of type (1, 1) with respect to J, while a generic 2-form containing only terms of the form (dz i ∧ dz j ) or (dz i ∧ dz j ) will be called of type (2, 0) or (0, 2), respectively. In addition a Calabi-Yau manifold is sympletic with respect to ω 6 , in other words dω 6 = 0. A complex manifold which is sympletic with respect to (2.1) is known as Kahler manifold, thus CY spaces are all Kahler. The Kahler condition itself implies that the holomy is reduced from SO [START_REF] Tian | [END_REF] to U (3). Furthermore, the fact that g 6 is Ricci-flat is equivalent to the existence of a 3-form

Ψ = ψ + + i ψ -, (2.5)
of type (3, 0) with respect to J, satisfying the following compatibility conditions [START_REF] Chiossi | Proc. conf. Differential Geometry[END_REF]

ω 6 ∧ ψ ± = 0, ψ + ∧ ψ -= 2 3 ω 6 ∧ ω 6 ∧ ω 6 dV (g 6 ), (2.6
)

ψ + ∧ ψ + = ψ -∧ ψ -= 0.
in such a way that dψ + = dψ -= 0.

(2.7)

In the formula (2.6) dV (g 6 ) denote the volume form of g 6 . In the situations described in (2.7) the holonomy is further reduced from U (3) to SU(3), thus CY manifolds are of SU(3) holonomy. The converse of these statements are also true, that is, for any Ricci flat Kahler metric in D=6 there will exist an SU(3) structure (ω 6 , Ψ) satisfying (2.6) and also

dω 6 = dψ + = dψ -= 0. (2.8)
The knowledge SU(3) structure determine univocally metric g 6 . In fact, the task to find complex coordinates for a given CY manifold may be not simple, but there always exists a tetrad basis e a with a = 1, .., 6 for which the SU(3) structure is expressed as and for which the metric is g 6 = e a ⊗ e a , where the Einstein summation is understood. The description given above just collect general facts about CY manifolds. In the following we will assume that our CY manifold M 6 is equipped with a metric g 6 in such a way that there is a Killing vector V preserving g 6 and the whole SU(3) structure (ω 6 , ψ + , ψ -). 1 For this situation there exists a local coordinate system (α, x i ) with i = 1, .., 5 for which V = ∂ α and for which the metric tensor g 6 take the following form

ω 6 = e 1 ∧
g 6 = (dα + A) 2 H 2 + Hg 5 .
(2.10)

where the function H, the one form A and the metric tensor g 5 are independent on the coordinate α. Thus these objects live in a 5-dimensional space which we denote M 5 . The metric g 5 appearing in (2.10) can be expressed as g 5 = e a ⊗ e a with a = 1, .., 5 for some basis of α-independent 1-forms e a . Then, if V also preserve the SU(3) structure (as we are assuming) one has the decomposition

ω 6 = ω 1 + 1 √ H e 5 ∧ (dα + A), (2.11
)

ψ + = H 3/2 ω 3 ∧ e 5 + ω 2 ∧ (dα + A), (2.12
)

ψ -= -H 3/2 ω 2 ∧ e 5 + ω 3 ∧ (dα + A). (2.13)
Here the 1-form e 5 and the two-forms ω 2 and ω 3 are by definition

e 5 √ H = i ∂α ω 6 , ( 2.14
)

ω 2 = i ∂α ψ + , ω 3 = i ∂α ψ -. (2.

15)

i V denoting the contraction with the vector field V . In fact the triplet of forms ω i may be represented as ω i = e 4 ∧ e i + ijk e j ∧ e k with i = 1, 2, 3 for some convenient choice of e a . By assumption V preserve (2.11)-(2.13) and therefore e 5 and ω i do not depend on the coordinate α, i.e, they are also defined in M 5 .

The task now is to derive the consequences of the CY relations (2.6) and (2.8) for the generic anzatz (2.10)-(2.13). From (2.6) it follows immediately that

ω 1 ∧ ω 2 = ω 1 ∧ ω 3 = ω 2 ∧ ω 3 = 0, (2.16
)

ω 1 ∧ ω 1 = H 2 ω 2 ∧ ω 2 = H 2 ω 3 ∧ ω 3 .
(2.17) Moreover (2.14) together with an elementary formula in differential geometry imply that

d 5 ( e 5 √ H ) = £ ∂α ω 6 -i ∂α d ω 6 .
(2.18)

Here d 5 = ∂ i dx i and £ ∂α is the Lie derivate along the vector ∂ α . But the vector ∂ α , by assumption, preserves ω 6 and ω 6 is closed, thus the right hand side of the last expression vanish and

d 5 ( e 5 √ H ) = 0. (2.19)
The last relation can be integrated, at least locally, to obtain that

e 5 = √ H dy, (2.20)
y being some function of the coordinates x i parameterizing M 5 , which is the momentum map of the isometry. At least locally, one can take the function y defined in (2.20) as one of the coordinates, which leads to the decomposition M 5 = M 4 × R y and d 5 = d 4 + ∂ y dy. The metric (2.10) in this coordinates becomes

g 6 = (dα + A) 2 H 2 + H 2 dy 2 + g(y), (2.21)
where g(y) will be determined below under certain additional assumptions. An analogous calculation taking into account (2.15) shows that

d 5 ω 2 = d 5 ω 3 = 0. (2.22)
In the remaining part of the paper it will be assumed that the forms ω i are defined on M 4 and that they depend on y as a parameter. We are not sure if this is the most general case, but is the one that we were able to deal with. Formally, this means that imply that ω 2 and ω 3 are y independent. Thus only ω 1 is allowed to depend on y. In addition g(y) can be interpreted as a four dimensional metric depending on y as a parameter. Taking into account this considerations, the SU(3) structure (2.11) and (2.13) takes the following form

ω i (X, ∂ y ) = 0. ( 2
ω 6 = ω 1 (y) + dy ∧ (dα + A), ψ + = H 2 ω 3 ∧ dy + ω 2 ∧ (dα + A), (2.24
)

ψ -= -H 2 ω 2 ∧ dy + ω 3 ∧ (dα + A).
The next task will be to find specific examples of this type of Calabi-Yau structures.

Solutions related to hyperkahler 4-dimensional spaces

The method we will employ in order to find particular Calabi-Yau structures of the form (2.24) is to start with an hyperkahler structure ω i defined over a 4-manifold M 4 . As is well known, a 4-dimensional hyperkahler manifold is a Kahler one which admits a Ricci flat metric g 4 . This automatically imply that the holonomy is in SU (2). In fact, these manifolds can be considered as 4-dimensional CY spaces; they are complex with respect to some endomorphism J 1 of the tangent space and that the corresponding form ω 1 is of type (1,1) with respect to J 1 and closed, i.e, d ω 1 = 0. The reduction of the holonomy to SU(2) imply the existence of a complex 2-form ω 2 + i ω 3 is of type (2,0) with respect to J 1 and so

ω 1 ∧ ω 2 = ω 1 ∧ ω 3 = 0, (2.25)
and which, in addition, satisfy

ω 1 ∧ ω 1 = ω 2 ∧ ω 2 = ω 3 ∧ ω 3 dV (g 4 ), (2.26
)

d 4 ω 2 = d 4 ω 3 = 0. (2.27)
The form ω 2 + i ω 3 is in fact the analog of the 3-form Ψ defined in (2.5) for the 4-dimensional case. The conditions stated above imply that all the endomorphisms J i defined through the usual relation

g 4 (X, J i Y ) = ω i (X, Y ), (2.28)
are all complex structures and that M 4 is Kahler with respect to any ω i . In fact any endomorphism of the form

J = a 1 J 1 + a 2 J 2 + a 3 J 3 , a 2 1 + a 2 2 + a 2 3 = 1, (2.29)
is a complex structure as well, so there is an S 2 -bundle of complex structures for any hyperkahler manifold. The triplet of Kahler forms ω i is enough to determine the hyperkahler metric g 4 univocally, as in the six dimensional case. Given an hyperkahler structure ω i , we deform ω 1 to a y-dependent two form

ω 1 (y) = ω 1 -d 4 d c 4 G, ( 2.30)
while keeping ω 2 and ω 3 intact. Here we have introduced the operator d c = J 1 d. In the expression (2.30) G denotes an unknown function which varies on M 4 and depends also on y.

In terms of this anzatz the SU(3) structure (2.24) take the following form

ω 6 = ω 1 -d 4 d c 4 G + dy ∧ (dα + A), ψ + = H 2 ω 3 ∧ dy + ω 2 ∧ (dα + A), (2.31) ψ -= -H 2 ω 2 ∧ dy + ω 3 ∧ (dα + A).
The reason for the choice (2.30) for ω 1 (y) is simple to explain. The form ω 1 in (2.30) is of type (1,1) with respect to the complex coordinates which diagonalize J 1 , and the term d 4 d c 4 G is also of this type. Thus the deformed form ω 1 (y) is of type (1,1) as well. As the form ω 2 + i ω 3 is kept intact and is of type (2, 0) the fundamental condition (2.16) is identically satisfied. Additionally (2.30) is closed with respect to d 4 , and this will simplify considerably the analysis given below.

Given the deformed structure (2.30), the compatibility condition (2.17) imply that

( ω 1 -d 4 d c 4 G) ∧ ( ω 1 -d 4 d c 4 G) = H 2 ω 2 ∧ ω 2 .
(2.32)

But the wedge products appearing in the last equality are all proportional to the volume form dV (g 4 ) of the initial hyperkahler metric g 4 , therefore the relation

( ω 1 -d 4 d c 4 G) ∧ ( ω 1 -d 4 d c 4 G) = M(G) ω 1 ∧ ω 1 , (2.33)
defines a non linear expression M(G) involving G. In the following we will loosely call M(G) a non linear operator, although the correct spelling is that M(G) is the action of a non linear operator over G. We will use this terminology in order to simplify the vocabulary, when there is no place for confusion.

dω 6 = dω 1 -dy ∧ (d 4 d c 4 G y + d 4 A) = 0,
together with the Kahler condition dω 1 = 0 imply that

A = -d c 4 G y , (2.35)
up to a gauge transformation that can be absorbed by a redefinition of the coordinate α in (2.31). Furthermore, the closure of ψ ± gives the following equations

ω 2 ∧ d 4 A = ω 3 ∧ d 4 A = 0, (2.36
)

ω 3 ∧ d 4 H 2 ∧ dy + ω 2 ∧ dy ∧ A y = 0. (2.37)
But the condition (2.36) is identically satisfied by use of (2.35) and therefore redundant, as

ω 2 ∧ d 4 A = -ω 2 ∧ d 4 d c 4 G y = 0,
because ω 2 contain only terms of type (2, 0) and (0, 2) with respect to J 1 while d 4 d c 4 G y is purely of type (1,1). Instead (2.37) give new constraints. The two forms ω 2 and ω 3 are related by

ω 2 (X, Y ) = ω 3 (J 1 X, Y ), (2.38)
and it is a general fact that an equation of the form

ω 2 ∧ α + β ∧ ω 3 = 0, is solved when α = J 1 β.
Taking into account this and the second (2.36) we conclude that

d c 4 H 2 = -∂ y A. ( 2
ω 6 = ω 1 -d 4 d c 4 G + dy ∧ (dα -d c 4 G y ), ψ + = G yy ω 3 ∧ dy + ω 2 ∧ (dα -d c 4 G y ), (2.42
)

ψ -= -G yy ω 2 ∧ dy + ω 3 ∧ (dα -d c 4 G y )
, which can be checked to be closed. The generic form of the 6-dimensional Calabi-Yau metric corresponding to this structure is given by

g 6 = g 4 (y) + G yy dy 2 + (dα -d c 4 G y ) 2 G yy , ( 2.43)
where g 4 (y) is the Kahler 4-dimensional metric corresponding to the deformed Kahler structure

ω 1 (y) = ω 1 -d 4 d c 4 G.
Note that g 4 (y) depends on y as a parameter, in other words, it is a four dimensional Kahler metric which varies as y take different values.

The Fayyazuddin linearization

It is important to remark that the family of SU(3) structures (2.42) and (2.43) found above are completely determined in terms of a single function G and derivates, therefore the task to find our CY metrics have been reduced to solve a single equation (2.41) defining G. This is a non linear equation and the general solution is not known, but it can be solved in some specific examples. We will focus now our attention in particular solutions in this equation. The source of the non linearity of the operator M(G) defined in (2.33) and (2.41) is the quadratic term

Q(G) = d 4 d c 4 G ∧ d 4 d c 4 G, (2.44)
therefore the operator M(G) will reduce to a linear one if Q(G) vanish. This will be the case when the function G is of the form G = G(w, w) where w = f (z 1 , z 2 ) is an holomorphic function of the coordinates (z 1 , z 2 ) which diagonalize the complex structure J 1 [START_REF] Bedford | [END_REF]. This affirmation may be justified as follows. From the formula

dd c G = 2 i G ij dz i ∧ dz j , (2.45) it follows that (2.44) is Q(G) = -8 (G 11 G 22 -G 12 G 21 ) dz 1 ∧ dz 1 ∧ dz 2 ∧ dz 2 .
(2.46)

But the functional dependence G = G(w, w) imply that

G ij = w i w j G ww ,
and by inserting this into (2.46) gives Q(G) = 0. This result may be paraphrased as follows.

If the function G depends only on two complex coordinates (w, w) then the quantity d 4 d c 4 G is essentially a 2-form in two dimensions, therefore the wedge product (2.44) vanish identically.

The situation described above is essentially the one considered by Fayyazuddin in the reference [9] and, if suitable boundary conditions are imposed, the resulting metrics give a dual description of D6 branes wrapping a complex submanifold in a hyperkahler manifold. A simple example is obtained when the initial hyperkahler structure is the flat metric on R 4

g 4 = dz 1 ⊗ dz 1 + dz 2 ⊗ dz 2 ,
(2.47) with the Kahler triplet which is expressed in complex form as

ω 1 = i 2 (dz 1 ∧ dz 1 + dz 2 ∧ dz 2 ), (2.48
)

ω 2 + i ω 3 = dz 1 ∧ dz 2 .
(2.49)

The volume 4-form for (2.47) in complex coordinates is simply

ω 1 ∧ ω 1 = - 1 2 dz 1 ∧ dz 1 ∧ dz 2 ∧ dz 2 .
(2.50)

In these terms the relation (2.33) for the flat metric in R 4 takes the following form

( ω 1 -dd c G) ∧ ( ω 1 -dd c G) = - 1 2 M(G) dz 1 ∧ dz 1 ∧ dz 2 ∧ dz 2 .
(2.51)

In addition the formula (2.45) obtained previously gives the linear term

2 dd c G ∧ ω 1 = 4 (G 11 + G 22 ) ω 1 ∧ ω 1 ,
and also the following quadratic one

Q(G) = dd c G ∧ dd c G = 16 (G 11 G 22 -G 12 G 21 ) ω 1 ∧ ω 1 .
By use of the last two identities it is obtained from (2.51) the following expression for M(G)

M(G) = 1 -4 (G 11 + G 22 ) + 16 (G 11 G 22 -G 12 G 21 ). ( 2

.52)

The fundamental equation defining the Calabi-Yau geometry is then obtained from (2.41) and (2.52), the result is

G yy = 1 -4 (G 11 + G 22 ) + 16 (G 11 G 22 -G 12 G 21 ). ( 2

.53)

The next problem is to find particular solutions of (2.53). As was discussed above if one assume that the solution depends on one complex coordinate w = f (z i ), then the non linear term in (2.53) vanish identically and the resulting equation is

G yy = 1 -4 (G 11 + G 22 ). (2.54)
This is a Laplace equation in five dimensions parameterized by (y, z 1 , z 2 , z 1 , z 2 ). We may derivate (2.54) twice with respect to y which gives a Laplace equation for the quantity V = G yy defining the radius of the "circle" in (2.43). This equation is

V yy + 4 (V 11 + V 22 ) = 0.
(2.55)

In the following we choose w = z 1 . A solution with good behavior at infinite is obtained when we put constant density charge at the planes z 1 = c i , in other words the right hand side of (2.55) is zero up to a delta term of the form δρ = N i=1 q δ(z 1 -c i ) δ(z 1 -c i ) δ(y -a i ). The resulting electrostatic potential is

V = 1 + N i=1 q |z 1 -c i | 2 + (y -a i ) 2 , (2.56)
and it is commonly said that the D6 branes are wrapped on the z 2 plane. We can also find the Kahler potential G explicitly through the relation V = G yy by a double integration

G = 2 q N i=1 (y -a i ) ln{(y -a i ) + (y -a i ) 2 + |z 1 -c i | 2 )} -(y -a i ) 2 + |z 1 -c i | 2 . (2.57)
The metric g 4 (y) corresponding to (2.57) is given by

g 1 1 = (1 + G 11 ) dz 1 ⊗ dz 1 + (1 + G 22 ) dz 2 ⊗ dz 2 = V dz 1 ⊗ dz 1 + dz 2 ⊗ dz 2 .
(2.58)

The full Calabi-Yau metric (2.43) take the following form

g 6 = dz 2 ⊗ dz 2 + V (dy 2 + dz 1 ⊗ dz 1 ) + 1 V (dψ + A 1 dz 1 + A1dz 1) 2 , (2.59)
where A is may be calculated by use of (2.35) and the result satisfy

∇ × A = ± ∇V. (2.60)
Therefore (2.59) is the direct sum of a flat metric in R 2 C and a general Gibbons-Hawking metric in dimension four [21]. As the Gibbons-Hawking metrics are hyperkahler the whole metric (2.59) is of holonomy SU(2), which is a subgroup of SU(3). Our aim in the following is to improve this situation and find Calabi-Yau metrics of holonomy exactly SU(3).

Non trivial Calabi-Yau extensions of the flat metric

As we have shown above, the Fayyazuddin linearization is easy to perform for the flat hyperkahler metric on R 4 . For other hyperkahler structures the task is much harder, and we did not manage to find an explicit solution in those cases. This does not means they do not exist, but that is difficult to find them, at least for us. The problem is that when the initial hyperkahler structure has non trivial curvature the laplace type equation (2.54) is generalized to some equation which is schematically

g ij G ij + G yy = 1, (3.61)
up to constants which are irrelevant for our discussion. Here g ij is the inverse metric of the initial hyperkahler structure. After writing this equation explicitly one assume a functional dependence of the form G(w, w) with some guess w = f (z i ) and then try to solve it. The existence of solutions of (3.61) of the form G(w, w) is insured only if

g ij w i w j = L(w, w), (3.62)
for our guess w, L(w, w) being a function which depends only on (w, w). But the inverse metric g ij of the initial hyperkahler structure depends in general on two complex variables, say (w, z 2 )

and their complex conjugates. Therefore it may be very hard to find a clever guess in order to assure that the left hand side of (3.62) is z 2 independent. That is the main technical problem to find Calabi-Yau metrics by means of the Fayyazuddin linearization. The difficulty described above should not be interpreted as a no-go theorem for the Fayyazuddin linearization. In fact we suspect that non trivial solutions may be found for some hyperkahler manifolds. In fact, it has been pointed out in [22] that the supergravity solution describing a D6-branes wrapping the S 2 in Eguchi-hanson gravitational instanton is described by the resolved conifold metric [2]. Arguably the Fayyazuddin linearization may work for this instanton, we hope to elaborate this point in a future, at the moment we have no answer.

The last alternative is to assume that G do not satisfy the constraint (3.62). In this case the equation to solve is the non linear one (2.41) and this may be even a harder task. The facts explained above seem to be discouraging when one attempts to find a 6-dimensional metric with holonomy exactly SU(3) by our methods. But in the remaining part of this work it will be shown that some solutions can be found explicitly. The idea behind is to tackle the full non linear equation (2.41) for the flat hyperkahler structure on R 4 . In fact, we worked out explicitly this equation in (2.53). Particular solutions will be found below and it will be shown that some of them correspond to non compact Ricci flat Kahler metrics with holonomy exactly SU(3).

The "radial" anzatz

A simple guess for solving the fundamental non linear equation (2.53) or equivalently

G yy -1 + 4 (G 11 + G 22 ) -16 (G 11 G 22 -G 12 G 21 ) = 0. (3.63)
is to take G is a function of y and the "radius" u = r 2 = z 1 z 1 + z 2 z 2 . In this situation the Fayyazuddin linearization does not work, as G is not of the form G(w, w) with w an holomorphic function of z i . For this anzatz for G the following identities are easy to prove

G i = z i G u , G i = z i G u , G ij = δ ij G u + z i z i G uu , (3.64)
from where it is obtained that

G 11 + G 22 = 2 G u + u G uu ,
and also that

G 11 G 22 -G 12 G 21 = (G u + z 1 z 1 G uu ) (G u + z 2 z 2 G uu ) -z 1 z 1 z 2 z 2 G uu = (G u ) 2 + u G uu G u .
With the help of the last equalities the equation (3.63) becomes

G yy -1 -16 (G u ) 2 -16 u G u G uu + 8 G u + 4 u G uu = 0, (3.65)
or, by rearranging terms

G yy -1 -8 (2 G u -1) (G u + u G uu ) -4 u G uu = 0. ( 3
a yy 2 u 2 + b yy u + c yy -8 (2 a u + 2 b -1) (2 a u + b) -4 a u -1 = 0. (3.68)
This is a polynomial expression in u and it will be identically zero if its coefficients vanish identically. This lead to three equations which can be written in the following form

a yy -64 a 2 = 0, (b - 1 4 ) yy -48(b - 1 4 ) a = 0, c yy -16 (b - 1 2 ) b -1 = 0.
By introducing the quantity ξ = b -1/4 the last system can be expressed as

a yy -64 a 2 = 0, ξ yy -48 ξ a = 0, c yy -16 ξ 2 = 0. (3.69)
The solutions of the system (3.69) will determine a family of Calabi-Yau geometries.

The general solution of the defining equations

The interesting fact about the procedure implemented above is that we have reduced a non linear equation in two variables (y, u) namely, (3.63), to a non linear system in one variable y with three unknown functions (a(y), b(y), c(y)) namely, (3.69). The unique non linear equation in (3.69) is the first one defining a(y) and this turns out to be very easy to solve in terms of elliptic functions. Once this equation is solved, the remaining ones are straightforward. The first equation (3.69) can be easily integrated to give

a y = ± m + 128 a 3 3 , ( 3.70)
with m a parameter with arbitrary values. We will consider two cases separately. For m > 0 the equation (3.70) implies that

y(a) = ± 1 √ m ∞ a dv 1 + 128 v 3 3 m + y 0 .
With the help of the formula

c -∞ dx √ 1 -x 3 = 1 √ 3 F (φ, k), (3.71)
with

φ = cos -1 c -1 + √ 3 c -1 - √ 3 , k = sin 5 π 12 ,
found in section 1.2.70 of [START_REF] Prudnikov | Integrals and series[END_REF] it follows after a short calculation that

y(a) = ± 1 √ 3 m 3 m 128 1 3 F (φ(a), k) + y 0 , (3.72) φ(a) = cos -1 (128) 1 3 a + (3 m) 1 3 -(3 m) 1 3 √ 3 (128) 1 3 a + (3 m) 1 3 + (3 m) 1 3 √ 3 , k = sin 5 π 12 .
Here F (φ, k) is an elliptic function of the first kind. The formula (3.72) gives y as a function of a. Note that the range of y is finite, as the function F take values in a finite range. The range of a is bounded from below but not from above, that is

-1 < 128 a 3 3 m , ( 3.73)
and it follows that a can take positive and negative values as well. When m < 0 the relation y(a) is given by

y(a) = ± 1 √ s ∞ a dv 128 v 3 3 s -1 + y 0 ,
where s is defined as s = -m. As for the case corresponding to m > 0, the value of a is bounded from below and not from above

1 < 128 a 3 3 s , ( 3.74)
but the difference is that a can take only positive values. The solution in this case is given by

y(a) = ± 1 √ 3 s 3 s 128 1 3 F (ψ(a), k) + y 0 , (3.75)
ψ(a) = cos -1 (128) 1 3 a + ( 3 s) 1 3 + (3 s) 1 3 √ 3 (128)
1 3 a + (3 s) 1 3 -(3 s) 1 3 √ 3 , k = sin π 12 .
The formulas (3.72) and (3.75) are the full solution of the first equation (3.69).

The second equation (3.69) gives ξ as function of y, but it seems to be a bit more problematic, as to solve it requires to invert (3.72) or (3.75) and to express a as a function of y. Instead of doing it is more direct to calculate ξ as a function of a, which implicitly give it as function of y by (3.72) or (3.75). From an elementary chain rule using (3.70) it is easy to show that, for an arbitrary function f (y) of the variable y, the following identities are true

f y = a y f a = ± m + 128 3 a 3 f a , f yy = (m + 128 3 a 3 ) f aa + 64 a 2 f a .
In these terms the second (3.69) is rewritten as The solutions of the hypergeometric equation are well known. This equation has three regular singular points at z = 0, z = 1 and at z → ∞.

(m + 128 3 a 3 ) ξ aa + 64 a 2 ξ a -48 a ξ = 0. ( 3
Once the functions a and b are determined, the third equation (3.69) is not difficult to solve. This equation gives c yy as a function of y and it is straightforward to find c(y) by a double integration. Nevertheless, as it will be seen below, this calculation is irrelevant and only the value of c yy is of importance.

Local form of the Calabi-Yau metric

After solving the system defining the functions a(y), b(y) and c(y) determining our Calabi-Yau geometries, the next problem is to find the local form of the generic CY metric (2.43) for this case. The Kahler form of the base 4-metric is given by

ω(y) = ω 1 -dd c G = i 2 (δ ij -4 G ij ) dz i ∧ dz j = i 2 (1 -4 G u -4 z i z i G uu ) dz i ∧ dz i -2 i G uu (z 1 z 2 dz 1 ∧ dz 2 + z 2 z 1 dz 2 ∧ dz 1 ). (3.80)
This expression may be simplified by use of (3.67) together with the definition ξ = b -1/4 to give

ω(y) = - i 2 (4 a u + 4 ξ + 4 a z i z i ) dz i ∧ dz i -2 i a (z 1 z 2 dz 1 ∧ dz 2 + z 2 z 1 dz 2 ∧ dz 1 ) (3.81)
The Kahler 4-metric corresponding to (3.81) is simply

g 4 (y) = -(4 a u + 4 ξ + 4 a z i z i ) dz i ⊗ dz i -4 a (z 1 z 2 dz 1 ⊗ dz 2 + z 2 z 1 dz 2 ⊗ dz 1 ). (3.82)
It is important to remark that when r 0 one has u 0 and z i z j 0, and therefore (3.82) behaves as g 4 (y) -4 ξ (dz 1 ⊗ dz 1 + dz 2 ⊗ dz 2 ).

(3.83)

Clearly (3.83) is, up to an a-dependent conformal factor, the flat metric on R 4 . Therefore near r = 0 no conical singularity appear. A more handy expression for the (3.82) can be obtained by introducing the polar coordinates (r, θ, ψ, φ) for R 4

z 1 = r cos θ 2 exp( i ψ + i φ 2 ), (3.84
)

z 2 = r sin θ 2 exp( i ψ -i φ 2 ). (3.85)
The expression for the metric (3.82) in these coordinates is

g 4 (y) = -4(2 a r 2 + ξ) ( dr 2 + r 2 σ 2 3 ) -4 (a r 2 + ξ) r 2 (σ 2 1 + σ 2 2 ), (3.86)
with σ i being the usual SO(3) Maurer-Cartan forms

σ 1 = 1 2 (sin ψ dθ -sin θ cos ψ dφ), σ 2 = 1 2
(-cos ψ dθsin θ sin ψ dφ), (3.87)

σ 3 = 1 2 (dψ + cos θ dφ),
for which the angles, a priori, take values in the following interval

0 ≤ θ ≤ π, 0 ≤ φ ≤ 2 π, (3.88) 0 ≤ ψ ≤ 4 π.
Clearly, when r 0 the metric (3.86) is approximated by Finally, the connection A defining our bundle is given by

g 4 (y) = -4 ξ dr 2 + r 2 (σ 2 1 + σ 2 2 + σ 2 3 ) , ( 3
A = d c G y = G yu d c u = i (a y u + b y ) (z 1 dz 1 -z 1 dz 1 + z 2 dz 2 -z 2 dz 2 ), ( 3
g 6 = 4 (2 a r 2 + ξ) ( dr 2 + r 2 σ 2 3 ) + 4 (a r 2 + ξ) r 2 (σ 2 1 + σ 2 2 ) + 16 (2 a u + ξ)(a u + ξ) m -128 a 3 3 da 2 + 1 16 (2 a u + ξ)(a u + ξ) dα + m - 128 a 3 3 (r 2 + ξ u ) r 2 σ 3 2 , ( 3.97)
where we made the redefinitions a → -a and ξ → -ξ. The curvature tensor is irreducible for all the cases described in (3.97), therefore these metrics are of holonomy exactly SU(3).

Global analysis of the new CY metrics

It will be convenient to give a consistency check about the correctness of (3.97). If we assume intuitively that the radius r has length dimensions [r] = L and that the metric has dimensions L 2 , then clearly the quantity a r 2 + ξ, appearing in the component g rr , should be dimensionless. This will be the case if [a] = L -2 . Under this assumption all the components of the Kahler base 4-metric of (

+ ξ y ) r 2 ] = L 2 [y] -1 = L.
Therefore if [α] = L the whole metric has dimensions L 2 , which give a consistency check for (3.97).

Let us also note that metrics (3.97) can never be compact if K = ∂ α is a global isometry. This is because a Killing vector satisfy the equation

-2K i + R ij K j = 0,
and by multiplying this equation by K j and integrating over the manifold gives that

M 6 (|∇ i K j | 2 -R ij K i K j ).
Here the first term is obtained by integration by parts and there is a boundary term which vanish if M 6 is compact. If the manifold is Ricci flat then |∇ i K j | 2 = 0 pointwise in the manifold, and such vectors do not exist in a compact manifold.

In order to make a global analysis of the metric (3.97) there are several options to consider, depending the signature of (3.97) and the sign of the parameter m defined in (3.70). For m > 0 the function a may take positive or negative values, as shown in (3.73), with maximal value a 0 given 128 a 3 0 = 3 m. In general the range of u = r 2 will be constrained unless a > 0 and ξ > 0, in this case the range of r is (0, ∞). Clearly there will be a power law singularity in the metric when (2 a u + ξ)(a u + ξ) = 0. In principle one may choose the hypergeometric function ξ(a) in such a way that is always positive and never zero in the range of values of a. It seems that in this case this singularity will be absent. The problem with this argument is that the metric (3.97) does not have an asymptotic region where its is locally flat, so it is should have a singularity, like a delta in the curvature acting as a cosmological constant. Therefore even if we manage to construct such a strictly positive function ξ(a) in our interval this does not mean that the singularity is not present, but that we are using a bad coordinate system. For instance, the metric dr 2 + r 3 dθ 2 is singular, but with a parametrization r = x + 1/x the singularity would not be seen for any positive x value. Arguably, the same situation is happening here. Therefore none of the constructed metrics should be geodesically complete.

Discussion

To conclude we remark the most salient features of the present work. We have developed a method to construct six dimensional Calabi-Yau metrics in terms of an initial 4-dimensional hyperkahler structure generalizing the situations considered in [9], the last ones correspond to D6 branes with two directions inside a complex submanifold of the hyperkahler space and one direction transversal to it parameterized with a coordinate y. In all the cases the resulting CY metrics posses a Killing vector which preserve the whole SU(3) structure.

It should be remarked that in general the CY metrics we described are not fibered over the initial hyperkahler metric. This situation in fact occurs in [9] and the physicist explanation is that when the D6 branes wrap the complex submanifold their back-reaction deform the initial hyperkahler geometry. The same consideration follows here and the resulting CY metrics are fibered over a 4-dimensional base which is just Kahler and depends on the coordinate y as a parameter. In other words it is Kahler for any fixed value of y.

We also showed the the entire geometry is described in terms of a single function G satisfying a non linear quadratic differential equation (2.41). That this function is enough to determine the full geometry is evident, as the whole SU(3) structure (2.42) is defined purely in terms of G and derivates. The function G depends on y and in fact can be interpreted as the Kahler potential for the base 4-dimensional metric for any fixed value of y. We studied the non linear problem associated to the flat hyperkahler metric in R 4 and we found some particular solutions, which allowed us to construct a family of CY examples with power law singularities.

An interesting open problem is the one arising by fact that the supergravity solution describing a D6-branes wrapping the S 2 in Eguchi-hanson gravitational instanton is described by the resolved conifold metric [22]- [24]. As the Eguchi-Hanson metric is hyperkahler, the method developed here may be suitable to make this link explicit. In particular it may be one of the situations for which the Fayyazuddin linearization works properly. We hope to find an answer to these questions in the near future.

. 23 )

 23 But the closure of ω 2 and ω 3 (2.22) together with the decomposition d 5 = d 4 + ∂ y dy and (2.23)

  e 2 + e 3 ∧ e 4 + e 5 ∧ e 6 , ψ + = (e 1 ∧ e 4 + e 2 ∧ e 3 ) ∧ e 5 + (e 1 ∧ e 3 + e 4 ∧ e 2 ) ∧ e 6 , (2.9) ψ -= -(e 1 ∧ e 3 + e 4 ∧ e 2 ) ∧ e 5 + (e 1 ∧ e 4 + e 2 ∧ e 3 ) ∧ e 6 ,

  This is the fundamental equation we will solve along the text. Finally, we have checked that the closure of ψ -gives no new constraints, but we omit this analysis. By introducing (2.37) and (2.40) into (2.31) it follows that the SU(3) structure is

	.39)
	From (2.39) combined with (2.35) it follows that
	G

yy = H 2 . (2.40) From the last equation together with (2.34) it is obtained a non linear differential equation determined the function G, namely G yy = M(G). (2.41)

  in which the coefficients a, b and c are assumed to be functions of y. By introducing this anzatz into (3.66) it is obtained that

					.66)
	A simple solution of (3.66) is found by further postulating the following quadratic expression
	for G	G =	a 2	u 2 + b u + c,	(3.67)

  .89) which gives another check that non conical singularity appears near r = 0. On the other hand, the fiber metric of the general Calabi-Yau metric (2.43) is determined in terms of G yy namely

	G yy =	a yy 2		u 2 + b yy u + c yy .	(3.90)
	The explicit value of the derivatives appearing in (3.90) are obtained by use of (3.69), the result
	is
	a yy = 64 a 2 ,	b yy = 48 a ξ.	(3.91)
	c yy = 16 (ξ -	1 4	) (ξ +	1 4	) + 1 = 16 ξ 2 .	(3.92)
	Inserting the expressions (3.91)-(3.92) into (3.90) gives
	G yy = 32 (a u) 2 + 48 ξ (a u) + 16 ξ 2 ,	(3.93)
	which can be expressed more neatly as
	G yy = 16 (2 a u + ξ)(a u + ξ).	(3.94)

  3.97) are of dimensions [L] 2 . Let us see if this is true for the fiber components. The component g yy is clearly dimensionless and therefore we should have [y] = L. On the other hand, the relation between y and a can be read off from (3.70) and is

	[y] = [a] -1 2 = L,
	which confirm what we postulated. Moreover the fiber contains the term
	(a

y r 2 + ξ y ) r 2 ,

and it is seen that [(a y r 2

In fact, a vector V preserving the whole SU (3) structure automatically preserve g 6 .

Acknowledgement:

The completion of the present work would not be possible without several important advices given by S. Cherkis, who also participated at early stages of this project. Also I sincerely acknowledge H. Lu, who pointed out a crucial numerical error in the initial form of this work. I have also been benefited with short but stimulating discussions with J. Vazquez-Poritz, M. Leston and C. Saemann. This work is supported by the Science Foundation Ireland Grant N06/RFP/MAT050.