On Bohr-Sommerfeld bases

This paper combines algebraic and Lagrangian geometry to construct a special basis in every space of conformal blocks, the Bohr-Sommerfeld (BS) basis. We use the method of [D. Borthwick, T. Paul and A. Uribe, Legendrian distributions with applications to the non-vanishing of Poincar\'e series of large weight, Invent. math, 122 (1995), 359-402, preprint hep-th/9406036], whereby every vector of a BS basis is defined by some half-weighted Legendrian distribution coming from a Bohr-Sommerfeld fibre of a real polarization of the underlying symplectic manifold. The advantage of BS bases (compared to bases of theta functions in [A. Tyurin, Quantization and ``theta functions'', Jussieu preprint 216 (Apr 1999), e-print math.AG/9904046, 32pp.]) is that we can use information from the skillful analysis of the asymptotics of quantum states. This gives that Bohr-Sommerfeld bases are unitary quasi-classically. Thus we can apply these bases to compare the Hitchin connection with the KZ connection defined by the monodromy of the Knizhnik-Zamolodchikov equation in combinatorial theory (see, for example, [T. Kohno, Topological invariants for 3-manifolds using representations of mapping class group I, Topology 31 (1992), 203-230; II, Contemp. math 175} (1994), 193-217]).Comment: 43 pages, uses: latex2e with amsmath,amsfonts,theore

Hemifield-specific visual recognition memory impairments in patients with unilateral temporal lobe removals.

Recent evidence on visual neglect suggests that each hemisphere maintains a retinotopically organized representation of the visual world contralateral to the current fixation point and that this representation is based not only on analysis of the current retinal input but, equally importantly, on information retrieved from memory. This idea predicts that unilateral damage to memory systems should produce a lateralized impairment of memory for the retinotopically contralateral visual world. To test this prediction we examined visual recognition memory performance in the left and right visual hemifields of patients who had undergone partial unilateral temporal lobe removals for the relief of epilepsy, either in the left hemisphere (n = 5) or the right (n = 5). The patients were given complex artificial scenes to remember, constructed of independent left and right halves, and were then tested for recognition of the left and the right halves separately. Stimuli were exposed tachistoscopically throughout and fixation was maintained on a central position. Patients made significantly more errors with half-scenes in the hemifield contralateral to their removal than in the ipsilateral hemifield, an increase of 50% in the error rate on average. The effect was seen equally in patients with left and right removals. This finding supports the idea that visual memory retrieval is retinotopically organized

The role of light scatter in the residual visual sensitivity of patients with complete cerebral hemispherectomy.

Various residual visual capacities have been reported for the phenomenally blind field of hemispherectomized patients, providing evidence for the relative roles of cortical and subcortical pathways in vision. We attempted to characterize these functions by examining the ability of five patients to detect, localize, and discriminate high-contrast flashed, flickering and moving targets. Dependent measures were verbal, manual, and oculomotor responses. As a control for light scatter, intensity thresholds for monocular detection of targets in the hemianopic field were compared with thresholds obtained when using an additional half eyepatch to occlude the blind hemiretina of the tested eye. One unilaterally destriate patient was tested on the same tasks. In photopic conditions, none of the hemispherectomized patients could respond to visual cues in their impaired fields, whereas the destriate patient could detect, discriminate, and point to targets, and appreciate the apparent motion of stimuli across his midline. Under reduced lighting, the threshold luminance required by hemispherectomized patients to detect stimuli presented monocularly was similar to that required for their detection when all visual information was occluded in the blind field, and only available to the visual system indirectly via light scatter. In contrast, the destriate patient's monocular threshold in his blind field was substantially lower than that for stimuli directly occluded in the blind field. As we found no range of stimuli which the hemispherectomized patients could detect or discriminate that was not also associated with discriminable scattered light, we conclude that the subcortical pathways which survive hemispherectomy cannot mediate voluntary behavioural responses to visual information in the hemianopic field