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Measuring small distances in \(N=2\) sigma models. (English) Zbl 0990.81689
Summary: We analyze global aspects of the moduli space of Kähler forms for \(N=(2,2)\) conformal \(\sigma\)-models. Using algebraic methods and mirror symmetry we study extensions of the mathematical notion of length (as specified by a Kähler structure) to conformal field theory and calculate the way in which lengths change as the moduli fields are varied along distinguished paths in the moduli space. We find strong evidence supporting the notion that, in the robust setting of quantum Calabi-Yau moduli space, string theory restricts the set of possible Kähler forms by enforcing “minimal length” scales, provided that topology change is properly taken into account. Some lengths, however, may shrink to zero. We also compare stringy geometry to classical general relativity in this context.

MSC:
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
32G81 Applications of deformations of analytic structures to the sciences
32J81 Applications of compact analytic spaces to the sciences
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
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