53 research outputs found

    Decoding the geometry of conformal field theories

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    To certain geometries, string theory associates conformal field theories. We discuss techniques to perform the reverse procedure: To recover geometrical data from abstractly defined conformal field theories. This is done by introducing appropriate notions of limits of conformal field theories and their degenerations, and by applying techniques from noncommutative geometry. This note is a summary of our work hep-th/0308143 , aimed to be less technical than the original paper, along with some new calculations confirming our interpretation of the rescaled limiting zero mode of the Virasoro field.Comment: 12 pages, contribution to the Proceedings of the 7th International Workshop "Lie Theory and Its Applications in Physics", Varna, Bulgari

    Mirror Symmetry on Kummer Type K3 Surfaces

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    We investigate both geometric and conformal field theoretic aspects of mirror symmetry on N=(4,4) superconformal field theories with central charge c=6. Our approach enables us to determine the action of mirror symmetry on (non-stable) singular fibers in elliptic fibrations of Z_N orbifold limits of K3. The resulting map gives an automorphism of order 4,8, or 12, respectively, on the smooth universal cover of the moduli space. We explicitly derive the geometric counterparts of the twist fields in our orbifold conformal field theories. The classical McKay correspondence allows for a natural interpretation of our results.Comment: 27 pages, no figures; references added, typos and equation (28) correcte

    Crystallographic Orbifolds: Towards a Classification of Unitary Conformal Field Theories with Central Charge c = 2

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    We study the moduli space C^2 of unitary two-dimensional conformal field theories with central charge c=2. We construct all the 28 nonexceptional nonisolated irreducible components of C^2 that may be obtained by an orbifold procedure from toroidal theories. The parameter spaces and partition functions are calculated explicitly, and all multicritical points and lines are determined. We show that all but four of the 28 irreducible components of C^2 corresponding to nonexceptional orbifolds are directly or indirectly connected to the moduli space of toroidal theories in C^2. We relate our results to those by Dixon, Ginsparg, Harvey on the classification of c=3/2 superconformal field theories and thereby give geometric interpretations to all nonisolated orbifolds discussed there.Comment: 47 pages, spelling mistakes corrected; final version for JHE

    Friendly giant meets pointlike instantons? On a new conjecture by John McKay

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    A new conjecture due to John McKay claims that there exists a link between (1) the conjugacy classes of the Monster sporadic group and its offspring, and (2) the Picard groups of bases in certain elliptically fibered Calabi-Yau threefolds. These Calabi-Yau spaces arise as F-theory duals of point-like instantons on ADE type quotient singularities. We believe that this conjecture, may it be true or false, connects the Monster with a fascinating area of mathematical physics which is yet to be fully explored and exploited by mathematicians. This article aims to clarify the statement of McKay's conjecture and to embed it into the mathematical context of heterotic/F-theory string-string dualities

    Not doomed to fail

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    In their recent manuscript “An uplifting discussion of T-duality ” [26], J. Harvey and G. Moore have reevaluated a mod two condition appearing in asymmetric orbifold constructions as an obstruction to the description of certain symmetries of toroidal conformal field theories by means of automorphisms of the underlying charge lattice. The relevant “doomed to fail” condition determines whether or not such a lattice automorphism g may lift to a symmetry in the corresponding toroidal conformal field theory without introducing extra phases. If doomed to fail, then in some cases, the lift of g must have double the order of g. It is an interesting question, whether or not “geometric” symmetries are affected by these findings. In the present note, we answer this question in the negative, by means of elementary linear algebra: “geometric” symmetries of toroidal conformal field theories are not doomed to fail. Consequently, and in particular, the symmetry groups involved in symmetry surfing the moduli space of K3 theories do not differ from their lifts
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