Higher-Level Appell Functions, Modular Transformations, and Characters
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We study modular transformation properties of a class of indefinite theta series involved in characters of infinite-dimensional Lie superalgebras. The level-ℓ Appell functionsOpen image in new window satisfy open quasiperiodicity relations with additive theta-function terms emerging in translating by the “period.” Generalizing the well-known interpretation of theta functions as sections of line bundles, the Open image in new window function enters the construction of a section of a rank-(ℓ+1) bundle Open image in new window. We evaluate modular transformations of the Open image in new window functions and construct the action of an SL(2,ℤ) subgroup that leaves the section of Open image in new window constructed from Open image in new window invariant.
Modular transformation properties of Open image in new window are applied to the affine Lie superalgebra Open image in new window at a rational level k>−1 and to the N=2 super-Virasoro algebra, to derive modular transformations of “admissible” characters, which are not periodic under the spectral flow and cannot therefore be rationally expressed through theta functions. This gives an example where constructing a modular group action involves extensions among representations in a nonrational conformal model.
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