Abstract
The Hilbert space of level q Chern-Simons theory of gauge group G of the ADE type quantized on T 2 can be represented by points that lie on the weight lattice of the Lie algebra g up to some discrete identifications. Of special significance are the points that also lie on the root lattice. The generating functions that count the number of such points are quasi-periodic Ehrhart polynomials which coincide with the generating functions of SU(q) representation of the ADE subgroups of SU(2) given by the McKay correspondence. This coincidence has roots in a string/M theory construction where D3(M5)-branes are put along an ADE singularity. Finally, a new perspective on the McKay correspondence that involves the inverse of the Cartan matrices is proposed.
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Acknowledgments
We would like to thank Emil Albrychiewicz, Ori Ganor, Yasunori Nomura, and Tong Zhou for useful discussions. In particular, we thank Ori Ganor for pointing out the potential significance of the states considered in this paper.
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Ju, C. Chern-Simons theory, Ehrhart polynomials, and representation theory. J. High Energ. Phys. 2024, 52 (2024). https://doi.org/10.1007/JHEP01(2024)052
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DOI: https://doi.org/10.1007/JHEP01(2024)052