IASM Preprint Series 2020-01:Quantum K-theory of Toric Varieties, Level Structures, and 3D Mirror Symmetry

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QUANTUM K-THEORY OF TORIC VARIETIES, LEVEL STRUCTURES, AND 3D MIRROR SYMMETRY


YONGBIN RUAN, YAOXIONG WEN, ZIJUN ZHOU


Abstract. We introduce a new version of 3d mirror symmetry for toric stacks, inspired by a 3d

N = 2 abelian mirror symmetry construction in physics. Given some toric data, we introduce

the K-theoretic I-function with effective level structure for the associated toric stack. When a

particular stability condition is chosen, it restricts to the I-function for the particular toric GIT

quotient. The mirror of a toric stack is defined by the Gale dual of the original toric data. We

then proved the mirror conjecture that the I-functions of a mirror pair coincide, under the

mirror map, which switches K¨ahler and equivariant parameters, and maps q !→ q−1.