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I am learning about mirror symmetry and having trouble understanding Givental's I- and J-functions. For example the J-function for the quintic threefold $X$ is defined by the formula $$ J:=e^{(t_0+t_1H)h}(1+h^{-2}\sum_{d\ge 0}N_dq^ddl-2h^{-3}\sum_{d \ge 0}N_dq^dpt), $$ where $q=e^{t_1}$ and $l$ is the class of a line, $pt$ is the class of a point, and $N_d$ is the GW invariant of degree $d$. Or some consider $J_X:=i_{!}(J)$ via the inclusion $i:X\rightarrow \mathbb{P}^4$. The $I$-function is defined in a similar manner. They are cohomology valued functions (or formal power series).

My questions are

  1. Where do these scary looking functions come from? It looks like the Gromov-Witten potential, which is simply the generating series of GW invariants, but has a strange factor $e^{(t_0+t_1H)h}$ for example.
  2. What is the advantage of considering cohomology valued functions? I think this is related to Givental's proof of mirror theorem, but I cannot not follow the proof.
  3. Are there any relation between Givental's I- and J- functions and Hosono et. al.'s cohomology valued GKZ hypergeometric series?
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1 Answer 1

  1. The better source for the $J$-function now would be Dubrovin-Zhang The idea of that is simple - the part of the genus=1 corellators comes from the three-points corellators only. From the point of view of the Frobenius manifold - from the algebra structure of it.

  2. The advantage of the cohomology-valued function is very simple. Everything Givental writes looks to ge thought of having certein singularity theory behind. In this case it is just natural.

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