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We define tautological relations for the moduli space of stable maps to a target variety. Using the double ramification cycle formula for target varieties of Janda–Pandharipande–Pixton–Zvonkine, we construct non-trivial tautological relations parallel to Pixton’s double ramification cycle relations for the moduli of curves. Examples and applications are discussed.

No abstract uploaded!

For projective conifold transitions between Calabi-Yau threefolds X and Y, with X close to Y in the moduli, we show that the combined information provided by the A model (Gromov--Witten theory in all genera) and B model (variation of Hodge structures) on X, linked along the vanishing cycles, determines the corresponding combined information on Y. Similar result holds in the reverse direction when linked with the exceptional curves.

The mirror theorem is generalized to any smooth projective variety X. That is, a fundamental relation between the Gromov-Witten invariants of X and Gromov-Witten invariants of complete intersections Y in X is established.