We define a generalization of Fan-Jarvis-Ruan-Witten theory, a "hybrid" model associated to a collection of quasihomogeneous polynomials of the same weights and degree, which is expected to match the Gromov-Witten theory of the Calabi-Yau complete intersection cut out by the polynomials. In genus zero, we prove that the correspondence holds for any such complete intersection of dimension three in ordinary, rather than weighted, projective space. These results generalize those of Chiodo-Ruan for the quintic threefold, and as in that setting, Givental's quantization can be used to yield a conjectural relation between the full higher-genus theories.

We prove a conjecture of Pixton, namely that his proposed formula for the double ramification cycle on M g,n vanishes in codimension beyond g. This yields a collection of tautological relations in the Chow ring of M g,n . We describe, furthermore, how these relations can be obtained from Pixton's 3-spin relations via localization on the moduli space of stable maps to an orbifold projective line.

We construct a version of r-spin theory in genus zero for Riemann surfaces with boundary and prove that its generating function is closely related to the wave function of the r-th Gelfand-Dickey integrable hierarchy. This provides an analogue of Witten's r-spin conjecture in the open setting. 1 1 When r = 2, our construction of the intersection numbers (1.3) recovers the construction from [32] when all a i are zero.i∈Iwhich follows directly from (2.1), one shows easily that

No abstract

We define a generalization of Fan-Jarvis-Ruan-Witten theory, a "hybrid" model associated to a collection of quasihomogeneous polynomials of the same weights and degree, which is expected to match the Gromov-Witten theory of the Calabi-Yau complete intersection cut out by the polynomials. In genus zero, we prove that the correspondence holds for any such complete intersection of dimension three in ordinary, rather than weighted, projective space. These results generalize those of Chiodo-Ruan for the quintic threefold, and as in that setting, Givental's quantization can be used to yield a conjectural relation between the full higher-genus theories.3 ) + 2Γ(d + 13 )ψ(1)where ψ is the digamma function, the logarithmic derivative of Γ.