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28
The Quantum Orbifold Cohomology of Weighted Projective Spaces
, 2007
"... We calculate the small quantum orbifold cohomology of arbitrary weighted projective spaces. We generalize Givental’s heuristic argument, which relates small quantum cohomology to S 1equivariant Floer cohomology of loop space, to weighted projective spaces and use this to conjecture an explicit for ..."
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Cited by 57 (20 self)
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We calculate the small quantum orbifold cohomology of arbitrary weighted projective spaces. We generalize Givental’s heuristic argument, which relates small quantum cohomology to S 1equivariant Floer cohomology of loop space, to weighted projective spaces and use this to conjecture an explicit formula for the small Jfunction, a generating function for certain genuszero Gromov–Witten invariants. We prove this conjecture using a method due to Bertram. This provides the first nontrivial example of a family of orbifolds of arbitrary dimension for which the small quantum orbifold cohomology is known. In addition we obtain formulas for the small Jfunctions of weighted projective complete intersections satisfying a combinatorial condition; this condition
Towards an enumerative geometry of the moduli space of twisted curves and rth roots
, 2008
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Localization in GromovWitten Theory and Orbifold GromovWitten Theory, Handbook of Moduli, Volume
 International Press and Higher Education Press
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THE SPACES OF LAURENT POLYNOMIALS, GROMOVWITTEN THEORY OF P 1ORBIFOLDS, AND INTEGRABLE HIERARCHIES
, 2007
"... Abstract. Let Mk,m be the space of Laurent polynomials in one variable x k +t1x k−1 +...tk+mx −m, where k, m ≥ 1 are fixed integers and tk+m ̸ = 0. According to B. Dubrovin [11], Mk,m can be equipped with a semisimple Frobenius structure. In this paper we prove that the corresponding descendent and ..."
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Cited by 9 (3 self)
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Abstract. Let Mk,m be the space of Laurent polynomials in one variable x k +t1x k−1 +...tk+mx −m, where k, m ≥ 1 are fixed integers and tk+m ̸ = 0. According to B. Dubrovin [11], Mk,m can be equipped with a semisimple Frobenius structure. In this paper we prove that the corresponding descendent and ancestor potentials of Mk,m (defined as in [16]) satisfy Hirota quadratic equations (HQE for short). Let Ck,m be the orbifold obtained from P 1 by cutting small discs D1 ∼ = {z  ≤ ǫ} and D2 ∼ = {z −1  ≤ ǫ} around z = 0 and z = ∞ and gluing back the orbifolds D1/Zk and D2/Zm in the obvious way. We show that the orbifold quantum cohomology of Ck,m coincides with Mk,m as Frobenius manifolds. Modulo some yettobeclarified details, this implies that the descendent (respectively the ancestor) potential of Mk,m is a generating function for the descendent (respectively ancestor) orbifold Gromov–Witten invariants of Ck,m. There is a certain similarity between our HQE and the Lax operators of the Extended bigraded Toda hierarchy, introduced by G. Carlet in [7]. Therefore, it is plausible that our HQE characterize the taufunctions of this hierarchy and we expect that the Extended bigraded Toda hierarchy governs the Gromov–Witten theory of Ck,m. 1.
GromovWitten theory of étale gerbes I: root gerbes, in preparation
"... Abstract. Let X be a smooth complex projective algebraic variety. Given a line bundle L over X and an integer r> 1 we study the GromovWitten theory of the stack rp L/X of rth root of L. We prove an exact formula expressing genus 0 GromovWitten invariants of rp L/X in terms of those of X. Assum ..."
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Cited by 9 (3 self)
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Abstract. Let X be a smooth complex projective algebraic variety. Given a line bundle L over X and an integer r> 1 we study the GromovWitten theory of the stack rp L/X of rth root of L. We prove an exact formula expressing genus 0 GromovWitten invariants of rp L/X in terms of those of X. Assuming that either rp L/X or X has semisimple quantum cohomology, we prove an exact formula between higher genus invariants. We also present constructions of moduli stacks of twisted stable maps to rp L/X starting from moduli stack of stable maps to X.
GROMOVWITTEN THEORY OF PRODUCT STACKS
, 905
"... Abstract. Let X1 and X2 be smooth proper DeligneMumford stacks with projective coarse moduli spaces. We prove a formula for orbifold GromovWitten invariants of the product stack X1 × X2 in terms of GromovWitten invariants of the factors X1 and X2. As an application, we deduce a decomposition resu ..."
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Cited by 8 (3 self)
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Abstract. Let X1 and X2 be smooth proper DeligneMumford stacks with projective coarse moduli spaces. We prove a formula for orbifold GromovWitten invariants of the product stack X1 × X2 in terms of GromovWitten invariants of the factors X1 and X2. As an application, we deduce a decomposition result for GromovWitten theory of trivial gerbes.
Logarithmic stable maps
"... Abstract. We introduce the notion of a logarithmic stable map from a minimal log prestable curve to a log twisted semistable variety of form xy = 0. We study the compactification of the moduli spaces of such maps and provide a perfect obstruction theory, applicable to the moduli spaces of (un)ramif ..."
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Cited by 5 (0 self)
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Abstract. We introduce the notion of a logarithmic stable map from a minimal log prestable curve to a log twisted semistable variety of form xy = 0. We study the compactification of the moduli spaces of such maps and provide a perfect obstruction theory, applicable to the moduli spaces of (un)ramified stable maps and stable relative maps. As an application, we obtain a modular desingularization of the main component of Kontsevich’s moduli space of elliptic stable maps to a projective space. 1.
Twisted GromovWitten rspin potential and Givental’s quantization
"... The universal curve π: C → M over the moduli space M of stable rspin maps to a target Kähler manifold X carries a universal spinor bundle L → C. Therefore the moduli space M itself carries a natural Ktheory class Rπ∗L. We introduce a twisted rspin Gromov–Witten potential of X enriched with Chern ..."
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Cited by 4 (1 self)
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The universal curve π: C → M over the moduli space M of stable rspin maps to a target Kähler manifold X carries a universal spinor bundle L → C. Therefore the moduli space M itself carries a natural Ktheory class Rπ∗L. We introduce a twisted rspin Gromov–Witten potential of X enriched with Chern characters of Rπ∗L. We show that the twisted potential can be reconstructed from the ordinary rspin Gromov–Witten potential of X via an operator that assumes a particularly simple form in Givental’s quantization formalism. 1