Symplectic geometry of Frobenius structures

By Alexander Givental. 2003

205 citations recorded by the source corpus.

The concept of a Frobenius manifold was introduced by B. Dubrovin [9] to capture in an axiomatic form the properties of correlators found by physicists (see [8]) in two-dimensional topological field theories “coupled to gravity at the tree level”. The purpose of these notes is to reiterate and expand the viewpoint, outlined in the paper [7] of T. Coates and the author, which recasts this concept in terms of linear symplectic geometry and exposes the role of the twisted loop group L (2) GL N of hidden svmmetries.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2d855f79456650d3a5ab9726bfc50bc11ec6a59a

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Citations in the corpus

The Crepant Transformation Conjecture for Toric Complete Intersections (2014, methodology): “It has the following special geometric properties [45]: it is a cone, and a tangent space T of LX is tangent to LX exactly along zT .”
The Abelian/Nonabelian correspondence and Frobenius manifolds (2006, methodology): “s that, with the identiﬁcation of cohomology spaces H∗ (X//T)a with H∗ X//G) by the map ˜σ∪ω→σ, the I-function generates the Lagrangian cone LX//G describing the rational Gromov-Witten theory of X//G [Giv3]. Since {z∂ ˜t i I(˜t, −z)}also form a basis of L/zL, where Lis the tangent space of LX//G at ”
Simple singularities and integrable hierarchies (2003, background): “According to a result from [12], when the Frobenius structure associated to the cone L is semisimple, one can identify L with the Cartesian product LA1 × .”
BCOV theory via Givental group action on cohomological field theories (2008, unknown): “Givental proved [20, 21] that the graph L of dF0 in V is the germ of a Lagrangian cone with the vertex at 0 such that its tangent spaces L are tangent to L exactly along zL.”
Identification of the Givental Formula with the Spectral Curve Topological Recursion Procedure (2012, methodology): “ Givental operator for a Frobenius manifold. Let Z({td,µ}) be the partition function of some N-dimensional semi-simple conformal cohomological ﬁeld theory. We recall the construction (due to Givental [18, 17, 19], see also Dubrovin [5]) of an operator series R(z) as in the previous section whose qua”
Quantum D-modules and generalized mirror transformations (2004, background): “∇ q n +pn· ∇ p n ). Let L (~)be the tangent space of L at q(~) +d F0M ∈ L. In [CG], Coates and Givental showed that these semi-inﬁnite subspaces L q(~) of H satisfy the following properties (see also [Giv4].) (1) The family {L q(~)} q(~)∈H + actually depends only on dimH∗(M)-parameters. More precise”
Higher genus quasimap wall-crossing for semi-positive targets (2013, background): “K2] that the formula (3.1.4) deﬁnes a family (with parameter t) of symplectic transformations on the symplectic space H = H∗(W//G,Λ){{z,z−1}} appearing in Givental’s formalism of GromovWitten theory, [G5]. The most general ε-wall-crossing formula in genus zero applies to the operators Sε t, see Conj”
Orbifold quantum Riemann–Roch, Lefschetz and Serre (2005, methodology): “. A more precise interpretation of D(c,F) will be given in Section 3. 3. Givental’s formalism Givental introduces a symplectic vector space formalism to describe Gromov-Witten theory (see [27], [16], [29]). In this formalism many properties of Gromov-Witten invariants can be expressed as application”
Moduli stacks of stable toric quasimaps (2009, background): “mponent and the big I-function of XΣ 7.1. Reminder on J-functions. For a smooth projective variety X (we assume for simplicity that the odd cohomology of X vanishes), a formalism due to Givental, see [G3], [CG], encodes the genus zero Gromov-Witten theory (with descendents) of Xinto a Lagrangian con”