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36
Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and GromovWitten invariants
, 2001
"... We present a project of classification of a certain class of bihamiltonian 1+1 PDEs depending on a small parameter. Our aim is to embed the theory of Gromov Witten invariants of all genera into the theory of integrable systems. The project is focused at describing normal forms of the PDEs and their ..."
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Cited by 44 (2 self)
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We present a project of classification of a certain class of bihamiltonian 1+1 PDEs depending on a small parameter. Our aim is to embed the theory of Gromov Witten invariants of all genera into the theory of integrable systems. The project is focused at describing normal forms of the PDEs and their local bihamiltonian structures satisfying certain simple axioms. A Frobenius manifold or its degeneration is associated to every bihamiltonian structure of our type. The main result is a universal loop equation on the jet space of a semisimple Frobenius manifold that can be used for perturbative reconstruction of the integrable hierarchy. We show that first few terms of the perturbative expansion correctly reproduce the universal identities between intersection numbers of Gromov Witten classes and their descendents.
On Hamiltonian perturbations of hyperbolic systems of conservation laws
, 2004
"... We study the general structure of formal perturbative solutions to the Hamiltonian perturbations of spatially onedimensional systems of hyperbolic PDEs vt + [φ(v)]x = 0. Under certain genericity assumptions it is proved that any bihamiltonian perturbation can be eliminated in all orders of the pert ..."
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Cited by 33 (5 self)
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We study the general structure of formal perturbative solutions to the Hamiltonian perturbations of spatially onedimensional systems of hyperbolic PDEs vt + [φ(v)]x = 0. Under certain genericity assumptions it is proved that any bihamiltonian perturbation can be eliminated in all orders of the perturbative expansion by a change of coordinates on the infinite jet space depending rationally on the derivatives. The main tools is in constructing of the socalled quasiMiura transformation of jet coordinates eliminating an arbitrary deformation of a semisimple bihamiltonian structure of hydrodynamic type (the quasitriviality theorem). We also describe, following [35], the invariants of such bihamiltonian structures with respect to the group of Miuratype transformations depending
Lie theory for nilpotent L∞algebras
 Ann. Math
"... Let R be a commutative algebra over a field K of characteristic 0. The spectrum Spec(R) of R is the set Hom(R, K) of all homomorphisms from R to K. Let Ω • be the simplicial differential graded (dg) commutative algebra whose nsimplices Ωn are the dg algebra of differential forms on the geometric n ..."
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Cited by 22 (0 self)
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Let R be a commutative algebra over a field K of characteristic 0. The spectrum Spec(R) of R is the set Hom(R, K) of all homomorphisms from R to K. Let Ω • be the simplicial differential graded (dg) commutative algebra whose nsimplices Ωn are the dg algebra of differential forms on the geometric nsimplex ∆ n. In [20], Sullivan reformulated Quillen’s
Deformations of semisimple bihamiltonian structures of hydrodynamic type
 J. Geom. Phys
"... We classify in this paper infinitesimal quasitrivial deformations of semisimple bihamiltonian structures of hydrodynamic type. 1 ..."
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Cited by 18 (6 self)
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We classify in this paper infinitesimal quasitrivial deformations of semisimple bihamiltonian structures of hydrodynamic type. 1
Deformation quantization of gerbes, I
"... This is the first in a series of articles devoted to deformation quantization of gerbes. Here we give basic definitions and interpret deformations of a given gerbe as MaurerCartan elements of a differential graded Lie algebra (DGLA). We classify all deformations of a given gerbe on a symplectic man ..."
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Cited by 9 (3 self)
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This is the first in a series of articles devoted to deformation quantization of gerbes. Here we give basic definitions and interpret deformations of a given gerbe as MaurerCartan elements of a differential graded Lie algebra (DGLA). We classify all deformations of a given gerbe on a symplectic manifold, as well as provide a deformationtheoretic interpretation of the first RozanskyWitten class. 1.
THE GROMOVWITTEN POTENTIAL ASSOCIATED TO A TCFT
, 2005
"... Abstract. This is the sequel to my preprint“TCFTs and CalabiYau categories”. Here we extend the results of that paper to construct, for certain CalabiYau A∞ categories, something playing the role of the GromovWitten potential. This is a state in the Fock space associated to periodic cyclic homolo ..."
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Cited by 8 (2 self)
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Abstract. This is the sequel to my preprint“TCFTs and CalabiYau categories”. Here we extend the results of that paper to construct, for certain CalabiYau A∞ categories, something playing the role of the GromovWitten potential. This is a state in the Fock space associated to periodic cyclic homology, which is a symplectic vector space. Applying this to a suitable A ∞ version of the derived category of sheaves on a CalabiYau yields the B model potential, at all genera. The construction doesn’t go via the DeligneMumford spaces, but instead uses the BatalinVilkovisky algebra constructed from the uncompactified moduli spaces of curves by Sen and Zwiebach. The fundamental class of DeligneMumford space is replaced here by a certain solution of the quantum master equation, essentially the “string vertices ” of Zwiebach. On the field theory side, the BV operator has an interpretation as the quantised differential on the Fock space for periodic cyclic chains. Passing to homology, something satisfying the master equation yields an element of the Fock space. 1. Notation We work throughout over a ground field K containing Q. Often we will use topological K vector spaces. All tensor products will be completed. All the topological vector spaces we use are inverse limits, so the completed tensor product is also an inverse limit. All the results remain true without any change if we work over a differential graded ground ring R, and use flat R modules. (An R module is flat if the functor of tensor product with it is exact). We could also have only a Z/2 grading on R. 2. Acknowledgements I would like to thank Tom Coates, Ezra Getzler, Alexander Givental and Paul Seidel for very helpful conversations, and Dennis Sullivan for explaining to me his ideas on the BatalinVilkovisky formalism and moduli spaces of curves. 3. Topological conformal field theories Let S be the topological category whose objects are the nonnegative integers, and whose morphism space S(n,m) is the moduli space of Riemann surfaces with n parameterised incoming and m parameterised outgoing boundaries, such that each connected component has at least one incoming boundary. These surfaces are not necessarily connected. Let Sχ(n,m) ⊂ S(n,m) be the space of surfaces of Euler characteristic χ.
A simple way of making a Hamiltonian system into a biHamiltonian one
, 2004
"... Given a Poisson structure (or, equivalently, a Hamiltonian operator) P, we show that its Lie derivative Lτ(P) along a vector field τ defines another Poisson structure, which is automatically compatible with P, if and only if [L 2 τ(P), P] = 0, where [·, ·] is the Schouten bracket. We further prove ..."
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Cited by 8 (2 self)
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Given a Poisson structure (or, equivalently, a Hamiltonian operator) P, we show that its Lie derivative Lτ(P) along a vector field τ defines another Poisson structure, which is automatically compatible with P, if and only if [L 2 τ(P), P] = 0, where [·, ·] is the Schouten bracket. We further prove that if dimkerP ≤ 1 and P is of locally constant rank, then all Poisson structures compatible with a given Poisson structure P on a finitedimensional manifold M are locally of the form Lτ(P), where τ is a local vector field such that L 2 τ(P) = L˜τ(P) for some other local vector field ˜τ. This leads to a remarkably simple construction of biHamiltonian dynamical systems. We also present a generalization of these results to the infinitedimensional case. In particular, we provide a new description for pencils of compatible local Hamiltonian operators of Dubrovin–Novikov type and associated biHamiltonian systems of hydrodynamic type.
On the quasitriviality deformations of bihamiltonian structures of hydrodynamic type
"... We prove in this paper the quasitriviality of a class of deformations of the one component bihamiltonian structures of hydrodynamic type. 1 ..."
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Cited by 5 (1 self)
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We prove in this paper the quasitriviality of a class of deformations of the one component bihamiltonian structures of hydrodynamic type. 1