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  • 1.
    Alm, Johan
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Merkulov, Sergei
    Stockholm University, Faculty of Science, Department of Mathematics.
    Grothendieck-Teichmuller group and Poisson cohomologies2015In: Journal of Noncommutative Geometry, ISSN 1661-6952, E-ISSN 1661-6960, Vol. 9, no 1, p. 185-214Article in journal (Refereed)
    Abstract [en]

    We study actions of the Grothendieck-Teichmuller group GRT on Poisson cohomologies of Poisson manifolds, and prove some go and no-go theorems associated with these actions.

  • 2.
    Merkulov, Sergei
    Stockholm University, Faculty of Science, Department of Mathematics.
    Classification of irreducible holonomies of torsion-free affine connections1999In: Annals of Mathematics, ISSN 0003-486X, Vol. 150, no 1, p. 77-Article in journal (Refereed)
  • 3.
    Merkulov, Sergei
    Stockholm University, Faculty of Science, Department of Mathematics.
    De Rham model for string topology2004In: International Mathematics Research Notices, ISSN 1073-7928, Vol. 55, p. 2955-Article in journal (Refereed)
  • 4.
    Merkulov, Sergei
    Stockholm University.
    Deformation quantization of the n-tuple point1999In: Communications in Mathematical Physics, Vol. 205, no 2, p. 369-Article in journal (Refereed)
  • 5.
    Merkulov, Sergei
    Stockholm University.
    Formality of canonical symplectic complexes and Frobenius manifolds1998In: International Mathematics Research Notices, Vol. 14, p. 727-Article in journal (Refereed)
  • 6.
    Merkulov, Sergei
    Stockholm University, Faculty of Science, Department of Mathematics.
    Frobenius-infinity invariants of homotopy Gerstenhaber algebras2001In: Duke Mathematical Journal, ISSN 0012-7094, Vol. 105, no 3, p. 411-Article in journal (Refereed)
  • 7.
    Merkulov, Sergei
    Stockholm University, Faculty of Science, Department of Mathematics.
    Nijenhius infinity and contractible dg manifolds2005In: Compositio Mathematica, ISSN 0010-437X, Vol. 141, p. 1238-1254Article in journal (Refereed)
  • 8.
    Merkulov, Sergei
    Stockholm University, Faculty of Science, Department of Mathematics.
    Operad of formal homogeneous spaces and Bernoulli numbers2008In: Algebra and Number Theory, ISSN 1937-0652, Vol. 2, no 4, p. 407-433Article in journal (Refereed)
  • 9.
    Merkulov, Sergei
    Stockholm University, Faculty of Science, Department of Mathematics.
    PROP profile of Poisson geometry2006In: Communications in Mathematical Physics, ISSN 0010-3616, Vol. 262, no 1, p. 117-135Article in journal (Refereed)
  • 10.
    Merkulov, Sergei
    Stockholm University, Faculty of Science, Department of Mathematics.
    Quantization of strongly homotopy Lie bialgebras2006Other (Other academic)
  • 11.
    Merkulov, Sergei
    Stockholm University, Faculty of Science, Department of Mathematics.
    Semi-infinite $A$-variations of Hodge structure over extended Kähler cone2001In: International Mathematics Research Notices, ISSN 1073-7928, Vol. 21, p. 1111-Article in journal (Refereed)
  • 12.
    Merkulov, Sergei
    Stockholm University, Faculty of Science, Department of Mathematics.
    Strong homotopy algebras of a Kähler manifold1999In: International Mathematics Research Notices, Vol. 3, p. 369-Article in journal (Refereed)
  • 13.
    Merkulov, Sergei
    Stockholm University, Faculty of Science, Department of Mathematics.
    The extended moduli space of special Lagrangian submanifolds2000In: Communications in Mathematical Physics, Vol. 209, no 1, p. 13-Article in journal (Refereed)
  • 14.
    Merkulov, Sergei A.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Formality Theorem for Quantizations of Lie Bialgebras2016In: Letters in Mathematical Physics, ISSN 0377-9017, E-ISSN 1573-0530, Vol. 106, no 2, p. 169-195Article in journal (Refereed)
    Abstract [en]

    Using the theory of props we prove a formality theorem associated with universal quantizations of Lie bialgebras.

  • 15.
    Merkulov, Sergei A.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Operads, configuration spaces and quantization2011In: Bulletin of the Brazilian Mathematical Society, ISSN 1678-7544, E-ISSN 1678-7714, Vol. 42, no 4, p. 683-781Article in journal (Refereed)
    Abstract [en]

    We review several well-known operads of compactified configuration spaces and construct several new such operads, (C) over bar, in the category of smooth manifolds with corners whose complexes of fundamental chains give us (i) the 2-coloured operad of A(infinity)-algebras and their homotopy morphisms, (ii) the 2-coloured operad of L(infinity)-algebras and their homotopy morphisms, and (iii) the 4-coloured operad of open-closed homotopy algebras and their homotopy morphisms. Two gadgets - a (coloured) operad of Feynman graphs and a de Rham field theory on (C) over bar - are introduced and used to construct quantized representations of the (fundamental) chain operad of (C) over bar which are given by Feynman type sums over graphs and depend on choices of propagators.

  • 16.
    Merkulov, Sergei A.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Permutahedra, HKR isomorphism and polydifferential Gerstenhaber-Schack complex2010In: Higher Structure in Geometry and Physics:  In Honor of Murray Gerstenhaber and Jim Stasheff / [ed] Cattaneo, Alberto S.; Giaquinto, Anthony; Xu, Ping, Boston: Birkhäuser , 2010, 1, p. 293-314Chapter in book (Other academic)
  • 17.
    Merkulov, Sergei A.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Wheeled Pro(p)file of Batalin-Vilkovisky Formalism2010In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 295, no 3, p. 585-638Article in journal (Refereed)
    Abstract [en]

    Using a technique of wheeled props we establish a correspondence between the homotopy theory of unimodular Lie 1-bialgebras and the famous Batalin-Vilkovisky formalism. Solutions of the so-called quantum master equation satisfying certain boundary conditions are proven to be in 1-1 correspondence with representations of a wheeled dg prop which, on the one hand, is isomorphic to the cobar construction of the prop of unimodular Lie 1-bialgebras and, on the other hand, is quasi-isomorphic to the dg wheeled prop of unimodular Poisson structures. These results allow us to apply properadic methods for computing formulae for a homotopy transfer of a unimodular Lie 1-bialgebra structure on an arbitrary complex to the associated quantum master function on its cohomology. It is proven that in the category of quantum BV manifolds associated with the homotopy theory of unimodular Lie 1-bialgebras quasi-isomorphisms are equivalence relations. It is shown that Losev-Mnev's BF theory for unimodular Lie algebras can be naturally extended to the case of unimodular Lie 1-bialgebras (and, eventually, to the case of unimodular Poisson structures). Using a finite-dimensional version of the Batalin-Vilkovisky quantization formalism it is rigorously proven that the Feynman integrals computing the effective action of this new BF theory describe precisely homotopy transfer formulae obtained within the wheeled properadic approach to the quantum master equation. Quantum corrections (which are present in our BF model to all orders of the Planck constant) correspond precisely to what are often called ""higher Massey products"" in the homological algebra.

  • 18.
    Merkulov, Sergei
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Huggett, Steven
    Twistor transform of vector bundles1999In: Mathematica Scandinavica, Vol. 85, no 2, p. 219-Article in journal (Refereed)
  • 19.
    Merkulov, Sergei
    et al.
    Stockholm University.
    Manin, Yuri
    Semisimple Frobenius (super)manifolds and quantum cohomology of $ P^r$1997In: Topological Methods in Nonlinear Analysis, ISSN 1230-3429, Vol. 9, no 1, p. 107-Article in journal (Refereed)
  • 20.
    Merkulov, Sergei
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Shadrin, Sergei
    University of Amsterdam.
    Markl, Martin
    Prague Institute for mathematics.
    Wheeled PROPs, graph complexes and the master equation2009In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 213, no 4, p. 496-535Article in journal (Refereed)
    Abstract [en]

    We introduce and study wheeled PROPs, an extension of the theory of PROPs which can treat traces and, in particular, solutions to the master equations which involve divergence operators. We construct a dg free wheeled PROP whose representations are in one-to-one correspondence with formal germs of SP-manifolds, key geometric objects in the theory of Batalin–Vilkovisky quantization. We also construct minimal wheeled resolutions of classical operads and as non-trivial extensions of the well-known dg operads and . Finally, we apply the above results to a computation of cohomology of a directed version of Kontsevich’s complex of ribbon graphs.

  • 21.
    Merkulov, Sergei
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Valette, Bruno
    Deformation theory of representations of prop(erad)s2007Other (Other academic)
  • 22.
    Merkulov, Sergei
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Vallette, Bruno
    University of Nice, France.
    Deformation theory of representations of operads I2009In: Journal für die Reine und Angewandte Mathematik, ISSN 0075-4102, E-ISSN 1435-5345, Vol. 634, p. 51-106Article in journal (Refereed)
    Abstract [en]

    In this paper and its follow-up [Merkulov and Vallette, J. reine angew. Math.], we study the deformation theory of morphisms of properads and props thereby extending Quillen's deformation theory for commutative rings to a non-linear framework. The associated chain complex is endowed with an L-algebra structure. Its Maurer-Cartan elements correspond to deformed structures, which allows us to give a geometric interpretation of these results.

    To do so, we endow the category of prop(erad)s with a model category structure. We provide a complete study of models for prop(erad)s. A new effective method to make minimal models explicit, that extends the Koszul duality theory, is introduced and the associated notion is called homotopy Koszul.

    As a corollary, we obtain the (co)homology theories of (al)gebras over a prop(erad) and of homotopy (al)gebras as well. Their underlying chain complex is endowed with an L-algebra structure in general and a Lie algebra structure only in the Koszul case. In particular, we make the deformation complex of morphisms from the properad of associative bialgebras explicit. For any minimal model of this properad, the boundary map of this chain complex is shown to be the one defined by Gerstenhaber and Schack. As a corollary, this paper provides a complete proof of the existence of an L-algebra structure on the Gerstenhaber-Schack bicomplex associated to the deformations of associative bialgebras.

  • 23.
    Merkulov, Sergei
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Willwacher, Thomas
    Grothendieck-Teichmuller and Batalin-Vilkovisky2014In: Letters in Mathematical Physics, ISSN 0377-9017, E-ISSN 1573-0530, Vol. 104, no 5, p. 625-634Article in journal (Refereed)
    Abstract [en]

    It is proven that, for any affine supermanifold M equipped with a constant odd symplectic structure, there is a universal action (up to homotopy) of the Grothendieck Teichmuller Lie algebra grt(1) on the set of quantum BV structures (i.e. solutions of the quantum master equation) on M.

1 - 23 of 23
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