Wheeled PROPs, graph complexes and the master equation

Markl, Martin; Merkulov, Sergeï; Shadrin, Sergey

doi:10.1016/j.jpaa.2008.08.007

Cited by 63 publications

(93 citation statements)

References 22 publications

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“…Similarly, the Deligne-Mumford boundary morphisms (2.2) give a PROP structure on H • (M g ). In this case, self-sewing of single surfaces enhances this to a wheeled PROP (a notion introduced in [26]). Cohomological field theories are algebras over the associated homology PROP of DM spaces, but to capture the full CohFT structure, we must add a cyclic structure, permuting inputs and outputs.…”

Section: Prop Descriptionmentioning

confidence: 99%

The structure of 2D semi-simple field theories

2011

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I classify all cohomological 2D field theories based on a semi-simple complex Frobenius algebra A. They are controlled by a linear combination of kappa-classes and by an extension datum to the Deligne-Mumford boundary. Their effect on the Gromov-Witten potential is described by Givental's Fock space formulae. This leads to the reconstruction of Gromov-Witten invariants from the quantum cup-product at a single semi-simple point and from the first Chern class, confirming Givental's higher-genus reconstruction conjecture. The proof uses the Mumford conjecture proved by Madsen and Weiss.Comment: Small errors corrected in v3. Agrees with published versio

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Section: Prop Descriptionmentioning

confidence: 99%

The structure of 2D semi-simple field theories

2011

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“…For part (i), here we are appealing to the description in [26,Section 3] of BBP in terms of 'metric trees' (whereas in [8] we used the modular analogue [12,Theorem 5.4]). The analogous combinatorial description of the double bar construction of a wheeled properad appears in Proof of [25,Theorem 4.2.5].…”

Section: )mentioning

confidence: 99%

Feynman diagrams and minimal models for operadic algebras

Chuang¹,

Lazarev

2010

Journal of the London Mathematical Society

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We construct an explicit minimal model for an algebra over the cobar-construction of a differential graded operad. The structure maps of this minimal model are expressed in terms of sums over decorated trees. We introduce the appropriate notion of a homotopy equivalence of operadic algebras and show that our minimal model is homotopy equivalent to the original algebra. All this generalizes and gives a conceptual explanation of well-known results for A ∞algebras. Furthermore, we show that these results carry over to the case of algebras over modular operads; the sums over trees get replaced by sums over general Feynman graphs. As a by-product of our work we prove gauge-independence of Kontsevich's 'dual construction' producing graph cohomology classes from contractible differential graded Frobenius algebras.

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“…Wheeled properads in the linear setting are heavily used in applications [KWZ12,MMS09,Mer09,Mer10a,Mer10b]. Foundational discussion of wheeled properads can be found in [YJ15,JY].…”

Section: (Wheeled) Properads As Generalized Categoriesmentioning

confidence: 99%

“…In the linear setting, 1-colored wheeled properads in biased form were introduced in [MMS09]. The explicit biased axioms can be found in [YJ15].…”

Section: Biased Wheeled Properadsmentioning

confidence: 99%

Infinity Properads and Infinity Wheeled Properads

Hackney

Robertson

Yau

2015

Lecture Notes in Mathematics

180

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PrefaceThis monograph fits in the intersection of two long and intertwined stories. The first part of our story starts in the mid-twentieth century, when it became clear that a new conceptual framework was necessary for the study of higher homotopical structures arising in algebraic topology. Some better known examples of these higher homotopical structures appear in work of J. F. Adams and S. Mac Lane [Mac65] on the coproduct in the bar construction and work of J. Stasheff [Sta63], J. M. Boardman and R. M. Vogt [BV68,BV73], and J. P. May [May72] on recognition principles for (ordinary, n-fold, or infinite) loop spaces. The notions of 'operad' and 'prop' were precisely formulated for the purpose of this work; the former is suitable for modeling algebraic or coalgebraic structures, while the latter is also capable of modeling bialgebraic structures, such as Hopf algebras. Operads came to prominence in other areas of mathematics beginning in the 1990s (but see, e.g. This renaissance in the world of operads [Lod96, LSV97] and the popularity of quantum groups [Dri83,Dri87] in the 1990s lead to a resurgence of interest in props, which had long been in the shadow of their little singleoutput nephew. Properads were invented around this time, during an effort of B. Vallette to formulate a Koszul duality for props [Val07]. Properads and props both model algebraic structures with several inputs and outputs, but properads govern a smaller class of such structures, those whose generating operations and relations among operations can be taken to be connected. This class includes most types of bialgebras that arise in nature, such as biassociative bialgebras, (co)module bialgebras, Lie bialgebras, and Hopf algebras. The use of 'colored' or 'multisorted' variants of operads or props [Bri00, BV73], where composition is only partially defined, allows one to address many other situations of interest. It allows one, for instance, to model morphisms of algebras associated to a given operad. There is a two-colored operad which encodes the data of two associative algebras as well as a map from one to the other; a resolution of this operad precisely gives the correct notion of morphism of A ∞ -algebras [Mar04,Mar02]. It also provides a unified way to treat operads, cyclic operads, modular operads, properads, and so on: for each there is a colored operad which controls the structure in question.The second part of our story is an extension of the theory of categories. Categories are pervasive in pure mathematics and, for our purposes, can be loosely described as tools for studying collections of objects up to isomorphism and comparisons between collections of objects up to isomorphism. When the objects we want to study have a homotopy theory, we need to generalize category theory to identify two objects which are, while possibly not isomorphic in the categorical sense, equivalent up to homotopy. For example, when discussing topological spaces we might replace 'homeomorphism' with 'homotopy equivalence'. This leads to the theory of ∞-ca...

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Wheeled PROPs, graph complexes and the master equation

Cited by 63 publications

References 22 publications

The structure of 2D semi-simple field theories

The structure of 2D semi-simple field theories

Feynman diagrams and minimal models for operadic algebras

Infinity Properads and Infinity Wheeled Properads

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