The Swiss-cheese operad

Meyer, J.; Morava, Jack; Wilson, W. Stephen; Voronov, Alexander A.

doi:10.1090/conm/239/03610

Cited by 80 publications

(96 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…It requires us to know the action of the modular transformation S in C V . Although the action of SL(2, Z) in a modular tensor category is explicitly known [MSei3,V,Ly,Ki,BK2], its relation to the modular transformation of the q-trace of the products of intertwining operators of V is not completely clear. This relation was first suggested by I. Frenkel and studied by Moore and Seiberg in [MSei3] but only on a physical level of rigor.…”

Section: Assumption 02 In This Work We Fix a Vertex Operator Algebmentioning

confidence: 99%

See 1 more Smart Citation

Cardy Condition for Open-Closed Field Algebras

Kong

2008

Commun. Math. Phys.

View full text Add to dashboard Cite

Let V be a vertex operator algebra satisfying certain reductivity and finiteness conditions such that C V , the category of V -modules, is a modular tensor category. We study open-closed field algebras over V equipped with nondegenerate invariant bilinear forms for both open and closed sectors. We show that they give algebras over a certain C-extension of the so-called Swiss-cheese partial dioperad, and we can obtain Ishibashi states easily in such algebras. The Cardy condition can be formulated as an additional condition on such open-closed field algebras in terms of the action of the modular transformation S : τ → − 1 τ on the space of intertwining operators of V . We then derive a graphical representation of S in the modular tensor category C V . This result enables us to give a categorical formulation of the Cardy condition and the modular invariance condition for 1-point correlation functions on the torus. Then we incorporate these two conditions and the axioms of the open-closed field algebra over V equipped with nondegenerate invariant bilinear forms into a tensor-categorical notion called the Cardy C V |C V ⊗V -algebra. In the end, we give a categorical construction of the Cardy C V |C V ⊗V -algebra in the Cardy case.

show abstract

Section: Assumption 02 In This Work We Fix a Vertex Operator Algebmentioning

confidence: 99%

“…If we restrict to {P(1, m|0, n)} m,n∈N and {Q(1, n)} n∈N , they simply gives a structure of 2-colored (partial) operad [V,Kont,Ko2]. Now we discuss an important example of the 2-colored partial nonassociative dioperad.…”

Section: The Family Of Maps γmentioning

confidence: 99%

Cardy Condition for Open-Closed Field Algebras

Kong

2008

Commun. Math. Phys.

View full text Add to dashboard Cite

show abstract

“…The unifying mathematical notion behind the above correspondences is the operads of little intervals and associated Swiss-Cheese operads, which relate associative algebras with 2-algebras (Deligne conjecture) [79,80,51]. On the other hand the Swiss-Cheese operad [86] is closely related to the tree level open-closed string field theory of Zwiebach [103]. Actually the path integral approach shows that the Deligne conjecture is just the bulk/boundary correspondence.…”

Section: A Sketch Of Our Programmentioning

confidence: 99%

TOPOLOGICAL OPEN P-BRANES

Park

2001

Symplectic Geometry and Mirror Symmetry

100

155

View full text Add to dashboard Cite

By exploiting the BV quantization of topological bosonic open membrane, we argue that flat 3-form C-field leads to deformations of the algebras of multi-vectors on the Dirichlet-brane world-volume as 2-algebras. This would shed some new light on geometry of M-theory 5-brane and associated decoupled theories. We show that, in general, topological open p-brane has a structure of (p + 1)-algebra in the bulk, while a structure of p-algebra in the boundary. The bulk/boundary correspondences are exactly as of the generalized Deligne conjecture (a theorem of Kontsevich) in the algebraic world of p-algebras. It also imply that the algebras of quantum observables of (p − 1)-brane are "close to" the algebras of its classical observables as p-algebras. We interpret above as deformation quantization of (p − 1)-brane, generalizing the p = 1 case. We argue that there is such quantization based on the direct relation between BV master equation and Ward identity of the bulk topological theory. The path integral of the theory will lead to the explicit formula. We also discuss some applications to topological strings and conjecture that the homological mirror symmetry has further generalizations to the categories of p-algebras.

show abstract

“…De nition 3.15. The E 1 -operad (or little intervals operad) consists of 1D Swiss cheese diagrams [Vor99] consisting of a large interval, several removed subintervals called holes, all considered up to di eomorphism. 5 These interval diagrams can be composed by plugging some new big intervals into the holes to get a new diagram.…”

mentioning

confidence: 99%

The Asaeda–Haagerup fusion categories

Grossman

Izumi

Snyder

2016

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

View full text Add to dashboard Cite

Abstract. The classification of subfactors of small index revealed several new subfactors. The first subfactor above index 4, the Haagerup subfactor, is increasingly well understood and appears to lie in a (discrete) infinite family of subfactors where the Z/3Z symmetry is replaced by other finite abelian groups. The goal of this paper is to give a similarly good description of the Asaeda-Haagerup subfactor which emerged from our study of its Brauer-Picard groupoid. More specifically, we construct a new subfactor S which is a Z/4Z × Z/2Z analogue of the Haagerup subfactor and we show that the even parts of the Asaeda-Haagerup subfactor are higher Morita equivalent to an orbifold quotient of S. This gives a new construction of the Asaeda-Haagerup subfactor which is much more symmetric and easier to work with than the original construction. As a consequence, we can settle many open questions about the Asaeda-Haagerup subfactor: calculating its Drinfel'd center, classifying all extensions of the Asaeda-Haagerup fusion categories, finding the full higher Morita equivalence class of the AsaedaHaagerup fusion categories, and finding intermediate subfactor lattices for subfactors coming from the Asaeda-Haagerup categories. The details of the applications will be given in subsequent papers.

show abstract

The Swiss-cheese operad

Cited by 80 publications

References 15 publications

Cardy Condition for Open-Closed Field Algebras

Cardy Condition for Open-Closed Field Algebras

TOPOLOGICAL OPEN P-BRANES

The Asaeda–Haagerup fusion categories

Contact Info

Product

Resources

About