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20
TFT construction of RCFT correlators I: Partition functions
, 2002
"... We formulate rational conformal field theory in terms of a symmetric special Frobenius algebra A and its representations. A is an algebra in the modular tensor category of MooreSeiberg data of the underlying chiral CFT. The multiplication on A corresponds to the OPE of boundary fields for a single ..."
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Cited by 72 (26 self)
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We formulate rational conformal field theory in terms of a symmetric special Frobenius algebra A and its representations. A is an algebra in the modular tensor category of MooreSeiberg data of the underlying chiral CFT. The multiplication on A corresponds to the OPE of boundary fields for a single boundary condition. General boundary conditions are Amodules, and (generalised) defect lines are AAbimodules. The relation with threedimensional TFT is used to express CFT data, like structure constants or torus and annulus coefficients, as invariants of links in threemanifolds. We compute explicitly the ordinary and twisted partition functions on the torus and the annulus partition functions. We prove that they satisfy consistency conditions, like modular invariance and NIMrep properties. We suggest that our results can be interpreted in terms of noncommutative geometry over the modular tensor category of MooreSeiberg data.
CATEGORY THEORY FOR CONFORMAL BOUNDARY CONDITIONS
, 2001
"... We study properties of the category of modules of an algebra object A in a tensor category C. We show that the module category inherits various structures from C, provided that A is a Frobenius algebra with certain additional properties. As a byproduct we obtain results about the FrobeniusSchur ind ..."
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Cited by 69 (17 self)
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We study properties of the category of modules of an algebra object A in a tensor category C. We show that the module category inherits various structures from C, provided that A is a Frobenius algebra with certain additional properties. As a byproduct we obtain results about the FrobeniusSchur indicator in sovereign tensor categories. A braiding on C is not needed, nor is semisimplicity. We apply our results to the description of boundary conditions in twodimensional conformal field theory and present illustrative examples. We show that module categories give rise to NIMreps of the fusion rules, and discuss a possible relation with the representation theory of vertex operator algebras.
The twisted Drinfeld double of a finite group via gerbes and finite groupoids
"... Abstract. The twisted Drinfeld double (or quasiquantum double) of a finite group with a 3cocycle is identified with a certain twisted groupoid algebra. The groupoid is the loop (or inertia) groupoid of the original group and the twisting is shown geometrically to be the loop transgression of the 3 ..."
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Cited by 24 (0 self)
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Abstract. The twisted Drinfeld double (or quasiquantum double) of a finite group with a 3cocycle is identified with a certain twisted groupoid algebra. The groupoid is the loop (or inertia) groupoid of the original group and the twisting is shown geometrically to be the loop transgression of the 3cocycle. The twisted representation theory of finite groupoids is developed and used to derive properties of the Drinfeld double, such as representations being classified by their characters. This is all motivated by gerbes and 3dimensional quantum field theory. In particular the representation category of the twisted Drinfeld double is viewed as the ‘space of sections ’ associated to a transgressed gerbe over the loop groupoid.
ON FROBENIUS ALGEBRAS IN RIGID MONOIDAL CATEGORIES
, 2008
"... We show that the equivalence between several possible characterizations of Frobenius algebras, and of symmetric Frobenius algebras, carries over from the category of vector spaces to more general monoidal categories. For Frobenius algebras, the appropriate setting is the one of rigid monoidal catego ..."
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Cited by 12 (4 self)
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We show that the equivalence between several possible characterizations of Frobenius algebras, and of symmetric Frobenius algebras, carries over from the category of vector spaces to more general monoidal categories. For Frobenius algebras, the appropriate setting is the one of rigid monoidal categories, and for symmetric Frobenius algebras it is the one of sovereign monoidal categories. We also discuss some properties of Nakayama automorphisms.
Equivariant operads, string topology, and Tate cohomology
, 2006
"... ABSTRACT. From an operad C with an action of a group G, we construct new operads using the homotopy fixed point and orbit spectra. These new operads are shown to be equivalent when the generalized GTate cohomology of C is trivial. Applying this theory to the little disk operad C2 (which is an S 1o ..."
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Cited by 11 (1 self)
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ABSTRACT. From an operad C with an action of a group G, we construct new operads using the homotopy fixed point and orbit spectra. These new operads are shown to be equivalent when the generalized GTate cohomology of C is trivial. Applying this theory to the little disk operad C2 (which is an S 1operad) we obtain variations on Getzler’s gravity operad, which we show governs the ChasSullivan string bracket. 1.
Orbifold genera, product formulas and power operations
 Adv. Math
, 2006
"... Moore, Verlinde and Verlinde, expressing the orbifold elliptic genus of the symmetric powers of an almost complex manifold M in terms of the elliptic genus of M itself. We show that from the point of view of elliptic cohomology an analogous ptypical statement follows as an easy corollary from the f ..."
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Cited by 7 (4 self)
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Moore, Verlinde and Verlinde, expressing the orbifold elliptic genus of the symmetric powers of an almost complex manifold M in terms of the elliptic genus of M itself. We show that from the point of view of elliptic cohomology an analogous ptypical statement follows as an easy corollary from the fact that the map of spectra corresponding to the genus preserves power operations. We define higher chromatic versions of the notion of orbifold genus, involving htuples rather than pairs of commuting elements. Using homotopy theoretic methods we are able to prove an integrality result and show that our definition is independent of the representation of the orbifold. Our setup is so simple, that it allows us to prove DMVVtype product formulas for these higher chromatic orbifold genera in the same way that the product formula for the topological Todd genus is proved. More precisely, we show that any genus induced by an H∞map into one of the MoravaLubinTate cohomology theories Eh has such a product formula and that the formula depends only on h and not on the genus. Since the complex H∞genera into Eh have been classified in [And95], a large family of genera to which our results apply is completely understood. Loosely speaking, our result says that some Borcherds
STRING TOPOLOGY OF CLASSIFYING SPACES
, 2007
"... Let G be a finite group or a compact connected Lie group and let BG be its classifying space. Let LBG: = map(S 1, BG) be the free loop space of BG i.e. the space of continuous maps from the circle S 1 to BG. The purpose of this paper is to study the singular homology H∗(LBG) of this loop space. We ..."
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Cited by 5 (1 self)
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Let G be a finite group or a compact connected Lie group and let BG be its classifying space. Let LBG: = map(S 1, BG) be the free loop space of BG i.e. the space of continuous maps from the circle S 1 to BG. The purpose of this paper is to study the singular homology H∗(LBG) of this loop space. We prove that when taken with coefficients in a field the homology of LBG is a homological conformal field theory. As a byproduct of our main theorem, we get on the cohomology H ∗ (LBG) a BValgebra structure.
A Model Structure on the Category of Small Categories for Coverings
, 2009
"... We define a new model structure on the category of small categories, which is intimately related to the notion of coverings and fundamental groups of small categories. Fibrant objects in the model structure coincide with groupoids, and the fibrant replacement is the groupoidification. ..."
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Cited by 5 (0 self)
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We define a new model structure on the category of small categories, which is intimately related to the notion of coverings and fundamental groups of small categories. Fibrant objects in the model structure coincide with groupoids, and the fibrant replacement is the groupoidification.
MULTICURVES AND EQUIVARIANT COHOMOLOGY
, 2008
"... Abstract. Let A be a finite abelian group. We set up an algebraic framework for studying Aequivariant complexorientable cohomology theories in terms of a suitable kind of equivariant formal groups. We compute the equivariant cohomology of many spaces in these terms, including projective bundles (a ..."
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Abstract. Let A be a finite abelian group. We set up an algebraic framework for studying Aequivariant complexorientable cohomology theories in terms of a suitable kind of equivariant formal groups. We compute the equivariant cohomology of many spaces in these terms, including projective bundles (and associated Gysin maps), Thom spaces, and infinite Grassmannians. 1.
FROBENIUS PAIRS AND ATIYAH DUALITY
"... Abstract. We define a notion of “Frobenius pair”, which is a mild generalization of the notion of “Frobenius object ” in a monoidal category. We then show that Atiyah duality for smooth manifolds can be encapsulated in the statement that a certain collection of structure obtained from a manifold for ..."
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Abstract. We define a notion of “Frobenius pair”, which is a mild generalization of the notion of “Frobenius object ” in a monoidal category. We then show that Atiyah duality for smooth manifolds can be encapsulated in the statement that a certain collection of structure obtained from a manifold forms a “commutative Frobenius pair ” in the stable homotopy