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Marco Mackaay

Publications and source records attributed to Marco Mackaay.

At least 37 records · Page 2Linked to original sources
arXiv2013

sl(N)-Web categories

In this paper we use colored sl(N)-matrix factorizations, due to Wu and Y.Y., in order to categorify part of the quantum skew Howe duality defined by Cautis, Kamnitzer and Morrison. In particular, we define web categories and 2-representations of Khovanov and Lauda's categorical quantum sl(m) on them. We show that each such web category is equivalent to the category of finite dimensional graded projective modules over a certain level N cyclotomic Khovanov-Lauda-Rouquier algebra.

math.QA↗
arXiv2013

The sl(N)-web algebras and dual canonical bases

In this paper, which is a follow-up to my paper with Yonezawa "sl(N)-web categories", I define and study sl(N)-web algebras for any N greater than one. For N=2 these algebras are isomorphic to Khovanov's arc algebras and for N=3 they are Morita equivalent to the sl(3)-web algebras which I defined and studied with Pan and Tubbenhauer. The main result of this paper is that the sl(N)-web algebras are Morita equivalent to blocks of certain level-N cyclotomic KLR algebras, for which I use the categorified quantum skew Howe duality from the aforementioned paper with Yonezawa. Using this Morita equivalence and Brundan and Kleshchev's work on cyclotomic KLR algebras, I show that there exists an isomorphism between a certain space of sl(N)-webs and the split Grothendieck group of the corresponding sl(N)-web algebra, which maps the dual canonical basis elements to the Grothendieck classes of the indecomposable projective modules (with a certain normalization of their grading).

math.QA↗
arXiv2012

A remark on Rasmussen's invariant of knots

We show that Rasmussen's invariant of knots, which is derived from Lee's variant of Khovanov homology, is equal to an analogous invariant derived from certain other filtered link homologies.

math.GT↗
arXiv2012

A diagrammatic categorification of the q-Schur algebra

In this paper we categorify the q-Schur algebra S(n,d) as a quotient of Khovanov and Lauda's diagrammatic 2-category U(sln). We also show that our 2-category contains Soergel's monoidal category of bimodules of type A, which categorifies the Hecke algebra H(d), as a full sub-2-category if d does not exceed n. For the latter result we use Elias and Khovanov's diagrammatic presentation of Soergel's monoidal category of type A.

math.QA↗
arXiv2010

Extended graphical calculus for categorified quantum sl(2)

A categorification of the Beilinson-Lusztig-MacPherson form of the quantum sl(2) was constructed in the paper arXiv:0803.3652 by the second author. Here we enhance the graphical calculus introduced and developed in that paper to include two-morphisms between divided powers one-morphisms and their compositions. We obtain explicit diagrammatical formulas for the decomposition of products of divided powers one-morphisms as direct sums of indecomposable one-morphisms; the latter are in a bijection with the Lusztig canonical basis elements. These formulas have integral coefficients and imply that one of the main results of Lauda's paper---identification of the Grothendieck ring of his 2-category with the idempotented quantum sl(2)---also holds when the 2-category is defined over the ring of integers rather than over a field.

math.QA↗
arXiv2010

The diagrammatic Soergel category and sl(N)-foams, for N > 3

For each N > 3, we define a monoidal functor from Elias and Khovanov's diagrammatic version of Soergel's category of bimodules to the category of sl(N) foams defined by Mackaay, Stosic and Vaz. We show that through these functors Soergel's category can be obtained from the sl(N) foams.

math.QA↗
arXiv2009

sl(3)-Foams and the Khovanov-Lauda categorification of quantum sl(k)

In this paper I define certain interesting 2-functors from the Khovanov-Lauda 2-category which categorifies quantum sl(k), for any k>1, to a 2-category of universal sl(3) foams with corners. For want of a better name I use the term "foamation" to indicate those 2-functors. I conjecture the existence of similar 2-functors to the 2-category of sl(n) foams with corners, for any n>1.

math.QA↗
arXiv2008

The 1,2-coloured HOMFLY-PT link homology

In this paper we define the 1,2-coloured HOMFLY-PT link homology and prove that it is a link invariant. We conjecture that this homology categorifies the coloured HOMFLY-PT polynomial for links whose components are labelled 1 or 2.

math.QA↗
arXiv2007

Sl(N) link homology using foams and the Kapustin-Li formula

We use foams to give a topological construction of a rational link homology categorifying the slN link invariant, for N>3. To evaluate closed foams we use the Kapustin-Li formula adapted to foams by Khovanov and Rozansky. We show that for any link our homology is isomorphic to Khovanov and Rozansky's.

math.GT↗
arXiv2006

The universal sl3-link homology

We define the universal sl3-link homology, which depends on 3 parameters, following Khovanov's approach with foams. We show that this 3-parameter link homology, when taken with complex coefficients, can be divided into 3 isomorphism classes. The first class is the one to which Khovanov's original sl3-link homology belongs, the second is the one studied by Gornik in the context of matrix factorizations and the last one is new. Following an approach similar to Gornik's we show that this new link homology can be described in terms of Khovanov's original sl2-link homology.

math.GT↗
arXiv2004

Categorical representations of categorical groups

The representation theory for categorical groups is constructed. Each categorical group determines a monoidal bicategory of representations. Typically, these categories contain representations which are indecomposable but not irreducible. A simple example is computed in explicit detail.

math.CT↗
arXiv2002

A note on the holonomy of connections in twisted bundles

Recently twisted K-theory has received much attention due to its applications in string theory and the announced result by Freed, Hopkins and Telemann relating the twisted equivariant K-theory of a compact Lie group to its Verlinde algebra. Rather than considering gerbes as separate objects, in twisted K-theory one considers a gerbe as being part of the data for a twisted vector bundle. There is also a notion of a connection in a twisted vector bundle and Kapustin has studied some aspects of the holonomy of such connections. In this note I study the holonomy of connections in twisted principal bundles and show that it can best be defined as a functor rather than a map. Even for the case which Kapustin studied the results in this paper give a more general picture.

math.DG↗
arXiv2001

Holonomy and parallel transport for Abelian gerbes

In this paper we establish a one-to-one correspondence between $S^1$-gerbes with connections, on the one hand, and their holonomies, for simply connected manifolds, or their parallel transports, in the general case, on the other hand. This result is a higher-order analogue of the familiar equivalence between bundles with connections and their holonomies for connected manifolds. The holonomy of a gerbe with group $S^1$ on a simply connected manifold $M$ is a group morphism from the thin second homotopy group to $S^1$, satisfying a smoothness condition, where a homotopy between maps from $[0,1]^2$ to $M$ is thin when its derivative is of rank $\leq 2$. For the non-simply connected case, holonomy is replaced by a parallel transport functor between two monoidal Lie groupoids. The reconstruction of the gerbe and connection from its holonomy is carried out in detail for the simply connected case. Our approach to abelian gerbes with connections holds out prospects for generalizing to the non-abelian case via the theory of double Lie groupoids.

math.DG↗
arXiv1999

Finite groups, spherical 2-categories, and 4-manifold invariants

In this paper we define a class of state-sum invariants of compact closed oriented piece-wise linear 4-manifolds using finite groups. The definition of these state-sums follows from the general abstract construction of 4-manifold invariants using spherical 2-categories, as we defined in a previous paper, although it requires a slight generalization of that construction. We show that the state-sum invariants of Birmingham and Rakowski, who studied Dijkgraaf-Witten type invariants in dimension 4, are special examples of the general construction that we present in this paper. They showed that their invariants are non-trivial by some explicit computations, so our construction includes interesting examples already. Finally, we indicate how our construction is related to homotopy 3-types. This connection suggests that there are many more interesting examples of our construction to be found in the work on homotopy 3-types, such as by Brown, for example.

math.QA↗