44 research outputs found
The Kapustin-Li formula revisited
We provide a new perspective on the Kapustin-Li formula for the duality
pairing on the morphism complexes in the matrix factorization category of an
isolated hypersurface singularity. In our context, the formula arises as an
explicit description of a local duality isomorphism, obtained by using the
basic perturbation lemma and Grothendieck residues. The non-degeneracy of the
pairing becomes apparent in this setting. Further, we show that the pairing
lifts to a Calabi-Yau structure on the matrix factorization category. This
allows us to define topological quantum field theories with matrix
factorizations as boundary conditions.Comment: 28 pages, 3 figures, comments welcom
Permutation branes and linear matrix factorisations
All the known rational boundary states for Gepner models can be regarded as
permutation branes. On general grounds, one expects that topological branes in
Gepner models can be encoded as matrix factorisations of the corresponding
Landau-Ginzburg potentials. In this paper we identify the matrix factorisations
associated to arbitrary B-type permutation branes.Comment: 43 pages. v2: References adde
Matrix Models for Supersymmetric Chern-Simons Theories with an ADE Classification
We consider N=3 supersymmetric Chern-Simons (CS) theories that contain
product U(N) gauge groups and bifundamental matter fields. Using the matrix
model of Kapustin, Willett and Yaakov, we examine the Euclidean partition
function of these theories on an S^3 in the large N limit. We show that the
only such CS theories for which the long range forces between the eigenvalues
cancel have quivers which are in one-to-one correspondence with the simply
laced affine Dynkin diagrams. As the A_n series was studied in detail before,
in this paper we compute the partition function for the D_4 quiver. The D_4
example gives further evidence for a conjecture that the saddle point
eigenvalue distribution is determined by the distribution of gauge invariant
chiral operators. We also see that the partition function is invariant under a
generalized Seiberg duality for CS theories.Comment: 20 pages, 3 figures; v2 refs added; v3 conventions in figure 3
altered, version to appear in JHE
B-type defects in Landau-Ginzburg models
We consider Landau-Ginzburg models with possibly different superpotentials
glued together along one-dimensional defect lines. Defects preserving B-type
supersymmetry can be represented by matrix factorisations of the difference of
the superpotentials. The composition of these defects and their action on
B-type boundary conditions is described in this framework. The cases of
Landau-Ginzburg models with superpotential W=X^d and W=X^d+Z^2 are analysed in
detail, and the results are compared to the CFT treatment of defects in N=2
superconformal minimal models to which these Landau-Ginzburg models flow in the
IR.Comment: 50 pages, 2 figure
Defect Perturbations in Landau-Ginzburg Models
Perturbations of B-type defects in Landau-Ginzburg models are considered. In
particular, the effect of perturbations of defects on their fusion is analyzed
in the framework of matrix factorizations. As an application, it is discussed
how fusion with perturbed defects induces perturbations on boundary conditions.
It is shown that in some classes of models all boundary perturbations can be
obtained in this way. Moreover, a universal class of perturbed defects is
constructed, whose fusion under certain conditions obey braid relations. The
functors obtained by fusing these defects with boundary conditions are twist
functors as introduced in the work of Seidel and Thomas.Comment: 46 page
The ABCDEF's of Matrix Models for Supersymmetric Chern-Simons Theories
We consider N = 3 supersymmetric Chern-Simons gauge theories with product
unitary and orthosymplectic groups and bifundamental and fundamental fields. We
study the partition functions on an S^3 by using the Kapustin-Willett-Yaakov
matrix model. The saddlepoint equations in a large N limit lead to a constraint
that the long range forces between the eigenvalues must cancel; the resulting
quiver theories are of affine Dynkin type. We introduce a folding/unfolding
trick which lets us, at the level of the large N matrix model, (i) map quivers
with orthosymplectic groups to those with unitary groups, and (ii) obtain
non-simply laced quivers from the corresponding simply laced quivers using a
Z_2 outer automorphism. The brane configurations of the quivers are described
in string theory and the folding/unfolding is interpreted as the
addition/subtraction of orientifold and orbifold planes. We also relate the
U(N) quiver theories to the affine ADE quiver matrix models with a
Stieltjes-Wigert type potential, and derive the generalized Seiberg duality in
2 + 1 dimensions from Seiberg duality in 3 + 1 dimensions.Comment: 30 pages, 5 figure
Adjunctions and defects in Landau-Ginzburg models
We study the bicategory of Landau-Ginzburg models, which has potentials as
objects and matrix factorisations as 1-morphisms. Our main result is the
existence of adjoints in this bicategory and a description of evaluation and
coevaluation maps in terms of Atiyah classes and homological perturbation. The
bicategorical perspective offers a unified approach to Landau-Ginzburg models:
we show how to compute arbitrary correlators and recover the full structure of
open/closed TFT, including the Kapustin-Li disk correlator and a simple proof
of the Cardy condition, in terms of defect operators which in turn are directly
computable from the adjunctions.Comment: 58 pages; v2: Fixed typos and references, removed comments about
graded matrix factorisations; v3: exposition improved and shortened, main
result now holds over an arbitrary ring k; v4: many improvements to
expositio
From Necklace Quivers to the F-theorem, Operator Counting, and T(U(N))
The matrix model of Kapustin, Willett, and Yaakov is a powerful tool for
exploring the properties of strongly interacting superconformal Chern-Simons
theories in 2+1 dimensions. In this paper, we use this matrix model to study
necklace quiver gauge theories with {\cal N}=3 supersymmetry and U(N)^d gauge
groups in the limit of large N. In its simplest application, the matrix model
computes the free energy of the gauge theory on S^3. The conjectured F-theorem
states that this quantity should decrease under renormalization group flow. We
show that for a simple class of such flows, the F-theorem holds for our
necklace theories. We also provide a relationship between matrix model
eigenvalue distributions and numbers of chiral operators that we conjecture
holds more generally. Through the AdS/CFT correspondence, there is therefore a
natural dual geometric interpretation of the matrix model saddle point in terms
of volumes of 7-d tri-Sasaki Einstein spaces and some of their 5-d
submanifolds. As a final bonus, our analysis gives us the partition function of
the T(U(N)) theory on S^3.Comment: 3 figures, 41 pages; v2 minor improvements, refs adde
Operator Counting and Eigenvalue Distributions for 3D Supersymmetric Gauge Theories
We give further support for our conjecture relating eigenvalue distributions
of the Kapustin-Willett-Yaakov matrix model in the large N limit to numbers of
operators in the chiral ring of the corresponding supersymmetric
three-dimensional gauge theory. We show that the relation holds for
non-critical R-charges and for examples with {\mathcal N}=2 instead of
{\mathcal N}=3 supersymmetry where the bifundamental matter fields are
nonchiral. We prove that, for non-critical R-charges, the conjecture is
equivalent to a relation between the free energy of the gauge theory on a three
sphere and the volume of a Sasaki manifold that is part of the moduli space of
the gauge theory. We also investigate the consequences of our conjecture for
chiral theories where the matrix model is not well understood.Comment: 27 pages + appendices, 5 figure
N=1 Supersymmetric Product Group Theories in the Coulomb Phase
We study the low-energy behavior of N=1 supersymmetric gauge theories with
product gauge groups SU(N)^M and M chiral superfields transforming in the
fundamental representation of two of the SU(N) factors. These theories are in
the Coulomb phase with an unbroken U(1)^(N-1) gauge group. For N >= 3, M >= 3
the theories are chiral. The low-energy gauge kinetic functions can be obtained
from hyperelliptic curves which we derive by considering various limits of the
theories. We present several consistency checks of the curves including
confinement through the addition of mass perturbations and other limits.Comment: 22 pages, LaTeX, minor changes. Eqs. (20) and (42) correcte