18 research outputs found
Mirror symmetry for circle compactified 4d SCFTs
We propose a mirror symmetry for 4d superconformal field
theories (SCFTs) compactified on a circle with finite size. The mirror symmetry
involves vertex operator algebra (VOA) describing the Schur sector (containing
Higgs branch) of 4d theory, and the Coulomb branch of the effective 3d theory.
The basic feature of the mirror symmetry is that many representational
properties of VOA are matched with geometric properties of the Coulomb branch
moduli space. Our proposal is verified for a large class of Argyres-Douglas
(AD) theories engineered from M5 branes, whose VOAs are W-algebras, and Coulomb
branches are the Hitchin moduli spaces. VOA data such as simple modules, Zhu's
algebra, and modular properties are matched with geometric properties like
-fixed varieties in Hitchin fibers, cohomologies, and some DAHA
representations. We also mention relationships to 3d symplectic duality.Comment: 53 pages, 6 figure
3d Modularity
We find and propose an explanation for a large variety of modularity-related
symmetries in problems of 3-manifold topology and physics of 3d
theories where such structures a priori are not manifest. These modular
structures include: mock modular forms, Weil
representations, quantum modular forms, non-semisimple modular tensor
categories, and chiral algebras of logarithmic CFTs.Comment: 119 pages, 10 figures and 20 table
An Algorithmic Approach to Operator Product Expansions, -Algebras and -Strings
String theory is currently the most promising theory to explain the spectrum
of the elementary particles and their interactions. One of its most important
features is its large symmetry group, which contains the conformal
transformations in two dimensions as a subgroup. At quantum level, the symmetry
group of a theory gives rise to differential equations between correlation
functions of observables. We show that these Ward-identities are equivalent to
Operator Product Expansions (OPEs), which encode the short-distance
singularities of correlation functions with symmetry generators. The OPEs allow
us to determine algebraically many properties of the theory under study. We
analyse the calculational rules for OPEs, give an algorithm to compute OPEs,
and discuss an implementation in Mathematica. There exist different string
theories, based on extensions of the conformal algebra to so-called W-algebras.
These algebras are generically nonlinear. We study their OPEs, with as main
results an efficient algorithm to compute the beta-coefficients in the OPEs,
the first explicit construction of the WB_2-algebra, and criteria for the
factorisation of free fields in a W-algebra. An important technique to
construct realisations of W-algebras is Drinfel'd- Sokolov reduction. The
method consists of imposing certain constraints on the elements of an affine
Lie algebra. We quantise this reduction via gauged WZNW-models. This enables us
in a theory with a gauged W-symmetry, to compute exactly the correlation
functions of the effective theory. Finally, we investigate the critical
W-string theories based on an extension of the conformal algebra with one
symmetry generator of dimension N. We clarify how the spectrum of this theory
forms a minimal model of the W_N-algebra.Comment: 127 pages, LaTex, shar-file including readme.txt, 12 latex files, 6
eps files and 6 pcx files, PhD. thesis KU Leuve
3d Modularity
We find and propose an explanation for a large variety of modularity-related symmetries in problems of 3-manifold topology and physics of 3d theories where such structures a priori are not manifest. These modular structures include: mock modular forms, Weil representations, quantum modular forms, non-semisimple modular tensor categories, and chiral algebras of logarithmic CFTs
3d modularity
We find and propose an explanation for a large variety of modularity-related symmetries in problems of 3-manifold topology and physics of 3d N = 2 theories where such structures a priori are not manifest. These modular structures include: mock modular forms, SL(2,â€) Weil representations, quantum modular forms, non-semisimple modular tensor categories, and chiral algebras of logarithmic CFTs
Conformal Field Theory Between Supersymmetry and Indecomposable Structures
This thesis considers conformal field theory in its supersymmetric extension as well as in its relaxation to logarithmic conformal field theory. Compactification of superstring theory on four-dimensional complex manifolds obeying the Calabi-Yau conditions yields the moduli space of N=(4,4) superconformal field theories with central charge c=6 which consists of two continuously connected subspaces. This thesis is concerned with the subspace of K3 compactifications which is not well known yet. In particular, we inspect the intersection point of the Z_2 and Z_4 orbifold subvarieties within the K3 moduli space, explicitly identify the two corresponding points on the subvarieties geometrically, and give an explicit isomorphism of the three conformal field theory models located at that point, a specific Z_2 and a Z_4 orbifold model as well as the Gepner model (2)^4. We also prove the orthogonality of the two subvarieties at the intersection point. This is the starting point for the programme to investigate generic points in K3 moduli space. We use the coordinate identification at the intersection point in order to relate the coordinates of both subvarieties and to explicitly calculate a geometric geodesic between the two subvarieties as well as its generator. A generic point in K3 moduli space can be reached by such a geodesic originating at a known model. We also present advances on the conformal field theoretic side of deformations along such a geodesic using conformal deformation theory. Since a consistent regularisation of the appearing deformation integrals has not been achieved yet, the completion of this programme is still an open problem. Moreover, we regard a relaxation of conformal field theory to logarithmic conformal field theory. The latter allows the indecomposable action of the L_0 Virasoro mode within a representation of the conformal symmetry. In particular, we study general augmented c_{p,q} minimal models which generalise the well-known (augmented) c_{p,1} model series. We calculate logarithmic nullvectors in both types of models. But most importantly, we investigate the low lying Virasoro representation content and fusion algebra of two general augmented c_{p,q} models, the augmented c_{2,3} = 0 model as well as the augmented Yang-Lee model at c_{2,5} = -22/5. These exhibit a much richer structure as the c_{p,1} models with indecomposable representations up to rank 3. We elaborate several of these new rank 3 representations in great detail and uncover astonishing features. Furthermore, we argue that irreducible representations corresponding to the Kac table domain of the proper minimal models cannot be included into the theory. In particular, the true vacuum representation is rather given by a rank 1 indecomposable but not irreducible subrepresentation of a rank 2 representation. We generalise these generic examples to give the representation content and the fusion algebra of general augmented c_{p,q} models as a conjecture. Finally, we open a new connection between logarithmic conformal field theory and quantum spin chains by relating some representations of the augmented c_{2,3} = 0 model to the representation content of a c=0 model which describes an XXZ quantum spin chain