Issues on Orientifolds: On the brane construction of gauge theories with SO(2n) global symmetry

We discuss issues related to orientifolds and the brane realization for gauge theories with orthogonal and symplectic groups. We specifically discuss the case of theories with (hidden) global SO(2n) symmetry, from three to six dimensions. We analyze mirror symmetry for three dimensional N=4 gauge theories, Brane Box Models and six-dimensional gauge theories. We also discuss the issue of T-duality for D_n space-time singularities. Stuck D branes on ON^0 planes play an interesting role.Comment: 38 pages, 23 figures, uses bibtex and (provided) utphys.bs

VOAs labelled by complex reflection groups and 4d SCFTs

We define and study a class of $\mathcal{N}=2$ vertex operator algebras $\mathcal{W}_{\mathcal{\mathsf{G}}}$ labelled by complex reflection groups. They are extensions of the $\mathcal{N}=2$ super Virasoro algebra obtained by introducing additional generators, in correspondence with the invariants of the complex reflection group $\mathcal{\mathsf{G}}$. If $\mathcal{\mathsf{G}}$ is a Coxeter group, the $\mathcal{N}=2$ super Virasoro algebra enhances to the (small) $\mathcal{N}=4$ superconformal algebra. With the exception of $\mathcal{\mathsf{G}} = \mathbb{Z}_2$, which corresponds to just the $\mathcal{N}=4$ algebra, these are non-deformable VOAs that exist only for a specific negative value of the central charge. We describe a free-field realization of $\mathcal{W}_{\mathcal{\mathsf{G}}}$ in terms of rank$(\mathcal{\mathsf{G}})$ $\beta \gamma bc$ ghost systems, generalizing a construction of Adamovic for the $\mathcal{N}=4$ algebra at $c = -9$. If $\mathcal{\mathsf{G}}$ is a Weyl group, $\mathcal{W}_{\mathcal{\mathsf{G}}}$ is believed to coincide with the $\mathcal{N}=4$ VOA that arises from the four-dimensional super Yang-Mills theory whose gauge algebra has Weyl group $\mathcal{\mathsf{G}}$. More generally, if $\mathcal{\mathsf{G}}$ is a crystallographic complex reflection group, $\mathcal{W}_{\mathcal{\mathsf{G}}}$ is conjecturally associated to an $\mathcal{N}=3$ $4d$ superconformal field theory. The free-field realization allows to determine the elusive `$R$-filtration' of $\mathcal{W}_{\mathcal{\mathsf{G}}}$, and thus to recover the full Macdonald index of the parent $4d$ theoryComment: 70 page

Isometric endomorphisms of free groups

An arbitrary homomorphism between groups is nonincreasing for stable commutator length, and there are infinitely many (injective) homomorphisms between free groups which strictly decrease the stable commutator length of some elements. However, we show in this paper that a random homomorphism between free groups is almost surely an isometry for stable commutator length for every element; in particular, the unit ball in the scl norm of a free group admits an enormous number of exotic isometries. Using similar methods, we show that a random fatgraph in a free group is extremal (i.e. is an absolute minimizer for relative Gromov norm) for its boundary; this implies, for instance, that a random element of a free group with commutator length at most n has commutator length exactly n and stable commutator length exactly n-1/2. Our methods also let us construct explicit (and computable) quasimorphisms which certify these facts.Comment: 26 pages, 6 figures; minor typographical edits for final published versio

Coset Realization of Unifying W-Algebras

We construct several quantum coset W-algebras, e.g. sl(2,R)/U(1) and sl(2,R)+sl(2,R) / sl(2,R), and argue that they are finitely nonfreely generated. Furthermore, we discuss in detail their role as unifying W-algebras of Casimir W-algebras. We show that it is possible to give coset realizations of various types of unifying W-algebras, e.g. the diagonal cosets based on the symplectic Lie algebras sp(2n) realize the unifying W-algebras which have previously been introduced as `WD_{-n}'. In addition, minimal models of WD_{-n} are studied. The coset realizations provide a generalization of level-rank-duality of dual coset pairs. As further examples of finitely nonfreely generated quantum W-algebras we discuss orbifolding of W-algebras which on the quantum level has different properties than in the classical case. We demonstrate in some examples that the classical limit according to Bowcock and Watts of these nonfreely finitely generated quantum W-algebras probably yields infinitely nonfreely generated classical W-algebras.Comment: 60 pages (plain TeX) (final version to appear in Int. J. Mod. Phys. A; several minor improvements and corrections - for details see beginning of file

Still Wrong Use of Pairings in Cryptography

Several pairing-based cryptographic protocols are recently proposed with a wide variety of new novel applications including the ones in emerging technologies like cloud computing, internet of things (IoT), e-health systems and wearable technologies. There have been however a wide range of incorrect use of these primitives. The paper of Galbraith, Paterson, and Smart (2006) pointed out most of the issues related to the incorrect use of pairing-based cryptography. However, we noticed that some recently proposed applications still do not use these primitives correctly. This leads to unrealizable, insecure or too inefficient designs of pairing-based protocols. We observed that one reason is not being aware of the recent advancements on solving the discrete logarithm problems in some groups. The main purpose of this article is to give an understandable, informative, and the most up-to-date criteria for the correct use of pairing-based cryptography. We thereby deliberately avoid most of the technical details and rather give special emphasis on the importance of the correct use of bilinear maps by realizing secure cryptographic protocols. We list a collection of some recent papers having wrong security assumptions or realizability/efficiency issues. Finally, we give a compact and an up-to-date recipe of the correct use of pairings.Comment: 25 page

Three-Index Symmetric Matter Representations of SU(2) in F-Theory from Non-Tate Form Weierstrass Models

We give an explicit construction of a class of F-theory models with matter in the three-index symmetric (4) representation of SU(2). This matter is realized at codimension two loci in the F-theory base where the divisor carrying the gauge group is singular; the associated Weierstrass model does not have the form associated with a generic SU(2) Tate model. For 6D theories, the matter is localized at a triple point singularity of arithmetic genus g=3 in the curve supporting the SU(2) group. This is the first explicit realization of matter in F-theory in a representation corresponding to a genus contribution greater than one. The construction is realized by "unHiggsing" a model with a U(1) gauge factor under which there is matter with charge q=3. The resulting SU(2) models can be further unHiggsed to realize non-Abelian G_2xSU(2) models with more conventional matter content or SU(2)^3 models with trifundamental matter. The U(1) models used as the basis for this construction do not seem to have a Weierstrass realization in the general form found by Morrison-Park, suggesting that a generalization of that form may be needed to incorporate models with arbitrary matter representations and gauge groups localized on singular divisors.Comment: 34 pages, 3 figure