Hanany-Witten effect and SL(2,Z) dualities in matrix models

We provide tests of dualities for three-dimensional N=4 quiver SCFTs with brane realizations in IIB string theory, by matching their exact partition functions on $S^3$. The dualities are generated by SL(2,Z) transformations and Hanany-Witten 5-brane moves. These contain mirror symmetry as well as dualities identifiying fixed points of Yang-Mills quivers and Chern-Simons theories. The partition function is given by a matrix model, that can be nicely rearranged into a sequence of factors mimicking the brane realization. Identities obeyed by these elementary factors can be used to match the partition functions of dual theories, providing tests for the full web of dualities. In particular we are able to check mirror symmetry for linear and circular quivers with gauge nodes of arbitrary ranks. Our analysis also leads to a proof of a conjectured formula evaluating the matrix models of linear quiver theories.Comment: 65 pages, 23 figures, v2, minor clarifications added, version published on JHE

Mirror Symmetry And Loop Operators

Wilson loops in gauge theories pose a fundamental challenge for dualities. Wilson loops are labeled by a representation of the gauge group and should map under duality to loop operators labeled by the same data, yet generically, dual theories have completely different gauge groups. In this paper we resolve this conundrum for three dimensional mirror symmetry. We show that Wilson loops are exchanged under mirror symmetry with Vortex loop operators, whose microscopic definition in terms of a supersymmetric quantum mechanics coupled to the theory encode in a non-trivial way a representation of the original gauge group, despite that the gauge groups of mirror theories can be radically different. Our predictions for the mirror map, which we derive guided by branes in string theory, are confirmed by the computation of the exact expectation value of Wilson and Vortex loop operators on the three-sphere.Comment: 92 pages, v2: minor clarifications in the introduction, to be published in JHE

Six-dimensional Origin of N=4\mathcal{N}=4 SYM with Duality Defects

We study the topologically twisted compactification of the 6d $(2,0)$ M5-brane theory on an elliptically fibered K\"ahler three-fold preserving two supercharges. We show that upon reducing on the elliptic fiber, the 4d theory is $\mathcal{N}=4$ Super-Yang Mills, with varying complexified coupling $\tau$, in the presence of defects. For abelian gauge group this agrees with the so-called duality twisted theory, and we determine a non-abelian generalization to $U(N)$. When the elliptic fibration is singular, the 4d theory contains 3d walls (along the branch-cuts of $\tau$) and 2d surface defects, around which the 4d theory undergoes $SL(2,\mathbb{Z})$ duality transformations. Such duality defects carry chiral fields, which from the 6d point of view arise as modes of the two-form $B$ in the tensor multiplet. Each duality defect has a flavor symmetry associated to it, which is encoded in the structure of the singular elliptic fiber above the defect. Generically 2d surface defects will intersect in points in 4d, where there is an enhanced flavor symmetry. The 6d point of view provides a complete characterization of this 4d-3d-2d-0d `Matroshka'-defect configuration.Comment: 62 pages, 4 figure

Partition functions of 3d D^\hat D-quivers and their mirror duals from 1d free fermions

We study the matrix models calculating the sphere partition functions of 3d gauge theories with $\mathcal{N}=4$ supersymmetry and a quiver structure of a $\hat D$ Dynkin diagram (where each node is a unitary gauge group). As in the case of necklace ($\hat A$) quivers, we can map the problem to that of free fermion quantum mechanics whose complicated Hamiltonian we find explicitly. Many of these theories are conjectured to be dual under mirror symmetry to certain unitary linear quivers with extra Sp nodes or antisymmetric hypermultiplets. We show that the free fermion formulations of such mirror pairs are related by a linear symplectic transformation. We then study the large N expansion of the partition function, which as in the case of the $\hat A$-quivers is given to all orders in 1/N by an Airy function. We simplify the algorithm to calculate the numerical coefficients appearing in the Airy function and evaluate them for a wide class of $\hat D$-quiver theories.Comment: 39 pages, 8 figure

M5-branes on S^2 x M_4: Nahm's Equations and 4d Topological Sigma-models

We study the 6d N=(0,2) superconformal field theory, which describes multiple M5-branes, on the product space S^2 x M_4, and suggest a correspondence between a 2d N=(0,2) half-twisted gauge theory on S^2 and a topological sigma-model on the four-manifold M_4. To set up this correspondence, we determine in this paper the dimensional reduction of the 6d N=(0,2) theory on a two-sphere and derive that the four-dimensional theory is a sigma-model into the moduli space of solutions to Nahm's equations, or equivalently the moduli space of k-centered SU(2) monopoles, where k is the number of M5-branes. We proceed in three steps: we reduce the 6d abelian theory to a 5d Super-Yang-Mills theory on I x M_4, with I an interval, then non-abelianize the 5d theory and finally reduce this to 4d. In the special case, when M_4 is a Hyper-Kahler manifold, we show that the dimensional reduction gives rise to a topological sigma-model based on tri-holomorphic maps. Deriving the theory on a general M_4 requires knowledge of the metric of the target space. For k=2 the target space is the Atiyah-Hitchin manifold and we twist the theory to obtain a topological sigma-model, which has both scalar fields and self-dual two-forms.Comment: 78 pages, 2 figure

UNDERSTANDING INFORMAL LEARNING IN VIRTUAL PROFESSIONAL COMMUNITIES OF TEACHERS IN KAZAKHSTAN

Reinforced internationally in the context of educational improvement, teachers’ professional networks, as a source of social capital, have been conceptualised as an integral part of teacher professionalism, as well as an essential element of successful educational change, and in the context of what Van Dijck, Poell and De Waal (2018) call “the platform society”, the use of social media platforms within professional networks of teachers has become an agenda for both research and practice. Therefore, with the overarching aim of understanding how to promote informal learning of teachers in virtual professional communities in Kazakhstan, this study explored this phenomenon within the conceptual framework identified by a review of related concepts, in particular a triangle of learning factors, namely, the need for professional connectedness, knowledge sharing self-efficacy, and knowledge sharing and receiving. This parallel mixed-method study was carried out in 29 schools of Kazakhstan by collecting teachers’ self-reported practice with the help of a paper-based questionnaires (n=440) and face-to-face interviews (n=41). An emergent trend within the identified findings is that teachers in Kazakhstan use social media within professional communities in order to obtain knowledge, which is manifested in an overlapping mixture of news, information, opinion, experience and resources, suggesting that virtual professional communities are one of the spaces for informal learning since they provide the opportunity to gain public and/or personal knowledge related to the teaching profession. In line with the identified conceptual framework, the results of the study provide a partial explanation for teachers’ engagement in virtual professional communities in the context of informal learning. The study suggests that both the need for professional connectedness, as part of professional identity and commitment for learning, and knowledge sharing self-efficacy are positively associated with knowledge sharing and receiving. As well as identifying contextual types of virtual professional communities, the study identifies some of the contextual factors associated with the need for professional connectedness in the research context, such as professional isolation of teachers in rural schools, the need for mentoring support, and the context of educational change, and contextual sources of knowledge sharing self-efficacy, such as professional comparison and sense of professional connectedness. Finally, in contribution to the growing body of research, the present study also argues for the importance of face-to-to face collaboration within and beyond schools in order to promote professional knowledge exchange within virtual professional communities. The research has clear implications for research and practice in the fields of teacher professional learning, particularly in Kazakhstan, hence it is believed that present study can help future efforts to support informal learning in virtual professional communities