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Register a new userA Web of 2d Dualities: ${bf Z}_2$ Gauge Fields and Arf Invariants
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We describe a web of well-known dualities connecting quantum field theories in $d=1+1$ dimensions. The web is constructed by gauging ${bf Z}_2$ global symmetries and includes a number of perennial favourites such as the Jordan-Wigner transformation, Kramers-Wannier duality, bosonization of a Dirac fermion, and T-duality. There are also less-loved examples, such as non-modular invariant $c=1$ CFTs that depend on a background spin structure.
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We study a model in $d = 2 + 1$ space-time dimensions with two sectors. One of them, which can be considered as the visible sector, contains just a $U(1)$ gauge field which acts as a probe for the other (hidden) sector, given by a second $U(1)$ gauge field and massive scalar and Dirac fermions. Covariant derivatives of these matter fields and a BF gauge mixing term couple the two sectors. Integration over fermion fields leads to an effective theory with Chern-Simons terms that support vortex like solutions in both sectors even if originally there was no symmetry breaking Higgs scalar in the visible sector. We study numerically the solutions which correspond to electrically charged magnetic vortices except for a critical value of the BF coupling constant at which solely purely magnetic vortices exist.
We develop a geometrical structure of the manifolds $Gamma$ and $hatGamma$ associated respectively to the gauge symmetry and to the BRST symmetry. Then, we show that ($hatGamma,hatzeta,Gamma$), where $hatzeta$ is the group of BRST transformations, is endowed with the structure of a principle fiber bundle over the base manifold $Gamma$. Furthermore, in this geometrical set up due to the nilpotency of the BRST operator, we prove that the effective action of a gauge theory is a BRST-exact term up to the classical action. Then, we conclude that the effective action where only the gauge symmetry is fixed, is cohomologically equivalent to the action where the gauge and the BRST symmetries are fixed.
We consider an analogue of Wittens $SL(2,mathbb{Z})$ action on three-dimensional QFTs with $U(1)$ symmetry for $2k$-dimensional QFTs with $mathbb{Z}_2$ $(k-1)$-form symmetry. We show that the $SL(2,mathbb{Z})$ action only closes up to a multiplication by an invertible topological phase whose partition function is the Brown-Kervaire invariant of the spacetime manifold. We interpret it as part of the $SL(2,mathbb{Z})$ anomaly of the bulk $(2k+1)$-dimensional $mathbb{Z}_2$ gauge theory.
In this paper, we exploit a subtle indeterminacy in the definition of the spherical Kervaire-Milnor invariant which was discovered by R. Stong to construct non-spin 4-manifolds with even intersection form and prescribed signature.
In arXiv:1906.11820 and arXiv:1907.05404 we proposed an approach based on graphs to characterize 5d superconformal field theories (SCFTs), which arise as compactifications of 6d $mathcal{N}= (1,0)$ SCFTs. The graphs, so-called combined fiber diagrams (CFDs), are derived using the realization of 5d SCFTs via M-theory on a non-compact Calabi--Yau threefold with a canonical singularity. In this paper we complement this geometric approach by connecting the CFD of an SCFT to its weakly coupled gauge theory or quiver descriptions and demonstrate that the CFD as recovered from the gauge theory approach is consistent with that as determined by geometry. To each quiver description we also associate a graph, and the embedding of this graph into the CFD that is associated to an SCFT provides a systematic way to enumerate all possible consistent weakly coupled gauge theory descriptions of this SCFT. Furthermore, different embeddings of gauge theory graphs into a fixed CFD can give rise to new UV-dualities for which we provide evidence through an analysis of the prepotential, and which, for some examples, we substantiate by constructing the M-theory geometry in which the dual quiver descriptions are manifest.
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