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Register a new userThe matrix regularization for Riemann surfaces with magnetic fluxes
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We consider the matrix regularization of fields on a Riemann surface which couple to gauge fields with a nonvanishing magnetic flux. We show that such fields are described as rectangular matrices in the matrix regularization. We construct the matrix regularization explicitly for the case of the sphere and torus based on the Berezin-Toeplitz quantization, and also discuss a possible generalization to cases with higher genera. We also discuss the matrix version of the Laplacian acting on the rectangular matrices.
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We quantize pure 2d Yang-Mills theory on an arbitrary Riemann surface in the gauge where the field strength is diagonal. Twisted sectors originate, as in Matrix string theory, from permutations of the eigenvalues around homotopically non-trivial loops. These sectors, that must be discarded in the usual quantization due to divergences occurring when two eigenvalues coincide, can be consistently kept if one modifies the action by introducing a coupling of the field strength to the space-time curvature. This leads to a generalized Yang-Mills theory whose action reduces to the usual one in the limit of zero curvature. After integrating over the non-diagonal components of the gauge fields, the theory becomes a free string theory (sum over unbranched coverings) with a U(1) gauge theory on the world-sheet. This is shown to be equivalent to a lattice theory with a gauge group which is the semi-direct product of S_N and U(1)^N. By using well known results on the statistics of coverings, the partition function on arbitrary Riemann surfaces and the kernel functions on surfaces with boundaries are calculated. Extensions to include branch points and non-abelian groups on the world-sheet are briefly commented upon.
Knizhnik-Zamolodchikov-Bernard equations for twisted conformal blocks on compact Riemann surfaces with marked points are written explicitly in a general projective structure in terms of correlation functions in the theory of twisted b-c systems. It is checked that on the moduli space the equations provide a flat connection with the spectral parameter.
We construct a consistent Kaluza-Klein reduction of $D=11$ supergravity on $Sigma_2times S^4$, where $Sigma_2=S^2,mathbb{R}^2$ or $H^2$, or a quotient thereof, at the level of the bosonic fields. The result is a gauged $N=4$, $D=5$ supergravity theory coupled to three vector multiplets, with the gauging lying in an $SO(2)times SE(3)subset SO(5,3)$ subgroup of the $SO(1,1)times SO(5,3)$ global symmetry group of the ungauged theory. For $Sigma_2=H^2$, the $D=5$ theory has a maximally supersymmetric $AdS_5$ vacuum which uplifts to the known solution of $D=11$ supergravity corresponding to M5-branes wrapping a Riemann surface with genus greater than one and dual to an $N=2$ SCFT in $d=4$. For $Sigma_2=S^2$, we find two $AdS_5$ solutions, one of which is new, and both of which are unstable. There is an additional subtruncation to an $N=2$ gauged supergravity coupled to two vector multiplets, with very special real manifold $SO(1,1)times SO(1,1)$, and a single hypermultiplet, with quaternionic Kahler manifold $SU(2,1)/S[U(2)times U(1)]$ and gauging associated with an $SO(2)timesmathbb{R}subset SU(2,1)$ subgroup.
We consider compactifications of $6d$ minimal $(D_{N+3},D_{N+3})$ type conformal matter SCFTs on a generic Riemann surface. We derive the theories corresponding to three punctured spheres (trinions) with three maximal punctures, from which one can construct models corresponding to generic surfaces. The trinion models are simple quiver theories with $mathcal{N}=1$ $SU(2)$ gauge nodes. One of the three puncture non abelian symmetries is emergent in the IR. The derivation of the trinions proceeds by analyzing RG flows between conformal matter SCFTs with different values of $N$ and relations between their subsequent reductions to $4d$. In particular, using the flows we first derive trinions with two maximal and one minimal punctures, and then we argue that collections of $N$ minimal punctures can be interpreted as a maximal one. This suggestion is checked by matching the properties of the $4d$ models such as `t Hooft anomalies, symmetries, and the structure of the conformal manifold to the expectations from $6d$. We then use the understanding that collections of minimal punctures might be equivalent to maximal ones to construct trinions with three maximal punctures, and then $4d$ theories corresponding to arbitrary surfaces, for $6d$ models described by two $M5$ branes probing a $mathbb{Z}_k$ singularity. This entails the introduction of a novel type of maximal puncture. Again, the suggestion is checked by matching anomalies, symmetries and the conformal manifold to expectations from six dimensions. These constructions thus give us a detailed understanding of compactifications of two sequences of six dimensional SCFTs to four dimensions.
We consider U(n+1) Yang-Mills instantons on the space Sigmatimes S^2, where Sigma is a compact Riemann surface of genus g. Using an SU(2)-equivariant dimensional reduction, we show that the U(n+1) instanton equations on Sigmatimes S^2 are equivalent to non-Abelian vortex equations on Sigma. Solutions to these equations are given by pairs (A,phi), where A is a gauge potential of the group U(n) and phi is a Higgs field in the fundamental representation of the group U(n). We briefly compare this model with other non-Abelian Higgs models considered recently. Afterwards we show that for g>1, when Sigmatimes S^2 becomes a gravitational instanton, the non-Abelian vortex equations are the compatibility conditions of two linear equations (Lax pair) and therefore the standard methods of integrable systems can be applied for constructing their solutions.
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