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We define the branched analog of SL(r,C)-opers and investigate their properties. For the usual SL(r,C)-opers, the underlying holomorphic vector bundle is independent of the opers. For the branched SL(r,C)-opers, the underlying holomorphic vector bundle depends on the oper. Given a branched SL(r,C)-oper, we associate to it another holomorphic vector bundle equipped with a logarithmic connection. This holomorphic vector bundle does not depend on the branched oper. We characterize the branched SL(r,C)-opers in terms of the logarithmic connections on this fixed holomorphic vector bundle.
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We study the branched holomorphic projective structures on a compact Riemann surface $X$ with a fixed branching divisor $S, =, sum_{i=1}^d x_i$, where $x_i ,in, X$ are distinct points. After defining branched ${rm SO}(3,{mathbb C})$--opers, we show that the branched holomorphic projective structures on $X$ are in a natural bijection with the branched ${rm SO}(3,{mathbb C})$--opers singular at $S$. It is deduced that the branched holomorphic projective structures on $X$ are also identified with a subset of the space of all logarithmic connections on $J^2((TX)otimes {mathcal O}_X(S))$ singular over $S$, satisfying certain natural geometric conditions.
We define and study the space of $q$-opers associated with Bethe equations for integrable models of XXZ type with quantum toroidal algebra symmetry. Our construction is suggested by the study of the enumerative geometry of cyclic quiver varieties, in particular, the ADHM moduli spaces. We define $(overline{GL}(infty),q)$-opers with regular singularities and then, by imposing various analytic conditions on singularities, arrive at the desired Bethe equations for toroidal $q$-opers.
Let X be a compact connected Riemann surface of genus g > 0 equipped with a nonzero holomorphic 1-form. Let M denote the moduli space of semistable Higgs bundles on X of rank r and degree r(g-1)+1; it is a complex symplectic manifold. Using the translation structure on the open subset of X where the 1-form does not vanish, we construct a natural deformation quantization of a certain nonempty Zariski open subset of M.
Let G be a finite subgroup of SL(n,C), then the quotient C^n/G has a Gorenstein canonical singularity. Bridgeland-King-Reid proved that the G-Hilbert scheme Hilb^G(C^3) gives a crepant resolution of the quotient C^3/G for any finite subgroup G of SL(3,C). However, in dimension 4, very few crepant resolutions are known. In this paper, we will show several examples of crepant resolutions in dimension 4 and show examples in which Hilb^G(C^4) is blow-up of certain crepant resolutions for C^4/G, or Hilb^G(C^4) has singularity.
The Gell-Mann grading, one of the four gradings of sl(3,C) that cannot be further refined, is considered as the initial grading for the graded contraction procedure. Using the symmetries of the Gell-Mann grading, the system of contraction equations is reduced and solved. Each non-trivial solution of this system determines a Lie algebra which is not isomorphic to the original algebra sl(3,C). The resulting 53 contracted algebras are divided into two classes - the first is represented by the algebras which are also continuous Inonu-Wigner contractions, the second is formed by the discrete graded contractions.
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