On the geometry of operator mixing in massless QCD-like theories

We revisit the operator mixing in massless QCD-like theories. In particular, we address the problem of determining under which conditions a renormalization scheme exists where the renormalized mixing matrix in the coordinate representation, $$Z(x, \mu )$$ Z ( x , μ ) , is diagonalizable to all perturbative orders. As a key step, we provide a differential-geometric interpretation of renormalization that allows us to apply the Poincaré-Dulac theorem to the problem above: We interpret a change of renormalization scheme as a (formal) holomorphic gauge transformation, $$-\frac{\gamma (g)}{\beta (g)}$$ - γ ( g ) β ( g ) as a (formal) meromorphic connection with a Fuchsian singularity at $$g=0$$ g = 0 , and $$Z(x,\mu )$$ Z ( x , μ ) as a Wilson line, with $$\gamma (g)=\gamma _0 g^2 + \cdots $$ γ ( g ) = γ 0 g 2 + ⋯ the matrix of the anomalous dimensions and $$\beta (g)=-\beta _0 g^3 +\cdots $$ β ( g ) = - β 0 g 3 + ⋯ the beta function. As a consequence of the Poincaré-Dulac theorem, if the eigenvalues $$\lambda _1, \lambda _2, \ldots $$ λ 1 , λ 2 , … of the matrix $$\frac{\gamma _0}{\beta _0}$$ γ 0 β 0 , in nonincreasing order $$\lambda _1 \ge \lambda _2 \ge \cdots $$ λ 1 ≥ λ 2 ≥ ⋯ , satisfy the nonresonant condition $$\lambda _i -\lambda _j -2k \ne 0$$ λ i - λ j - 2 k ≠ 0 for $$i\le j$$ i ≤ j and k a positive integer, then a renormalization scheme exists where $$-\frac{\gamma (g)}{\beta (g)} = \frac{\gamma _0}{\beta _0} \frac{1}{g}$$ - γ ( g ) β ( g ) = γ 0 β 0 1 g is one-loop exact to all perturbative orders. If in addition $$\frac{\gamma _0}{\beta _0}$$ γ 0 β 0 is diagonalizable, $$Z(x, \mu )$$ Z ( x , μ ) is diagonalizable as well, and the mixing reduces essentially to the multiplicatively renormalizable case. We also classify the remaining cases of operator mixing by the Poincaré–Dulac theorem.

Extending meromorphic connections to coadmissible ̑𝒟-modules

We investigate when a meromorphic connection on a smooth rigid analytic variety 𝑋 gives rise to a coadmissible D ⏜ X \overparen{\mathcal{D}}_{X} -module, and show that this is always the case when the roots of the corresponding 𝑏-functions are all of positive type. We also use this theory to give an example of an integrable connection on the punctured unit disk whose pushforward is not a coadmissible module.

Energy of sections of the Deligne–Hitchin twistor space

We study a natural functional on the space of holomorphic sections of the Deligne–Hitchin moduli space of a compact Riemann surface, generalizing the energy of equivariant harmonic maps corresponding to twistor lines. We show that the energy is the residue of the pull-back along the section of a natural meromorphic connection on the hyperholomorphic line bundle recently constructed by Hitchin. As a byproduct, we show the existence of a hyper-Kähler potentials for new components of real holomorphic sections of twistor spaces of hyper-Kähler manifolds with rotating $$S^1$$ S 1 -action. Additionally, we prove that for a certain class of real holomorphic sections of the Deligne–Hitchin moduli space, the energy functional is basically given by the Willmore energy of corresponding equivariant conformal map to the 3-sphere. As an application we use the functional to distinguish new components of real holomorphic sections of the Deligne–Hitchin moduli space from the space of twistor lines.

On the Notion of Conductor in the Local Geometric Langlands Correspondence

Under the local Langlands correspondence, the conductor of an irreducible representation of Gln(F) is greater than the Swan conductor of the corresponding Galois representation. In this paper, we establish the geometric analogue of this statement by showing that the conductor of a categorical representation of the loop group is greater than the irregularity of the corresponding meromorphic connection.

Stokes factors and multilogarithms

Let G be a complex, affine algebraic group and a meromorphic connection on the trivial G-bundle over , with a pole of order 2 at zero and a pole of order 1 at infinity. We show that the map taking the residue of at zero to the corresponding Stokes factors is given by an explicit, universal Lie series whose coefficients are multilogarithms. Using a non-commutative analogue of the compositional inversion of formal power series, we show that the same holds for the inverse of , and that the corresponding Lie series coincides with the generating function for counting invariants in abelian categories constructed by D. Joyce.

Semistable reduction for overconvergent F-isocrystals, IV: local semistable reduction at nonmonomial valuations

We complete our proof that given an overconvergent F-isocrystal on a variety over a field of positive characteristic, one can pull back along a suitable generically finite cover to obtain an isocrystal which extends, with logarithmic singularities and nilpotent residues, to some complete variety. We also establish an analogue for F-isocrystals overconvergent inside a partial compactification. By previous results, this reduces to solving a local problem in a neighborhood of a valuation of height 1 and residual transcendence degree zero. We do this by studying the variation of some numerical invariants attached to p-adic differential modules, analogous to the irregularity of a complex meromorphic connection. This allows for an induction on the transcendence defect of the valuation, i.e., the discrepancy between the dimension of the variety and the rational rank of the valuation.

On the Quantization of Isomonodromic Deformations on the Torus

The quantization of isomonodromic deformation of a meromorphic connection on the torus is shown to lead directly to the Knizhnik–Zamolodchikov–Bernard equations in the same way as the problem on the sphere leads to the system of Knizhnik–Zamolodchikov equations. The Poisson bracket required for a Hamiltonian formulation of isomonodromic deformations is naturally induced by the Poisson structure of Chern–Simons theory in a holomorphic gauge fixing. This turns out to be the origin of the appearance of twisted quantities on the torus.