Periodic Reporting for period 1 - QuantMod (Quantisation of moduli spaces: Hitchin connections and isomonodromic deformations)
Reporting period: 2023-10-02 to 2025-10-01
This is a project in (pure) Mathematics, intimately tied with theoretical/mathematical Physics. The main goal is to generalize previous constructions in complex-algebraic Poisson geometry, representation theory, and low-dimensional topology; all related with certain mathematica; models in quantum field theory (QFT).
Namely, geometric quantization & deformation quantization are two important mathematically-rigorous entry points into QFT, and they both famously lead to linear actions of mapping class/braid groups. The latter have found important usage, and we aim at:
(i) extending the definitions of mapping class/braid groups;
(ii) defining `quantum' spaces on which they will act;
and (iii) computing the actions of the new groups, both before and after quantization.
The potential applications include the definition of `quantum' topological invariants, the proof of Kohno--Drinfel'd type theorems, and the mathematical formalization of QFTs with topological/conformal symmetries: notably, the construction of `irregular' conformal blocks in the WZNW model, complementing the (nowadays) much-studied irregular Liouville theory.
The new ingredient are moduli spaces of irregular-singular connections, defined on principal bundles over Riemann surfaces, extending the regular-singular case. We put particular emphasis on nongeneric irregular singularities, as well as twisted ones (which cover the general case): this is current frontier of this much-active line of research.
Then the main background result is that the moduli spaces fit into flat Poisson/symplectic fibre bundles, over isomonodromic families of `wild' Riemann surfaces, which in turn generalize ordinary pointed Riemann surfaces.
Thus:
(i) the fundamental groups of the stacks of wild Riemann surfaces generalize mapping class/braid groups;
(ii) the quantization of the moduli spaces leads to new flat (projective) bundles of irregular nonabelian theta-functions & conformal blocks;
and (iii) their monodromy yields a new `quantum' action of wild mapping class groups, whose semiclassical limit corresponds to braiding the Stokes data of irregular-singular connections.
Namely, geometric quantization & deformation quantization are two important mathematically-rigorous entry points into QFT, and they both famously lead to linear actions of mapping class/braid groups. The latter have found important usage, and we aim at:
(i) extending the definitions of mapping class/braid groups;
(ii) defining `quantum' spaces on which they will act;
and (iii) computing the actions of the new groups, both before and after quantization.
The potential applications include the definition of `quantum' topological invariants, the proof of Kohno--Drinfel'd type theorems, and the mathematical formalization of QFTs with topological/conformal symmetries: notably, the construction of `irregular' conformal blocks in the WZNW model, complementing the (nowadays) much-studied irregular Liouville theory.
The new ingredient are moduli spaces of irregular-singular connections, defined on principal bundles over Riemann surfaces, extending the regular-singular case. We put particular emphasis on nongeneric irregular singularities, as well as twisted ones (which cover the general case): this is current frontier of this much-active line of research.
Then the main background result is that the moduli spaces fit into flat Poisson/symplectic fibre bundles, over isomonodromic families of `wild' Riemann surfaces, which in turn generalize ordinary pointed Riemann surfaces.
Thus:
(i) the fundamental groups of the stacks of wild Riemann surfaces generalize mapping class/braid groups;
(ii) the quantization of the moduli spaces leads to new flat (projective) bundles of irregular nonabelian theta-functions & conformal blocks;
and (iii) their monodromy yields a new `quantum' action of wild mapping class groups, whose semiclassical limit corresponds to braiding the Stokes data of irregular-singular connections.
The technical/scientific part of the work essentially consisted in:
(i) learning (more) relevant techniques, particularly in complex (symplectic) geometry and (geometric) representation theory;
(ii) studying/reviewing (many) past technical articles;
(iii) establishing connections (and collaborations) with some of their authors;
(iv) properly defining/constructing the mathematical objects which are relevant to the main goals of the action;
(v) making precise statements about these new objects;
(vi) proving these statements;
(vii) organizing the original results/theorems into several interconnected papers;
(viii) presenting these texts to the relevant scientific community, by posting to (standard) open-access on-line repositories;
and (ix) submitting the papers to reputable scientific journals, aiming for publications.
While (iv)--(vii) constitute the technical/scientific core, all these activities have been performed nonstop throughout the whole action.
The main achievements have been:
(i) understanding the needed bits of the theory of Deligne--Mumford stacks, category-O modules for truncated-current Lie algebras, complex reflection groups, the confluence/unfolding of irregular singularities, etc.;
(ii) gathering the relevant background about the scientific area of interest, being able to isolate unsolved problems and properly report on state-of-the-art (with thorough bibliographic references);
(iii) reinforcing and/or starting various intercontinental collaborations;
(iv)--(vi) cf. the next subsection `Results beyond the state of the art';
(vii) writing up 4 papers, one of which might lead to 2 separate publications, totalling more than 340 pages which carefully frame and explain all the new results;
(viii) posting them to the arXiv and/or HAL;
and (ix) submitting the above to good journals, and getting acceptance/publication of 5 older articles, upon successfully addressing the referee reports: in Pacific Journal of Mathematics, Annales de l'Institut Fourier, Journal of Lie Theory, Transformation Groups, and the Publications of the Research Institute for Mathematical Sciences (Kyoto University).
(i) learning (more) relevant techniques, particularly in complex (symplectic) geometry and (geometric) representation theory;
(ii) studying/reviewing (many) past technical articles;
(iii) establishing connections (and collaborations) with some of their authors;
(iv) properly defining/constructing the mathematical objects which are relevant to the main goals of the action;
(v) making precise statements about these new objects;
(vi) proving these statements;
(vii) organizing the original results/theorems into several interconnected papers;
(viii) presenting these texts to the relevant scientific community, by posting to (standard) open-access on-line repositories;
and (ix) submitting the papers to reputable scientific journals, aiming for publications.
While (iv)--(vii) constitute the technical/scientific core, all these activities have been performed nonstop throughout the whole action.
The main achievements have been:
(i) understanding the needed bits of the theory of Deligne--Mumford stacks, category-O modules for truncated-current Lie algebras, complex reflection groups, the confluence/unfolding of irregular singularities, etc.;
(ii) gathering the relevant background about the scientific area of interest, being able to isolate unsolved problems and properly report on state-of-the-art (with thorough bibliographic references);
(iii) reinforcing and/or starting various intercontinental collaborations;
(iv)--(vi) cf. the next subsection `Results beyond the state of the art';
(vii) writing up 4 papers, one of which might lead to 2 separate publications, totalling more than 340 pages which carefully frame and explain all the new results;
(viii) posting them to the arXiv and/or HAL;
and (ix) submitting the above to good journals, and getting acceptance/publication of 5 older articles, upon successfully addressing the referee reports: in Pacific Journal of Mathematics, Annales de l'Institut Fourier, Journal of Lie Theory, Transformation Groups, and the Publications of the Research Institute for Mathematical Sciences (Kyoto University).
During the action, the core of our technical/scientific work has led to two main axes of results.
First:
(1.i) a definition of spaces of irregular conformal blocks à la WZNW, in terms of parabolic Verma modules for truncated-current Lie algebras (TCLAs);
(1.ii) using the same modules, a deformation quantization of coadjoint orbits for dual TCLAs;
(1.iii) building on (1.i) the definition of a new flat `irregular' version of the KZ connection;
(1.iv) building on (1.ii) a construction of quantized (de Rham) moduli spaces of irregular-singular connections;
Second:
(2.i) a construction of moduli stacks of wild Riemann surfaces, in the untwisted case;
(2.ii) the definition and study of local WMCGs, also in the twisted case;
(2.iii) building on (2.i) a definition of global wild mapping class groups (WMCGs);
(2.iv) building on (2.ii) new modular interpretations of complex reflection groups in meromorphic 2d gauge theory.
Besides its own mathematical interest, the first axis yields a large new infinite family of (finite-dimensional) quantum representations of braid groups. Importantly, this setup now covers the complete local untwisted case, including the nongeneric one, which was essentially unexplored.
In addition, the second axis concludes the topological study in the most general setup, beyond the above: it yields in particular symplectic dynamics of the braid groups of many complex reflection arrangements, such as those of the generalized symmetric groups, on moduli spaces of ramified irregular-singular connections.
Further research and collaborations are key to relate with geometric quantization, and to construct more general quantum actions.
First:
(1.i) a definition of spaces of irregular conformal blocks à la WZNW, in terms of parabolic Verma modules for truncated-current Lie algebras (TCLAs);
(1.ii) using the same modules, a deformation quantization of coadjoint orbits for dual TCLAs;
(1.iii) building on (1.i) the definition of a new flat `irregular' version of the KZ connection;
(1.iv) building on (1.ii) a construction of quantized (de Rham) moduli spaces of irregular-singular connections;
Second:
(2.i) a construction of moduli stacks of wild Riemann surfaces, in the untwisted case;
(2.ii) the definition and study of local WMCGs, also in the twisted case;
(2.iii) building on (2.i) a definition of global wild mapping class groups (WMCGs);
(2.iv) building on (2.ii) new modular interpretations of complex reflection groups in meromorphic 2d gauge theory.
Besides its own mathematical interest, the first axis yields a large new infinite family of (finite-dimensional) quantum representations of braid groups. Importantly, this setup now covers the complete local untwisted case, including the nongeneric one, which was essentially unexplored.
In addition, the second axis concludes the topological study in the most general setup, beyond the above: it yields in particular symplectic dynamics of the braid groups of many complex reflection arrangements, such as those of the generalized symmetric groups, on moduli spaces of ramified irregular-singular connections.
Further research and collaborations are key to relate with geometric quantization, and to construct more general quantum actions.