Periodic Reporting for period 4 - DerSympApp (Derived Symplectic Geometry and Applications)
Período documentado: 2023-03-01 hasta 2024-02-29
On the one hand, symplectic geometry is a natural setting for classical mechanics: phase spaces appear to be symplectic manifolds (or variations of these). Indeed, a symplectic (or, rather, Poisson) structure is precisely what is needed to associate a dynamical system on the given space to each energy function. On the other hand, derived algebraic geometry has been invented in order to deal in a satisfying way with spaces that are singular, which is the case for many spaces appearing in algebraic geometry (moduli spaces) and in classical physics (spaces of solutions of equations of motion).
The theory of symplectic structures on derived stacks gave birth to what is now called derived symplectic geometry. Since then we have seen a rapid evolution of knowledge, and several applications as well (such as general existence theorems for symplectic structures and symmetric obstruction theories on moduli spaces, for instance). This project aims at providing new theoretical developments (shifted symplectic groupoids, for example) and new applications of derived symplectic geometry (to classical field theories, for instance).
On the foundational side (Main Goal A), our aim was to introduce the new notion of shifted symplectic groupoids and prove that they provide an alternative approach to shifted Poisson structures. Along the way, we intended be able to prove several conjectures that have recently been formulated by the PI and other people.
Applications are related to mathematical physics: interpretation of the Batalin–Vilkovisky formalism in terms of derived symplectic reduction (Main Goal C), constuction of fully extended topological field theories (Main Goal B). Quantization problems will also be discussed at the end of the proposal.
This project proposal lies at the crossroads of algebraic geometry, mathematical physics (in its algebraic and geometric aspects) and higher algebra.
Conclusion for the Main Goal A: as intended, we have developped the theory of shifted symplectic groupoids, explained its relation to lagrangian morphisms, and applied this to the construction of a lagrangian version of the deformation to normal cone.
Conclusion for the Main Goal B: as intended, we constructed a fully extended topological field theory with values in a higher symmetric monoidal category of iterated lagrangian correspondences. There are also other results about field theories that have been obtained (both of general nature and for specific theories).
Conclusion for Main Goal C : as intended, we developped the theory of derived symplectic reduction, and explained how it explains some aspects of the classical BV formalism.
A few results on quantization of some very specific constructions of topological field theories have been ontained.
The theory of symplectic structures on derived stacks gave birth to what is now called derived symplectic geometry. Since then we have seen a rapid evolution of knowledge, and several applications as well (such as general existence theorems for symplectic structures and symmetric obstruction theories on moduli spaces, for instance). This project aims at providing new theoretical developments (shifted symplectic groupoids, for example) and new applications of derived symplectic geometry (to classical field theories, for instance).
On the foundational side (Main Goal A), our aim was to introduce the new notion of shifted symplectic groupoids and prove that they provide an alternative approach to shifted Poisson structures. Along the way, we intended be able to prove several conjectures that have recently been formulated by the PI and other people.
Applications are related to mathematical physics: interpretation of the Batalin–Vilkovisky formalism in terms of derived symplectic reduction (Main Goal C), constuction of fully extended topological field theories (Main Goal B). Quantization problems will also be discussed at the end of the proposal.
This project proposal lies at the crossroads of algebraic geometry, mathematical physics (in its algebraic and geometric aspects) and higher algebra.
Conclusion for the Main Goal A: as intended, we have developped the theory of shifted symplectic groupoids, explained its relation to lagrangian morphisms, and applied this to the construction of a lagrangian version of the deformation to normal cone.
Conclusion for the Main Goal B: as intended, we constructed a fully extended topological field theory with values in a higher symmetric monoidal category of iterated lagrangian correspondences. There are also other results about field theories that have been obtained (both of general nature and for specific theories).
Conclusion for Main Goal C : as intended, we developped the theory of derived symplectic reduction, and explained how it explains some aspects of the classical BV formalism.
A few results on quantization of some very specific constructions of topological field theories have been ontained.
On the theoretical/foundational side, we have made several progresses towards the use of shifted symplectic groupoids in derived Poisson geometry, while on the application side we have focused on derived critical loci and relative version thereof. En route, we have developped operadic tools that are essential for our purpose of using formal derived geometry, as well as chigher categorical tools designed for our applications towards Topological Field Theories. Members of the project have also explored derived geomety in the differentiable and complex analytic contexts.
The main results we would like to highligt are:
- Related to Main Goal A:
(1) Prove of the equivalence between Lie algebroids and L-infinity-spaces, which is an essential step toward the study of shifted symplectic and lagrangian stuctures on formal derived stacks [work of PI with postdoc Joost Nuiten - relevant for Tasks A.2 and A.2' in the project]
(2) Set up the theory of shifted symplectic groupoids and oriented co-groupoids, extension of the AKSZ construction to this context, and application to a symplectic/lagrangian variant of the deformation to normal cone [work of PI and collaborator - completely solve Tasks A.1 A.3 and A.4]
- Related to main Goal B:
(3) The construction (via AKSZ) of an oriented fully extended topological field theory taking its values in a higher symmetric monoidal category of iterated lagrangian correspondences [work of PI and collaborators - completely solves Task B.1]
(4) The foundations of derived differential geometry, leading to a representability theorem for moduli of solutions to elliptic PDEs [work of PhD student Pelle Steffens - goes beyond Task B.1']
(5) The quantization of twisted and dynamical character varieties, using factorization homology, in relation to Chern-Simons theory with sources [work of PhD student Corina Keller - unexpected result, that sits in between Main Goal B and Main Goal C]
(6) Steps toward a noncommutative version of the AKSZ fully extended topological field theory [work of PI with postdoc Tristan Bozec - unexpected result, that sits in between Main Goal B and Main Goal C]
- Related to Main Goal C:
(7) Proof that shifted symplectic reduction commutes taking (absolute and relative) derived critical loci [completely solves Tasks C.1 and C.1']
(8) Construction of a derived lagrangian fibration structure on the derived critical locus of a function, and in interpretation of the classical BV formalism using derived symplectic reduction and equivariant derived geometry [work of PhD student Albin Grataloup - goes beyond Tasks C.1 and C.1']
(9) Proof that deformed relative Calabi-Yau completions are non-commutative analogues of relative derived critical loci, as well as the use of the relative critical locus constuction to produce new lagrangian subvarieties in Hilbert schemes of points in the plane [work of PI with postdoc Tristan Bozec - unexpected result]
(10) Construction of derived enhancement of moduli spaces and showing that two constructions of virtual classes give equivalent results (work of postdoc David Kern - unexpected result, related to Task A.2]
The main results we would like to highligt are:
- Related to Main Goal A:
(1) Prove of the equivalence between Lie algebroids and L-infinity-spaces, which is an essential step toward the study of shifted symplectic and lagrangian stuctures on formal derived stacks [work of PI with postdoc Joost Nuiten - relevant for Tasks A.2 and A.2' in the project]
(2) Set up the theory of shifted symplectic groupoids and oriented co-groupoids, extension of the AKSZ construction to this context, and application to a symplectic/lagrangian variant of the deformation to normal cone [work of PI and collaborator - completely solve Tasks A.1 A.3 and A.4]
- Related to main Goal B:
(3) The construction (via AKSZ) of an oriented fully extended topological field theory taking its values in a higher symmetric monoidal category of iterated lagrangian correspondences [work of PI and collaborators - completely solves Task B.1]
(4) The foundations of derived differential geometry, leading to a representability theorem for moduli of solutions to elliptic PDEs [work of PhD student Pelle Steffens - goes beyond Task B.1']
(5) The quantization of twisted and dynamical character varieties, using factorization homology, in relation to Chern-Simons theory with sources [work of PhD student Corina Keller - unexpected result, that sits in between Main Goal B and Main Goal C]
(6) Steps toward a noncommutative version of the AKSZ fully extended topological field theory [work of PI with postdoc Tristan Bozec - unexpected result, that sits in between Main Goal B and Main Goal C]
- Related to Main Goal C:
(7) Proof that shifted symplectic reduction commutes taking (absolute and relative) derived critical loci [completely solves Tasks C.1 and C.1']
(8) Construction of a derived lagrangian fibration structure on the derived critical locus of a function, and in interpretation of the classical BV formalism using derived symplectic reduction and equivariant derived geometry [work of PhD student Albin Grataloup - goes beyond Tasks C.1 and C.1']
(9) Proof that deformed relative Calabi-Yau completions are non-commutative analogues of relative derived critical loci, as well as the use of the relative critical locus constuction to produce new lagrangian subvarieties in Hilbert schemes of points in the plane [work of PI with postdoc Tristan Bozec - unexpected result]
(10) Construction of derived enhancement of moduli spaces and showing that two constructions of virtual classes give equivalent results (work of postdoc David Kern - unexpected result, related to Task A.2]
The project is now ended, but there are still a few preprints that will be released in the next few months, that are related to it (as they are prolongations of some work that was started during the project):
- PhD student Corina Keller started a project about Poisson structures and factorization homology, that should be finished soon. Like her previous work, this sits at the intersection of Main Goals B and C.
- PhD students Corina Keller and Albin Grataloup had been working on Task B.2 ut they couldn't make enough progress before they left the ERC project. There is an ongoing work of the PI with former postdoc Tristan Bozec that should be released with in a year from now, and that will completely solve Task B.2.
- we could not complete Tasks A.2 and A.2' but we have set up all the tools to achieve these tasks. It is now just a matter of time, and there is an ongoing work of the PI with a new PhD student, that will in the end complete these tasks.
The progress beyond the state of the art that were made during the project are the ones summarized in the overview of results we have previous given:
- new notions of shifted symplectic groupoids and of oriented co-groupoids have emerged, allowing a derived symplectic/lagrangian deformation to normal cone.
- considerable progress in the area of formal derived geometry.
- construction of new fully extended topological field theories using derived symplectic geometry (and non-commutative versions).
- foundations of derived differential geometry and proof of a representability theorem for derived moduli spaces of solutions to an elliptic non linear PDE.
- full mathematical understanding of the quantization of Chern-Simons theory with sources, through factorization homology and quantum dynamical character varieties.
- understanding of shifted symplectic reduction in various contexts, and its relation to the BV formalism.
- relating relative critical loci and deformed Calabi-Yau completions.
- PhD student Corina Keller started a project about Poisson structures and factorization homology, that should be finished soon. Like her previous work, this sits at the intersection of Main Goals B and C.
- PhD students Corina Keller and Albin Grataloup had been working on Task B.2 ut they couldn't make enough progress before they left the ERC project. There is an ongoing work of the PI with former postdoc Tristan Bozec that should be released with in a year from now, and that will completely solve Task B.2.
- we could not complete Tasks A.2 and A.2' but we have set up all the tools to achieve these tasks. It is now just a matter of time, and there is an ongoing work of the PI with a new PhD student, that will in the end complete these tasks.
The progress beyond the state of the art that were made during the project are the ones summarized in the overview of results we have previous given:
- new notions of shifted symplectic groupoids and of oriented co-groupoids have emerged, allowing a derived symplectic/lagrangian deformation to normal cone.
- considerable progress in the area of formal derived geometry.
- construction of new fully extended topological field theories using derived symplectic geometry (and non-commutative versions).
- foundations of derived differential geometry and proof of a representability theorem for derived moduli spaces of solutions to an elliptic non linear PDE.
- full mathematical understanding of the quantization of Chern-Simons theory with sources, through factorization homology and quantum dynamical character varieties.
- understanding of shifted symplectic reduction in various contexts, and its relation to the BV formalism.
- relating relative critical loci and deformed Calabi-Yau completions.