Categorical Kähler Geometry

Transcription

1 Categorical Kähler Geometry Pranav Pandit joint work with Fabian Haiden, Ludmil Katzarkov, and Maxim Kontsevich University of Vienna June 6, 2018 Pranav Pandit (U Vienna) Categorical Kähler Geometry June 6, / 18

2 I. Physical Theory = Geometric Space Le temps et l espace... Ce n est pas la nature qui nous les impose, c est nous qui les imposons à la nature parce que nous les trouvons commodes. Time and space... it is not Nature which imposes them upon us, it is we who impose them upon Nature because we find them convenient. - Henri Poincaré Pranav Pandit (U Vienna) Categorical Kähler Geometry June 6, / 18

3 Idea: View geometric features of a spacetime as emerging from observations of scattering processes for strings propogating in that space-time. Pranav Pandit (U Vienna) Categorical Kähler Geometry June 6, / 18

4 II. Take Symmetry Seriously! Slogan: Never ask if two entities are equal; instead provide an identification of one with the other. Symmetries = identifications of an object with itself Symmetries can have symmetries, and so on ad infinitum! This is modeled by 8-groupoids Grothendieck s Homotopy Hypothesis: π ď8 : Top Spaces Ñ 8-groupoids implements an equivalence of homotopy theories for any good model of 8-groupoids Pranav Pandit (U Vienna) Categorical Kähler Geometry June 6, / 18

5 Homotopical Mathematics Classical Entity Sets Categories Groups Abelian groups modules / field k Associative rings assoc. k-algebras Commutative rings Topoi Algebraic Spaces Symplectic structures Abelian Categories Homotopical Analogue Spaces 8-categories Loop spaces Spectra chain complexes / k A 8 /E 1 -rings dg-algebras / k E n -rings E 8 -rings 8-topoi n-geometric 8-stacks 0-shifted symplectic structures n-shifted symplectic structures Stable 8-categories Pranav Pandit (U Vienna) Categorical Kähler Geometry June 6, / 18

6 - Algebras of observables in classical and quantum field theories are factorization algebras. Simplest case: Observables in TFTs are E d -algebras. - Solutions to equations of motion = (-1)-shifted symplectic space - BV-quantization is a natural construction in derived geometry - Boundary conditions (branes) can naturally be organized into 8-categories - Various moduli spaces in math and physics are naturally derived 8-stacks - The philosophy deformation problems are controlled by dg-lie algebras becomes a theorem in the homotopical world - Derived moduli spaces automatically have the expected dimension - Intersection theory is better behaved Pranav Pandit (U Vienna) Categorical Kähler Geometry June 6, / 18

7 Classification of TFTs Topological Field Theories (TFTs) of dim d are physical theories that assign invariants to manifolds of dimension ď d. Theorem (Lurie) A TFT Z is completely determined by Z(pt) In topological string theory: d 2, and Zpptq is a k-linear stable 8-category over k C. Definition (Kontsevich) A derived noncommutative space (nc-space) over k is a k-linear stable 8-category Examples: FukpX, ωq, DCohpX, I q, DReppQq Pranav Pandit (U Vienna) Categorical Kähler Geometry June 6, / 18

8 nc-geometry There are well-developed nc-analogues of various notions from complex algebraic geometry and symplectic geometry: - Properties, such as smoothness and compactness - Structures, such as orientations (Calabi-Yau structures) - Invariants, such as K-theory, Betti and de Rham cohomology - Hodge theory - Gromov-Witten theory (curve-counting) - Donaldson-Thomas theory (counting BPS states) Pranav Pandit (U Vienna) Categorical Kähler Geometry June 6, / 18

9 III. Harmonic representatives Study isomorphism classes of mathematical objects by finding canonical good representatives in each isomorphism class. Schema: - E isomorphism class of object - MetpE q = space of representatives in the isomorphism class - Auxillary data: convex function S : MetpEq Ñ R Definition - Unstable: S is not bounded below - Semistable: S is bounded below - Polystable: S attains a (unique) minimum Fixed point of flow generated by grads = Minimizer of S (harmonic representative) Pranav Pandit (U Vienna) Categorical Kähler Geometry June 6, / 18

10 Linear example: finite dimensions 1 W Ă V finite dimensional vector spaces; E rvs P V {W MetpEq v ` W ; an isomorphism v Ñ v 1 is an element w P W such that v v 1 w. Auxillary structure: inner product on V Spvq }v} 2. W K X Metprvsq = minimizers of S isomorphism V {W» W K 2 Hodge theory: infinite dimensional analogue V Ω k cl px q, W d drω k 1 px q Riemannian metric on X gives inner product on Ω k px q HdR k px q» Harmonic k-forms := tα α 0u (BPS states) Pranav Pandit (U Vienna) Categorical Kähler Geometry June 6, / 18

11 Nonlinear examples: I Complex reductive G ü V inducing G ü X Ă PpV q Auxillary structure: h hermitian metric on V G K C ; K ü pv, hq preserving X E rxs P X {G MetpEq : G orbit» G{K Spxq }x} 2 Φ, the moment map, is essentially the derivative of S. X ps {G» Φ 1 p0q{k GIT quotient» symplectic quotient (Kempf-Ness theorem) Pranav Pandit (U Vienna) Categorical Kähler Geometry June 6, / 18

12 Nonlinear examples: II Infinite dimensional analogue of Kempf-Ness (Donaldson-Uhlenbeck-Yau): X space of connections on a smooth complex vector bundle on a complex manifold Y with F p2,0q 0; K = compact gauge group Auxillary structure: Kähler metric on Y S is given by Bott-Chern secondary characteristic classes Moment map is given by curvature (Polystable holomorphic bundles)» (Hermitian-Yang-Mills connections). RHS = connections satisfying a certain PDE (BPS branes) Gradient flow for S is Donaldson s heat flow Problem: Generalize this to complexes of vector bundles Pranav Pandit (U Vienna) Categorical Kähler Geometry June 6, / 18

13 Categorical Kähler geometry N = (2,2) superconformal field theory path integral emergent geometry Kähler geometry px, I, ω 1,1 q A{B twist 2d-topological field theories f.d. k-linear stable 8-categories cob. hyp. nc geom Fuk/DCoh forget I {ω 1,1 ˆ symplectic/complex geometry Pranav Pandit (U Vienna) Categorical Kähler Geometry June 6, / 18

14 Categorical Kähler geometry N = (2,2) superconformal field theory path integral emergent geometry Kähler geometry px, I, ω 1,1 q?? A{B twist f.d. k-linear stable 8-categories +?? nc geom Fuk/DCoh forget I {ω 1,1 cob. hyp. 2d-topological ˆ field theories symplectic/complex geometry +π-stability Pranav Pandit (U Vienna) Categorical Kähler Geometry June 6, / 18

15 nc-kähler metrics Kähler classes : Kähler metrics :: nc-kähler classes :?? Pranav Pandit (U Vienna) Categorical Kähler Geometry June 6, / 18

16 nc-kähler metrics Kähler classes : Kähler metrics :: nc-kähler classes :?? Long-term goals of the program: 1 Find a notion of nc-kähler metric on C that gives rise to An underlying nc-kähler class (Bridgeland stability structure) on C A Kähler metric on the moduli of polystable objects of C. A Donaldson-Uhlenbeck-Yau correspondence: MC ps θ» Mharm C Pranav Pandit (U Vienna) Categorical Kähler Geometry June 6, / 18

17 nc-kähler metrics Kähler classes : Kähler metrics :: nc-kähler classes :?? Long-term goals of the program: 1 Find a notion of nc-kähler metric on C that gives rise to An underlying nc-kähler class (Bridgeland stability structure) on C A Kähler metric on the moduli of polystable objects of C. A Donaldson-Uhlenbeck-Yau correspondence: MC ps θ» Mharm C 2 Construct natural nc-kähler metrics on FukpX, ωq and D coh px, I q coming from Ω n,0 and ω 1,1 respectively. Pranav Pandit (U Vienna) Categorical Kähler Geometry June 6, / 18

18 nc-kähler metrics Kähler classes : Kähler metrics :: nc-kähler classes :?? Long-term goals of the program: 1 Find a notion of nc-kähler metric on C that gives rise to An underlying nc-kähler class (Bridgeland stability structure) on C A Kähler metric on the moduli of polystable objects of C. A Donaldson-Uhlenbeck-Yau correspondence: MC ps θ» Mharm C 2 Construct natural nc-kähler metrics on FukpX, ωq and D coh px, I q coming from Ω n,0 and ω 1,1 respectively. 3 Develop local-to-global principles for studying Fukaya categories and stability structures on them, and study applications to mirror symmetry, higher Teichmüller theory, non-abelian Hodge theory, etc. Pranav Pandit (U Vienna) Categorical Kähler Geometry June 6, / 18

19 nc-kähler classes = Bridgeland stability structures A Bridgeland stability structure on C consists of A family of full subcats tc ss θ u θpr of semistable objects of phase θ. A homomorphism Z : K 0 pcq Ñ C, the central charge. Pranav Pandit (U Vienna) Categorical Kähler Geometry June 6, / 18

20 nc-kähler classes = Bridgeland stability structures A Bridgeland stability structure on C consists of A family of full subcats tc ss θ u θpr of semistable objects of phase θ. A homomorphism Z : K 0 pcq Ñ C, the central charge. Such that 1 E P Cθ ss then ZpEq P R ą0 expp? 1θq 2 MappCθ ss, Css θ q» 0 for θ ą θ Cθ ss r1s» Css θ`π. Pranav Pandit (U Vienna) Categorical Kähler Geometry June 6, / 18

21 nc-kähler classes = Bridgeland stability structures A Bridgeland stability structure on C consists of A family of full subcats tc ss θ u θpr of semistable objects of phase θ. A homomorphism Z : K 0 pcq Ñ C, the central charge. Such that 1 E P Cθ ss then ZpEq P R ą0 expp? 1θq 2 MappCθ ss, Css θ q» 0 for θ ą θ Cθ ss r1s» Css θ`π. 4 Every E P C admits a Harder-Narasimhan filtration : 0» E 0 Ñ E 1 Ñ Ñ E n» E with gr i E P C ss θ i for some θ 1 ą θ 2 ą ą θ n Pranav Pandit (U Vienna) Categorical Kähler Geometry June 6, / 18

22 nc-kähler classes = Bridgeland stability structures A Bridgeland stability structure on C consists of A family of full subcats tc ss θ u θpr of semistable objects of phase θ. A homomorphism Z : K 0 pcq Ñ C, the central charge. Such that 1 E P Cθ ss then ZpEq P R ą0 expp? 1θq 2 MappCθ ss, Css θ q» 0 for θ ą θ Cθ ss r1s» Css θ`π. 4 Every E P C admits a Harder-Narasimhan filtration : 0» E 0 Ñ E 1 Ñ Ñ E n» E with gr i E P C ss θ i for some Polystable objects of phase θ: C ps θ θ 1 ą θ 2 ą ą θ n : pc ss θ qsemisimple Pranav Pandit (U Vienna) Categorical Kähler Geometry June 6, / 18

23 Category FukpX, ωq ReppQq Kähler data hol vol form Ω tz v P Hu vpvertpqq Object Lag upto isotopy pte v u v, tt α u αparrpqq q Metrized object Lagrangian pe v, h v q, h v hermitian metric Operator Ω P : ř z v pr v ` řrt α, T α s Flow F L 9 ArgΩL h 1 h 9 ArgP Kähler potential ds C pf q ş L Ωf S C ř log det h v ` ř T αt α Harmonic metric Fixed points of F Fixed points of F/rescaling CritpS C q CritpS C q = special Lagrangian DUY theorem?? King s theorem Theorem (King) There is a stability structure on DReppQq for which the polystable objects are shifts of objects E P ReppQq that admit a harmonic metric. Pranav Pandit (U Vienna) Categorical Kähler Geometry June 6, / 18

24 Metrized objects: non-archimedean case - K nonarchimedean field with ring of integers O K and residue field k. - C met a O K -linear stable 8-category - C sp : C met b OK k and C gen : C met b OK K. - Stability structure ptc ss sp,θ u θ P R, Z sp q on the special fiber. Definition Let E P C gen. 1 A metrization of E is an object Ẽ P C met and an equivalence α : Ẽ b O K K Ñ E. 2 A metrization pẽ, αq is harmonic of phase θ if Ẽ b O K k P C ps sp,θ. MetpEq : space metrizations of E. Pranav Pandit (U Vienna) Categorical Kähler Geometry June 6, / 18

25 A nonarchimedean categorical DUY theorem Theorem (Haiden-Katzarkov-Kontsevich-P.) There is a natural Bridgeland stability structure tcgen,θ ss u θpr, Z gen on the generic fiber C gen, such that E P C ps gen,θ if and only if E admits a harmonic metrization. Pranav Pandit (U Vienna) Categorical Kähler Geometry June 6, / 18

26 A nonarchimedean categorical DUY theorem Theorem (Haiden-Katzarkov-Kontsevich-P.) There is a natural Bridgeland stability structure tcgen,θ ss u θpr, Z gen on the generic fiber C gen, such that E P C ps gen,θ if and only if E admits a harmonic metrization. Key idea: Given E P C gen + pẽ, αq P MetpEq, HN-filtraion of Ẽ b O K k in C sp defines a tangent vector to MetpEq flow on the generalized building Met The flow converges to a fixed point iff the object is polystable. More generally it converges to the HN-filtration, which is a point in a compactification of MetpE q. Pranav Pandit (U Vienna) Categorical Kähler Geometry June 6, / 18