Categorical Kähler Geometry - Duke University › scshgap › files › 2018 › 05 ›...

Categorical Kahler Geometry

Pranav Pandit

joint work with Fabian Haiden, Ludmil Katzarkov,and Maxim Kontsevich

University of Vienna

June 6, 2018

Pranav Pandit (U Vienna) Categorical Kahler Geometry June 6, 2018 1 / 18

I. Physical Theory = Geometric Space

Le temps et l’espace... Ce n’est pas lanature qui nous les impose, c’est nousqui les imposons a la nature parce quenous les trouvons commodes.

Time and space ... it is not Naturewhich imposes them upon us, it is wewho impose them upon Nature becausewe find them convenient.

- Henri Poincare


Idea: View geometric features of a spacetime as “emerging” fromobservations of scattering processes for strings propogating in thatspace-time.


II. Take Symmetry Seriously!

Slogan: Never ask if two entities are equal; instead provide anidentification of one with the other.

Symmetries = identifications of an object with itselfSymmetries can have symmetries, and so on ad infinitum!This is modeled by 8-groupoids

Grothendieck’s Homotopy Hypothesis:

πď8: Top Spaces Ñ 8-groupoids

implements an equivalence of homotopy

theories for any good model of

8-groupoids


Homotopical Mathematics

Classical Entity Homotopical Analogue

Sets SpacesCategories 8-categories

Groups Loop spacesAbelian groups Spectra

modules / field k chain complexes / kAssociative rings A8/E1-ringsassoc. k-algebras dg-algebras / k

En-ringsCommutative rings E8-rings

Topoi 8-topoiAlgebraic Spaces n-geometric 8-stacks

Symplectic structures 0-shifted symplectic structuresn-shifted symplectic structures

Abelian Categories Stable 8-categories


- Algebras of observables in classical and quantum field theories arefactorization algebras.Simplest case: Observables in TFTs are Ed -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


Classification of TFTs

Topological Field Theories (TFTs) of dim d are physical theories thatassign 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 Z pptq is a k-linear stable8-category over k “ C.

Definition (Kontsevich)

A derived noncommutative space (nc-space) over k is a k-linear stable8-category

Examples: FukpX , ωq, DCohpX , I q, DReppQq


nc-geometry

There are well-developed nc-analogues of various notions from complexalgebraic 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)


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 : MetpE q 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)


Linear example: finite dimensions

1 W Ă V finite dimensional vector spaces; E “ rv s P V W

MetpE q “ v `W ; an isomorphism v Ñ v 1 is an element w P W suchthat v ´ v 1 “ w .

Auxillary structure: inner product on V Spvq “ v2.WK XMetprv sq = minimizers of S

isomorphism V W » WK

2 Hodge theory: infinite dimensional analogueV “ Ωk

clpX q, W “ ddRΩk´1pX q

Riemannian metric on X gives inner product on ΩkpX q

HkdRpX q » Harmonic k-forms := tα|∆α “ 0u (BPS states)


Nonlinear examples: I

Complex reductive G ü V inducing G ü X Ă PpV qAuxillary structure: h hermitian metric on VG “ KC; K ü pV , hq preserving X

E “ rxs P X GMetpEq :“ G´orbit » GKSpxq “ x2

Φ, the moment map, is essentially the derivative of S .

X psG » Φ´1p0qKGIT quotient » symplectic quotient (Kempf-Ness theorem)


Nonlinear examples: II

Infinite dimensional analogue of Kempf-Ness (Donaldson-Uhlenbeck-Yau):

X “ space of connections on a smooth complex vector bundle on acomplex manifold Y with F p2,0q “ 0;K = compact gauge group

Auxillary structure: Kahler metric on YS is given by Bott-Chern secondary characteristic classesMoment 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


Categorical Kahler geometry

¨

˚

˚

˝

N = (2,2)super-conformalfield theory

˛

‹

‹

‚ “emergent” geometry//

AB twist

¨

˝

KahlergeometrypX , I , ω1,1q

˛

‚

forget I ω1,1

path integral

ss

¨

˚

˝

f.d. k-linearstable8-categories

+ ??

˛

‹

‚

77„

cob. hyp.wwnc´geom

''

??

;;

¨

˝

2d-topologicalfield theories

+π-stability

˛

‚ //ˆ

symplectic/complexgeometry

˙

Fuk/DCohll


Categorical Kahler geometry

¨

˚

˚

˝

N = (2,2)super-conformalfield theory

˛

‹

‹

‚ “emergent” geometry//

AB twist

¨

˝

KahlergeometrypX , I , ω1,1q

˛

‚

forget I ω1,1

path integral

ss

¨

˚

˝

f.d. k-linearstable8-categories+ ??

˛

‹

‚

77„

cob. hyp.wwnc´geom

''

??

;;

¨

˝

2d-topologicalfield theories+π-stability

˛

‚ //ˆ

symplectic/complexgeometry

˙

Fuk/DCohll


nc-Kahler metrics

Kahler classes : Kahler metrics :: nc-Kahler classes : ??

Long-term goals of the program:

1 Find a notion of nc-Kahler metric on C that gives rise to

§ An underlying nc-Kahler class (Bridgeland stability structure) on C§ A Kahler metric on the moduli of polystable objects of C.§ A Donaldson-Uhlenbeck-Yau correspondence: MCps

θ»Mharm

C

2 Construct natural nc-Kahler metrics on FukpX , ωq and DcohpX , I qcoming from Ωn,0 and ω1,1 respectively.

3 Develop local-to-global principles for studying Fukaya categories andstability structures on them, and study applications to mirrorsymmetry, higher Teichmuller theory, non-abelian Hodge theory, etc.


nc-Kahler classes = Bridgeland stability structures

A Bridgeland stability structure on C consists of

A family of full subcats tCssθ uθPR of semistable objects of phase θ.

A homomorphism Z : K0pCq Ñ C, the central charge.

Such that

1 E P Cssθ then Z pE q P Rą0expp

?´1θq

2 MappCssθ , Css

θ1 q » 0 for θ ą θ1.

3 Cssθ r1s » Css

θ`π.

4 Every E P C admits a Harder-Narasimhan “filtration”:

0 » E0 Ñ E1 Ñ ¨ ¨ ¨ Ñ En » E

with griE P Cssθi

for some

θ1 ą θ2 ą ¨ ¨ ¨ ą θn

Polystable objects of phase θ: Cpsθ :“ pCss

θ qsemisimple


Category FukpX , ωq ReppQq

Kahler data hol vol form Ω tzv P HuvPVertpQqObject Lag upto isotopy ptEvuv , tTαuαPArrpQqq

Metrized object Lagrangian pEv , hv q, hv hermitian metric

Operator Ω P :“ř

zvprv `ř

rT ˚α ,Tαs

Flow F 9L “ ArgΩL h´1 9h “ ArgP

Kahler potential dSCpf q “ş

L Ωf SC “ř

log det hv `ř

T ˚αTα

Harmonic metric Fixed points of F Fixed points of F/rescaling“ CritpSCq “ CritpSCq

= special Lagrangian

DUY theorem ?? King’s theorem

Theorem (King)

There is a stability structure on DReppQq for which the polystable objectsare shifts of objects E P ReppQq that admit a harmonic metric.


Metrized objects: non-archimedean case

- K nonarchimedean field with ring of integers OK and residue field k .

- Cmet a OK -linear stable 8-category

- Csp :“ Cmet bOKk and Cgen :“ Cmet bOK

K .

- Stability structure ptCsssp,θuθ P R,Zspq on the special fiber.

Definition

Let E P Cgen.

1 A metrization of E is an object E P Cmet and an equivalenceα : E bOK

K Ñ E .

2 A metrization pE , αq is harmonic of phase θ if E bOKk P Cps

sp,θ.

MetpE q :“ space metrizations of E .


A nonarchimedean categorical DUY theorem

Theorem (Haiden-Katzarkov-Kontsevich-P.)

There is a natural Bridgeland stability structure tCssgen,θuθPR,Zgen on the

generic fiber Cgen, such that E P Cpsgen,θ if and only if E admits a harmonic

metrization.

Key idea: Given E P Cgen + pE , αq P MetpE q,

HN-filtraion of E bOKk in Csp defines a “tangent vector” to MetpE q

flow on the generalized building Met

The flow converges to a fixed point iff the object is polystable. Moregenerally it converges to the HN-filtration, which is a point in acompactification of MetpE q.


Categorical Kähler Geometry - Duke University › scshgap › files › 2018 › 05 ›...

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