physics aspects of d-branes from matrix-type...

$: Physics aspects of D-branes from matrix-type ...web.phys.ntu.edu.tw/string/files2013/20130510_Liu.pdf · { An invitation to Azumaya D-geometry for string-theorists. Outline Matrix-type/Azumaya$
Physics aspects of D-branesfrom matrix-type noncommutative geometry

on D-brane world-volume

Chien-Hao LiuJoint work with Shing-Tung Yau

in part with Si Li, Ruifang Song, Baosen Wu(Beamer file prepared with help from Yu-jong Tzeng)

Yau’s group at Harvard University

String Theory Seminar, National Taiwan UniversityMay 10, 2013

Azumaya geometry Example D-brane as morphism Azumaya geometry at work D-string world-sheet instanton

– An invitation to Azumaya D-geometry for string-theorists.

Outline

Matrix-type/Azumaya noncommutative geometryon D-brane world-volume

How to describe a D-brane (world-volume)on a target-space(-time) : Two examples

Supersymmetric D-branes on a Calabi-Yau spacethrough four equivalent aspects of morphisms

Azumaya geometry at work for D-branes in string theory

D-string world-sheet instantons (in progress)

Chien-Hao Liu Physics aspects of D-branes through Azumaya geometry 1/78


1 Matrix-type/Azumaya noncommutative geometry on D-braneworld-volume

2 How to describe a D-brane (world-volume) in atarget-space(-time): Two examples

3 Supersymmetric D-branes on a Calabi-Yau space through fourequivalent aspects of morphisms

4 Azumaya geometry at work for D-branes in string theory

5 D-string world-sheet instantons (in progress)



An open-string induced phenomenon and a fundamental question

Question

When D-branes are stacked, the scalar fields describing theircollective position/deformations are enhanced by additionalmassless scalar fields created by open strings and becomematrix-valued, what happens to the D-brane? And to thespace-time?

Answer to this question reflects what we take as fundamentalnature of D-brane or of space-time (cf. remark in[Polchinski: String theory, vol.1: Sec. 8.7, p. 272] ).



Lesson from Quantum Mechanics

Pointlike particle in a space(-time) Y with coordinates (yi )i .

Classical mechanics: (yi )i describes the position (and hencedeformations) of a point in Y .

Quantization of the particle: yi become operator-valued,reflecting the fact that the nature of the particle is changed fromclassical mechanics to quantum mechanics. Nothing is changed forthe space-time Y; i.e. Y remains classical.



Emergence of Matrix/Azumaya-type noncommutativityon D-brane world-volume

Substitution from or parallel reasoning to quantum mechanics

· particle ⇒ D-brane

· quantization ⇒ stackification

· operator-valued ⇒ matrix-valued

· quantized ⇒ noncommutatized

⇓

Stacked D-branes as Azumaya space

Stacking changes the nature of D-branes from an ordinary space toa noncommutative space with a matrix/Azumaya-typenoncommutative structure.



Original work of Ho and Wu and its coming back

This was observed by Pei-Ming Ho and Yong-Shi Wu in 1996:

[H-W] P.-M. Ho and Y.-S. Wu, Noncommutative geometry and D-branes, Phys.Lett. B398 (1997), 52–60. (arXiv:hep-th/9611233)

but somehow was overlooked by the major stringy community(likely due to competing comments from Polchinski/Douglas whoemphasizes more on the target space-time noncommutativityaspect).

Re-picked up ten years later (!) in December 2006 (cf. [L-Y1]D(1)) from re-reading Polchinski from Grothendiek’s viewpoint ofalgebraic geometry and his theory of schemes, after discussionswith Duiliu-Emanuel Diaconescu on open-string world-sheetinstantons that drove me to re-think about D-branes.



D-brane world-volume vs. target space

stacked D-brane X space-time Y

(1) Ho-Wu (1996) / Grothendieck’s AG (2006)

(2)

X nc

Y nc

Y

X



Issues that need to be answered first

Thus, a door, once opened but shut, is re-opened, but where doesit lead to? How can one re-understand D-branes through it? Moreimportantly and constructively: What new features/ properties ofD-branes can this help us understand?

Before we can address these high level questions, we have toanswer first two low level, yet fundamental, questions:

What is an Azumaya space with a fundamental module?

What is a morphism from such a space to a (commutative ornoncommutative) space(-time)?

Quantum field theory on D-brane world-volume is supposed tocover a field theory for such morphisms. (CAUTION: There’reother fields on D-brane world-volume.)



Example 1: D0-brane on the complex line A1C 1/4

Question

What is a D0-brane on a complex line A1C?

Answer: A map/morphism

ϕ : (Space (Mr (C)),Cr ) −→ A1C

from an Azumaya point with a fundamental module to A1C.

Question

These are just symbols/words. What exactly do they mean?

Answer: ϕ is defined by a C-algebra homomorphism

Mr (C)←− C[z ] : ϕ] .

contravariantlyChien-Hao Liu Physics aspects of D-branes through Azumaya geometry 10/78



Geometry of an Azumaya point over C

Spec C( )A2

Spec C( )A1

M ( ) noncommutative cloudr

Spec

NC cloudA1

NC cloudA2

Spec C( )A

A NC cloud

Many novel features of D-branes turn out to be originated fromthe richness of the “structure” of an Azumaya pointover C.




ϕ : (Space (Mr (C)),Cr ) −→ A1C

Mr (C) ←− C[z ] : ϕ]

Kerϕ] = (f (z)) = ((z − a1)d1 · · · (z − ak )dk )

⇒ Imϕ = fuzzy points located at z = a1, . . . , ak

Cr fundamental representation fo Mr (C)

⇒ C[z ]-module via ϕ]

⇒ Coherent sheaf on A1C, supported on Imϕ.




Morphisms from Azumaya point illustrated

open-string target-space(-time) Y

Spec

D0-brane of rank r

M ( ) NC cloudr

r

ϕ 1

ϕ 2ϕ 3

ϕ 2

un-Higgsing

Higgsing

Note that the Higgsing-unHiggsing of D-branes is an outcome ofdeformations of morphisms.



Example 2: Spectral covers from D-branes

“Pile/Smear” Example 1 along a complex variety/manifold X⇒ Spectral cover X in a complex line bundle LX over X .I.e. X = image of a morphism over X :

(X ,OAzX , E)

ϕ //

$$

LX

xxX .

Spec

D0-brane of rank r

M ( ) NC cloudr

r ϕ

X

Remark. This is an important example; cf. Gaiotto-Moore-Neitzke’s

work on counting BPS states in d=4, or d=3 SQFT.

[G-M-N] D. Gaiotto,G.W. Moore, and A. Neitzke, Wall-crossing, Hitchin systems,and the WKB approximation, arXiv:0907.3987 [hep-th].



Supersymmetric D-branes on a Calabi-Yau space 1/3

• Effective space-time aspect:

[B-B-S] K. Becker, M. Becker, and A. Strominger, Fivebranes, membranes andnon-perturbative string theory , Nucl. Phys. B456 (1995), 130 – 152.(arXiv:hep-th/9507158)

• Open string world-sheet aspect:

[O-O-Y] H. Ooguri, Y. Oz, and Z. Yin, D-branes on Calabi-Yau spaces and theirmirrors, Nucl. Phys. B477 (1996), 407 – 430. (arXiv:hep-th/9606112)




• Symplectic/differential/calibrated geometry for A-branes:

· Constructible sheaves on special Lagrangian submanifolds.

· Fukaya A∞-category Fuk (Y ) of graded Lagrangian submanifoldson Y .

· Open Gromov-Witten theory to define the A∞-structureon Fuk (Y ).

• Algebraic geometry for B-branes:

· Derived category Coh b(Y ) of coherent sheaves on Y .

· Stability conditions on objects of Coh b(Y ).

• Mirror symmetry : (Y , A-branes)⇐⇒ (Y , B-branes).

· From Strominger-Yau-Zaslow construction to tropicalgeometry and Gross-Siebert program.




Where we are in the Wilson’s theory-space of string theory:Solitonic vs. soft D-branes

Superstring [ ]

T

D-branesmallT

D-branelargeT

D-brane tension

NLSM [ ]Wilsond=2, SQFTb

Wilsond=4, SQFT



D-branes as morphism: Proto-defintion

Proto-definition of D-branes

A D-brane on Y is a morphism ϕ : (X ,OAzX , E)→ (Y ,OY ).

Here,

· X = (X ,OX ): ringed-space (commutative to begin with),(Y ,OX ) commutative or noncommuttative ringed space;

· E is a locally free OX -module of finite rank, and

· OAzX = EndOX

(E), as sheaf of (noncommutative) OX -algebras.

Question [morphism]

What does a “morphism” mean in this context?



Morphism between “spaces” as contravariant gluing systemof ring-homomorphisms 1/2

Grothendieck’s theory of schemes:

Space as a gluing system of ringsSpace (Rαα)

Morphism Space (Rαα)→ Space (Sββ) as contravariantgluing system of ring-homomorphisms Rαα←Sββ.

Key point: In commutative case, there is really a way to sepecfy apoint-set-with-topology Spec R to ring R in a functorial way, whichglue to a scheme, while in our case, we have to abandon such apoint-set-with-topology notion to make our definition reallydescribe D-branes in string theory.



Morphism between “spaces” as contravariant gluing systemof ring-homomorphisms 2/2

In other words,

ϕ : (X ,OAzX , E) −→ (Y ,OY )

is only a symbol. Its real content is in

OAzX ←− OY : ϕ]

that represents an equivalence class of contravariant gluingsystems of ring-homomorphisms. In particular, in generalthere is no morphism X → Y that underlies ϕ.



Four aspects of such morphisms when target Y is commutative.Aspect 1: Fundamental

Summarizing the discussion up to now:

Definition [morphism-fundamental]

A morphismϕ : (X ,OAz

X , E) −→ (Y ,OY )

is defined byOAz

X ←− OY : ϕ]

that represents an equivalence class of contravariant gluingsystems of ring-homomorphisms.



Aspect 2: Graph of morphism/Fourier-Mukai transform 1/3

An observation: A morphism ϕ : (X ,OAzX , E)→ (Y ,OY )

determines and is determined by the following diagram:

OAzX = EndOX

(E)

Aϕ := Imϕ]?

OO

OYϕ]

oo

OX

?

OO

.

This is exactly the data of a coherent sheaf Eϕ on X × Y that isflat and relative dimension-0 over X .




Definition [graph of morphism]

The coherent sheaf Eϕ on X × Y , which is flat and of relativedimension 0 over X is called the graph of the morphismϕ : (X ,OAz

X , E)→ (Y ,OY ).

Thus, morphisms from Azumaya spaces X Az with a fundamentalmodule to Y form a subclass of Fourier-Mukai transforms from Xto Y .




X

πϕ

Xϕ

ndX( )

Azumaya cloudYϕ

fϕ

X

Y

ϕΓSupp ( ) =

=

X Y

pr2

pr1

Azumaya, morphism

Fourier-Mukai transform

Xϕ



Aspect 3: Morphism to moduli stack of D0-branes

M0Azf

r (Y ):= moduli stack of morphisms from an Azumaya pointwith a fundamental module of rank r to Y . Then,

M0Azf

r (Y ) is the same as the stack of 0-dimensional coherentsheaf of length r on Y ;

Aspect 2 can be translated immediately to:

[special role played by stack of D0-branes]

A morphism ϕ : (X ,OAzX , E)→ Y is the same as a morphism

X →M0Azf

r (Y ).

open stringsD0-branes

p-cycle Dp-brane

Smearing D0-branesalong a p-cycleto get a Dp-brane



Aspect 4: Equivariant map to atlas of moduli stack of D0-branes

From morphisms to stack to G -morphisms to atlas:

Isom (φ, π)φ //

pr1

Atlas

π

Xφ //M 0Azf

r (Y )

Choose Atlas to be the representation-theoretical atlas

Quot H0(O⊕r

Y , r) :=

O⊕r

Y → E → 0 , length E = r ,

H0(O⊕rY )→ H0(E)→ 0

Then, pr1: principal GLr (C)-bundle, φ: GLr (C)-morphism.



Remark: From algebraic to sympletic

We begin with the fundamental proto-defintion of D-branes, whichis more akin to algebraic geometry. But when the target Y iscommutative, in particular, a Calabi-Yau space, then followingthese four aspects of morphisms from Azumaya spaces, we getcloser and closer to a language that is accommodatable insympletic/differential geometry, with Quot-schemes replaced by(the analytic) Douady spaces. It becomes thus more and moreaccesssible to string-theorists as well.



B-branes on a Calabi-Yau space from stable morphisms

• The defintion of morphisms from Azumaya spaces with a fundamentalmodule in the realm of complex algebraic geometry is by natureholomorphic. It gives thus holomorphic D-branes.

• The special connection on the Chan-Paton module is supposed to bespecified by a stability condition via a Donaldson-Uhlenbeck-Yau-typetheorem, though in practice this is very hard to come by. Thus, we takestable morphisms ϕ to a Calabi-Yau space Y as giving a B-brane on Y , ifan apprapriate notion of central charge Z (ϕ) – and hence the associatednotion of stability condition on ϕ – can be defined.

• Subtlety. For a fixed target Calabi-Yau space Y , if we consider onlymorphisms from a fixed (X ,OAs

X , E), it’s easier to adapt thesolitonic-brane situation to the current situation. However, when the(X ,OAs

X , E) are allowed to vary, some technical subtlety (if notconceptual ones) may arise. This is a similar, yet more involved, subtletywhen mathematicians tried to construct moduli space of stable bundlesover all stable curves. We’ll make this concrete in Sec. 5 when addressingthe construction of D-string world-sheet instantons in our setting.



A-branes on a Calabi-Yau space from special Lagrangian morphisms 1/4

Notation [Calabi-Yau n-fold]. Y = (Y , J, ω,Ω, ), where

Y : smooth 2n-manifold;

J : complex structure on Y ;

ω : (Ricci-flat) Kahler 2-form on (Y , J);

Ω : nowhere-vanishing holomorphic n-form on (Y , J).




Definition [A-branes as morphisms: Aspect 1 - fundamental]

Given a Calabi-Yau n-fold Y = (Y , J, ω,Ω, ), an A-brane on Y is amorphism ϕ : (X ,OAz

X , E ,∇)→ Y such that

ϕ∗ω = ϕ∗(Im Ω) = 0 .

Here, E is a locally free OX ,C(:= O∞X ⊗R C)-module,OAz

X := EndOX ,C(E) the sheaf of OX ,C-module endomorphisms,and ∇ is a flat connection on E , possibly with singularities.

ϕ is defined by ϕ] : O∞Y ,C → OAzX . The pull-back operation ϕ∗ on

differential forms should be defined accordingly; we’ll use Aspect 2,Fourier-Mukai transform, to understand this.




Aspect 2 [Fourier-Mukai transform]:

X

Y

ϕΓSupp ( ) =

=

X Y

pr2

pr1

Xϕ

· Xϕ : stratifiedpiecewise-smooth

· E : C-constructible sheaf(“flat” & dim-0)/X

· (pr∗2 ω)|Xϕ= 0

· (pr∗2 Ω)|Xϕ = 0

Issues to be understood: (cf. [D(6): Sec.4.2]) Deformation theory for such special

Lagrangian morphisms w/ a constructible sheaf to a Calabi-Yau space that remembershow they degenerate and collide as in the theory of schemes and coherent sheaves.

Cf. The vertical complex/scheme structure on X × Y .




A special class of such morphisms from Azumaya manifold with afundamental module, before a general theory is developed:

roof diagram

(X , V)f−→ Y

c ↓X ,

where

· c : X → Xbranched covering of smooth

manifolds along a codimR-2 submanifold,

· f : X → Y · V complex vector bundle on X

special Lagrangian morphism with a singular flat connection

V

X

h(c,f )

!!

f

((

c

Xϕ(c,f ) fϕ(c,f )

//

πϕ(c,f )

Y

X



Remark: Effect of B-field to D-branes

The challenge to understand B-field, (cf. curvingon a gerbe, SL(2,Z)-twisted cohomology, .... ?)

From fundamental module to twisted fundamental module,and from trivial Azumaya structure to non-trivial Azumayastructure (associated to a non-zero elementof a Brauer group).

An additional deformation-quantization-typenoncommutativity on D-brane world-volume.

This is a topic in its own right. I refer you to D(5)(cf. [L-Y4]1) formore detailed discussions on some of the issues.

1[L-Y4]: C.-H. Liu and S.-T. Yau, Nontrivial Azumaya noncommutative schemes, morphisms therefrom,

and their extension by the sheaf of algebras of differential operators: D-branes in a B-field background a laPolchinski-Grothendieck Ansatz, arXiv:0909.2291 [math.AG]. (D(5))



D-project in progress 1/4

After brewing for a decade, a project on D-branes is finally set upin early spring 2007 to (re-)understand them mathematicallyas fundamental/soft objects (rather than solitonic objects)in string theory via morphisms from this matrix-type/Azumayanoncommutative geometry, reviewed in Sec. 1, Sec. 2, Sec. 3.

[L-Y1] C.-H. Liu and S.-T. Yau, Azumaya-type noncommutative spaces andmorphism therefrom: Polchinski’s D-branes in string theory fromGrothendieck’s viewpoint, arXiv:0709.1515 [math.AG]. (D(1))

[L-L-S-Y] S. Li, C.-H. Liu, R. Song, S.-T. Yau, Morphisms from Azumaya prestablecurves with a fundamental module to a projective variety: TopologicalD-strings as a master object for curves, arXiv:0809.2121 [math.AG].(D(2))




[L-Y2] C.-H. Liu and S.-T. Yau, Azumaya structure on D-branes and resolutionof ADE orbifold singularities revisited: Douglas-Moore vs.Polchinski-Grothendieck, arXiv:0901.0342 [math.AG]. (D(3))

[L-Y3] ——–, Azumaya structure on D-branes and deformations and resolutionsof a conifold revisited: Klebanov-Strassler-Witten vs.Polchinski-Grothendieck, arXiv:0907.0268 [math.AG]. (D(4))

[L-Y4] ——–, Nontrivial Azumaya noncommutative schemes, morphismstherefrom, and their extension by the sheaf of algebras of differentialoperators: D-branes in a B-field background a la Polchinski-GrothendieckAnsatz, arXiv:0909.2291 [math.AG]. (D(5))




[L-Y5] ——–, D-branes and Azumaya noncommutative geometry: FromPolchinski to Grothendieck, arXiv:1003.1178 [math.SG]. (D(6))

[L-Y6] ——–, D-branes of A-type, their deformations, and Morse cobordism ofA-branes on Calabi-Yau 3-folds under a split attractor flow:Donaldson/Alexander-Hilden-Lozano-Montesinos-Thurston/Hurwitz/Denef-Joyce meeting Polchinski-Grothendieck, arXiv:1012.0525[math.SG]. (D(7))

[L-Y7] ——–, A natural family of immersed Lagrangian deformations of abranched covering of a special Lagrangian 3-sphere in a Calabi-Yau 3-foldand its deviation from Joyce’s criteria: Potential image-support rigidity ofA-branes that wrap around a sL S3, arXiv:1109.1878 [math.DG]. (D(8.1))




[L-Y8] ——– (with Baosen Wu), D0-brane realizations of the resolution of areduced singular curve, arXiv:1111.4707 [math.AG]. (D(9.1))

[L-Y9] ——–, A mathematical theory of D-string world-sheet instantons, I:Compactness of the stack of Z-semistable Fourier-Mukai transforms froma compact family of nodal curves to a projective Calabi-Yau 3-fold ,arXiv:1302.2054 [math.AG]. (D(10.1))

• A terse review that emphasizes underlying concepts, whys & examples:

[L] C.-H. Liu, Azumaya noncommutative geometry and D-branes -an origin of the master nature of D-branes, lecture at Simons Center forGeometry and Physics, Stony Brook University,arXiv:1112.4317 [math.AG].

Sec. 4 and Sec. 5 next contain some more highlight of thisD-project up to December 2012, based on these works.



Mathematical D-geometry through Azumaya geometry

D-brane in superstring theory

Azumaya geometry: morphisms from Azumay spaces witha fundamental module

Purely mathematicalgeneralization

New theory/problemin its own right ornew meaning to oldtheory/problem

Quantun field theory+ Supersymmetrymethod

Statements in algebraicor symplectic/differentialgeometry

feedback (ideally)



Mathematical D-geometry through Azumaya geometry

In this section, I’ll discuss how morphisms from Azumaya spaceswith a fundamental module, and their deformations, are at work instring-theory literature from our point of view. For this purpose,each theme is assigned a title of the related stringy work. On themathematical side, each theme gives rise to a distinct topic in itsown. These are samples from a large pool.

Remark. It’ll ring better if you are already familiar with these string-theorywork since I didn’t prepare to review them here. However, such familiarity isnot required to understand the underlying Azumaya D-geometry in the section.



Higgsing/un-Higgsing of D-branes

Higgsing/uin-Higgsing, the most fundamental behavior, ofD-branes is simply realized as deformations of morphisms, asalready seen in Sec. 2, Example 1 (pp. 10-13):

open-string target-space(-time) Y

Spec

D0-brane of rank r

M ( ) NC cloudr

r

ϕ 1

ϕ 2ϕ 3

ϕ 2

un-Higgsing

Higgsing



Bershadsky-Vafa-Sadov: D-manifold (1995)

[B-V-S1] M. Bershadsky, C. Vafa, and V. Sadov, D-strings on D-manifolds, Nucl.Phys. B463 (1996), 398–414. (arXiv:hep-th/9510225)

[B-V-S2] ——–, D-branes and topological field theories, Nucl. Phys. B463 (1996),420–434. (arXiv:hep-th/9511222)

[Va] C. Vafa, Gas of D-branes and Hagedorn density of BPS states, Nucl.Phys. B463 (1996), 415–419. (arXiv:hep-th/9511088)

• Given a variety Y over C, the Hilbert scheme Y [r ] of 0-dimensionalsubschemes of Y of length r is tautological a substack of the stack

M0Azf

r (Y ) of morphisms from Azumaya points with a fundamentalmodule of rank r to Y .• From our point of view, one should first look at either the stack

M0Azf

r (Y ) or the Quot-scheme Quot Y (O⊕rY , r) of 0-dimensional

quotient sheaves of O⊕rY of length r ; and then lead the way to

Y [r ], though this looks a very difficult task at the moment.



Douglas-Moore/Johnson-Myers:D-brane probe to ADE surface singularity (1996) 1/4

[D-M] M.R. Douglas and G.W. Moore, D-branes, quivers, and ALE instantons,arXiv:hep-th/9603167.

[J-M] C.V. Johnson and R.C. Myers, Aspects of type IIB theory on ALE spaces,Phys. Rev. D55(1997), 6382–6393.(arXiv:hep-th/9610140)

• Basic idea/set-up in [D-M]:

· Type IIB superstring model on R5+1 ×(C2/Γ

), where Γ ⊂ SU(2) discrete

subgroup; i.e. IIB compactified on the singular local Calabi-Yau space C2/Γ.

· D5-brane world-volume (6d) sitting at the singular locus R5+1 × 0of the 10d space-time R5+1 ×

(C2/Γ

);

· The d=6, N=1 supersymmetric QFT on D5-brane world-volumein such configuration has space of vacua M~ζ•

, depending on

the vacuum expectation value ~ζ• of the salar fields ~φ• in the theory.

· For appropriate ~ζ•, M~ζ•gives a resolution of C2/Γ.




D-brane probe resolutionof ADE surfacesingularities in [D-M] canbe interpreted directlyas a resolutionof the singularityby aggregationsof D0-branesin our sense.

Spec

D0-brane of type r

M ( ) NC cloudr

r

ϕ 1ϕ 2

ϕ 3

ϕ 4

2 G/[ ]

Chan-Paton modulefrom push-forwardsitting over image D-brane

2 : atlas of orbifold

fundamental module on pt Az




• A mathematical abstraction in birational geometry:

· Y : projective variety over C,

· M0Azfp

r (Y ): the stack of punctual D0-branes of rank r on Y ,

· πY : M0Azf

pr (Y )→ Y the built-in canonical morphism.

Conjecture [abundance] ([L-Y8] D(9.1))

Let f : Y ′ → Y be a birational morphism from a projective varietyY ′ to Y . Then, f factors through an embedding

f : Y ′ →M0Azf

pr (Y ), for some r . M

0Azfp

r (Y )

πY

Y ′*

f

77

f // Y .

In particular, any resolutionρ : Y → Y of Y

factors through some M0Azf

pr (Y ).




• A non-Calabi-Yau compact example:

Proposition [reduced curve] ([L-Y8] D(9.1), with Baosen Wu)

Conjecture holds for any resolution ρ : C ′ → C of a reduced curve.Namely, there exists an r0 ∈ N depending only on the tuple(np′)ρ(p′)∈Csing

and a (possibly empty) setb.i.i.(p) : p ∈ Csing , C has multiple branches at p , bothassociated to the germ of Csing in C , such that, for any r ≥ r0,

there exists an embedding ρ : C ′ →M0Azf

pr (C )

that makes the following diagram commute: M0Azf

pr (C )

πC

C ′*

ρ

77

ρ // C .

Here, np′ and b.i.i.(p) are somecharacterization indices forp ∈ Csing , p′ ∈ ρ−1(p).



Douglas: D-geometry (1997)

(Frame set to be completed.)

[Do] M.R. Douglas, D-branes in curved space, Adv. Theor. Math. Phys. 1(1997), 198–209.(arXiv:hep-th/9703056)

[D-K-O] M.R. Douglas, A. Kato, and H. Ooguri, D-brane actions on Kahlermanifolds, Adv. Theor. Math. Phys. 1 (1997), 237–258.(arXiv:hep-th/9708012)

• Distinguished geometry on the moduli stack of D0-branes onCalabi-Yau space.

Cf. [L: Sec. 1, last theme] , lecture at Simons Center for Geometry andPhysics, 2011.



Klebanov-Strassler-Witten: D-brane probe to conifold (1998, 2000) 1/3

[K-W] I.R. Klebanov and E. Witten, Superconformal field theory on threebranesat a Calabi-Yau singularity, Nucl. Phys. B536 (1999),199–218.(arXiv:hep-th/9807080)

[K-S] I.R. Klebanov and M.J. Strassler, Supergravity and a confining gaugetheory: duality cascade and χSB-resolution of naked singularities, J. HighEnergy Phys. (2000) 052, 35 pp.(arXiv:hep-th/0007191)

Resolution and deformation of a conifold singularity via D-branes:

Y YY

0 0

a D-brane congurationwithout fractional branes

a D-brane congurationwith a fractional brane

the moduli space of its supersymmetric vacua

,




A resolution Y ofthe conifoldY := Spec

(C[x,y,u,v ](xy−uv)

)remains achieved bya D0-brane aggregrationthat gives an embedding

Y →M0Azf

2 (Y ).

Y

0

Spec

D0-brane of rank 2M ( ) NC cloud2

2

U Space

Y

0

ΛcSpace

τ

~ϕ




Superficially infinitesimal deformations ofa morphism between noncommutativespaces can be “materialized” to a truedeformation. This is the mathematicalreason behind the phenomenon ofdeformation of the conifold singularityby D-brane probe.At the momentof this lecture,unlike the situationof D-brane proberesolution of singularities,D-brane probe deformation of conifoldseems to be generalizable in terms ofAzumaya geometry only to a special classof singularties.See [L-Y3: Sec. 2] (D(4)).

Spec

D0-brane of rank 2M ( ) NC cloud2

2

RΞSpace

4z1 z2 z3 z4[ , , , ]

π Ξ

Y Y

4ξ1 ξ ξ ξ2 3 4[ , , , ] M ( )2Space

p0 p

ϕ

~ϕ~ϕ δ2 η1 ( , , , )δ1 η2

ϕ δ2 η1 ( , , , )δ1 η2



Gomez-Sharpe: Information-preserving geometry on D-branes (2000)

[G-S] T. Gomez and E. Sharpe, D-branes and scheme theory,arXiv:hep-th/0008150.

• This work was propelled by the same quest that propelled us andwas unfortunately also ignored by the main D-community.

• From our point of view, the “information geometry” Gomez andSharpe tried to capture for D-branes is realized as the“commutative leftover” when one tries to “squeeze/condense” thenoncommutative cloud OAz

X of an Azumay space X Az into acommutative space Y .

Cf. [L-Y5: Sec. 2.4 (4)] (D(6)).



Sharpe: B-field, gerbes, D-brane bundle (2001)


[Sh] E. Sharpe, Stacks and D-brane bundles, Nucl. Phys. B610 (2001),595–613.(arXiv:hep-th/0102197)

Cf. [L-Y4] (D(5)).



Denef: (Dis)assembling of A-branes under split attractor flow (2001)(Frame set to be completed.)

[De] F. Denef, (Dis)assembling special Lagrangians, arXiv:hep-th/0107152.

an inverse split attractor flowon the complex moduli space Mof Calabi-Yau 3-fold Y

the wall of marginal stabilityassociated to the decompositionΓ Γ + Γ in H (Y; )1 2

Γ Γ2Γ1 H (Y; )3

3

Γ Γ 2Γ 1

complex

M complex

Deformation/degeneration ofspecial Lagrangian submanifolds(in classes , , )driven by deformation of complex stuctures on Yalong the attractor flows , , andassociated to , , respectively

γΓ2

γΓ2

γΓ

γΓ

γΓ1

γΓ1

Cf. [L-Y6: Sec. 3] (D(7)).Chien-Hao Liu Physics aspects of D-branes through Azumaya geometry 54/78


D-brane, spectral cover, and Hitchin system

A spectral cover data in [B-N-R], [Hi], [Ox] is given by a morphismof the form

(X ,OAzX , E)

ϕ //

$$

L

yyX ,

where L is the total space of a line bundle L on X and both(X ,OAz

X )→ X and L→ X are built-in morphisms. L can bereplaced by any fibration over X . Cf. [L-Y5: Sec. 2.4] (D(6))andSec. 2, Example 2 (p. 14) of this lecture.

[B-N-R] A. Beauville, M.S. Narasimhan, and S. Ramanan, Spectral curves and thegeneralized theta divisor, J. reine angew. Math. 398 (1989), 169–179.

[Hi] N. Hitchin, Stable bundles and integrable systems, Duke Math. J. 54(1987), 91–114.

[Ox] W.M. Oxbury, Spectral curves of vector bundle endomorphisms, KyotoUniversity preprint, 1988; private communication, spring 2002.



Dijkgraaf-Hollands-Su lkowski-Vafa:Quantum spectral curve (2007, 2008)


[D-H-S-V] R. Dijkgraaf, L. Hollands, P. Su lkowski, and C. Vafa, Supersymmetricgauge theories, intersecting branes and free fermions, J. High EnergyPhys. 0802 (2008)106, 57pp. (arXiv:0709.4446 [hep-th])

[D-H-S] R. Dijkgraaf, L. Hollands, and P. Su lkowski, Quantum curves andD-modules, arXiv:0810.4157 [hep-th].

Cf. [L-Y5: Sec. 2.4] (D(6)).



Cecotti-Cordava-Vafa:Recombination of A-branes under RG-flow (2011) 1/3

[C-C-V] S. Cecotti, C. Cordova, and Cumrun Vafa, Braids, walls, and mirrors,arXiv:1110.2115 [hep-th].

• Deformation of branes :(De)amalgamationof branes.

While what’sdisplayed here andthe next two pagesis only for Azumayacircles, similarbehaviors occur fordeformations ofmorphisms fromgeneral Azumayamanifold as well.

Cf. [L-Y5: Sec. 4.3] (D(6)).




• Deformation of branes :Large- vs. small-branewrapping

Cf. [L-Y5: Sec. 4.3] (D(6));

[L-Y6: Sec. 2.3] (D(7)).




• Deformation of branes :Brane-anti-branecancellation

Cf. [L-Y5: Sec. 4.3] (D(6)).



Quantum field theory on D-brane world-volume that takes morphismsas the basic fields

• Remark. Morphisms from Azumaya manifolds with a fundamentalmodule could be a starting point of the following topics, which arelargely unknown/undeveloped:

A perturbative D-brane theory in a similar spirit to theperturbative string theory.

A non-linear sigma model for D-branes, i.e. field theory on theD-brane world-volume for maps from D-brane world-volume toa Calabi-Yau space.

A (re-)derivation of Dirac-Born-Infeld-type action for stackedD-branes from a more fundamental, “Azumaya differentialgeometry” aspect.

Cf. Comments in Polchinski’s textbook on actions for stackedD-branes; discussions with Li-Sheng Tseng (2010).



Berman-Perry: Genus expansion of membrane world-volumes (2006) 1/5

[B-P] D.S. Berman and M.J. Perry, M-theory and the string genus expansion,Phys. Lett. B635 (2006), 131–135. (arXiv:hep-th/0601141)

Just like in the case of perturbative string theory, in consideringperturbative D-brane theory and its amplitudes one wouldunavoidably have to address the following question:

Question [sum over D-brane world-volume]

How do we sum over D-brane world-volumes? Unlike the stringworld-sheet case, whose topology is classified by genus, in higherdimensional case, we don’t have such a simple classification.

Answer: For D3-branes, Azumaya geometry singles out theAzumaya 3-sphere S3,Az , thus we may consider only sum over thegenus of graph 4-manifolds, which are connected sums ofcollections of S3 × S1.




Alexander-Hilden-Lozano-Montesinos-Thurston meetingPolchinski-Grothendieck/Ho-Wu

• Background. Classical works on 3-manifolds from branched coversof the 3-sphere S3:

[Al] J.W. Alexander, Note on Riemann surfaces, Bull. Amer. Math. Soc. 26(1920), 370–372.

[Hil] H.M. Hilden, Three-fold branched coverings of S3, Amer. J. Math. 98(1976), 989–997.

[H-L-M] H.M. Hilden, M.T. Lozano, and J.M. Montesinos, Universal knots, inKnot theory and manifolds, D. Rolfsen ed., 25–59, Lect. NotesMath. 1144, Springer, 1985.

[Mon] J.M. Montesinos, 3-manifolds as 3-fold branched covers of S3, Quart. J.Math. Oxford (2) 27 (1976), 84–94.

[Thu] W.P. Thurston, Universal links, preprint, 1982.




Theorem [branched covering].(Alexander [Al], 1920.)

Any closed, connected, orientable 3-manifold is realizable as a branchedcovering of S3.

Theorem [3-fold enough].(Hilden [Hil] and Montesinos [Mon], 1976.)

Any closed, connected, orientable 3-manifold is realizable as a 3-fold (i.e.degree-3) irregular branched covering of S3 with the branch locus in S3 a knot.

Theorem [universal link].(Thurston [Thu], 1982.)

There exists a (6-component) link L1 in S3 such that any closed, connected,orientable 3-manifold is realizable as a branched covering of S3 that is branchedonly over L1.

Theorem [universal knot]. (Hilden-Lozano-Montesinos [H-L-M], 1985.)

There exists a knot K 1 in S3 such that any closed, connected, orientable3-manifold is realizable as a branched covering of S3 that is branched only overK 1.




Theorem [S3,Az and fundamental D3-brane] ([L-Y6: Sec. 2.4.2] (D(7))).

Let Y be a spatial slice of a target space-time of string theory andN ⊂ Y be the image of any smooth map f : N ′ → Y , where N ′ is asmooth 3-manifold. Then there exists a morphism ϕ : S3,Az → Y froman Azumaya 3-sphere such that the image ϕ(S3,Az ) of ϕ is exactly N.Furthermore, one can require that the rank of the fundamental module Eof S3,Az be 3 · (number of irreducible components of N) . Or one mayrequire that πϕ : S3

ϕ → S3 be a branched-covering map over a universal

knot or a universal link in S3.

This specifies morphisms from (S3,Az , E) as most fundamental D3-branesfrom the viewpoint of Azumaya geometry.Similar result works on (S2,Az , E) for D2-branes.




Alexander-Hilden-Lozano-Montesinos-Thurston meetingPolchinski-Grothendieck/Ho-Wu illustrated:

• A possible feedback of Azumaya geometry to string theory: A genus-likeexpansion of the path integral for D3-branes (resp. D2-branes) based ongraph manifolds M, which is a connected sum of a collection of S3 × S1

(resp. S2 × S1).

time

ϕ

M

M Az

S Az3,

S 3

Space-time



Two more guiding quesions

Question [Douglas] (December, 2011)

What problems for D-branes in string theory can such a settingsolve?

Question [Vafa] (February, 2013)

A general morphism in the setting may create a double-stackingeffect to D-branes. That seems to lead to, e.g., nonlocalunphysical effects on D-branes. How to either clarify/explain thisor remove them by confining to a consistent, physical, yet stillabundant enough, subclass of morphisms?



Mathematical theory of D-string world-sheet instantons (in progress) 1/10

Various instantons in superstring theory:When a Eulcideanized brane world-volumewraps around a cycle in the internalCalabi-Yau space (or a cycle inG2 7-manifold in the caseof M-theory), the wholebrane-world-volumelooks like a pointin the effectivespace-time,i.e. localizedboth in spaceand in itime.Such brane configuration creates thusa (brane-world-volume) instantonon the lower-dimensional effective space-time.

Euclideanized/Wick-rotatedbrane world-volume

lower-dimensional (e.g. 4d)eective space-timefrom a compacticationof a 10d superstring modelon a Calabi-Yau space

the internal Calabi-Yau spaceover p

cycle

p

low-dimensionaleffective QFT

10d superstring model

homomorphism / wrapping




Example [fundamental string].Fundamental-string world-sheet instanton⇒ Stable maps from Riemann surfaces to a Calabi-Yau space⇒ Gromov-Witten theory.

Q. How can we describe D-string world-sheet instantonsin our setting?

(√) (C ,OAzC , E), ⇐⇒ (Euclidean) D-string world-sheet

C : nodal curve/C with open-string-induced structure

(√) ϕ : (C ,OAzC , E) −→ Y ⇐⇒ wrapping on a holomorphic 1-cycle

in a Calabi-Yau space Y

(?) stability condition ⇐⇒ special connection + “good wrapping”




• Lesson from solitonic B-branes: Use the slope/phase functionassociated to (BPS) central charge of a supersymmetric D-braneto define stability of the brane. E.g.:

[Dou1] M.R. Douglas, D-branes on Calabi-Yau manifolds, arXiv:math/0009209.

[Dou2] ——–, Dirichlet branes, homological mirror symmetry, and stability,arXiv:math/0207021 [math.AG]

[Br] T. Bridgeland, Stability conditions on triangulated categories, Ann. Math.166 (2007), 317 – 345. (arXiv:math.AG/0212237)

• (BPS) central charge formula of F• ∈ Db(Coh (Y )): E.g.:

[C-Y] Y.-K.E. Cheung and Z. Yin, Anomalies, branes, and currents, Nucl. Phys.B517 (1998), 69 – 91. (arXiv:hep-th/9710206)

• Adjustment. A twisting from a polarization class on C is needed ifone wants to obtain a bounded moduli space of stable morphisms.




Let C : nodal curve with polarization class L; Y : projectiveCalabi-Yau manifold with complexified Kahler class B +

√−1J.

Definition [twisted central charge] (in large fundamental-string tension limit)

Let ϕ : (C ,OAzC , E)→ Y be a morphism. Recall its graph a

1-dimensional coherent sheaf Eϕ on C × Y . Then, the twistedcentral charge of ϕ associated to the data (B +

√−1J, L) is

defined to be

Z B+√−1J,L(ϕ) := Z B+

√−1J,L(Eϕ)

:=

∫C×Y

pr ∗2

(e−(B+

√−1J)√

td(TY )

)· pr ∗1 e−

√−1L · τC×Y (Eϕ) .

Here, τC×Y (Eϕ) := ch (Eϕ) · td (TC×Y ) is the τ -class of Eϕ andpr1 : C × Y → C , pr2 : C × Y → Y are projection maps.




Note.

Z B+√−1J,L : Coh 1(C × Y ) −→ H− :=

z ∈ C

∣∣∣∣ either Im z < 0or Im z = 0 with Re z > 0

.

Thus, its associated phase function φZ : Coh 1(C × Y )→ (−π, 0].

Definition [Z -semistable, -stable, -unstable; strictly Z -semistable].

A 1-dimensional coherent sheaf F on C × Y is said to beZ -semistable (resp. Z -stable) if F is pure and φZ (F ′) ≤ (resp. <)φZ (F) for any nonzero proper subsheaf F ′ ⊂ F . Such F is calledZ -unstable if it is not Z -semistable, and is called strictlyZ -semistable if it is Z -semistable but not Z -stable. When thecentral charge functional Z is known and fixed either explicitly orimplicitly, we may use the terminology: semistable, stable,unstable, strictly semistable, for simplicity.




• Remark [space of stability conditions]. Under this setting, a stabilitycondition on morphisms ϕ : (C ,OAz

C , E) to Y depends on

(B +√−1J, L) ∈ Stab 1,[0](C × Y ) := H2(Y ;R)×

√−1 · KCone (Y )× DCG +(C) .

Here, DCG (C ) is the degree class group of the nodal curve C ,DCG +(C ) ⊂ DCG (C ) the semigroup of effective classes, and KCone (Y )the Kahler cone of Y . It’s natural to take this as a subspace of

Stab 1,[0](C × Y )R := H2(Y ;R)×√−1 · KCone (Y )× DCG +(C )R ,

where DCG (C )R := DCG (C )⊗Z R, DCG +(C )R ⊂ DCG (C )R the coneof effective classes spanned by R>0-rays through elements inDCG +(C ) ⊂ DCG (C )R. There is a natural chamber structure onStab 1,[0](C × Y )R defined by a locally finite collection fo quadratichypersurfaces. Stability conditions that lie in the same chamber definethe same moduli space of stable morphisms.




Main technical issue.

Goal. As in Gromov-Witten theory for stable maps, the domain of our stablemorphisms is not fixed. To have a good mathematical theory of D-string world-sheetinstantons, we want our moduli stack of stable morphisms from Azumaya nodalcurves with a fundamental module to Y to be compact (a minimal requirement).And then, we want to show that it has a good tangent-obstruction theory at least insome imporant cases to define D-string world-sheet instanton numbers from a purelyD-string world-sheet aspect. This would give us a parallel theory for D-strings asGromov-Witten theory for fundamental strings.

Problems from P1-tree bubbling. Also as in Gromov-Witten theory,degenerations ET of stable morphisms in our problem may give rise to objects not inour class of morphisms; e.g. Supp (E0) may have vertical components, E0 may nolonger be locally free. Just considering morphisms, all such bad degenerations can becorrected by P1-bubbling trees added to nodal curves, as in Gromov-Witten theory.However, as there is no known universal estimate to relate the complexity ofdegenerations of stable morphisms to a bound on the complexity of the P1-treesneeded to absorb the bad degeneration. Thus, there is no way to select beforehanda large enough polarization class on nodal curves to guarantee that we never use upits positivity condition to define stability condition on morphisms when extending thepolarization class on a nodal curve to its cousin with P1-bubbling trees.




The way out. Need to separate the underlying domain nodalcurve C of a morphism ϕ : (C ,OAz

C , E)→ Y in our problem intothe union of a major nodal subcurve C0 and a minor subcurve C1.C0 takes up all the imaginary part of the central charge of ϕ andcontribute to the stability of ϕ by the standard definition (p. 65)while C1 consists only of P1-trees, whose contributation to thestability of ϕ is imposed by hand.

An auxiliary moduli stack. To ease this process, for a givengenus g , let M be a compact stack of nodal curves of genus g ,(in application, we woud require M to be large enough to contain,e.g., Mg for our purpose); CM/M the associated universal nodalcurve over M; L a relative polarization class on CM/M. Considerfirst the following auxiliary moduli stack:(continued on the next page)




Let (Y ,B +√−1J) be a projective Calabi-Yau 3-fold with a fixed

complexified Kahler class and

FM 1,[0];Z-ssCM/M (Y ; c) =

Z B+√−1J,L-semi-stable Fourier-Mukai transforms

of dimension 1, width [0], and central charge c ∈ H−from fibers of CM/M to Y

Theorem [compactness of FM 1,[0];Z-ss

CM/M (Y ; c)] ([L-Y9](D(10.1))).

The moduli stack FM 1,[0];Z-ssCM/M (Y ; c) is compact.

Proof

· Kleiman’s Ampleness Criterion.

· Kleiman’s Boundedness Criterion for a family of coherent sheaves.

· Langton’s argument through elementary modifications of coherentsheaves over a discrete valuation ring.




To be completed:

Completion of the construction of stability conditions:D(10.2) in preparation.

Next: Tangent-obstruction theory of the moduli problem andnew invariants of a projective variety through D-stringworld-sheet instantons along this line.



Epilogue. A reflection3 on strings, branes, and dualities:

Mystery and beyond mystery, door to all magics.

So large that it has no bounds; so big that it takes a long time to make;so harmonious that it fits no tunes; so beautiful that it takes no shapes.

A reflection on the Azumaya noncommutativity:

What’s naught could be the most useful. Thank you.

Lecture dedicated to my numerous teachers and L.-M..————————————————3From Lau-Tzu (600 B.C.) Tao-te Ching ; translation by Ling-Miao Chou.


physics aspects of d-branes from matrix-type...

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