Physics aspects of D-branes from matrix-type noncommutative geometry on D-brane world-volume

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from matrix-type noncommutative geometry on D-brane world-volume

Chien-Hao Liu

Joint 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 University May 10, 2013

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

Outline

Matrix-type/Azumaya noncommutative geometry on 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 space through four equivalent aspects of morphisms

Azumaya geometry at work for D-branes in string theory D-string world-sheet instantons (in progress)

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1 Matrix-type/Azumaya noncommutative geometry on D-brane world-volume

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

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

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

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

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An open-string induced phenomenon and a fundamental question

Question

When D-branes are stacked, thescalar fields describing their collective position/deformations areenhancedby additional massless scalar fields created by open strings andbecome matrix-valued,what happens to the D-brane? And to the space-time?

Answer to this question reflects what we take as fundamental nature of D-brane or of space-time (cf. remark in

[Polchinski: String theory, vol.1: Sec. 8.7, p. 272]).

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Lesson from Quantum Mechanics

Pointlike particlein a space(-time)Y with coordinates (yi)i. Classical mechanics: (y_i)_i describes the position(and hence deformations) of a point in Y.

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

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Emergence of Matrix/Azumaya-type noncommutativity on 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-branesfrom an ordinary space to a noncommutative space with a matrix/Azumaya-type

noncommutative structure.

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Original work of Ho and Wu and its coming back

This was observed byPei-Ming HoandYong-Shi Wu in1996:

[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 who emphasizes more on the target space-time noncommutativity aspect).

Re-picked upten years later (!) in December 2006 (cf. [L-Y1]

D(1)) fromre-reading Polchinski from Grothendiek’s viewpoint of algebraic geometryand his theory of schemes, after discussions withDuiliu-Emanuel Diaconescu on open-string world-sheet instantons that drove me to re-think about D-branes.

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D-brane world-volume vs. target space

stacked D-brane X space-time Y

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

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X^nc

Y^nc Y

X

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Issues that need to be answered first

Thus, a door, once opened but shut, is re-opened, butwhere does it lead to? How can one re-understand D-branes through it? More importantly and constructively: What new features/ properties of D-branes can this help us understand?

Before we can address these high level questions, we have to answer first two low level, yetfundamental,questions:

What is an Azumaya space with a fundamental module?

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

Quantum field theory on D-brane world-volumeis supposed to cover afield theory for such morphisms. (CAUTION: There’re other fields on D-brane world-volume.)

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Example 1: D0-brane on the complex line A¹_C ^1/4

Question

What is aD0-brane on a complex line A¹_C? Answer: Amap/morphism

ϕ : (Space (Mr(C)), C^r) −→ A¹_C

from an Azumaya point with a fundamental module to A¹_C. Question

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

Answer: ϕ is defined byaC-algebra homomorphism Mr(C) ←− C[z] : ϕ^].

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Example 1: D0-brane on the complex line A¹_C ^2/4 Geometry of an Azumaya point over C

Spec C( )A2

Spec C( )A1

M ( ) noncommutative cloudr

Spec NC cloud

A1

NC cloud A2

Spec C( )A A NC cloud

Many novel features of D-branes turn out to be originated from therichness of the “structure” of an Azumaya point

over C.

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Example 1: D0-brane on the complex line A¹_C ^3/4

ϕ : (Space (M_r(C)), C^r) −→ A¹_C

Mr(C) ←− C[z] : ϕ^] Ker ϕ^]= (f (z)) = ((z − a₁)^d¹· · · (z − a_k)^d^k)

⇒Im ϕ = fuzzy points located at z = a1, . . . , a_k C^r fundamental representation fo M_r(C)

⇒C[z ]-module via ϕ^]

⇒Coherent sheaf on A¹_C, supported on Im ϕ.

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Example 1: D0-brane on the complex line A¹_C ^4/4 Morphisms from Azumaya point illustrated

open-string target-space(-time) Y Spec

D0-brane of rank r M ( ) NC cloud_r

r

ϕ₁

ϕ2

ϕ3

ϕ2

un-Higgsing Higgsing

Note that theHiggsing-unHiggsing of D-branesis an outcome of deformations of morphisms.

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Example 2: Spectral covers from D-branes

“Pile/Smear” Example 1 alonga complex variety/manifold X

⇒ Spectral cover eX in a complex line bundle L_X over X. I.e.X = imagee of a morphism over X :

(X , O_X^Az, E ) ^ϕ ^//

$$

LX

xxX .

Spec D0-brane of rank r

M ( ) NC cloudr

r ϕ

X Remark. This is animportant example; cf.Gaiotto-Moore-Neitzke’s work oncounting 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].

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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 and non-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 their mirrors, Nucl. Phys. B477 (1996), 407 – 430. (arXiv:hep-th/9606112)

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Supersymmetric D-branes on a Calabi-Yau space ^2/3

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

·

Constructible sheavesonspecial Lagrangian submanifolds.

·

^{Fukaya A}∞-category Fuk (Y )ofgraded Lagrangian submanifolds on Y .

·

Open Gromov-Witten theoryto define theA_∞-structure on Fuk (Y ).

• Algebraic geometry for B-branes:

·

Derived category Coh^b(Y )ofcoherent sheaveson Y .

·

Stability conditionson objects of Coh^b(Y ).

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

·

FromStrominger-Yau-Zaslow construction to tropical geometry andGross-Siebert program.

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Supersymmetric D-branes on a Calabi-Yau space ^3/3 Where we are in the Wilson’s theory-space of string theory:

Solitonic vs. soft D-branes

Sup

erstrin

g [ ]

T

D-brane small T D-brane

large T

D-brane tension NLSM [ ]

Wilsond=2, SQFTb

Wilson d=4, SQFT

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D-branes as morphism: Proto-defintion

Proto-definition of D-branes

AD-brane on Y is amorphism ϕ : (X , O^Az_X , E ) → (Y , OY).

Here,

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

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

· O_X^Az = EndO_X(E ), as sheaf of (noncommutative)O_X-algebras.

Question [morphism]

What does a “morphism” mean in this context?

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Morphism between “spaces” as contravariant gluing system

of ring-homomorphisms ^1/2

Grothendieck’s theory of schemes:

Space as a gluing system of rings Space ({Rα}_α)

Morphism Space ({Rα}_α) → Space ({Sβ}_β) ascontravariant gluing system of ring-homomorphisms {R_α}_α←{S_β}_β. Key point: In commutative case, there is really a way to sepecfy a point-set-with-topology Spec R to ring R in a functorial way, which glue to ascheme, while inour case, we have toabandon such a point-set-with-topology notionto make our definition really describe D-branes in string theory.

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Morphism between “spaces” as contravariant gluing system

of ring-homomorphisms ^2/2

In other words,

ϕ : (X , O^Az_X , E ) −→ (Y , O_Y)

isonly a symbol. Itsreal contentis in O_X^Az ←− O_Y : ϕ^]

that representsan equivalence class of contravariant gluing systems of ring-homomorphisms. In particular,in general there is no morphism X → Y that underlies ϕ.

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Four aspects of such morphisms when target Y is commutative.

Aspect 1: Fundamental

Summarizing the discussion up to now:

Definition [morphism-fundamental]

Amorphism

ϕ : (X , O^Az_X , E ) −→ (Y , O_Y) isdefined by

O_X^Az ←− O_Y : ϕ^]

that representsan equivalence class of contravariant gluing systems of ring-homomorphisms.

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

An observation: Amorphism ϕ : (X , O_X^Az, E ) → (Y , O_Y) determines and is determined by the followingdiagram:

O_X^Az = EndO_X(E )

A_ϕ := Im ϕ^? ^]

OO

O_Y

ϕ^]

oo

O_X

?OO

. This is exactly the data of acoherent sheaf eE_ϕ on X × Y that is flat and relative dimension-0 over X.

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Definition [graph of morphism]

Thecoherent sheaf eE_ϕ on X × Y, which isflat and of relative dimension 0 over X is called the graphof the morphism ϕ : (X , O^Az_X , E ) → (Y , O_Y).

Thus,morphisms from Azumaya spaces X^Az with a fundamental module to Y form a subclass of Fourier-Mukai transformsfrom X to Y.

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X πϕ Xϕ

nd_X( ) Azumaya cloud

ϕ Y

fϕ

X

Y Γϕ

Supp ( ) =

=

X Y

pr₂

pr₁ Azumaya, morphism Fourier-Mukai transform

Xϕ

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Aspect 3: Morphism to moduli stack of D0-branes

M⁰_r^Azf(Y ):= moduli stackof morphisms froman Azumaya point with a fundamental module of rank r to Y. Then,

M⁰_r^Azf(Y ) is the same asthe stack of 0-dimensional coherent sheaf of length r on Y;

Aspect 2 can be translated immediately to:

[special role played by stack of D0-branes]

Amorphism ϕ : (X , O^Az_X , E ) → Y is the same as a morphism X → M⁰_r^Azf(Y ).

open strings D0-branes

p-cycle Dp-brane

Smearing D0-branes along a p-cycle to get a Dp-brane

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

Frommorphisms to stackto G -morphisms to atlas:

Isom (φ, π) ^φ^e ^//

pr1

Atlas

π

X ^φ ^//M⁰_r^Azf(Y ) ChooseAtlas to be therepresentation-theoretical atlas

Quot^H⁰(O^⊕r_Y , r ) :=

( O_Y^⊕r → eE → 0, length eE = r, H⁰(O^⊕r_Y ) → H⁰( eE) → 0

)

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

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Remark: From algebraic to sympletic

We begin withthe fundamental proto-defintion of D-branes, which ismore akin to algebraic geometry. But when the target Y is commutative, in particular, a Calabi-Yau space, thenfollowing these four aspects of morphismsfrom Azumaya spaces, we get closer and closer to a language that is accommodatable in sympletic/differential geometry, with Quot-schemes replaced by (the analytic) Douady spaces. It becomes thusmore and more accesssible to string-theoristsas well.

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

•The defintion of morphisms from Azumaya spaces with a fundamental module in the realm of complex algebraic geometry is by nature holomorphic. It gives thusholomorphic D-branes.

•The special connection on the Chan-Paton module is supposed to be specified by astability conditionvia a Donaldson-Uhlenbeck-Yau-type theorem, though in practice this is very hard to come by. Thus, we take stable morphismsϕ to a Calabi-Yau space Y as giving a B-brane on Y , if an apprapriate notion ofcentral charge Z (ϕ)– and hence theassociated notion of stability conditionon ϕ – can be defined.

• Subtlety. For a fixed target Calabi-Yau space Y , if we consider only morphisms from a fixed (X , O^As_X , E ), it’s easier to adapt the

solitonic-brane situation to the current situation. However,when the (X , O^As_X , E ) are allowed to vary, some technical subtlety (if not

conceptual ones) may arise. This is a similar, yet more involved, subtlety when mathematicians tried to construct moduli space of stable bundles over all stable curves. We’ll make this concrete inSec. 5when addressing theconstruction of D-string world-sheet instantons in our setting.

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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 onY;

ω : (Ricci-flat)K¨ahler 2-formon(Y , J);

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

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

Given aCalabi-Yau n-fold Y = (Y , J, ω, Ω, ), an A-brane on Y is a morphismϕ : (X , O_X^Az, E , ∇) → Y such that

ϕ^∗ω = ϕ^∗(Im Ω) = 0 .

Here,E is a locally freeO_{X ,C}(:= O_X^∞⊗_RC)-module,

O^Az_X := EndO_{X ,C}(E )the sheaf of O_{X ,C}-module endomorphisms, and∇is a flat connection onE,possibly with singularities.

ϕ is defined by ϕ^]: O^∞_{Y ,C}→ O_X^Az. Thepull-back operationϕ^∗ on differential formsshould be defined accordingly; we’ll useAspect 2, Fourier-Mukai transform, to understand this.

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Aspect 2 [Fourier-Mukai transform]:

X

Y

Γϕ Supp ( ) =

=

X Y

pr₂

pr₁ Xϕ

·

^Xϕ : stratified

piecewise-smooth

·

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

·

^(pr^∗2ω)|X_ϕ = 0

·

^(pr^∗2Ω)|X_ϕ= 0

Issues to be understood: (cf.[D(6): Sec.4.2])Deformation theoryfor such special Lagrangian morphisms w/ a constructible sheaf to a Calabi-Yau spacethat remembers how they degenerate and collideas in the theory of schemes and coherent sheaves.

Cf.Thevertical complex/scheme structureonX × Y.

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A-branes on a Calabi-Yau space from special Lagrangian morphisms ^4/4 Aspecial class of such morphismsfrom Azumaya manifold with a fundamental module, before a general theory is developed:

roof diagram

( eX , eV)−→ Y^˜^f

c↓

X ,

where

·

^{c : e}^{X → X}

branched coveringof smooth

manifolds along a codim_R-2 submanifold,

·

^{f : e}^˜ ^{X → Y}

·

^V^e complex vector bundle on eX special Lagrangian morphism with asingular flat connection

Ve

Xe

h_(c,˜_{f )}

!!

˜f

((

c

X_ϕ

(c,˜f ) f_ϕ

(c,˜f )

//

π_ϕ

(c,˜f )

Y

X

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Remark: Effect of B-field to D-branes

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

From fundamental module to twisted fundamental module, and from trivial Azumaya structure to non-trivial Azumaya structure(associated to a non-zero element

of a Brauer group).

An additional deformation-quantization-type noncommutativity on D-brane world-volume.

This is a topic in its own right. I refer you to D(5)(cf. [L-Y4]¹) for more 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 la

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D-project in progress ^1/4

After brewing for a decade, a project onD-branesis finally set up inearly spring 2007 to (re-)understand them mathematically as fundamental/soft objects(rather than solitonic objects) in string theoryvia morphisms from this matrix-type/Azumaya noncommutative geometry, reviewed in Sec. 1, Sec. 2, Sec. 3.

[L-Y1] C.-H. Liu and S.-T. Yau,Azumaya-type noncommutative spaces and morphism therefrom: Polchinski’s D-branes in string theory from Grothendieck’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 prestable curves with a fundamental module to a projective variety: Topological D-strings as a master object for curves, arXiv:0809.2121 [math.AG].

(D(2))

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[L-Y2] C.-H. Liu and S.-T. Yau,Azumaya structure on D-branes and resolution of 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 resolutions of a conifold revisited: Klebanov-Strassler-Witten vs.

Polchinski-Grothendieck, arXiv:0907.0268 [math.AG]. (D(4)) [L-Y4] ——–,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 la Polchinski-Grothendieck Ansatz, arXiv:0909.2291 [math.AG]. (D(5))

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[L-Y5] ——–,D-branes and Azumaya noncommutative geometry: From Polchinski to Grothendieck, arXiv:1003.1178 [math.SG]. (D(6)) [L-Y6] ——–,D-branes of A-type, their deformations, and Morse cobordism of

A-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 a branched covering of a special Lagrangian 3-sphere in a Calabi-Yau 3-fold and its deviation from Joyce’s criteria: Potential image-support rigidity of A-branes that wrap around a sL S³, arXiv:1109.1878 [math.DG]. (D(8.1))

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[L-Y8] ——– (with Baosen Wu),D0-brane realizations of the resolution of a reduced 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 from a 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 for Geometry and Physics, Stony Brook University,

arXiv:1112.4317 [math.AG].

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

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Mathematical D-geometry through Azumaya geometry

D-brane in superstring theory

Azumaya geometry:

morphisms from Azumay spaces with a fundamental module

Purely mathematical generalization

New theory/problem in its own right or new meaning to old theory/problem Quantun field theory

+ Supersymmetry method

Statements in algebraic or symplectic/differential geometry

feedb

ack (ideally)

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Mathematical D-geometry through Azumaya geometry

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

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

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Higgsing/un-Higgsing of D-branes

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

open-string target-space(-time) Y Spec

D0-brane of rank r M ( ) NC cloud_r

r

ϕ₁

ϕ2

ϕ3

ϕ2

un-Higgsing Higgsing

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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, theHilbert scheme Y^{[r ]} of 0-dimensional subschemes of Y of length r is tautological asubstack ofthe stack M⁰_r^Azf(Y ) of morphisms from Azumaya points with a fundamental module of rank r to Y .

• From our point of view, one should first look at either the stack M⁰_r^Azf(Y ) or the Quot-scheme Quot_Y(O_Y^⊕r, r )of 0-dimensional quotient sheaves of O^⊕r_Y of length r ; and then lead the way to Y^{[r ]}, though this looks a very difficult task at the moment.

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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 onR⁵⁺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 thesingular locus R⁵⁺¹× {0}

of the 10d space-timeR⁵⁺¹× C²/Γ

;

·

^Thed=6, N=1 supersymmetric QFT on D5-brane world-volume in such configuration hasspace of vacua M_~_ζ

•, depending on thevacuum expectation value ~ζ_• of the salar fields ~φ_• in the theory.

·

For appropriate ~ζ_•, M_~_ζ

• gives a resolution of C²/Γ.

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D-brane probe resolution ofADE surface

singularitiesin[D-M]can be interpreted directly as aresolution of the singularity by aggregations of D0-branes in our sense.

Spec D0-brane of type r M ( ) NC cloudr

r

ϕ1

ϕ2

ϕ3

ϕ4

2/G [ ] Chan-Paton module from push-forward sitting over image D-brane

2: atlas of orbifold fundamental module

on pt^Az

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• A mathematical abstraction in birational geometry:

·

Y : projective variety over C,

·

^M⁰r^Azf^p(Y ): the stack of punctual D0-branes of rank r on Y ,

·

π_Y : M⁰

Azfp

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

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

Let f : Y⁰ → Y be a birational morphism from a projective variety Y⁰ to Y . Then, f factors through an embedding

˜f : Y⁰ ,→ M⁰

Azfp

r (Y ), for some r . M⁰

Azfp

r (Y )

πY

Y⁰ ^*

˜f

77

f // Y .

In particular, any resolution ρ : eY → Y of Y

factors through some M⁰

Azfp

r (Y ).

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• A non-Calabi-Yau compact example:

Proposition [reduced curve] ([L-Y8] D(9.1), with Baosen Wu) Conjecture holds for any resolution ρ : C⁰ → C of a reduced curve.

Namely, there exists an r0 ∈ N depending only on the tuple (n_p⁰)_ρ(p⁰_)∈C_sing and a (possibly empty) set

{b.i.i.(p) : p ∈ C_sing, C has multiple branches at p }, both associated to the germ of C_sing in C , such that, for any r ≥ r0, there exists an embedding ˜ρ : C⁰,→ M⁰

Azf

r p(C )

that makes the following diagram commute: M⁰

Azfp

r (C )

πC

C⁰ ^*

˜ ρ

77

ρ // C .

Here, n_p⁰ and b.i.i.(p) are some characterization indices for p ∈ C_sing, p⁰ ∈ ρ⁻¹(p).

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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 K¨ahler manifolds, Adv. Theor. Math. Phys. 1 (1997), 237–258.

(arXiv:hep-th/9708012)

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

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

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

[K-W] I.R. Klebanov and E. Witten,Superconformal field theory on threebranes at 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 gauge theory: duality cascade and χSB-resolution of naked singularities, J. High Energy Phys. (2000) 052, 35 pp.(arXiv:hep-th/0007191)

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

Y Y

Y

0 0

a D-brane configuration

without fractional branes a D-brane configuration

with a fractional brane the moduli space of its supersymmetric vacua

,

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Aresolution eY of theconifold Y := Spec

C[x ,y ,u,v ] (xy −uv )

remains achieved by a D0-brane aggregration that gives anembedding Y ,→ Me ⁰₂^Azf(Y ).

Y 0

Spec

D0-brane of rank 2

M ( ) NC cloud2 2

U Space

Y 0

Λc Space τ ϕ~

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Superficially infinitesimal deformations of a morphism between noncommutative spaces can be “materialized” to a true deformation.This is the mathematical reason behind the phenomenon of deformation of the conifold singularity by D-brane probe.

At the moment of this lecture, unlike the situation of D-brane probe resolution of singularities,

D-brane probe deformation of conifold seems to be generalizable in terms of Azumaya geometry only to a special class of singularties.

See [L-Y3: Sec. 2] (D(4)).

Spec

D0-brane of rank 2

M ( ) NC cloud2

2

RΞ Space

4[ , , , ]z₁z2z3z4

π^Ξ

Y Y

4[ , , , ]ξ1ξ ξ ξ2 3 4 SpaceM ( )2

0 p p

ϕ ϕ~

ϕ~ ( , , , )δ₁δ2η1η2

ϕ ( , , , )δ1δ2η1η2

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G´omez-Sharpe: Information-preserving geometry on D-branes (2000)

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

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

• From our point of view, the “information geometry” G´omez and Sharpe tried to capture for D-branes is realized asthe

“commutative leftover” when one tries to “squeeze/condense” the noncommutative cloud O_X^Az of an Azumay space X^Az into a commutative space Y.

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

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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)).

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Denef: (Dis)assembling of A-branes under split attractor flow (2001)

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

an inverse split attractor flow on the complex moduli space M of Calabi-Yau 3-fold Y

the wall of marginal stability associated to the decomposition Γ Γ₁ + Γ2 in H (Y; )

Γ Γ₁ Γ₂ H ₃(Y; )

3

Γ Γ₁ Γ₂

complex

Mcomplex

Deformation/degeneration of special Lagrangian submanifolds (in classes , , ) driven by deformation of complex stuctures on Y along the attractor flows , , and associated to , , respectively

γ_Γ

2

γ_Γ

2

γ_Γ

1

γ_Γ

1

Cf. [L-Y6: Sec. 3] (D(7)).

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D-brane, spectral cover, and Hitchin system

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

(X , O^Az_X , E ) ^ϕ ^//

$$

L

yyX ,

where L is the total space of a line bundle L on X and both (X , O_X^Az) → X and L → X are built-in morphisms. L can be replaced by any fibration over X. Cf. [L-Y5: Sec. 2.4] (D(6))and Sec. 2, Example 2 (p. 14) of this lecture.

[B-N-R] A. Beauville, M.S. Narasimhan, and S. Ramanan,Spectral curves and the generalized 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, Kyoto University preprint, 1988; private communication, spring 2002.

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Dijkgraaf-Hollands-Su lkowski-Vafa:

Quantum spectral curve (2007, 2008)

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

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

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

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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)amalgamation of branes.

While what’s displayed here and the next two pages is only for Azumaya circles, similar behaviors occur for deformations of morphisms from general Azumaya manifold as well.

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

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• Deformation of branes : Large- vs. small-brane wrapping

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

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

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• Deformation of branes : Brane-anti-brane

cancellation

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

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

• Remark. Morphisms from Azumaya manifolds with a fundamental module could be a starting point of the following topics, which are largely unknown/undeveloped:

A perturbative D-brane theoryin a similar spirit to the perturbative string theory.

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

A (re-)derivation of Dirac-Born-Infeld-type action for stacked D-branesfrom a more fundamental, “Azumaya differential geometry” aspect.

Cf. Comments in Polchinski’s textbook on actions for stacked D-branes; discussions withLi-Sheng Tseng (2010).

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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 considering perturbative D-brane theoryand itsamplitudesone would unavoidably have to address the following question:

Question [sum over D-brane world-volume]

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

Answer: For D3-branes, Azumaya geometry singles out the Azumaya 3-sphere S^3,Az, thus we may consider only sum over the genus of graph 4-manifolds, which areconnected sums of

collections of S³× S¹.

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Berman-Perry: Genus expansion of membrane world-volumes (2006) ^2/5 Alexander-Hilden-Lozano-Montesinos-Thurston meeting

Polchinski-Grothendieck/Ho-Wu

• Background. Classical works on3-manifolds from branched covers of 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, in Knot theory and manifolds, D. Rolfsen ed., 25–59, Lect. Notes Math. 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.

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

Any closed, connected, orientable 3-manifold is realizable as a branched covering 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 S³with the branch locus in S³a 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 S³that is branched only over L¹.

Theorem [universal knot]. (Hilden-Lozano-Montesinos [H-L-M], 1985.) There exists a knot K¹in S³such that any closed, connected, orientable 3-manifold is realizable as a branched covering of S³that is branched only over K¹.

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Theorem [S^3,Az and fundamental D3-brane] ([L-Y6: Sec. 2.4.2] (D(7))).

LetY be aspatial slice of a target space-time of string theoryand N ⊂ Y be theimage of any smooth map f : N⁰→ Y, whereN⁰ is a smooth 3-manifold. Thenthere exists a morphism ϕ : S^3,Az → Y from an Azumaya 3-sphere such that the image ϕ(S^3,Az) of ϕ is exactly N.

Furthermore, one can require that the rank of the fundamental module E of S^3,Az be 3 · (number of irreducible components of N) . Or one may require that πϕ: S_ϕ³→ S³be a branched-covering map over a universal knot or a universal link in S³.

This specifiesmorphisms from (S^3,Az, E )asmost fundamental D3-branes from the viewpoint of Azumaya geometry.

Similar result works on (S^2,Az, E ) for D2-branes.

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

•A possible feedback of Azumaya geometry to string theory: Agenus-like expansion of the path integral for D3-branes(resp. D2-branes) based on graph manifoldsM, which is a connected sum of a collection of S³× S¹ (resp. S²× S¹).

time

ϕ

M M^Az

S^3,^Az

S³

Space-time

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Two more guiding quesions

Question [Douglas] (December, 2011)

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

Question [Vafa] (February, 2013)

A general morphism in the setting may create adouble-stacking effectto D-branes. That seems to lead to, e.g., nonlocal

unphysical effects on D-branes. How to either clarify/explain this or remove them by confining to a consistent, physical, yet still abundant enough, subclass of morphisms?

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

Various instantons in superstring theory:

When aEulcideanized brane world-volume wraps around a cycle in the internal Calabi-Yau space(ora cycle in G27-manifoldin the case of M-theory), the whole brane-world-volume looks like a point in the effective space-time, i.e. localized both in space and in itime.

Such brane configuration creates thus a(brane-world-volume) instanton

on the lower-dimensional effective space-time.

Euclideanized/Wick-rotated brane world-volume

lower-dimensional (e.g. 4d) effective space-time from a compactification of a 10d superstring model on a Calabi-Yau space the internal Calabi-Yau space over p cycle

p

low-dimensional effective QFT 10d superstring model

homomorphism / wrapping

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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 instantons in our setting?

(^√) (C , O^Az_C , E ), ⇐⇒ (Euclidean) D-string world-sheet C : nodal curve/C with open-string-induced structure (^√) ϕ : (C , O_C^Az, E ) −→ Y ⇐⇒ wrappingon a holomorphic 1-cycle

in a Calabi-Yau space Y

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

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• Lesson from solitonic B-branes: Use theslope/phase function associated to(BPS) central chargeof a supersymmetric D-brane to 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^• ∈ D^b(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. Atwisting from a polarization class on C is needed if one wants to obtain a bounded moduli space of stable morphisms.

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LetC: nodal curve with polarization classL; Y: projective Calabi-Yau manifold withcomplexified K¨ahler classB +√

−1J.

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

Letϕ : (C , O_C^Az, E ) → Y be a morphism. Recall its graph a 1-dimensional coherent sheafEe_ϕ on C × Y . Then, thetwisted central charge of ϕassociated to the data (B +√

−1J, L) is defined to be

Z^B+

√−1J,L(ϕ) := Z^B+

√−1J,L( eE_ϕ)

:=

Z

C ×Y

pr₂^∗

e^−(B+

√−1J)

√

td (TY)

· pr₁^∗e⁻

√−1L · τ_{C ×Y}( eE_ϕ) .

Here, τ_{C ×Y}( eE_ϕ) := ch ( eE_ϕ) · td (T_{C ×Y}) is the τ -class of eE_ϕ and pr₁ : C × Y → C , pr₂ : C × Y → Y are projection maps.

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Note.

Z^B+

√−1J,L

:Coh₁(C × Y )−→ ˆH−:=

z ∈ C

either Im z < 0

or Im z = 0 with Re z > 0

. Thus, its associatedphase function φ^Z : Coh₁(C × Y ) → (−π, 0].

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

A 1-dimensional coherent sheaf ˜F on C × Y is said to be

Z -semistable(resp.Z -stable)if ˜F is pure and φ^Z( ˜F⁰) ≤(resp.<) φ^Z( ˜F ) for any nonzero proper subsheaf ˜F⁰ ⊂ ˜F. Such ˜F is called Z -unstableif it is not Z -semistable, and is called strictly

Z -semistableif it is Z -semistable but not Z -stable. When the central charge functional Z is known and fixed either explicitly or implicitly, we may use the terminology: semistable, stable, unstable, strictly semistable, for simplicity.