K3 Compactifications and a Moonshine for M

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K3 Compactifications and a Moonshine for M 24 (I)

Miranda Cheng Harvard University

Taiwan String Workshop 2011

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Based on

M.C., E. Verlinde 0706.2363 M.C., E. Verlinde 0806.2337 M.C., A. Dabholar 0809.4258 M.C., L. Hollands 0901.1758 M.C. 1005.5415

the work of MANY, especially A. Sen and his group,

A. Danholkar, S. Govindarajan and their groups

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Outline

• Motivations

• K3 Compactifications (i): the K3 CFT

• K3 Compactifications (ii): type II string on K3xT

2

and their orbifolds

• A Group of Crossing the Walls

• A Generalised Kac-Moody Symmetry

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I. Motivations

for studying K3 compactification of string theory

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Opening the Black Box

I. Motivations

1. Gravity Beyond GR

Macroscopic Gravity Physics:

BH have thermodynamical entropy SBH(P,Q)

given by Area Law

Microscopic Physics:

Statistical entropy log(d(P,Q))

given by the dim. of q. Hilbert space

e.g. Strominger-Vafa ’96: agreement to leading order in 1/P,1/Q for D1-D5-p string black holes

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I. Motivations

1. Gravity Beyond GR

N=4 d=4 theories are the only cases where the full answer

(not just asymptotic growth) has been proposed on the microscopic side.

Hence provides excellent testing ground for ideas in QG.

eg.

higher-order corrections

[Cardoso Kappeli, Mohaupt, de Wit ’04, David, Jatkar, Sen 0510147,0605210, Kraus-Larsen 0506176]

eg2.

Prescription Euclidean path integral (Quantum Entropy Functions)

[A. Sen 0903.1477, 0911.1563]

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2. Microscopic Theory of BPS States

I. Motivations

Microscopic theory of black holes

= Computing spectrum of non-pert. BPS states

= Quantising D-brane moduli space in general complicated

In our proposal, just a simple theory of generalised Harmonic Oscillators!

Our answer sheds lights on the structure of q. D-brane moduli space

and gives predictions for hard questions such as the counting of sLags in the internal manifolds.

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3. Wall-Crossing Physics

eg2. supergravity

[Denef, Denef-Moore]

r12

(Q1, P1) (Q2, P2)

I. Motivations

eg. SU(2) Seiberg-Witten

Change the parameters in the Lagrangian of the theory. In general the spectrum changes.

Nevertheless, the supersymmetric index is protected and cannot change unless some states disappear to spatial infinity.

On the subspace in the parameter space where this happens, the index can and in general does jump.

BPS states can (dis)appear at the “walls of marginal stability” in moduli space.

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3. Wall-Crossing Physics

Question: What is the structure of the change of the BPS index?

Where/when/how does it jump?

In our case, we found a simple description of wall-crossing in terms of simple group theory.

Furthermore, an exact matching has been found between microscopic and macroscopic pictures of wall-crossing.

I. Motivations

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4. Quantum Symmetries of K3 Surfaces

Being the unique CY2, many well-studied string compactifications involve K3 manifolds. K3 also play an important role in string dualities.

Geometric symmetries of K3 at various enhancement points are known

(Nikulin’s involution).

Recently a larger symmetry has been suggested when B-fields/mirror symmetry are taken into account. This should be relevant for all string compactifications involving K3 manifolds.

If true this symmetry should also act on the non-perturbative spectrum when further compactified down to 4d. By studying the 4d BPS partition function, new evidence for the existence of such a new symmetry is

provided and new predictions for twisted BPS spectra are made.

Mathematically, the presence of this symmetry would imply many new insights on various algebraic geometric problems involving K3 surfaces.

I. Motivations

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5. Mathematical Motivations

Discrete Mathematics

Generalised Kac-Moody

Algebra

Sporadic Groups

Automorphic Forms

Hyperbolic Reflection Groups

moonshine

denominator formula Weyl Group

I. Motivations

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5. Mathematical Motivations

Discrete Mathematics

Generalised Kac-Moody

Algebra

Sporadic Groups

Automorphic Forms

Hyperbolic Reflection Groups

moonshine

denominator formula Weyl Group

I. Motivations

Spectrum-Generating Algebra

Partition Function

Symmetry Group Wall-Crossing

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II. K3 Compactifications (ii) : the K3 CFT’s

the worldsheet, perturbative perspective

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II. K3 Compactification (i)

N=(2,2) SCFT and Elliptic Genus of Calabi-Yau manifolds

N=(2,2) 2d sigma model on

L ∼

Σ

�|∂φ|2 − B�

+ fermions

when the target space is CY,

conformal with conserved currents

J, G

±

, T

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II. K3 Compactification (i)

N=(2,2) SCFT and Elliptic Genus of Calabi-Yau manifolds

N=(2,2) 2d sigma model on

L ∼

Σ

�|∂φ|2 − B�

+ fermions

when the target space is CY,

conformal with conserved currents

J, G

±

, T

Define the Elliptic Genus of X Z(τ, z; X) = Tr

RR

( −1)

JL+JR

y

2πiJL

q

L0−c/24

q ¯

L¯0−c/24

q = e

2πiτ

, y = e

2πiz

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II. K3 Compactification (i)

Elliptic Genus of CY:

Z(τ, z; X) � � �

z=0

= χ(X) Z(τ, z; X) � � �

z=1/2

= σ(X)

• Generalisation of various topological invariants e.g.

• Transforms nicely (weak Jacobi form) under SL(2,Z) Can be understood as a consequence of the equivalence

between the Feynmann path integral formalism and canonical quantisation of QFT.

• These two conditions are often enough to determine

uniquely. Z(τ, z; X)

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II. K3 Compactification (i)

Elliptic Genus of K3 surfaces:

K3 is also hyper-Kähler

N=(4,4) superconformal symmatry

The elliptic genus can be decomposed into characters of representations of N=4

superconformal algebra

[Eguchi-Ooguri-Taormina-Yang ‘89]

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II. K3 Compactification (i)

Elliptic Genus of K3 surfaces:

K3 is also hyper-Kähler

N=(4,4) superconformal symmatry

The elliptic genus can be decomposed into characters of representations of N=4

superconformal algebra

[Eguchi-Ooguri-Taormina-Yang ‘89]

Z(τ, z; K3) = (2y + 20 + 2y

−1

) + q( ·) + q

2

( ·) + ·

= �

n≥0,�

c(4n − �

2

)q

n

y

= χ θ

12

(τ, z)

η

3

(τ ) +

n=0

A

n

q

n−1/8

θ

12

(τ, z)

η

3

(τ )

χ, A

n

∈ Z

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II. K3 Compactification (i)

M

S1

Fig. 1: The string configuration corresponding to a twisted sec- tor by a given permutation g ∈ SN. The string disentangles into seperate strings according to the factorization of g into cyclic per- mutations.

2. The Proof

The Hilbert space of an orbifold field theory [6] is decomposed into twisted sectors Hg, that are labelled by the conjugacy classes [g] of the orbifold group, in our case the symmetric group SN. Within each twisted sector, one only keeps the states invariant under the centralizer subgroup Cg of g. We will denote this Cg invariant subspace by HCgg. Thus the total orbifold Hilbert space takes the form

H(SNM) = !

[g]

HCgg. (2.1)

For the symmetric group, the conjugacy classes [g] are characterized by partitions {Nn} of N

"

n

nNn = N, (2.2)

where Nn denotes the multiplicity of the cyclic permutation (n) of n elements in the decomposition of g

[g] = (1)N1(2)N2 . . . (s)Ns. (2.3)

4

Elliptic Genus of Symmetric Product : S

N

X = X

N

/S

N

The loop space of an orbifold factors according to “boundary condition”L(M/G)

X(σ + 2π) = g · X(σ) , g ∈ G

e.g.

G=S

9

g= (2)(4)(3)

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II. K3 Compactification (i)

M

S1

Fig. 1: The string configuration corresponding to a twisted sec- tor by a given permutation g ∈ SN. The string disentangles into seperate strings according to the factorization of g into cyclic per- mutations.

2. The Proof

The Hilbert space of an orbifold field theory [6] is decomposed into twisted sectors Hg, that are labelled by the conjugacy classes [g] of the orbifold group, in our case the symmetric group SN. Within each twisted sector, one only keeps the states invariant under the centralizer subgroup Cg of g. We will denote this Cg invariant subspace by HCgg. Thus the total orbifold Hilbert space takes the form

H(SNM) =!

[g]

HCgg. (2.1)

For the symmetric group, the conjugacy classes [g] are characterized by partitions {Nn} of N

"

n

nNn = N, (2.2)

where Nn denotes the multiplicity of the cyclic permutation (n) of n elements in the decomposition of g

[g] = (1)N1(2)N2. . . (s)Ns. (2.3)

4

Elliptic Genus of Symmetric Product : S

N

X = X

N

/S

N

[Dijkgraaf-Moore-Verlinde2 ‘97]

N

p

N

Z(τ, z; S

N

X) = 2nd quantized string partition function on X × S

1

= �

n,m,�

1

(1 − p

n

q

m

y

)

c(4nm−�2)

Counting susy ground states of D1-D5 string on K3XS

1

with N=Q1Q5+1

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III. K3 Compactifications (ii) : type II on K3xT 2

the spacetime, non-perturbative perspective

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III. K3 Compactification (ii)

het/T

6

∼ = II/K3 × T

2

(with Z

N

-orbifold) N = 4, d = 4 String Theory

massless: 1 × gravity- + n × vector-multiplet vectors: Q, P ∈ Γ

6,n

scalars: SL(2)

U (1) × SO(6, n)

SO(6) × SO(n)

axion-dilaton λ T -moduli: Q, P → QR, PR

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duality : U = S × T

inv.: Q·Q, Q·P, P ·P

�Q P

�a b c d

� �Q P

, λ → aλ + b

cλ + d , N|c

S-duality=Rotation SL(2, Z) ∼ SO(2, 1; Z) Z

2

Parity=Reflection GL(2, Z) ∼ O(2, 1; Z)

Built-in R

2,1

:

X =

�x+ x x x

, |X|2 = detX = det(γXγT ) , γ =

�a b c d

III. K3 Compactification (ii)

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x+ x

Moduli

x

Charges

Λ

Q,P

=

� Q · Q Q · P Q · P P · P

S B-H = π|Λ

Q,P

|

T = 1

QR,PR

| Λ

QR,PR

, S = 1 Imλ

� |λ|

2

Reλ Reλ 1

M

Q,P

= |Z|

Z = �

QR,PR

|(T + S)

S-duality : X → γXγ

T

Λ

Q,P

Z

Attractor Point : Z|

att.

∼ Λ

Q,P

III. K3 Compactification (ii)

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x+ x

Moduli

x

Charges

Λ

Q,P

=

� Q · Q Q · P Q · P P · P

S B-H = π|Λ

Q,P

|

T = 1

QR,PR

| Λ

QR,PR

, S = 1 Imλ

� |λ|

2

Reλ Reλ 1

M

Q,P

= |Z|

Z = �

QR,PR

|(T + S)

S-duality : X → γXγ

T

Λ

Q,P

Z

Attractor Point : Z|

att.

∼ Λ

Q,P

III. K3 Compactification (ii)

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x+ x

Moduli

x

Charges

Λ

Q,P

=

� Q · Q Q · P Q · P P · P

S B-H = π|Λ

Q,P

|

T = 1

QR,PR

| Λ

QR,PR

, S = 1 Imλ

� |λ|

2

Reλ Reλ 1

M

Q,P

= |Z|

Z = �

QR,PR

|(T + S)

S-duality : X → γXγ

T

Λ

Q,P

Z

Attractor Point : Z|

att.

∼ Λ

Q,P

III. K3 Compactification (ii)

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Microscopic Spectrum: the1/2-BPS States

States preserving 1/2 of supersymmetries have S

B−H

= π |Λ

Q,P

| = 0

S-dual to purely electric states with P=0

and dual to perturbative heterotic string states

with the supersymmetric side on its ground states

ground states counted by the bosonic string partition function

n=−1

d

n

q

n

= 1 q �

k=1

(1 − q

k

)

24

= 1

η

24

(τ ) D(Q, 0) = d

Q2/2

(on the light-cone)

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Microscopic Spectrum: the1/4-BPS States

States preserving 1/4 of supersymmetries have (inside the light-cone) S

B−H

= π |Λ

Q,P

| > 0

Can be realized as D1-D5-P-Taub-NUT bound states in type IIB frame, with partition function

1

Φ(Ω) = 1 pqy

n,m,�

1

(1 − p

n

q

m

y

)

c(4nm−�2)

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SL(2, Z) ∼ Sp(1, Z)

modular form of weight w

τ, Imτ > 0

(upper-half plane)

f (aτ + b

cτ + d) = (cτ + d)wf (τ ) ad − bc = 1

Ω =

�ρ ν ν σ

, ImΩ ∈ V + (Siegel upper-half plane)

Sp(2, Z)

Siegel modular form of weight w

Φ( AΩ + B

CΩ + D ) = (CΩ + D)wΦ(Ω)

AD

T

− BC

T

= 1 (−1)

Q·P

D(Q, P ) =

dΩ e

iπ(ΛQ,P ,Ω)

Φ(Ω)

Proposed:

Dijkgraaf, Verlinde, Verlinde ’96 Jaktar, Sen ’05

Derived:

Shih, Strominger, Yin ‘05 David, Sen ’06

Challenged:

Dabholkar, Gaiotto, Nampuri ’07 Sen ’07

Completed:

M.C., Verlinde ’07 Re-Derived:

Banerjee, Sen, Srivastava ’08 M.C., L. Hollands ‘09

Re-Interpreted:

M.C., Verlinde ’08, M.C., Dabholar ’08

d(Q) =

dτ e−πiQ2 f (τ )

III. K3 Compactification (ii)

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IV. A Group of

Crossing the Walls

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IV. A Group of Crossing the Walls

MQ,P ≤ MQ1,P1 + MQ2,P2 (always true)

MQ,P = MQ1,P1 + MQ2,P2 (W.M.S. condition)

C ZQ1,P1

ZQ2,P2

x+ x

x

N = 4 : co-linearity

Z

ΛQ1,P1 ΛQ2,P2

R2,1

N = 2 : central charge aligned

Study of Wall-Crossing Idea: W.M.S =>

Discrete Moduli Dependence of BPS Spectrum

(Q1, P1), (Q2, P2) : 1/2-BPS charges

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IV. A Group of Crossing the Walls

MQ,P ≤ MQ1,P1 + MQ2,P2 (always true)

MQ,P = MQ1,P1 + MQ2,P2 (W.M.S. condition)

C ZQ1,P1

ZQ2,P2

x+ x

x

N = 4 : co-linearity

Z

ΛQ1,P1 ΛQ2,P2

R2,1

N = 2 : central charge aligned

Study of Wall-Crossing Idea: W.M.S =>

Discrete Moduli Dependence of BPS Spectrum

(Q1, P1), (Q2, P2) : 1/2-BPS charges

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IV. A Group of Crossing the Walls

MQ,P ≤ MQ1,P1 + MQ2,P2 (always true)

MQ,P = MQ1,P1 + MQ2,P2 (W.M.S. condition)

C ZQ1,P1

ZQ2,P2

x+ x

x

N = 4 : co-linearity

Z

ΛQ1,P1 ΛQ2,P2

R2,1

N = 2 : central charge aligned

Study of Wall-Crossing Idea: W.M.S =>

Discrete Moduli Dependence of BPS Spectrum

(Q1, P1), (Q2, P2) : 1/2-BPS charges

(34)

IV. A Group of Crossing the Walls

MQ,P ≤ MQ1,P1 + MQ2,P2 (always true)

MQ,P = MQ1,P1 + MQ2,P2 (W.M.S. condition)

C ZQ1,P1

ZQ2,P2

x+ x

x

N = 4 : co-linearity

Z

ΛQ1,P1 ΛQ2,P2

R2,1

N = 2 : central charge aligned

Study of Wall-Crossing Idea: W.M.S =>

Discrete Moduli Dependence of BPS Spectrum

(Q1, P1), (Q2, P2) : 1/2-BPS charges

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The Group of Attractor Flow Idea:

W.M.S.

=> Partition of Moduli Space

Compartment~ Fund. Domain of a Reflection Group W

=> W= Group of Wall-Crossing

Ordering of elements of W ~ Streamlines of Attr. Flow => W= Group of (Discrete) Attr. Flow

III. A Group of Crossing the Walls

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project onto

III. A Group of Crossing the Walls

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Z2

Z3 ΛQ,PW

Z ∈ w(W)

III. A Group of Crossing the Walls

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Z2

Z3 ΛQ,PW

Z ∈ w(W)

III. A Group of Crossing the Walls

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Z2

Z3 ΛQ,PW

Z ∈ w(W)

III. A Group of Crossing the Walls

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Z2

Z3 ΛQ,PW

Z ∈ w(W)

III. A Group of Crossing the Walls

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Z2

Z3 ΛQ,PW

Z ∈ w(W)

III. A Group of Crossing the Walls

(42)

Z2

Z3 ΛQ,PW

Z ∈ w(W)

III. A Group of Crossing the Walls

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Properties of W

• Microscopic: Weyl Group of the Algebra of the BPS Spectrum

• Gives a discrete metric on moduli space

• Macroscopic: Group of Attr. Flow

• Use: Catalogue of BPS states @ All Moduli

• S-duality = W � Sym (W)

e.g. GL(2, Z) ∼ {group generated by 3 reflections} � Sym( )

for finite Lie algebras, W = {reflections w.r.t roots}

e.g. su(2)

roots = {α, −α}

W = Z2

III. A Group of Crossing the Walls

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IV. A Generalised

Kac-Moody Symmetry

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Generalised Kac-Moody Algebras as the Algebra for Physical States

Idea:

The Denominator for the GKM’s

= Generating Function for the Physical Degeneracies

=> The algebra generates the BPS Spectrum

In particular, the Weyl Groups = Groups of Wall-Crossing

=> Deep understanding of the structure of N=4 Quantum Gravity

This is our conjecture/proposal. It has passed all tests so far in different physically interesting limit.

IV. A GKM Symmetry

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Checks

Dualities

• BH entropy (leading and corrections)

• Twisted Quantum Entropy Function

• Wall-Crossing Formula from Supergravity

• Gauge Theory Limit

IV. A GKM Symmetry

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Generalised-Kac-Moody Superalgebra

• non-positive definite Cartan matrices (K-M)

• imaginary simple roots in (the closure of) the light-cone (G-)

• bosonic and fermionic (super-)

• denominator formulas often given by automorphic forms

IV. A GKM Symmetry

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denominator identity:

root multiplicity <-> imaginary simple roots

Weyl vector � : (α, �) = −1

2(α, α) for all simple real roots α

e(�) �

α∈∆+

(1 − e(α) )multα = �

w∈W

det(w) w(e(�)Ξ)

Ξ = Σα∈∆ims M (α)e(α) graded degeneracy of simple imaginary root α

=

�M (α) (α, α) < 0 M (α)˜ (α, α) = 0

k>0

(1 − qk)M (kα)˜ = 1 − �

n>0

M (nα)qn

IV. A GKM Symmetry

Here, the root multiplicity multα is given by

the Fourier coefficients of the elliptic genus ZK3(τ, z) of K3

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Full Microscopic Degeneracy Formula

e(−2�)

α∈∆+(1 − e(α) )2 multα = �

w∈W, Λ∈W

D(w(Λ)) e(w(Λ))

D(Q, P )|Z= D(w−1Q,P )) , Z ∈ w(W)

IV. A GKM Symmetry

1 pqy �

(n,m,�)>0(1 − pnqmy)c(4nm−�2) = �

N,M,L

pN qMyLD(N, M, L)

(50)

Full Microscopic Degeneracy Formula

vacuum

free oscillators moduli dependence W= group of wall-crossing

e(−2�)

α∈∆+(1 − e(α) )2 multα = �

w∈W, Λ∈W

D(w(Λ)) e(w(Λ))

D(Q, P )|Z= D(w−1Q,P )) , Z ∈ w(W)

IV. A GKM Symmetry

1 pqy �

(n,m,�)>0(1 − pnqmy)c(4nm−�2) = �

N,M,L

pN qMyLD(N, M, L)

(51)

Full Microscopic Degeneracy Formula

e(−2�)

α∈∆+(1 − e(α) )2 multα = �

w∈W, Λ∈W

D(w(Λ)) e(w(Λ))

D(Q, P )|Z= D(w−1Q,P )) , Z ∈ w(W)

Equivalently, we have the map V + × V + → wt spc

˜Λ(Λ, Z) = ρ + 1

2w−1(Λ)

and D(Λ)|Z can be obtained by considering Verma module w. highest wt ˜Λ

IV. A GKM Symmetry

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Attractor Flow and Hierarchy of Stability

Moduli Z ∈ wk(W) w = wn → wn−1 →· · · → 1 Oscillation Level wn−1(Λ) > wn−1−1(Λ) >· · · > Λ

Verma Module M(˜Λ

wn) ⊃ M(˜Λ

wn−1) ⊃· · · ⊃ M(˜Λ

1)

dominant weight module

IV. A GKM Symmetry

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Summary so far

We started with CFT on K3 (the perturbative picture) and considered the elliptic genus of K3.

Using its “lift” to string theory on K3xS

1

and string

duality, we obtain the non-perturbative partition function for 1/4 BPS states in the N=4, d=4 theory obtained from compactifying type II string on K3xT

2

.

The orbifolded theories can also be considered.

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Summary so far

Using supergravity analysis, the wall-crossing of the 1/4 BPS indices have been studied and a group

structure was found to underlie the moduli dependence of the spectra.

Surprisingly, the 1/4 BPS spectrum exhibits a GKM

algebra symmetry, whose chamber structure encodes the wall-crossing structure.

In Part II we will see how all of the above structures

appear to be tied to a sporadic group M

24

!

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Thank you!

謝謝!

Figure

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