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Wall Crossing and an Entropy Enigma

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Wall Crossing and an Entropy Enigma. Work done with Frederik Denef. hep-th/0702146 ... Single-centered black holes always cross the wall. ... – PowerPoint PPT presentation

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Title: Wall Crossing and an Entropy Enigma


1
Wall Crossing and an Entropy Enigma
Strings 2007, Madrid, June 28
  • Work done with Frederik Denef
  • hep-th/0702146
  • arXiv0705.2564

and with Emanuel Diaconescu arXiv0706.3193
TexPoint fonts used in EMF AAAAAAAAAAAA
2
Related References

3
Outline
  • 1. Setting the Scene
  • 2. Wall-Crossing Formula
  • 3. Mathematical Applications
  • 4. D6-D2-D0 Partition Functions
  • 5. Derivation of OSV
  • 6. The Entropy Enigma
  • 7. A Nonperturbative ?
  • 8. A Concluding Riddle

4
Setting the Scene

Setting Type IIA strings on a compact Calabi-Yau
3-fold X. BPS States in the d4, N2 SUGRA are
wrapped
D6-D4-D2-D0 branes in X
They carry RR charge
But existence of states also depends on boundary
conditions on VM scalars
5
Space of BPS States
Is finite dimensional and depends on
(Seiberg Witten)
These state spaces are interesting!
Accounts for black hole entropy (Strominger
Vafa Maldacena, Strominger, Witten)
Plays a key role in the OSV conjecture
Plays a key role in quantum corrections to
effective Sugra
Possibly carry algebraic structures generalizing
GKM (Harvey Moore)
Possibly categorify knot invariants (Gukov, Vafa,
et. al. )
6
N2 Central Charges
Duality invariant product of electricmagnetic
charges
7
Wall-Crossing Formula
Marginal Stability Wall
tms
t
t-
8
Remarks I
9
Remarks - II
D-brane moduli space changes topology by
blowing-down and blowing-up projective spaces.
Macroscopic 2-centered solutions
Microscopic Quiver Quantum Mechanics
10
Macro-proof Denef Multicenter Solutions
Micro-Proof Quiver Quantum Mechanics (Denef,
QQHHH)
11
Nonprimitive Wall-Crossing

Denefs Halos
FOCK SPACE
Gives products such as McMahon, similar to DT, GV
infinite products
12
Mathematical Tests/Applications
(with Emanuel Diaconescu)
D4 wraps rigid surface
with holomorphic bundle
Change of J will induce decay
Hodge Polynomial of
Reproduces nontrivial results of Gottsche,
Yoshioka
Makes new math predictions
Moduli of D4 branes is NOT the moduli of coherent
sheaves!

13
D6-D2-D0 PartitionFunctions
Stable
UNSTABLE
z
Marginal Stability Wall
14
Relation to Donaldson-Thomas
15
Core of the Matter
z
ONLY CORE STATES
HALOS CORES
HALOS CORES
16
OSV
Still has wall-crossing as a function of J !!
Diaconescu-Moore
Denef-Van den Bleeken
17
A Refined OSV Formula
Has no wall-crossing. In fact is B-independent
18
Polar States
For D4-D2-D0 Charge
Define
(slightly lying here)
Then
is modular
U-duality implies
Black hole degeneracies (Fareytail story)
19
Microscopic Polar States
X
S
20
Macroscopic Polar States
Attractor formula
Polar states are realized as Denefs
multi-centered solutions
Extreme Polar States
D6
21
Sketch of Derivation of OSV
22
Limitations on the Derivation
The derivation crucially depends on using only
the extreme polar states
the derivation is only valid at LARGE coupling
These states can have large entropy!
23
Entropy Enigma
24
Magical Cancellations?
Contradicts OSV
and even black hole dominance of the
asymptotic degeneracy of states!
Closely related question
  • k3 indicates the entropy enigma, k2 suggests
    there are magical cancellations
  • Huang,Klemm,Marino,Tavanfar find tentative
    evidence for k2, and not k3!
  • The issue is open and important.

25
Degeneracy Dichotomy
Either there are no magical cancellations, and we
have the entropy enigma,
or, there, are magical cancellations. In that
case we must worry about
  • Physically, the dimension determines the entropy.
  • All successful microstate entropy computations
    have used the index.
  • We expect in the full theory dimensionindex.


landscape of metastable states
26
Remark on the nonperturbative topological string
  • One interesting point of OSV was the promise of a
    nonperturbative
  • definition of the topological string.
  • The Dondaldson-Thomas product formula natrually
    splits as a spin
  • zero and positive spin factor

27


28
Concluding Riddle
Why did the BPS state cross the wall?
  • We want to understand black hole entropy
  • We found lots of irrelevant stuff Halos,
    Multi-Centered
  • Core states, Swing States,
  • Single-centered black holes always cross the
    wall.
  • Multi-centered solutions might or might not but
    the
  • multicentered scaling solutions which cross
    the wall
  • have macroscopic entropy.)
  • So we really want to count the states which
    cross the wall
  • So we need an answer to our riddle at the
    microscopic level!!

29
Related References
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