BPS Dyons, Wall Crossing and Borcherds Algebra
Transcript of BPS Dyons, Wall Crossing and Borcherds Algebra
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BPS Dyons, Wall Crossing and Borcherds Algebra
Erik Verlinde
University of Amsterdam
Simons Workshop on Geometry and Physics, 2008
Based on work with Miranda Cheng
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Outline
• N=4 Dyonic Black Holes and Walls of Marginal Stability
• Hyperbolic Kac Moody Algebra and its Weyl Group
• Discrete Attractor Flow and Arithmetic Decay
• Microscopic Description: Borcherds Algebra.
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Heterotic Strings on TMomentum and winding form an even Lorentzian lattice
Given a point on the moduli space
one can write
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n
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Heterotic Strings on TThe leftmoving sector contains 16 supersymmetry charges.
8 of these annihilate the left-moving ground states.
Such states are called 1/2 BPS states. Their mass equals
Note that
Equality holds when , this is an attractor point.
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n
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Heterotic Strings on TMomentum and winding correspond to electric charges
Left ground states: 1/2 BPS states, counting of rightmovers
with
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BPS Dyons for Heterotic String on TElectric and magnetic charge
1/4 BPS states obey
Z = largest eigenvalue of the 4x4 anti-symm. matrix
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6
P,Q
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Dyonic Black HolesEntropy
S-duality
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S-duality + Parity: SL(2,Z) => PGL(2,Z)
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Attractor Flow
At the horizon the moduli reach fixed point.
Flow of Narain moduli
Flow of axion dilaton
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Walls of Marginal StabilityIn general one has
Hence,
At the wall of marginal stability
=> real co-dimension 1 subspace of moduli space.
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P
Q
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Walls of Marginal StabilityFrom the central charge matrix it follows that at
a 1/4 BPS state may decay in a pair of 1/2 BPS states
Supergravity analysis: bound state is stable when
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Walls of Marginal StabilityOther walls are obtained by using that the matrix
transforms under PGL(2,Z) as
this leads to
for the decay
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The Walls
Let us introduce the space of symmetric matrices
with norm
The locations of the walls can be written as
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Walls and Weyl Chambers
The basis
obeys
= Cartan matrix of a Generalized Kac Moody algebra
α1
α2α3
α1
α2 α3 α1α2
α3
Dihedral group of outer automorphisms
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∞
∞∞
Weyl Group and ChambersCoxeter diagram
Weyl chambers can be visualized on Poincare disk
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Walls and Weyl Chambers
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Matrix of T-duality invariants => integral weight vector
Attractor flow
BPS counting jumps at walls
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Discrete Attractor flow
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Arithmetic DecayDecay labeled bypair of fractions
(Farey series)
s1
s1
s2
s2
s3
s3
−10
−21
−11
−12
01
13
12
23
11
32
21
31
10
−10
−11
01
12
11
21
10
−10
01
11
10
1
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Arithmetic DecayDecay labeled by
decomposition along root
Degeneracy should jump byα−
α
α+
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Weyl-Kac-Borcherds Formula
In addition to real roots there are null and imaginary roots.
The product
with c determined by the elliptic genus of K3 is a
Sp(2,Z)-modular form of weight k =10.
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The Counting Formula The # of dyonic BPS states with charge (P,Q) equals
Reproduces entropy
and has double poles
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(DVV, 1996)
P Q
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Walls in the Space of ContoursHence, the counting formula is contour dependent!
The contours are determined by the moduli
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Poles coincide with walls of
Weyl Chambers
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The Wall Crossing Formula
The contribution at the pole at v =0 is given by
which can be further evaluated to give
This describes exactly the contribution due to the decay of 1/2 BPS bound states!
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Towards Microscopic Description
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Defining relations of Generalized Kac Moody algebra
No representations are known.
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Towards Microscopic Description
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Weyl-Kac-Borcherds formula counts the degeneracy of highest weight modules
The Weyl group acts on the highest weight
One can show that for a dominant weight
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Conclusions
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• The occurence of a Hyperbolic KM Algebra understood from supergravity perspective (cf. Cosmic Billiards).
• Wall crossings are related to Weyl reflections.
• Microscopic counting in terms of Borcherds Formula incorporates wall crossing through moduli dept. contour.
• Can one obtain representation of Borcherds Algebra?
• Counting is more elaborate for non-primitive charges.
• Can one generalize this to all N=4 theories? Or N=2?
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