Non-Supersymmetric String Theory Savdeep Sethi University of Chicago Great Lakes 2008
6/19/2008
Outline I
Motivation
II
A Pair of Non-Supersymmetric Backgrounds
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hep-th/0804.XXXX, Emil Martinec, Daniel Robbins & S. S. Work in progress! Motivated by the following papers: hep-th/0603104, Daniel Robbins, Emil Martinec + S. S. hep-th/0601062, Ben Craps, Arvind Rajaraman + S. S.
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I Motivation All known consistent theories of quantum gravity are supersymmetric at the Planck scale.
How then is supersymmetry broken?
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The standard picture of SUSY breaking proceeds as follows: Compactify to D=4 on some space perhaps with fluxes. Typically preserve N=1 SUSY and then study SUSY breaking in effective field theory.
Alternatively, one can try to consider non-SUSY metastable compactifications. In all cases, quantum gravity (and cosmology) is decoupled from SUSY breaking. 5
This is typically a semi-classical two step process. 1. Find a static solution of string theory. 2. Break SUSY using the resulting four-dimensional effective theory.
When is this valid?
The Universe is not static!
Neglecting the physics of singularities near the beginning or end of time. Essentially neglecting cosmology.
Upshot: we will find indications that this is not a valid procedure and that quantum gravity does not decouple from SUSY breaking. 6
What we will see suggests that we need the entire history of a string solution to determine whether it is a solution.
It is likely that some compactifications are sensible and others are not but you cannot tell from semi-classical considerations. Supersymmetric and non-supersymmetric string theory are quite different regardless of how low the SUSY breaking scale might appear to be.
correlation
graviton dynamics
SUSY breaking
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II A Pair of Non-Supersymmetric Backgrounds To date, there is an extremely poor understanding of non-supersymmetric string theory. We need some simple examples where we can try to understand the physics. This includes a better understanding of tachyon condensation.
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Let us consider type IIA string theory on an orbifold
L
When L is large, the effects of SUSY breaking are suppressed. (Scherk-Schwarz; Rohm) 9
All space-time fermions get mass from the anti-periodic boundary conditions on the circle. String modes on the circle are characterized by winding and momentum: (n,w) Even winding w – conventional GSO
(NS+, NS+)
(R+, NS+)
(NS+, R-)
(R+, R-)
(NS-, R+)
(R-, R+)
Odd winding – modified GSO (NS-, NS-)
(R-, NS-)
tachyon!
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Level matching (including zero point energies): Bosons
NL = NR + nw
Femions
NL = NR + (n – ½) w
Spacetime bosons: m2 = n2 /L2 + w2 L2 / (®’)2 + 2(NL + NR)/®’ Spacetime fermions: m2 = (n-1/2)2 /L2 + w2 L2 / (®’)2 + 2(NL + NR)/®’
(NS-, NS-) sector with odd winding then NL = NR = -1/2, n=0 (®’)2 m2 = (2w+1)2 L2 – 2.
No tachyons for L2 > 2 ®’. 11
Lastly note that as L ! 0, all space-time fermions decouple and we are left with the sectors:
(NS+, NS+)
(NS-, NS-)
(R+, R-)
(R-, R+)
This is type 0A string theory.
This twisted circle compactification is a good static tree level solution of string theory with controlled SUSY breaking. Let us call this the A-twist. At 1 string loop, a potential is generated for the dilaton and the radius L. The solution becomes cosmological driving L ! 0 where tachyons develop. 12
This is the first background. The second background is simply the T-dual of this one!
We will call this the B-twist. It is a quotient that acts on winding modes. n
w
So in the T-dual theory there is a momentum mode tachyon when the circle L’ becomes too large! No tachyons when (L’ )2 < ®’/2. These theories are identical to all orders in string perturbation theory since the CFTs are identical.
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Non-perturbatively? How do we define and understand these backgrounds.
We will use the DLCQ approach to find a non-perturbative Matrix Model description. X- ~ X- + 2¼ R
P+ = N/R
Relate this light-like compactification to a space-like one in a decoupling limit. The N units of momentum become D-branes.
Let us recall how T-duality worked in the holographic description.
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Consider type IIA string theory in the DLCQ description. This is governed by the theory of N D1-strings on a world-volume circle of size ®’.
Matrix Strings
(DVV, Motl, …)
N D1-strings on ¾3
String parameters (®’, gs)
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If we compactify IIA string on T2 we expect to see T-duality in the SUSY case. Let the sides of the torus be (L1, L2). Each circle compactification promotes the Dp-brane to a D(p+1)-brane. So we consider D3-branes on a T^3 with sides of length (¾1, ¾2, ¾3) and coupling constant:
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If we now send L1 ! ®’/L1 and L2 ! ®’/L2 we expect to recover the same physics. Repeating the parameter map gives SYM compactifed on a T3 with parameters:
T-duality ) S-duality
(Susskind; Ganor, Ramgoolam, Taylor) 17
We certainly have no reason to expect S-duality to survive in the non-SUSY case!
So let’s turn to the A and B twists of type IIA on a circle.
The A twist was studied by Banks and Motl. In the decoupling limit one is considering IIA string with N D0-branes on a very small T2 of sides ¾3 (setting the string scale) and ¾1 (the twisted circle).
This is studying string theory in a highly tachyonic regime! 18
The Matrix Model is a kind of quiver gauge theory with gauge group U(N) £ U(N) and fermions in the (N, N*) and (N*,N) reps. This is a non-SUSY theory with no supersymmetry in the ultraviolet spectrum. Classically, one recovers long strings with the correct behavior to describe the A-twist.
At two-loops, however, there is a UV divergence that generates a confining potential. Regardless of the regularization scheme, there appear to be no approximate flat directions. V(b) ~ |b|
where b is the graviton separation.
Flat directions are critical for a picture of semi-classical gravity. 19
We can interpret this result many ways: 1. Failure of decoupling? 2. Non-existence of the A-twisted background?
Whichever interpretation you choose, the B-twist has strikingly different physics.
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For the B-twist, we follow exactly the same procedure. However, after boosting we are considering IIA string theory with N D0-branes on a very small T2. This is now far from the tachyonic regime!
What we find is maximal SYM on a T2 with sides ¾3 (setting the IIA string scale), ¾1 (the twisted circle). After the T-duality taking us from D0-branes to D2-branes, the B-twist is replaced by the A-twist.
So we have anti-periodic B.C. for fermions on the ¾1 circle. This is like a thermal compactification of YM. 21
The low-energy limit of D2-branes
¾1 twisted b.c.
¾3 (string scale)
Compactifying further would add dimensions and supersymmetric circles.
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Is there ever a regime that looks like semi-classical gravity? For this we require approximate flat directions. So let us try to compute the potential in this theory. First note that the breaking is an IR effect so all the good UV properties of the theory are preserved.
Coulomb branch:
Tr( ©2) » b2 b
[©i, ©j]=0
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The effective potential is computed by summing the 1-loop vacuum energies.
Exponential suppression of UV physics and of (b, ¾1) dependence.
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This is potential between gravitons which should correspond to some cosmology which we are attempting to unravel.
It is computed in the weak coupling regime and we would like to understand what happens in the Matrix String regime at strong coupling.
Let us turn to an even simpler case.
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Consider N D1-strings on a circle with twisted B.C. This corresponds to M-theory on a strange orbifold that acts on membrane wrapping modes. Nevertheless, it should define a non-supersymmetric string theory in ten dimension.
Why?
Energy
gym
(Matrix String regime)
1/¾3
(SUSY breaking scale)
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So we expect a matrix string picture but the it turns out that the light-cone GS fermions have anti-periodic B.C. on the ¾1 circle. This corresponds to type OA string theory. We propose this theory as the non-perturbative definition of type 0A which always has a closed string tachyon. This is a different definition in M-theory than the proposal of Bergman and Gaberdiel which involves M-theory on the A-twisted circle.
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Similarly: D2-branes on T^2 in the IR should define non-SUSY type IIB strings.
We can twist along one or both directions. ¾2
One should correspond to standard type 0B.
¾1
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Lots of open puzzling issues:
Where is the time-dependence? Is there a good large N limit? Can we recover string perturbation theory?
Does this define closed string tachyon condensation? Can we correlate SUSY breaking with cosmological observations? etc.
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