Dualities & Supersymmetry Breaking

I˜nakiGarc´ıa-Etxebarria

10th March, 2014 A bit of a pickle for string theory: much of the work so far has been on (mostly) supersymmetric string theories (particularly on the formal side, but pheno is also often build on mostly susy backgrounds). Perhaps a good opportunity to revisit some old, and not so old, ideas on non-supersymmetric string backgrounds.

Introduction S-duality Tachyons USp(32) Conclusions

Motivation

Despite good theoretical reasons to assume the existence of supersymmetry, we do not have any direct experimental evidence for it. Perhaps a good opportunity to revisit some old, and not so old, ideas on non-supersymmetric string backgrounds.

Introduction S-duality Tachyons USp(32) Conclusions

Motivation

Despite good theoretical reasons to assume the existence of supersymmetry, we do not have any direct experimental evidence for it. A bit of a pickle for string theory: much of the work so far has been on (mostly) supersymmetric string theories (particularly on the formal side, but pheno is also often build on mostly susy backgrounds). Perhaps a good opportunity to revisit some old, and not so old, ideas on non-supersymmetric string backgrounds. All aspects of a unified theory, so things look good... but plethora of vacua, so finding our vacuum is difficult.

No similar classiﬁcation for non-supersymmetric theories. How many higher dimensional string theories? Dualities between them? How many vacua?

Introduction S-duality Tachyons USp(32) Conclusions

Classiﬁcation

In 10d we have 5 susy string theories (6? [Sethi: 1304.1551]). but plethora of vacua, so ﬁnding our vacuum is diﬃcult.

No similar classiﬁcation for non-supersymmetric theories. How many higher dimensional string theories? Dualities between them? How many vacua?

Introduction S-duality Tachyons USp(32) Conclusions

Classiﬁcation

In 10d we have 5 susy string theories (6? [Sethi: 1304.1551]).

All aspects of a uniﬁed theory, so things look good... No similar classiﬁcation for non-supersymmetric theories. How many higher dimensional string theories? Dualities between them? How many vacua?

Introduction S-duality Tachyons USp(32) Conclusions

Classiﬁcation

In 10d we have 5 susy string theories (6? [Sethi: 1304.1551]).

All aspects of a unified theory, so things look good... but plethora of vacua, so finding our vacuum is difficult. Introduction S-duality Tachyons USp(32) Conclusions

Classiﬁcation

In 10d we have 5 susy string theories (6? [Sethi: 1304.1551]).

All aspects of a unified theory, so things look good... but plethora of vacua, so finding our vacuum is difficult.

No similar classiﬁcation for non-supersymmetric theories. How many higher dimensional string theories? Dualities between them? How many vacua? This classiﬁcation typically is not physical, T-duality can mix 1 ↔ 2 and the 2 ↔ 3 distinction can be an artifact of weak coupling.

Introduction S-duality Tachyons USp(32) Conclusions

Non-supersymmetric string theories

One can imagine diﬀerent classes of non-susy string backgrounds 1. Defect susy breaking: a susy bulk, with (mutually) non-susy defects. 2. Bulk susy breaking: non-susy bulk, but built from susy ingredients which are non-susy when taken together. 3. Intrinsically non-supersymmetric: no susy in sight in the bulk. All of these come in stable and unstable ﬂavors. Introduction S-duality Tachyons USp(32) Conclusions

Non-supersymmetric string theories

One can imagine different classes of non-susy string backgrounds 1. Defect susy breaking: a susy bulk, with (mutually) non-susy defects. 2. Bulk susy breaking: non-susy bulk, but built from susy ingredients which are non-susy when taken together. 3. Intrinsically non-supersymmetric: no susy in sight in the bulk. All of these come in stable and unstable flavors. This classification typically is not physical, T-duality can mix 1 ↔ 2 and the 2 ↔ 3 distinction can be an artifact of weak coupling. Introduction S-duality Tachyons USp(32) Conclusions

Outline

1 Introduction

2 Strong/weak duality for realistic 4d theories

3 Tachyon condensation and stable vacua in non-susy theories

4 Into the wilderness

5 Conclusions Introduction S-duality Tachyons USp(32) Conclusions

Outline

1 Introduction

2 Strong/weak duality for realistic 4d theories Montonen-Olive duality A new class of 4d dualities with N = 1 N = 0 dualities

3 Tachyon condensation and stable vacua in non-susy theories

4 Into the wilderness

5 Conclusions Introduction S-duality Tachyons USp(32) Conclusions

Montonen-Olive N = 4 duality

Given a 4d N = 4 ﬁeld theory with gauge group G and gauge coupling τ = θ + i/g2, there is a completely equivalent description with gauge group G∨ and coupling −1/τ (for θ = 0 this is g ↔ 1/g). Examples:

G G∨ U(1) U(1) U(N) U(N) SU(N) SU(N)/ZN SO(2N + 1) Sp(2N)

Very non-perturbative duality, exchanges gauge bosons with monopoles! (So, the usual field theory tools are not particularly illuminating here.) For example, N = 4 U(N) theory is the low energy description of N D3s on flat space. Using the duality dictionary, one gets ∨ 2 U(N) = U(N).(gYM = gs) More interestingly, SO(2N + 1) is the low energy theory for 2N D3s on top of a Og3−. Applying the duality dictionary, this is 2N D3s on top of a O3+, which at low energies gives SO(2N + 1)∨ = Sp(2N). Beautiful field theory insights follow trivially from the duality dictionary. For example, the gauge boson ! monopole map follows easily from the F 1 ! D1 duality dictionary entry.

Introduction S-duality Tachyons USp(32) Conclusions

Montonen-Olive duality from string theory

Just “engineer” the ﬁeld theory one wants in string theory, and apply the IIB S-duality dictionary to the construction. More interestingly, SO(2N + 1) is the low energy theory for 2N D3s on top of a Og3−. Applying the duality dictionary, this is 2N D3s on top of a O3+, which at low energies gives SO(2N + 1)∨ = Sp(2N). Beautiful ﬁeld theory insights follow trivially from the duality dictionary. For example, the gauge boson ! monopole map follows easily from the F 1 ! D1 duality dictionary entry.

Introduction S-duality Tachyons USp(32) Conclusions

Montonen-Olive duality from string theory

Introduction S-duality Tachyons USp(32) Conclusions

Montonen-Olive duality from string theory

Just “engineer” the ﬁeld theory one wants in string theory, and apply the IIB S-duality dictionary to the construction. For example, N = 4 U(N) theory is the low energy description of N D3s on ﬂat space. Using the duality dictionary, one gets ∨ 2 U(N) = U(N).(gYM = gs) More interestingly, SO(2N + 1) is the low energy theory for 2N D3s on top of a Og3−. Applying the duality dictionary, this is 2N D3s on top of a O3+, which at low energies gives SO(2N + 1)∨ = Sp(2N). Introduction S-duality Tachyons USp(32) Conclusions

Montonen-Olive duality from string theory

Just “engineer” the field theory one wants in string theory, and apply the IIB S-duality dictionary to the construction. For example, N = 4 U(N) theory is the low energy description of N D3s on flat space. Using the duality dictionary, one gets ∨ 2 U(N) = U(N).(gYM = gs) More interestingly, SO(2N + 1) is the low energy theory for 2N D3s on top of a Og3−. Applying the duality dictionary, this is 2N D3s on top of a O3+, which at low energies gives SO(2N + 1)∨ = Sp(2N). Beautiful field theory insights follow trivially from the duality dictionary. For example, the gauge boson ! monopole map follows easily from the F 1 ! D1 duality dictionary entry. Introduction S-duality Tachyons USp(32) Conclusions

Montonen-Olive duality from string theory There are four versions of the O3 plane in string theory, distinguished by discrete RR and NSNS 2-form ﬂuxes B2,C2 in the transverse space: [Witten:hep-th/9805112]

3 5 H (S /Z2, Ze) = Z2 .

(0, 0) : O3− + ND3s −→ SO(2N)

(0, 1) : Og3− + ND3s −→ SO(2N + 1) (1, 0) : O3+ + ND3s −→ USp(2N)

(1, 1) : Og3+ + ND3s −→ USp(2N)

IIB SL(2, Z) exchanges the conﬁgurations. Under S-duality

Og3− ←→ O3+ : SO(2N + 1) ←→ USp(2N) Introduction S-duality Tachyons USp(32) Conclusions

Discrete torsion argument for N = 4

The charge of the system is classified by the cohomology on the 5 5 S /Z2 = RP that surrounds the configuration. For fields even under the orientifold action, we have:

• 5 H (RP , Z) = {Z, 0, Z2, 0, Z2, Z} ,

while for ﬁelds odd under the orientifold action:

• 5 H (RP , Ze) = {0, Z2, 0, Z2, 0, Z2} .

This is (co)homology with local coefficients. Working on the S5 covering space k ⊗ C ' γk ⊗ γC. For coefficients in Z we have γk = k while for coefficients in Ze we have γk = −k. Ordinary (co)homology theory otherwise: H• = ker ∂/ im ∂. Introduction S-duality Tachyons USp(32) Conclusions

Discrete torsion argument for N = 4 In particular, H3 = dBNSNS and F3 = dC2 belong to 3 5 H (RP , Ze) = Z2, classifying the orientifold types. New N < 4 dualities Engineer certain N < 4 theories in IIB, develop the S-duality dictionary as needed, and read the eﬀect of strong/weak duality on N < 4 theories.

Introduction S-duality Tachyons USp(32) Conclusions

Beyond N = 4

Montonen-Olive is deﬁned for N = 4, but IIB S-duality is believed to hold in general. Can we get some mileage out of this? Introduction S-duality Tachyons USp(32) Conclusions

Beyond N = 4

Montonen-Olive is deﬁned for N = 4, but IIB S-duality is believed to hold in general. Can we get some mileage out of this? New N < 4 dualities Engineer certain N < 4 theories in IIB, develop the S-duality dictionary as needed, and read the eﬀect of strong/weak duality on N < 4 theories. Introduction S-duality Tachyons USp(32) Conclusions

3 Orientifolding C /Z3 Orbifolding N = 4 duality down to N = 1

Work with B. Heidenreich and T. Wrase

Consider the orientifold action with generators {R, I Ω(−1)FL }:

R :(x, y, z) −→ (ωx, ωy, ωz) I :(x, y, z) −→ (−x, −y, −z)

with ω = exp(2πi/3). Introduction S-duality Tachyons USp(32) Conclusions

A proposed N = 1 duality

USp(Ne + 4) SU(Ne) SU(3) U(1)R Z3 Ai 2 − 2 1 3 Ne Bi 1 2 + 4 −2 3 Ne

(here Ne ∈ 2Z) is dual to

SO(N − 4) SU(N) SU(3) U(1)R Z3 i 2 2 A 3 + N 1 i 2 4 B 1 3 − N −2

1 i j k in both cases with W = 2 ijkTr A A B . Global anomalies, the moduli spaces and the spectrum of operators match if Ne = N − 3 . (As far as we have been able to check so far.) H•(X, Ze) = {Z2, 0, Z2, 0, Z2, 0}

22 = 4 choices of torsion =⇒ SL(2, Z) singlet plus triplet.

Introduction S-duality Tachyons USp(32) Conclusions

Discrete torsion

3 We did the computation for the orientifold of C /Z3, with horizon 5 5 manifold X = RP /Z3 ∼ (S /Z3)/Zf2. It is easier to work in homology and use Poincare duality

i H (X, Ze) = Hdim(X)−i(X, Ze) .

We are thus looking for elements of H2(X, Ze). Can be conveniently computed using a long exact sequence: [Hatcher]

i p∗ ... Hi(X, Ze) Hi(Y, Z) Hi(X, Z) i−1 p∗ Hi−1(X, Ze) Hi−1(Y, Z) Hi−1(X, Z) ... Introduction S-duality Tachyons USp(32) Conclusions

Discrete torsion

3 We did the computation for the orientifold of C /Z3, with horizon 5 5 manifold X = RP /Z3 ∼ (S /Z3)/Zf2. It is easier to work in homology and use Poincare duality

i H (X, Ze) = Hdim(X)−i(X, Ze) .

We are thus looking for elements of H2(X, Ze). Can be conveniently computed using a long exact sequence: [Hatcher]

i p∗ ... Hi(X, Ze) Hi(Y, Z) Hi(X, Z) i−1 p∗ Hi−1(X, Ze) Hi−1(Y, Z) Hi−1(X, Z) ...

H•(X, Ze) = {Z2, 0, Z2, 0, Z2, 0}

22 = 4 choices of torsion =⇒ SL(2, Z) singlet plus triplet. Introduction S-duality Tachyons USp(32) Conclusions

Superconformal index matching A very powerful and reﬁned indicator of duality comes from 3 putting the theory on S × R, and computing the index [Romelsberger:hep-th/0510060,0707.3702], [Kinney, Maldacena, Minwalla, Raju:hep-th/0510251]: Z I(t, x, f) = dg Tr (−1)F e−βHtRx2J 3 fg , (1)

with 2H = {Q, Q†}. Romelsberger gave a procedure for computing the index from weak coupling quantities. Start with the “letter”: (2t2 − t(x + x−1))χ (g) i (t, x, g, f) = Adj T (1 − tx)(1 − tx−1) P ri 2−ri i t χRi (g) χRi (f) − t χRi (g) χRi (f) + G F G F . (1 − tx)(1 − tx−1) and then take the plethystic exponential: " ∞ # Z X 1 I (t, x, f) = dg exp i (tk, xk, gk, f k) . T k T k=1 A conjecture about elliptic hypergeometric functions (See Spiridonov et al.)

IUSp = ISO

Introduction S-duality Tachyons USp(32) Conclusions

Superconformal index matching For SO(3) × SU(7) ↔ USp(8) × SU(4) we get:

2 ISO/USp(t, x, f) = 1 + t 3 χ0,2(f) + χ4,0(f) 4 + t 3 2χ0,4(f) + 2χ2,0(f) + χ3,1(f) + 2χ4,2(f) + χ8,0(f) 5 −1 + t 3 (x + x ) χ0,2(f) + χ4,0(f) 2 + t 3χ0,6(f) + χ12,0(f) + χ1,4(f) + 5χ2,2(f) + 3χ3,3(f)

+ 2χ4,1(f) + 3χ4,4(f) + χ5,2(f) + 4χ6,0(f) + χ6,3(f) + χ7,1(f) + 2χ8,2(f) + 4 + ...

We have checked up to order t3 for this value of N, higher orders for other values of N, and to all orders in the large N limit: Introduction S-duality Tachyons USp(32) Conclusions

Superconformal index matching For SO(3) × SU(7) ↔ USp(8) × SU(4) we get:

+ 2χ4,1(f) + 3χ4,4(f) + χ5,2(f) + 4χ6,0(f) + χ6,3(f) + χ7,1(f) + 2χ8,2(f) + 4 + ...

We have checked up to order t3 for this value of N, higher orders for other values of N, and to all orders in the large N limit: A conjecture about elliptic hypergeometric functions (See Spiridonov et al.)

IUSp = ISO For example, for

3 2,1 H Y /Z2, Ze = Z2 ⊕ Z2

so there are 22·2 = 16 orientifold types: 1 SL(2, Z) singlet, 3 triplets, 1 sextet. 10 diﬀerent weakly coupled limits.

Introduction S-duality Tachyons USp(32) Conclusions

General case By a cellular homology computation one can see that for a toric O3/O7 orientifold of a toric CY3 cone, with k sides isolated conical singularity of the cone ﬁxed points of the orientifold only at the conical singularity

3 k−2 H (X, Ze) = Z2 Introduction S-duality Tachyons USp(32) Conclusions

3 k−2 H (X, Ze) = Z2 For example, for

3 2,1 H Y /Z2, Ze = Z2 ⊕ Z2

so there are 22·2 = 16 orientifold types: 1 SL(2, Z) singlet, 3 triplets, 1 sextet. 10 diﬀerent weakly coupled limits. (Branes at singularities are somewhat special, in that step 4 can be done using perturbative ingredients.)

None of the above requires that the brane conﬁguration at the singularity preserves susy.

Introduction S-duality Tachyons USp(32) Conclusions

Recapitulation

Our philosophy for finding duals: 1 Build a configuration of branes at singularities. 2 Measure its conserved charges, including torsion. 3 Apply IIB S-duality to the charges. 4 Construct the brane configuration in the same geometry with the dual charges. D3 charge conservation explains Ne = N − 3. None of the above requires that the brane configuration at the singularity preserves susy.

Introduction S-duality Tachyons USp(32) Conclusions

Recapitulation

Our philosophy for finding duals: 1 Build a configuration of branes at singularities. 2 Measure its conserved charges, including torsion. 3 Apply IIB S-duality to the charges. 4 Construct the brane configuration in the same geometry with the dual charges. D3 charge conservation explains Ne = N − 3. (Branes at singularities are somewhat special, in that step 4 can be done using perturbative ingredients.) Introduction S-duality Tachyons USp(32) Conclusions

Recapitulation

Our philosophy for finding duals: 1 Build a configuration of branes at singularities. 2 Measure its conserved charges, including torsion. 3 Apply IIB S-duality to the charges. 4 Construct the brane configuration in the same geometry with the dual charges. D3 charge conservation explains Ne = N − 3. (Branes at singularities are somewhat special, in that step 4 can be done using perturbative ingredients.)

None of the above requires that the brane conﬁguration at the singularity preserves susy. Getting close to QCD (not SQCD) at strong coupling!

Non-susy Seiberg duality can be another handle on the problem, also with interesting string theory structure. [Armoni],...

Introduction S-duality Tachyons USp(32) Conclusions

N = 0 dualities

Exact dualities for non-susy theories: Montonen-Olive + D3s: [Uranga], [Sugimoto], [Hook,Torroba]. N = 1 duals + D3s: [Hook,Torroba],... N = 0 duals? Non-susy Seiberg duality can be another handle on the problem, also with interesting string theory structure. [Armoni],...

Introduction S-duality Tachyons USp(32) Conclusions

N = 0 dualities

Exact dualities for non-susy theories: Montonen-Olive + D3s: [Uranga], [Sugimoto], [Hook,Torroba]. N = 1 duals + D3s: [Hook,Torroba],... N = 0 duals?

Getting close to QCD (not SQCD) at strong coupling! Introduction S-duality Tachyons USp(32) Conclusions

N = 0 dualities

Exact dualities for non-susy theories: Montonen-Olive + D3s: [Uranga], [Sugimoto], [Hook,Torroba]. N = 1 duals + D3s: [Hook,Torroba],... N = 0 duals?

Getting close to QCD (not SQCD) at strong coupling!

Non-susy Seiberg duality can be another handle on the problem, also with interesting string theory structure. [Armoni],... Introduction S-duality Tachyons USp(32) Conclusions

Outline

1 Introduction

2 Strong/weak duality for realistic 4d theories

3 Tachyon condensation and stable vacua in non-susy theories Supercritical string theory Stable solitons of the supercritical string

4 Into the wilderness

5 Conclusions Introduction S-duality Tachyons USp(32) Conclusions

Supercritical string theory

[Hellerman, Swanson, Liu, . . . ]

String theory makes perfect sense in d > 10, if one includes an appropriate linear dilaton proﬁle r D − 10 Φ = X . 4α0 0 There is typically a tachyon, and the full theory has solutions with a bubble of tachyon condensation expanding at the speed of light: ! β r 0 √ (X0+X1) 2α T = e 2 β ≡ D − 10 with the tachyon vev “destroying” some of the supercritical dimensions 2 L = ... + T Xi . Introduction S-duality Tachyons USp(32) Conclusions

Supercritical string theory Introduction S-duality Tachyons USp(32) Conclusions

HetSO(32) as a stable soliton of a supercritical string

[Hellerman: th/0405041] discusses the spectrum of a particular supercritical version of the heterotic SO(32) string, which he dubs HO+(n)/. In ﬂat space it has SO(32 + n) × SO(9 + n, 1) gauge symmetry, and spectrum: A tachyon in ( , 1).

GMN , BMN , Φ.

AM in (Adj, ). No fermions. In flat space this has no stable vacuum. Net effect: only the dynamics on the fixed locus survives after tachyon condensation, and it is HetSO(32).

Introduction S-duality Tachyons USp(32) Conclusions

HetSO(32) as a stable soliton of a supercritical string 9,1 n An interesting modiﬁcation: put the theory on R × (R /Z2). Due to the orbifold projection one has

SO(32 + n) × SO(9 + n, 1) → SO(32 + n) × SO(n) × SO(9, 1)

and there are some fermions living at the ﬁxed locus of the orbifold, restoring susy (HetSO(32)). Gravitino and dilatino. Chiral fermion in ( , 1). Anti-chiral fermion in ( , ). Chiral fermion in (1, ).

In addition, at the orbifold point, Gmµ = Bmµ = T = 0. Introduction S-duality Tachyons USp(32) Conclusions

HetSO(32) as a stable soliton of a supercritical string 9,1 n An interesting modiﬁcation: put the theory on R × (R /Z2). Due to the orbifold projection one has

SO(32 + n) × SO(9 + n, 1) → SO(32 + n) × SO(n) × SO(9, 1)

In addition, at the orbifold point, Gmµ = Bmµ = T = 0.

Net eﬀect: only the dynamics on the ﬁxed locus survives after tachyon condensation, and it is HetSO(32). Supported by a [Polchinski-Witten] argument, the “weird” Z2 orbifold is simply the Z2 = O(1) gauge symmetry on the D1. It acts on the D1-D9 and D1-D9 strings, associated with the extra dimensions.

Introduction S-duality Tachyons USp(32) Conclusions

S-duality for Type I with D9-D9 pairs?

The resulting structure before tachyon condensation is very reminiscent of type I with D9-D9 pairs.

S-duality between unstable theories? [Hellerman] Introduction S-duality Tachyons USp(32) Conclusions

S-duality for Type I with D9-D9 pairs?

The resulting structure before tachyon condensation is very reminiscent of type I with D9-D9 pairs.

S-duality between unstable theories? [Hellerman]

Supported by a [Polchinski-Witten] argument, the “weird” Z2 orbifold is simply the Z2 = O(1) gauge symmetry on the D1. It acts on the D1-D9 and D1-D9 strings, associated with the extra dimensions. Introduction S-duality Tachyons USp(32) Conclusions

General lesson

We can “geometrize” the search for string theories, if we have a good understanding of supercritical strings. Introduction S-duality Tachyons USp(32) Conclusions

Outline

1 Introduction

2 Strong/weak duality for realistic 4d theories

3 Tachyon condensation and stable vacua in non-susy theories

4 Into the wilderness

5 Conclusions What is the S-dual of this? (Stable non-susy heterotic?) [Polchinski, Witten] argument says it’s a worldsheet with SU(2) = USp(2) gauge group and matter content

USp(2) SO(8) USp(32) Aµ 1 1 i X 8v 1 a S+ 8s 1 aˆ S− 8c 1 I λ− 1

Introduction S-duality Tachyons USp(32) Conclusions

Sugimoto’s stable non-susy theory A few perturbatively stable non-supersymmetric 10d (or orbifold) theories have been explicitly constructed: [Sagnotti] [Angelantonj] [Blumenhagen, Font, L¨ust], [Antoniadis, Dudas, Sagnotti], [Sugimoto: th/9905159]. We focus on Sugimoto’s construction: O9+ + 32D9s. Introduction S-duality Tachyons USp(32) Conclusions

Sugimoto’s stable non-susy theory A few perturbatively stable non-supersymmetric 10d (or orbifold) theories have been explicitly constructed: [Sagnotti] [Angelantonj] [Blumenhagen, Font, L¨ust], [Antoniadis, Dudas, Sagnotti], [Sugimoto: th/9905159]. We focus on Sugimoto’s construction: O9+ + 32D9s. What is the S-dual of this? (Stable non-susy heterotic?) [Polchinski, Witten] argument says it’s a worldsheet with SU(2) = USp(2) gauge group and matter content

USp(2) SO(8) USp(32) Aµ 1 1 i X 8v 1 a S+ 8s 1 aˆ S− 8c 1 I λ− 1 Introduction S-duality Tachyons USp(32) Conclusions

Outline

1 Introduction

2 Strong/weak duality for realistic 4d theories

3 Tachyon condensation and stable vacua in non-susy theories

4 Into the wilderness

5 Conclusions S-duality for orientifolds seems important. Non-critical theories may provide a unifying framework for some cases, geometrizing the search for stable theories, but... no trace of a “non-critical M-theory” yet. We need to go beyond free worldsheets. Non-abelian worldsheets start making a natural appearance! (recent appearances in the susy context too: [Hori,Tong], [Hori] (susy breaking), [Gomis, Lee],...)

Introduction S-duality Tachyons USp(32) Conclusions

Conclusions

No “big picture” yet, but some suggestive observations. Introduction S-duality Tachyons USp(32) Conclusions

Conclusions

No “big picture” yet, but some suggestive observations.

S-duality for orientifolds seems important. Non-critical theories may provide a unifying framework for some cases, geometrizing the search for stable theories, but... no trace of a “non-critical M-theory” yet. We need to go beyond free worldsheets. Non-abelian worldsheets start making a natural appearance! (recent appearances in the susy context too: [Hori,Tong], [Hori] (susy breaking), [Gomis, Lee],...)