Adventures in Non-supersymmetric String Theory
Justin Kaidi
East Asian Strings Seminar Series
December 4, 2020[2010.10521] JK[1908.04805] JK, Parra-Martinez, Tachikawa[1911.11780] JK, Parra-Martinez, Tachikawa

Justin Kaidi

Exploring non-SUSY Strings

Spectrum of String Theories• In the 80s, it was discovered that SUSY string theories are di↵erent limitsof a single underlying theory:

Justin Kaidi

Exploring non-SUSY Strings

Spectrum of String Theories• In the 80s, it was discovered that SUSY string theories are di↵erent limitsof a single underlying theory.
• But there also exist non-SUSY string theories– Type 0A/B, unoriented Type 0 strings, and 7 heterotic strings

Justin Kaidi

Exploring non-SUSY Strings

Spectrum of String Theories• In the 80s, it was discovered that SUSY string theories are di↵erent limitsof a single underlying theory.
• But there also exist non-SUSY string theories– Type 0A/B, unoriented Type 0 strings, and 7 heterotic strings• There are also string theories in dimensions lower than 10d. In particular,much study on 2d strings [Takayanagi, Toumbas ‘03; Douglas, Klebanov, Kutasov,Maldacena, Martinec, Seiberg ‘03]

– 2d Type 0A/B, unoriented 2d Type 0 strings, and 3 heterotic strings

Justin Kaidi

Exploring non-SUSY Strings

Guiding Principle• String theory is a unique theory of Quantum Gravity

Justin Kaidi

Exploring non-SUSY Strings

Tachyons• non-SUSY string theories generically have closed string tachyons– Exceptions: Pin+ Type 0, O(16) ⇥ O(16) heterotic, Type I˜• But the presence of a tachyon doesn’t mean the theory is inconsistent.
Rather, it means we’re expanding about the wrong vacuum.
• As long as there exists one stable vacuum, the theories are consistent• The stable vacua may be lower-dimensional– In many cases, they will be known two-dimensional string theories• Goal: Classify the non-SUSY 10d strings and find stable vacua for themPart 1: Classify unoriented Type 0 strings
Part 2: Find stable vacua for non-SUSY heterotic strings

Justin Kaidi

Exploring non-SUSY Strings

Part I

Part I: Classifying unoriented Type 0 strings

Justin Kaidi

Exploring non-SUSY Strings

GSO Projectioni• Work in NS-R formalism in lightcone gauge; (XL,R,iL,R ) with i = 1, . . . , 8• The spectrum of the physical string is obtained by doing a GSO projection[Gliozzi, Scherk, Olive ‘77]

– Allowed GSO projections must project onto subsectors satisfying e.g. closureof OPE, mutual locality, and modular invariance of torus amplitudes.
• GSO projection can also be thought of as a sum over spin structures [Seiberg,Witten ‘86]

– Di↵erent GSO projections correspond to possible phases assigned to spin
structures in a way compatible with cutting/gluing of worldsheet.
– For example for Type IIA/B, can havewith ↵(NS, NS) = ↵(NS, R) = ↵(R, NS) = 1 and ↵(R, R) = ±1.

Justin Kaidi

Exploring non-SUSY Strings

SPT Phases• For our purposes, an SPT phase is a gapped theory with unique groundstate on a manifold without boundary,
Z[X] 2 U (1)@X = ;• The partition function will only depend on the bordism class of X,[Kapustin ‘14; Kapustin, Thorngren, Turzillo, Wang ‘15]

Z : ⌦sd(BG) ! U (1)s = structure of tangent bundle, G = structure of principal bundle.
• Hence d-dimensional SPT phases are classified by0ds(BG) := Hom (⌦sd(BG), U (1))• In the presence of boundary, we can have gapless edge modes.

Justin Kaidi

Exploring non-SUSY Strings

Kitaev Chain and Arf Invariant• We’ll mainly be concerned with fermionic SPTs. The simplest case is theKitaev chain/Majorana wire [Kitaev ’01]
02Spin(pt) = Z2

• Has been realized experimentally [Mourik et al ’12]
• The partition function of this phase on a manifold with spin structureZ[⌃, ] = e⇡iArf(⌃, )• One can calculateisJustin Kaidi

Exploring non-SUSY Strings

Type 0 Strings• Indeed, consider possible SPT phases we can add to the worldsheet,DE2Arf(⌃, )0Spin(pt) = Z2 = ( 1)
• Two di↵erent worldsheet theories with following torus partition functions• So two di↵erent points of view
(1) No SPT phase, but use di↵erent projectors, i.e. P0B vs. P0A.
(2) Sum over spin structure with non-trivial SPT phase.

Justin Kaidi

Exploring non-SUSY Strings

Unoriented Type 0 Strings• Let’s now allow worldsheets to be unoriented.
– Two options:

Pin+ : w2 = 0

Pin : w2 + w12 = 0• Possible SPT phases:
02Pin (pt) = Z8 =02Pin+ (pt) = Z2

De⇡iABK( )/4DE= ( )Arf(ˆ )Ewith ˆ the spin structure on the oriented double cover.
• Prediction: an 8 + 2(?) = 10-fold classification of unoriented Type 0strings, matching with the Altland-Zirnbauer classification of topologicalsuperconductors.
• Some scattered results in the literature [Bergman, Gaberdiel ’99; Blumenhagen, Font,
Lüst ’99], but nothing complete.

Justin Kaidi

Exploring non-SUSY Strings

D-branes and K-theory• This leads to a rich spectrum of stable D-branes:eKe1KDKO
DKO 1DKO 2DKO 3DKO 4DKO 5DKO 6DKO 71Z
02Z2Z22Z0002ZZ200Z
2Z2Z Z20Z
0Z0Z Z21Z
02Z
Z2Z202Z0Z2Z220Z0Z
Z2Z Z20Z Z2Z2Z

3Z
0002Z
Z22Z2Z22Z
040Z
0Z
0Z

Z22Z2Z Z20Z

5Z
02Z
0Z2Z22Z
Z2Z2060Z
0Z

Z2Z2Z
0Z
Z2Z Z27Z
02Z2Z22Z0002Z
Z280Z2Z2Z Z20Z0Z0Z Z29Z02ZZ2Z202Z0Z2Z2• Additional information:– None of the unoriented Type 0 strings has spacetime SUSY
– All RR tadpoles can be cancelled. NSNS tadpoles can be cancelled by adding
branes, or via Fischler-Susskind mechanism– The Pin strings have closed string tachyons, but Pin+ strings aretachyon-freeJustin Kaidi

Exploring non-SUSY Strings

Part IIPart II: Stable vacua for heterotic strings

Justin Kaidi

Exploring non-SUSY Strings

Tachyonic Heterotic Strings• Tachyonic heterotic strings are constructed as follows:i• Start with (XL,R,iL ) with i = 1, . . . , 8 anda
R with a = 1, . . . , 32

• Simplest partition function is• This theory has 32 tachyons, 4032 massless bosons– graviton (35), B-field (28), dilaton (1), and 496 gauge bosons of SO(32)[Kawai, Lewellen, Tye ‘86; Dixon, Harvey ‘86]

Justin Kaidi

Exploring non-SUSY Strings

Tachyonic Heterotic Strings• The worldsheet theory studied before has a (Z2)5 symmetry,
g1 =3 ⌦ 12 ⌦ 12 ⌦ 12 ⌦ 12 ,g 3 = 12 ⌦ 12 ⌦3 ⌦ 12 ⌦ 12 ,g 5 = 12 ⌦ 12 ⌦ 12 ⌦ 12 ⌦• Each Z2 acts as

g 2 = 12 ⌦g 4 = 12 ⌦ 12 ⌦ 12 ⌦3 .

1 on 16 of theatachyons
32168421

massless fermions
0256
384
448
480
496

3 ⌦ 12 ,and as +1 on others• We can now gauge (Z2)n for 0  n  5.
– This breaks SO(32) 6! SO(25 n) ⇥ SO(32n0123453 ⌦ 12 ⌦ 12 ⌦ 12 ,25 n)gauge bosons
496
368
304
272
256
248

gauge group
SO(32)
O(16) ⇥ E8O(8) ⇥ O(24)(E7 ⇥ SU (2))2U (16)E8

Justin Kaidi

Exploring non-SUSY Strings

Tachyon Condensation• All of the heterotic strings above have tachyons. We now try to condensethem [Hellerman, Swanson ‘06; ‘07]
• Say ˜a are subset ofsuperpotentialainvariant under SO(25 n). Condensation produces

W =5 n2X

a=1

˜aT a(X)• Linearized equation of motion for T a(X),µ@ @µ Ta2@

µ2 a@µ T + 0 T = 0↵a• One solution=3 n22p↵0X,aT =mr

2 X + a+1eX0↵

Justin Kaidi

Exploring non-SUSY Strings

Condensation to d > 2• Previous solution gives the following scalar potential
V = Ae2 X+5 n2X(Xa+1 2)

a=1

BeX+5 n2X

˜aa+1+ ...

a=15 n• As X + ! 1, fluctuations along X 2, . . . , X 2 +1 are suppressed, and we
5 n
225 n +1
get a theory in d = 10 2
localized at X = · · · = X
= 0.
n
3
4
5d6
8
9

massless fermions
112
240
248

gauge bosons
266
255
248

gauge groupE7 ⇥ E7SU (16)E8

• Low-energy gravity+gauge theories can be checked to be anomaly-free!

Justin Kaidi

Exploring non-SUSY Strings

Condensation to d = 2• For n < 2, then d = 10

25 n < 0 so this doesn’t work.

• In these cases we simply condense to d = 2, where dilaton background lifts
remaining tachyons:
n0
1
2d2
2
2

massless bosons
2480massless fermions
08
12

gauge group
O(24)O(8) ⇥ E8O(24)• The three theories obtained in this way are precisely the three 2d heterotic
strings known in the literature! [Davis, Larsen, Seiberg ‘05]
• We have thus connected the known 2d theories with non-SUSY 10dtheories via dynamical transitions.

Justin Kaidi

Exploring non-SUSY StringsConclusion
I) The worldsheets of di↵erent string theories can di↵er by subtle topologicalterms. These terms explain the di↵erent GSO projections, D-brane
spectra, and orientifoldings allowed in the theories.
2) Tachyonic strings admit lower-dimensional stable vacua. Many of these
are known 2d strings.
• Possible future extensions:
1) SPT phases for heterotic worldsheets, e.g. |02Spin(BZ52)| = 65, 536!
2) Worldsheet domain walls?
3) Orbifolds: beyond discrete torsion• There is still much to explore in perturbative string theory!

Justin Kaidi

Exploring non-SUSY Strings

The End (for now)Thank you!