t Hooft Anomaly & Modular Bootstrap Shu-Heng Shao Institute for - - PowerPoint PPT Presentation

‘t Hooft Anomaly & Modular Bootstrap

Shu-Heng Shao

Institute for Advanced Study

Joint ICTP/SISSA String Seminar April 3rd , 2019

Based on work with Ying-Hsuan Lin (Caltech) ArXiv: 1904.xxxxx

Two Major Non-Perturbative Tools for QFT

1

‘t Hooft Anomaly

• Obstruction to gauging 𝐻
• Invariant under RG and duality

[See Zohar’s Lectures]

Conformal Bootstrap

• Consistency of CFT
• Constrain operator spectrum

[See Leonardo’s Talk]

Wouldn’t it be nice if we can combine the two techniques?

Questions We’d Like to Ask:

In CFT with Global Symmetry 𝐻 and Anomaly 𝛽:

1. Is there an upper bound on the lightest 𝐻 charged

• perator?

2. How does the bound depend on the anomaly 𝛽?

Weak Gravity Conjecture like questions [See Matt’s Lectures]

2

Today: Bootstrap with Anomaly

3

Setup: 2D Bosonic CFT with Global Symmetry 𝐻 = ℤ& Question: Is there an upper bound on the lightest ℤ𝟑 odd operator?

2D CFT with Global Symmetry ℤ𝟑

4

Non-Anomalous Anomalous Bound on Charged Operator

NO 😮 YES 😎

Moral: It is harder to “hide” a symmetry if it is anomalous

Reminder on ‘t Hooft Anomalies

• A global symmetry with ‘t Hooft anomaly is still a perfectly

healthy symmetry in a consistent QFT.

• You just cannot gauge it.
• Different from the ABJ anomaly, where the axial “symmetry”

is not a true global symmetry.

5

Outline

• Symmetry and Anomaly in Two Dimensions
• Modular Bootstrap

6

Outline

• Symmetry and Anomaly in Two Dimensions
• Modular Bootstrap

7

Symmetry and Topological Defect

• Continuous global symmetry → Noether charge
• More generally, a 0-form global symmetry 𝑕 ∈ 𝐻 (continuous or

discrete) is associated to a codimension-1 topological defect 𝑀,.

• Topological defect acts on local operators by symmetry transformation.

8

=

ϕ (±1) ϕ

0-form Codimension-1

Global Symmetry ⟶ Topological Defect

9

Basic Properties of Topological Lines

• They are topological

vAll physical observables are invariant under continuous deformation of topological lines. vThey commute with both Virasoro algebras.

=

10

Defect Hilbert Space ℋ/

• Followed from the topological

property, states in ℋ/ are in representations of both Virasoro algebras. ℋ/ = 0 (𝑜/)4,4

6 𝑊𝑗𝑠4 ⊗ 𝑊𝑗𝑠4 6

• 4,4

6∈ℋ<

(𝑜/)4,4

6 ∈ ℕ

𝑀 ℋ/ 11

Operator-State Map for ℋ/

12

|𝜈〉 ∈ ℋ/ 𝜈(𝑦)

Non-local operator living at the end of the defect line E.g. Electron in QED 𝑀 𝑀

Crossing Relations of the ℤ𝟑 Line

13

= α

Do it twice: 𝛽& = 1 (the cocycle condition)

Crossing and Anomaly

• 𝛽 = +1: Non-Anomalous (can be gauged)
• 𝛽 = −1: Anomalous (can not be gauged)

14

Indeed, the bosonic, unitary ℤ𝟑 anomaly is classified by

𝛽 ∈ 𝐼F(ℤ𝟑,U(1))=ℤ𝟑

= α

Crossing and Anomaly

• Consider the torus partition function of the would-be ℤ𝟑 orbifold theory:

15

= α

𝑎HIJ =

K & (

+ ) +

K & (

) +

Untwisted sector Twisted sector

“ ”

16

= 𝛽

When 𝛽 = −1, this is an ambiguity ⟹ anomaly

Crossing and Anomaly (Recap)

• 𝛽 = +1: Non-Anomalous
• 𝛽 = −1: Anomalous

17

= α

Outline

• Symmetry and Anomaly in Two Dimensions
• Modular Bootstrap

18

Modular Bootstrap

• Positivity: Expansion on Virasoro characters
• Crossing: Modular 𝑇 Transformation

19

Torus Partition Function

20

• The torus partition function can

be expanded on the Virasoro characters with positive coefficients 𝑎 𝜐, 𝜐̅ = 𝑈𝑠

ℋ [𝑟4ST/&V𝑟

W4

6ST/&V]

= 0 𝑜4,4

6

• 4,4

6∈ℋ

𝜓4(𝜐) 𝜓4

6(𝜐̅)

𝑜4,4

6 ∈ ℕ

𝜓4 𝜐 = 𝑟4STSK

&V

𝜃(𝜐)

ℋ

Torus Partition Function with ℤ𝟑 Line

21

𝑎/ 𝜐, 𝜐̅ = 𝑈𝑠

ℋ [𝑀

[ 𝑟4ST/&V𝑟 W4

6ST/&V]

= 0 (𝑜4,4

6 \

− 𝑜4,4

6 S )

• 4,4

6∈ℋ

𝜓4(𝜐) 𝜓4

6(𝜐̅)

𝑜4,4

6 ±

∈ ℕ

𝑀

ℋ

Defect Hilbert Space 𝓘𝑴

𝑎/ 𝜐, 𝜐̅ = 𝑈𝑠

ℋ< [𝑟4S T &V𝑟

W4

6S T &V]

= 0 (𝑜/)4,4

6 𝜓4(𝜐) 𝜓4 6(𝜐̅)

• 4,4

6∈ℋ<

(𝑜/)4,4

6∈ ℕ

𝑀 ℋ/ 22

Positivity

23

𝑎± 𝜐, 𝜐̅ = 𝑜4,4

6 ± 𝜓4 𝜐 𝜓4 6 𝜐̅

• 4,4

6 ∈ℋ±

𝑎/(𝜐, 𝜐̅) = 0 (𝑜/)4,4

6 𝜓4(𝜐) 𝜓4 6(𝜐̅)

• 4,4

6∈ℋ<

∨ ∨

=

K & (

) ±

𝑎/ 𝑎

=

Anomaly and Modular Transformation

24

𝑈SK 𝑈 ∥ 𝛽

𝑎/(𝜐, 𝜐̅)

𝑎/(𝜐, 𝜐̅) is invariant under b 𝑈& 𝑈V If 𝛽 = +1 If 𝛽 = −1

Spin Selection Rule in ℋ/

• The spin 𝑡 = ℎ − ℎ

W of a state in the defect Hilbert space ℋ/ is constrained by the anomaly 𝛽 [Chang-Lin-SHS-Wang-Yin]:

25

If 𝛽 = +1 (Non-Anomalous) 𝑡 ∈ ℤ 2 1 4 + ℤ 2 If 𝛽 = −1 (Anomalous)

• Now we see that the anomaly controls the spin of non-local
• perators living at the end of the line.
• How do we convert this information to constraints on local
• perators?
• Modular Transformation

26

Spin Selection Rule in ℋ/

Crossing

27

𝑎/ 𝑎

𝑇 𝑇

𝑎 𝑎/ The spins here depend on the anomaly

Positivity

28

±

Crossing

𝑇 𝑇

The spins here and there depend on the anomaly

2D CFT with Global Symmetry ℤ𝟑

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Non-Anomalous Anomalous Bound on Charged Operator

NO 😮 YES 😎

Moral: It is harder to “hide” a symmetry if it is anomalous

Upper Bounds on ℤ𝟑 Odd Operators 𝚬S

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2 4 6 8 0.5 1.0 1.5 2.0 2.5 3.0 ΔS(anomalous)

𝑑 𝑡𝑣(2)K

WZW

Why is this result interesting? (at least to

me…)

• As a Bootstrapper…
• As an Anomaler...

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For a Bootstrapper…

• Global symmetry helps us target the CFT we want to bootstrap.

E.g. 3d O(2) bootstrap [See Zohar’s and Leonardo’s Talks]

• ‘t Hooft anomaly is a more refined information for the global

symmetry.

• Even the very existence of a bound might depend on the anomaly!

32

For an Anomaler…

• In a gapped phase, discrete anomalies imply that:

vThe symmetry is either spontaneously broken, or vThere is a TQFT matching the anomaly.

• Rather surprisingly, discrete anomalies also constrain the

spectrum of local operators in a gapless CFT phase.

33

Generalization

• 2d CFT with 𝑉(1) Global Symmetry
• Higher Dimensions

34

Generalization

• 2d CFT with 𝑉(1) Global Symmetry
• Higher Dimensions

35

2D CFT with Global Symmetry 𝑽(𝟐)

36

• A U(1) Noether current has a holomorphic component 𝑲(𝒜) and an

antiholomorphic component 𝑲̅(𝒜 W).

• The topological line generating the U(1) is

𝑉‚ = 𝑓𝑦𝑞 𝑗𝜄 † 𝑒𝑨 𝐾(𝑨)

• − 𝑗𝜄 † 𝑒𝑨̅ 𝐾̅(𝑨̅)
• While 𝑲(𝒜) and 𝑲̅(𝒜

W) are separately conserved, 𝜖 𝐾 6 𝑨̅ = 𝜖̅ 𝐾 𝑨 = 0, each of them generically generates an ℝ symmetry, instead of 𝑉 1 .

2D CFT with Global Symmetry 𝑽(𝟐)

37

Non-Anomalous Anomalous Bound on U(1) Charged Operator

NO 💀 YES 👽

Previous Work

• The authors of [Benjamin-Dyer-Fitzpatrick-Kachru] and [Montero-

Shiu-Soler] derived a bound on the lightest U(1) charged

• perator for a holomorphic U(1).
• A holomorphic U(1) is always anomalous (chiral anomaly),

which is consistent with our observation.

• However, there is no bound for a non-anomalous U(1) (free

boson example).

38

Example: 2d Compact Boson

[See Leonardo’s Discussion Session]

• At any radius 𝑆, there are two U(1)’s: the winding 𝑽(𝟐)𝒙 and

the momentum 𝑽(𝟐)𝒐.

• Both U(1)’s are non-anomalous (and non-holomorphic), but

there is a mixed anomaly between the two.

• Hence 𝑽(𝟐)𝒆𝒋𝒃𝒉 = 𝑒𝑗𝑏𝑕(𝑽(𝟐)𝒙 x 𝑽(𝟐)𝒐) is anomalous.

39

2d c=1 Compact Boson at Radius 𝑺

40

𝑽(𝟐)𝒙 𝑽(𝟐)𝒐 𝑽(𝟐)𝒆𝒋𝒃𝒉

Lightest Charged Op. Winding mode KK mode Winding or KK mode

∆S

𝑆& 2 1 2𝑆& Min[

›œ & , K &›œ]

Bound? NO😮 NO😮 YES😎 Anomaly? NO NO YES

Generalization

• 2d CFT with 𝑉(1) Global Symmetry
• Higher Dimensions

41

Higher Dimensions

• Is there such an anomaly-dependent bound on local
• perators in higher than 2 dimensions?
• NO! It was shown in [Wang-Wen-Witten 2016] that given a

discrete, unitary, bosonic symmetry 𝐻 and its anomaly 𝛽 in 𝑒 dimensions, there is a 𝑒-dimensional TQFT carrying this symmetry and anomaly.

42

• In 𝒆 > 𝟑, these TQFTs have a unique vacuum:

Trivial local operators, but non-trivial anomalies.

• Hence, discrete, unitary, bosonic anomalies do not constrain local
• perators in 𝒆 > 𝟑.
• In 𝒆 = 𝟑, those TQFTs have degenerate vacua (spontaneous

symmetry breaking): Non-trivial local operators, non-trivial anomalies.

43

Higher Dimensions

Turning to the non-anomalous case…

• So we have derived a

bound for an anomalous ℤ&.

• What can we say

about a non- anomalous ℤ&?

44

2 4 6 8 0.5 1.0 1.5 2.0 2.5 3.0 ΔS(anomalous)

𝑑

Order-Disorder Bound

• For a non-anomalous ℤ𝟑, even though there is no bound on the

ℤ𝟑 odd operator, there is a bound on the lightest operator in

ℋS⨁ℋ/

45

ℤ𝟑 odd (order) Defect Hilbert space (disorder)

• The “order” and “disorder” operators cannot both be too

heavy (analogous to [Levin 2019]).

2d Ising CFT [See Leonardo’s Talk]

46

𝜏 𝑦

• rder

ℎ = ℎ W = 1 16 𝜈 𝑦 disorder ℎ = ℎ W = 1 16

ℤ& line

Order-Disorder Bound

47

2 4 6 8 10 0.0 0.5 1.0 1.5 2.0 2.5 3.0 ΔHI¢/¢£¤(non-anomalous)

Ising^2

Conclusion

• Weak Gravity Conjecture Question in CFT: Is there a bound
• n the lightest charged operator?
• In 2d CFT, there is a bound if the symmetry is anomalous, but

not otherwise.

• Discrete ‘t Hooft anomalies constrain local operators in the

gapless phase.

48

Outlook

• Generalize to anomalies involving spacetime action, e.g. time-reversal

anomaly.

• For anomalies that cannot be carried by TQFT, do they constrain

bound on charged operators? E.g. Continuous global symmetry in 4d.

• Implications on the Weak Gravity Conjecture in AdS.

Chern-Simons terms and Confinement.

49

Thank You!

50