Strings 2018, OIST
June 25th , 2018

Beyond Symmetry:

Topological Lines in 2D
Shu-Heng Shao
Institute for Advanced Study
Based on work with Chi-Ming Chang, Ying-Hsuan Lin, Yifan Wang, Xi Yin
ArXiv: 1802.04445

The Power of Symmetry• Global symmetries are powerful in constraining RG flows,
correlation functions, and other physical observables in QFT.
• ’t Hooft anomaly matching, Ward identity of correlators, checking
dualities…

• More recently, higher form global symmetries have also shown
to be useful in constraining the dynamics of QFT [Kapustin-Seiberg,
Gaiotto-Kapustin-Seiberg-Willet…].

• Global symmetries can be defined in terms of their chargeoperators, i.e. topological defects.
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Symmetry and Topological Defect• Continuous global symmetry → Noether charge operator• More generally, a 0-form global symmetry 𝑔 ∈ 𝐺 (continuous ordiscrete) is associated to a codimension-1 topological defect 𝐿& .
• Topological defect acts on local operators by symmetry transformation.

𝐿&𝑔∈𝐺
ϕ=gϕ2

0-form

Codimension-1Global Symmetry ⟶ Topological Defect⟵?
NO3

The Power of Non-Symmetry• There are codimension-1 topological defects that are NOT
the charge operators of any 0-form global symmetries.
• These non-symmetry topological defects are equally
powerful in constraining the dynamics of QFT.
• We don’t have to look far for such defects – they areubiquitous in 2D CFT (e.g. 2D Ising model).
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Roadmap
q-form Global Symmetry& Symmetry Codim-(q+1) Defect↑0-form Global Symmetry ⟶& Symmetry Codim-1 DefectNon-Symmetry Codim-1 Defect[Bhardwaj-Tachikawa]

Today: (Non-)Symmetry Topological Lines in 2D CFT
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Theorem• In 2D, if certain non-symmetry topological line ispreserved along a RG flow, then the IR theory can NOT
be trivially gapped [Chang-Lin-SHS-Wang-Yin].
• Similar in spirit to ‘t Hooft anomaly matching6

Basic Properties of Topological Lines
1. They are topological• All physical observables areinvariant under continuous
deformation of topologicallines.

=• They commute with both
Virasoro algebras.
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Basic Properties of Topological Lines
2. Defect Hilbert Space ℋ+• Followed from the topologicalproperty, states in ℋ+ are inrepresentations of both Virasoro
algebras.
ℋ+ = -1/,/𝐿𝑛/,/1 𝑉𝑖𝑟/ ⊗ 𝑉𝑖𝑟/1𝑛/,/1 ∈ ℕℋ+8

Basic Properties of Topological Lines
3. Action on the Hilbert Space• Topological line maps a local operator
to another of the same conformal
weights (ℎ, ℎ;).

𝐿

ϕ=ϕ'

• The map might not be invertible, inwhich case 𝐿 is a non-symmetry line.
9Basic Properties of Topological Lines• In a RG flow, a topological line
𝐿 is said to be preserved by therelevant deformation 𝜙(𝑥) if
they commute with each other.

ϕ• That is, the topological line
𝐿 doesn’t “feel” the insertion of𝜙(𝑥) and remains topologicalafter the deformation.

=ϕ10

Vacuum Expectation Value 𝑳• Symmetry lines:
𝐿 =1• Non-Symmetry lines:
𝐿 ≠1
𝐿 is RG invariant!

𝐿1

= 𝐿

1

Assumption: The theory has aunique vacuum |1⟩
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Topological Lines in RCFT
• In a diagonal RCFT, there is a simple class of topological lines, theVerlinde lines, that are in 1-to-1 correspondence with the primary
operators and commute with both chiral algebras [Verlinde, Petkova-Zuber,Moore-Seiberg].

𝒪Primary→𝐿𝒪Verlinde Line• The action on the local operators is determined by the modular S matrix.
12Example: 2D Ising Model• Three topological lines in the Ising model [PetkovaZuber,Frohlich-Fuchs-Runkel-Schweigert…], corresponding to thethree primary operators
Primary:𝟏(𝟎,𝟎)𝜺( 𝟏 , 𝟏 )𝟐𝟐

𝝈 𝟏,𝟏𝟏𝟔 𝟏𝟔↓
Topological Line:𝑰𝑳𝜺𝑳𝝈13

Example: 2D Ising Model
Symmetry Line?

Vev

Trivial LineIYes

1ℤO Line

𝐿P

Yes

1Duality Defect 𝐿Q

No214

Symmetry Line in the Ising Model• 𝐿P implements the ℤO global symmetry.
LϵLϵ1=1ϵLϵ=ϵ

σ= (-1) σ
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Non-Symmetry Line in the Ising Model• The action of 𝐿Q on local operators is not invertible→ Non-Symmetry line [Petkova-Zuber, Frohlich-Fuchs-Runkel-Schweigert…]
Lσ1

Lσ= 21ϵLσ= (- 2 ) ϵσ=016

Theorem• In 2D, if a non-symmetry topological line 𝐿 with𝐿 ∉ ℤTU
is preserved along a RG flow, then the IR theory can NOT betrivially gapped [Chang-Lin-SHS-Wang-Yin].
Remark:• Degenerate vacua that are not consequences of spontaneous symmetry
breaking → Spontaneous Non-Symmetry Breaking• Similar in spirit to ‘t Hooft anomaly matching: the IR theory has to be
nontrivial to match certain degrees of freedom in the UV.
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Proof• Prove by contradiction. Assume that the IR theory is trivially gapped,ℋVW = 1 .
• Next, consider the torus partition function of the IR theory with 𝐿
inserted horizontally at a constant time slice.

𝐿= 𝑇𝑟ℋZ[ 𝐿 = 𝐿ℋVW18

Proof: Modular Invariance𝐿modularinvariance∥= 𝑇𝑟ℋZ[ 𝐿 = 𝐿 ∉ ℤTUℋVWContradiction!
Q.E.D.

= 𝑇𝑟ℋ] 1 ∈ ℤTU

𝐿ℋ+19

Example: Tri-Critical Ising Modelca• Relevant deformation by 𝜀(_,_) in the 𝑐 = tri-critical Ising model.
dU``• A non-symmetry topological line of vev 𝟐 is preserved along the flow
⟹ The IR theory can NOT be trivially gapped.

−𝜀 aTri-Critical Ising CFT+𝜀 a

Gapped Phase
Ising CFTwith 3 Vacua [Huse, Zamolodchikov, Kastor-Martinec-Shenker,Cardy-Ludwig, Fateev-Zamolodchikov]

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Example: Tri-Critical Ising Model• Relevant deformation by 𝜎(a i , i ) in the tri-critical Ising model.
jk jk𝟏l 𝟓• A non-symmetry topological line of vev
is preserved along the flow
𝟐

⟹ The IR theory can NOT be trivially gapped.
• The ℤ𝟐 global symmetry is explicitly broken by 𝜎 a , and yet the vacuum istwo-fold degenerate → Spontaneous Non-Symmetry Breaking

Tri-Critical Ising CFT
𝜎aGapped Phasewith 2 Vacua[Zamolodchikov, Ellem-Bazhanov]21

Conclusion• The Power of Symmetry: Global symmetry, especially when it has ‘t
Hooft anomaly, is powerful in constraining RG flows.
• The Power of Non-Symmetry: Non-symmetry topological defects areequally powerful in constraining the dynamics of QFT.
• Certain non-symmetry topological lines imply the IR theory can NOT
be trivially gapped [Chang-Lin-SHS-Wang-Yin].
• Consideration of topological lines and modular invariance sometimes
determines the IR TQFT completely.
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Outlook• Given a set of topological lines,• Can they be coupled to a non-degenerate vacuum?
• If so, how many different ways?
• More generally, is there a TQFT/CFT that realizes these topologicallines?
[Moore-Segal, Bhardwaj-Tachikawa]• Non-symmetry codimension-one topological defects in higher
dimensions.
• For example, the non-symmetry surface defect in the 3𝑑 𝑈(1)q
Chern-Simons theory [Fuchs-Runkel-Schweigert, Kapustin-Saulina].
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Thank You!

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