Joint ICTP/SISSA String Seminar April 3 rd , 2019 ‘t Hooft Anomaly & Modular Bootstrap Shu-Heng Shao Institute for Advanced Study Based on work with Ying-Hsuan Lin (Caltech) ArXiv: 1904.xxxxx
Two Major Non-Perturbative Tools for QFT ‘t Hooft Anomaly Conformal Bootstrap • Obstruction to gauging 𝐻 • Consistency of CFT • Invariant under RG and duality • Constrain operator spectrum [See Zohar’s Lectures] [See Leonardo’s Talk] Wouldn’t it be nice if we can combine the two techniques? 1
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 operator? 2. How does the bound depend on the anomaly 𝛽 ? Weak Gravity Conjecture like questions [See Matt’s Lectures] 2
Today: Bootstrap with Anomaly Setup: 2D Bosonic CFT with Global Symmetry 𝐻 = ℤ & Question: Is there an upper bound on the lightest ℤ 𝟑 odd operator? 3
2D CFT with Global Symmetry ℤ 𝟑 Non-Anomalous Anomalous Bound on Charged Operator NO 😮 YES 😎 Moral: It is harder to “hide” a symmetry if it is anomalous 4
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. ϕ ϕ = (± 1 ) 8
0-form Codimension-1 Global Symmetry ⟶ Topological Defect 9
Basic Properties of Topological Lines • They are topological v All physical observables are invariant under continuous deformation of = topological lines. v They 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. 𝑀 ℋ / = 0 (𝑜 / ) 4,4 6 𝑊𝑗𝑠 4 ⊗ 𝑊𝑗𝑠 4 6 6∈ℋ < 4,4 (𝑜 / ) 4,4 6 ∈ ℕ ℋ / 11
Operator-State Map for ℋ / Non-local operator living at the end of the defect line E.g. Electron in QED 𝑀 𝑀 𝜈(𝑦) |𝜈〉 ∈ ℋ / 12
Crossing Relations of the ℤ 𝟑 Line = α Do it twice: 𝛽 & = 1 (the cocycle condition) 13
Crossing and Anomaly = α • 𝛽 = +1 : Non-Anomalous (can be gauged) • 𝛽 = −1 : Anomalous (can not be gauged) Indeed, the bosonic, unitary ℤ 𝟑 anomaly is classified by 𝛽 ∈ 𝐼 F (ℤ 𝟑 ,U(1))= ℤ 𝟑 14
Crossing and Anomaly = α • Consider the torus partition function of the would-be ℤ 𝟑 orbifold theory: “ ” K K + + ) + ) 𝑎 HIJ = & ( & ( Twisted sector Untwisted sector 15
When 𝛽 = −1 , this is an ambiguity ⟹ anomaly = 𝛽 16
Crossing and Anomaly (Recap) • 𝛽 = +1 : Non-Anomalous = α • 𝛽 = −1 : Anomalous 17
Outline • Symmetry and Anomaly in Two Dimensions • Modular Bootstrap 18
Modular Bootstrap • Positivity : Expansion on Virasoro characters • Crossing : Modular 𝑇 Transformation 19
� 𝜓 4 𝜐 = 𝑟 4STSK &V Torus Partition Function 𝜃(𝜐) • The torus partition function can be expanded on the Virasoro characters with positive coefficients 6ST/&V ] ℋ [𝑟 4ST/&V 𝑟 W 4 𝑎 𝜐, 𝜐̅ = 𝑈𝑠 = 0 𝑜 4,4 𝜓 4 (𝜐) 𝜓 4 6 (𝜐̅) 6 6∈ℋ 4,4 𝑜 4,4 6 ∈ ℕ ℋ 20
� Torus Partition Function with ℤ 𝟑 Line 𝑎 / 𝜐, 𝜐̅ = 𝑈𝑠 6ST/&V ] [ 𝑟 4ST/&V 𝑟 W 4 ℋ [𝑀 S ) 𝑀 \ = 0 (𝑜 4,4 − 𝑜 4,4 𝜓 4 (𝜐) 𝜓 4 6 (𝜐̅) 6 6 6∈ℋ 4,4 ± 𝑜 4,4 ∈ ℕ 6 ℋ 21
� Defect Hilbert Space 𝓘 𝑴 ℋ < [𝑟 4S T 6S T W 4 𝑎 / 𝜐, 𝜐̅ = 𝑈𝑠 &V 𝑟 &V ] 𝑀 = 0 (𝑜 / ) 4,4 6 𝜓 4 (𝜐) 𝜓 4 6 (𝜐̅) 6∈ℋ < 4,4 (𝑜 / ) 4,4 6 ∈ ℕ ℋ / 22
� � Positivity 𝑎 / 𝑎 ± 𝜓 4 𝜐 𝜓 4 K 𝑎 ± 𝜐, 𝜐̅ = ) ± = & ( 0 𝑜 4,4 6 𝜐̅ 6 6 ∈ℋ ± ∨ 4,4 0 𝑎 / (𝜐, 𝜐̅) = 0 (𝑜 / ) 4,4 6 𝜓 4 (𝜐) 𝜓 4 6 (𝜐̅) = 6∈ℋ < ∨ 4,4 0 23
Anomaly and Modular Transformation 𝑎 / (𝜐, 𝜐̅) is 𝑈 invariant under ∥ If 𝛽 = +1 𝑈 & 𝛽 b 𝑈 SK 𝑈 V If 𝛽 = −1 𝑎 / (𝜐, 𝜐̅) 24
Spin Selection Rule in ℋ / W of a state in the defect Hilbert space ℋ / • The spin 𝑡 = ℎ − ℎ is constrained by the anomaly 𝛽 [Chang-Lin-SHS-Wang-Yin] : ℤ If 𝛽 = +1 (Non-Anomalous) 2 𝑡 ∈ 1 4 + ℤ If 𝛽 = −1 (Anomalous) 2 25
Spin Selection Rule in ℋ / • Now we see that the anomaly controls the spin of non-local operators living at the end of the line . • How do we convert this information to constraints on local operators ? • Modular Transformation 26
Crossing 𝑎 𝑎 𝑇 The spins here depend on the 𝑇 anomaly 𝑎 / 𝑎 / 27
Positivity Crossing 𝑇 ± 𝑇 The spins here and there depend on the anomaly 28
2D CFT with Global Symmetry ℤ 𝟑 Non-Anomalous Anomalous Bound on Charged Operator NO 😮 YES 😎 Moral: It is harder to “hide” a symmetry if it is anomalous 29
Upper Bounds on ℤ 𝟑 Odd Operators 𝚬 S 3.0 Δ S (anomalous) 2.5 2.0 1.5 1.0 𝑡𝑣(2) K 0.5 WZW 𝑑 30 2 4 6 8
Why is this result interesting? (at least to me…) • As a Bootstrapper… • As an Anomaler... 31
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: v The symmetry is either spontaneously broken, or v There 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) • 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) 6 𝑨̅ = 𝜖̅ 𝐾 𝑨 = 0, • While 𝑲(𝒜) and 𝑲̅(𝒜 W) are separately conserved, 𝜖 𝐾 each of them generically generates an ℝ symmetry, instead of 𝑉 1 . 36
2D CFT with Global Symmetry 𝑽(𝟐) Non-Anomalous Anomalous Bound on U(1) Charged Operator NO 💀 YES 👽 37
Previous Work • The authors of [Benjamin-Dyer-Fitzpatrick-Kachru] and [Montero- Shiu-Soler] derived a bound on the lightest U(1) charged operator 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 𝑽(𝟐) 𝒙 𝑽(𝟐) 𝒐 𝑽(𝟐) 𝒆𝒋𝒃𝒉 Lightest Winding mode KK mode Winding or KK mode Charged Op. ∆ S 1 𝑆 & › œ K Min[ & , &› œ ] 2𝑆 & 2 NO 😮 NO 😮 YES 😎 Bound? Anomaly? NO NO YES 40
Generalization • 2d CFT with 𝑉(1) Global Symmetry • Higher Dimensions 41
Higher Dimensions • Is there such an anomaly-dependent bound on local operators 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
Higher Dimensions • 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 operators in 𝒆 > 𝟑 . • In 𝒆 = 𝟑 , those TQFTs have degenerate vacua (spontaneous symmetry breaking): Non-trivial local operators, non-trivial anomalies. 43
Turning to the non-anomalous case… • So we have derived a 3.0 Δ S (anomalous) bound for an 2.5 anomalous ℤ & . 2.0 • What can we say 1.5 about a non- 1.0 anomalous ℤ & ? 0.5 𝑑 2 4 6 8 44
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