Aspects of geometric phases in QFT Vasilis Niarchos Durham - - PowerPoint PPT Presentation

Aspects of geometric phases in QFT

Vasilis Niarchos

Annual Theory Meeting Durham, December 19, 2017 Durham University based on work with Marco Baggio and Kyriakos Papadodimas

Parameter-spaces and
 geometric phases in QM
 (review I)

2


 Frequently in QM the Hamiltonian/spectrum depends on continuous parameters —external couplings 
 
 can be components of a fixed or slowly varying background 
 The space of λ’s is endowed with rich geometric properties that encode important physical effects

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(λ1, λ2, . . .)

Berry phase is one of the most characteristic examples

under adiabatic cycles on space of λ’s quantum states can pick up a phase 
 
 
 
 
 much like a frame dragged around a curved manifold

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|Ψ(~ )iT = eiγ e− i

~

R T

0 dt EΨ(t) |Ψ(~

)i0

Berry phase

• Intrinsic property of the quantum system 


Phases depend only on path C in λ-space
 
 
 


• r


• For D degenerate states the U(1) phase upgrades to a U(D) transform


⇒ Berry connection is non-abelian

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= i I

C

d~ hΨ(~ )|@~

|Ψ(~

)i

Pancharatnam-Berry connection
 


connection on a vector bundle 


• f Hilbert spaces

= i I

C

d~ A(~ )

• there is a corresponding curvature


• for which one can write a general spectral formula



 
 
 
 
 in sector with energy En and degenerate states 
 


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|n, ai

⇣ F (n)

µν

⌘

ab =

X

m6=n

X

c,d

1 (En Em)2 hn, b|∂µH|m, cigcd

(m)hm, d|∂νH|n, ai (µ $ ν)

gab = ha|bi Fµν = ∂Aν ∂λµ − ∂Aµ ∂λν + [Aµ, Aν]

Berry phase is a physically detectable effect with many applications… 
 The curvature is a very interesting observable


• typically hard to evaluate analytically, but
• if known it helps constrain the parametric dependence of state
• verlaps and potentially other observables 


(examples soon)

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⇣ F (n)

µν

⌘

ab

Geometric phases in (continuum) QFT?

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tt* equations: a beautiful example from the 90s (Cecotti-Vafa ’91)

• 2d QFTs with 4 real supersymmetries
• parameter space: complex (superpotential) couplings
• topologically twisted, spectrum truncated to 1/2-BPS states


(chiral - antichiral)
 
 corresponding operators close among themselves under the OPE (chiral ring)

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φi(x)φj(0) ∼ Ck

ij φk(0) + regular

Cecotti & Vafa computed the Berry curvature of these ground states and found an expression that closes on the chiral ring
 
 
 


• RHS is algebraic functional of OPE coefficients and 2-point functions



 
 
 chiral-antichiral overlap → all SUSIES broken


(topological-antitopological → tt*)

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• Fi¯

j

L

K = [Ci, ¯

Cj]L

K

≡ CP

iKgP ¯ QC∗ ¯ Q ¯ j ¯ R g ¯ RL − gK ¯ NC∗ ¯ N ¯ j ¯ U g ¯ UV CL iV

gK ¯

L = h¯

L|Ki

• LHS is curvature, involves derivatives wrt complex couplings

• In a set of conventions LHS is expressed only in terms of derivatives
• f the 2-point overlaps


 Leads to PDEs for coupling dependence of non-SUSY quantities which Cecotti & Vafa solved in several examples

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λi

Unfortunately these ideas were not pushed much further in the context

• f higher-dimensional QFTs

• tt* equations in 4d QFTs were unknown until recently and


results of analogous power were beyond reach

• Berry phase as an intrinsic quantum property was not explored

systematically in QFT

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Parallel ideas in QFT
 (review II)

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The idea to think about the properties of the space of parameters (theory space) seriously has been discussed over the years in (continuum) QFT in various contexts

• Wilsonian RG
• Zamolodchikov c-theorem in 2d QFTs



 Zamolodchikov metric

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gij = x4hOi(x)Oj(0)i

• x2=x2

Early ‘90s: much interest in 2d CFTs / worldsheet description of strings 
 In that context: 2d QFT couplings = spacetime fields Interest to understand the theory space of exactly marginal couplings (deformations that preserve conformal invariance) conformal manifolds
 It was understood (Kutasov, Sonoda, Zwiebach...) that besides the Zamolodchikov metric the conformal manifolds also possess natural notions of parallel transport, connection

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Conformal perturbation theory: a connection in theory-space tells us how to compare operators in near-by CFTs
 A covariant derivative incorporates regularization prescriptions

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rµhϕ1(z1) · · · ϕn(zn)i ⇠  Z dd+1x hOµ(x)ϕ1(z1) . . . ϕn(zn)i

• regularised

the curvature of this connection involves integrated correlation functions
 
 Roughly:
 
 
 
 
 
 integrations lead to divergences
 regularisation leads to non-vanishing commutators ⇒ curvature

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Z dd+1x Z dd+1y hO[µ(x)Oν](y)ϕ1(z1)ϕ2(z2)i

}

antisymmetry

[rµ, rν]12 ⇠

Ranganathan-Sonoda-Zwiebach (’93) prescription
 (cut out small balls around operators, do not allow collisions, remove divergent pieces) 4-point operator formula for the curvature
 
 
 
 
 
 
 
 


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(Fµν)kl = 1 (2π)4 Z

|x|≤1

d4x Z

|y|≤1

d4y hϕl(1)O[µ(x)Oν](y)ϕk(0)i

• n R4

Connection in conformal perturbation theory vs Berry connection in QM 
 
 They are obviously related concepts… 
 Q: Can one go from one to the other? 
 Q: Can one import the geometric phases of QM systematically & more generally in QFT? Computable? 
 Q: Will this lead to new lessons?

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Berry phases in (non-perturbative) QFT
 (old meets modern)

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Technical approach



 Berry phase by quantizing QFT in the Hamiltonian framework
 
 Obvious issues:

• UV divergences: 


☞ renormalize

• IR issues: e.g. continuous spectra


☞ put theory on a hypercylinder : , compact

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R × Q Q

Baggio-VN-Papadodimas ’17

• Consider electromagnetism with a θ interaction


• θ has non-trivial implications if it varies in spacetime, or if there are

boundaries/walls

• interested in adiabatic changes of θ in time

L = − 1 4e2 FµνF µν + θ 64π2 Fµν ˜ F µν

e.g. Wilczek, PRL ‘89

Example 1: Berry phase of photons

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• this theory is free: as we change e and θ adiabatically the spectrum is

unchanged, but the energy eigenstates can rotate


• Hamiltonian deformations


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@e2H = 1 e4 Z d3x ⇣ ~ E2 − ~ B2⌘ , @θH = 1 8⇡2 Z d3x ~ E · ~ B

• Place on and quantize in Coulomb gauge


• Evaluate the spectral sum in the formula for Berry curvature



 The curvature has non-vanishing components only for identical external states


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R × T3

~ A(t, ~ x) = X

~ k

X

✏=±

s ~e2 2!kV ⇣ ~ e✏(~ k)a~

k,✏e−i!kt+i~ k·~ x + ¯

~ e✏(~ k)a†

~ k,✏ei!kt−i~ k·~ x⌘

!k = c|~ k| , ki = 2⇡ni R , ni ∈ Z , V = R3

• for a general multi-photon state with 


positive helicity photons, negative helicity photons
 
 
 
 
 
 
 
 ☞ a non-trivial Berry phase for photons ☞ independent of momentum

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n+ n− (Fτ ¯

τ)(n+,n−) = n+ − n−

8 1 (Imτ)2 |n+, n−i

τ = θ 2π + i4π e2

☞ Implications:
 linearly polarized light changes polarization under (e,θ) variation
 
 
 
 
 
 
 
 
 
 ☞ experimentally visible? (e.g. topological insulators…)


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|pz, ˆ xi = 1 p 2 ⇥ |pz, +i + |pz, i ⇤ |pz, ˆ φi = 1 p 2 ⇥ eiφ|pz, +i + e−iφ|pz, i ⇤

e.g. Essin-Moore-Vanderbilt, PRL ’09

Example 2: conformal manifolds, conformal pert. theory

• In CFT there is a natural correspondence between states and operators



 OPERATOR-STATE correspondence
 
 in radial quantization or on 
 
 
 


• The Berry connection for states should map to a natural connection

for operators in Conformal Perturbation Theory

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R × Sd |Ii ⇠ OI(0)|0i

• Two seemingly different expressions for curvature



 
 
 
 
 
 
 
 
 
 These expressions compute the same object !

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(Fµν)IJ = X

n/ ∈HI

X

a,b∈Hn

1 (∆I ∆n)2 hJ|∂µH|n, aigab

(n)hn, b|∂νH|Ii (µ $ ν)

(Fµν)IJ = Z

|x|≤1

dd+1x Z

|y|≤1

dd+1yhOJ(1)Oµ(x)Oν(y)OI(0)i (µ $ ν) Berry CFT

double d-dim integral double (d+1)-dim integral Baggio-VN-Papadodimas ’17

Example 3: SUPERconformal manifolds, tt* equations

Conformal manifolds are rare in non-SUSY QFTs In SUSY QFTs they are abundant: superconformal manifolds 
 Depending on amount of SUSY, their existence can be argued (non- perturbatively) in QFT, their dimension is known etc...
 
 
 Also examples in large-N limit in AdS/CFT from (super)gravity solutions


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 Here focus on 4d N=2 SCFTs --8 Poincare supersymmetries (very similar statements for 2d N=(2,2) --4 Poincare SUSIES) 
 


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• 4d N=2 has 8 supercharges 


& superconformal partners

• N=2 chiral primaries: (+ complex conjugate)



 
 chiral ring under the Operator Product Expansion (OPE)
 
 
 Exactly marginal interactions are descendants of N=2 chiral primaries
 


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Qi

α, ¯

Qi, ˙

α

Sα

i , ¯

Si, ˙

α

(i = 1, 2 α = ±) [ ¯ Qi

˙ α, φI] = 0

∆ = R 2

Oi = Z d4θ φi , ¯ Oj = Z d4¯ θ ¯ φj

φI(x)φJ(0) = CK

IJ φK(0) + . . .

Details of 4d N=2 SUSY

Similar to the chiral ring in the 2d Cecotti-Vafa case above Compute (non-perturbatively) the Berry curvature of chiral primary states If the computation closes on the chiral ring we get a 4d version of the 
 tt* equations 
 Important note: unlike Cecotti-Vafa we consider the physical theory without topological twist!

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Berry curvature in sector of N=2 chiral primary states
 
 
 
 


conveniently recast as


Insert expressions for δΗ and compute…

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(Fµν)I ¯

J = lim x→0

⇣ ˜ Fµν ⌘

I ¯ J

(Fµν)IJ = X

n/ ∈HI

X

a,b∈Hn

1 (∆I ∆n)2 hJ|δµH|n, aigab

(n)hn, b|δνH|Ii (µ $ ν)

⇣ ˜ Fµν ⌘

IJ = hJ|δµH(H + ˆ

R x)−2δνH|Ii (µ $ ν)

N=2 superconformal deformations

After a few elementary steps using SCA relations
 
 
 
 


• regulate by separating integrated insertions in time
• take limit
• use OPEs

and the result is…


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H + ˆ R = H − R 2

Dolan-Osborn conventions

⇣ ˜ Fµν ⌘

I ¯ J = i

" h ¯ J| I φk (H + ˆ R)4 (H + ˆ R x)2 I ¯ φ¯

l|Ii h ¯

J| I ¯ φ¯

l

(H + ˆ R)4 (H + ˆ R x)2 I φk|Ii # 4i " h ¯ J| I φk (H + ˆ R)2 (H + ˆ R x)2 I ¯ φ¯

l|Ii h ¯

J| I ¯ φ¯

l

(H + ˆ R)2 (H + ˆ R x)2 I φk|Ii #

x → 0

φk φ ¯ φ ∼ . . .

tt* equations for 4d N=2 SCFTs
 
 
 
 
 
 
 Derived earlier by Papadodimas ’09 in conformal perturbation theory 
 with the use of superconformal Ward identities 
 
 Rederived here as a non-perturbative Berry phase Are these equations useful?


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• Fi¯

j

• K ¯

L = −[Ci, ¯

Cj]K ¯

L + gi¯ jgK ¯ L

✓ 1 + R c ◆

gK ¯

L = h¯

φL(1)φK(0)i

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Example: SU(2) N=2 SCQCD
 
 chiral primary operators 
 
 in normalization 
 
 non-trivial info in 2-point functions
 
 
 tt* equations
 
 
 


φ2n ∝

• Tr

⇥ ϕ2⇤n

φ2(x)φ2n(0) = φ2n+2(0) + . . .

hφ2n(x)¯ φ2n(0)i = g2n(τ, ¯ τ) |x|4n ∂τ∂¯

τ log g2n = g2n+2

g2n − g2n g2n−2 − g2

g0 = 1 , n = 1, 2, . . .

semi-infinite Toda

Baggio-VN-Papadodimas ’14

τ = θ 2π + 4πi g2

Y M

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the unique physically relevant solution requires further input
 
 for example, knowledge of the Zamolodchikov metric 
 
 
 
 Recent developments in SUSY QFT have allowed access to this quantity
 
 A beautiful result by Gerschkovitz-Gomis-Komargodski ’14 relates g2 to the 4-sphere partition function 
 
 
 4-sphere PF reduced to a matrix integral using SUSY localisation

g2 = ∂τ∂¯

τZS4

g2 ⇠ hTr(ϕ2)Tr( ¯ ϕ2)i

Pestun ‘07

Baggio-VN-Papadodimas ’14

SU(2) N=2 SCQCD

+ ALL instanton
 corrections

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h

• Trϕ2n1

Trϕ2n2 Tr ¯ ϕ2n1+n2i

τ = θ 2π + 4πi g2

Y M

This study led to the first exact non-perturbative computation of non-trivial 3-point functions in 4d QFTs
 
 
 In fact, supersymmetric localization now allows the complete solution of extremal N-point functions in the N=2 chiral ring
 
 > an in principle complete solution of the 4d tt* equations


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Baggio-VN-Papadodimas ’14 Gerschkovitz-Gomis-Ishtiaque-Karasik-Komargodski-Pufu ’16

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Berry phase, tt* equations Supersymmetric
 localization on spheres Non-perturbative
 correlation functions
 in flat space 4d N=2 SCFTs
 1/2-BPS sectors

Baggio-VN-Papadodimas ’16
 (also Gomez-Russo ’16, ‘17)

Large-N ’t Hooft limit
 SU(N) N=2 SCQCD

hTrϕn1Trϕn2Tr ¯ ϕn1+n2i

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4 N

4 √ 3 N 8 √ 2 N

4 √ 15 N

• useful even at tree level: compact expressions for complicated

Wick contractions

• vast continuous families of interacting SCFTs with access to highly

non-trivial (non-SUSY) data that probe global structure in 
 theory-space, dualities etc…

• new detailed information beyond large-N limit that could be useful in 


further probes of gauge-gravity dualities

• QFTs can exhibit non-trivial Berry phases 


(examples in this talk: E&M, CFTs)
 
 Possibly new physically interesting effects? 


• Evaluating the Berry phase of QFTs by putting them on different

curved manifolds appears to be a useful strategy
 
 revisit Hamiltonian methods in QFT?

Main messages - outlook

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• The Berry phase of 1/2-BPS states in 4d N=2 & 2d N=(2,2) SCFTs are

non-trivial examples where non-perturbative computations are possible


• Very interesting to extend these results to: 


• lower SUSY, 


• non-conformal theories…



 where methods like SUSY localization are not yet available…

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