Introduction all QFTs that we use in physics are in some sense - PowerPoint PPT Presentation

The TT Deformation of Quantum Field Theory John Cardy University of California, Berkeley All Souls College, Oxford ICMP , Montreal, July 2018 TT deformation

Introduction all QFTs that we use in physics are in some sense effective field theories, valid over only some range of energy/length scales if they are (perturbatively) renormalizable, this range of scales may be very large and they have more predictive power, but eventually new physics should enter if they are non-renormalizable (the action involves operators with dimension > d ) they may still be useful up to some energy scale ∼ UV cut-off Λ even so they may still make sense at higher energies if they have a ‘UV completion’, eg. if they flow from a non-trivial RG fixed point – ‘asymptotic safety’ TT deformation

However another possibility is that the UV limit is not a conventional UV fixed point corresponding to a local QFT, but is something else ( eg string theory): IR UV ? TT deformation

The TT deformation of 2d QFT is an example of a non-renormalizable deformation of a local QFT for which, however, many physical quantities make sense and are finite and calculable in terms of the data of the undeformed theory. However this deformation is very special – this has been termed ‘asymptotic fragility,’ which could be used as a constraint on physical theories. TT deformation

What is TT ? consider a sequence of 2d euclidean field theories T ( t ) ( t ∈ R ) in a domain endowed with a flat euclidean metric η ij , each with a local stress-energy tensor T ( t ) ij ( x ) ∼ δ S ( t ) /δ g ij ( x ) T ( 0 ) is a conventional local QFT (massive, or massless (CFT)) deformation is defined formally by � S ( t + δ t ) = S ( t ) − δ t det T ( t ) d 2 x det T ( t ) d 2 x into correlation � equivalently by inserting functions ⇒ note that this uses T ( t ) , not T ( 0 ) ⇐ TT deformation

2 ǫ ik ǫ jl T ij T kl ∝ T zz T ¯ z − T 2 det T = 1 in complex coordinates z ¯ z ¯ z for a CFT this is T zz T ¯ z ≡ TT z ¯ since this has dimension 4 , we would expect � det T � ∼ Λ 4 Zamolodchikov (2004) pointed out that by conservation ∂ i T ij = 0 that ∂ ǫ ik ǫ jl T ij ( x ) T kl ( x + y ) = ∂ ǫ mk ǫ jl T ij ( x ) T kl ( x + y ) ∂ y m ∂ x i so that in any translationally invariant state � ǫ ik ǫ jl T ij ( x ) T kl ( x ) � = � ǫ ik ǫ jl T ij ( x ) T kl ( x + y ) � finite, and calculable in terms of matrix elements of T ij so the deformation is in some sense ‘solvable’ TT deformation

TT as a topological deformation In this talk I’ll describe a different approach which also works in non-translationally invariant geometries � ǫ ik ǫ jl T ij ( x ) T kl ( x ) d 2 x = � � ǫ ik ǫ jl h ij ( x ) h kl ( x ) d 2 x + � h ij T ij d 2 x e ( δ t / 2 ) [ dh ij ] e − ( 1 / 2 δ t ) h ij = O ( δ t ) may be viewed as an infinitesimal change in the metric g ij = η ij + h ij in 2d we can always write h ij = a i , j + a j , i + δ ij Φ where a i is an infinitesimal diffeomorphism x i → x i + a i ( x ) and e Φ ∼ 1 + Φ is the conformal factor TT deformation

however, at the saddle point h = h ∗ [ T ] (which is sufficient since the integral is gaussian) T ij ∝ ǫ ik ǫ jl h ∗ kl conservation ∂ i T ij = 0 then implies that � Φ = 0 , so the metric is flat Φ can be absorbed into the diffeomorphism moreover we can take a i , j = a j , i TT deformation

the action is then � � ǫ ik ǫ jl ( ∂ i a j )( ∂ k a l ) d 2 x − 2 ( ∂ i a j ) T ij d 2 x ( 2 /δ t ) � � ∂ i ( ǫ ik ǫ jl a j ∂ k a l ) d 2 x − 2 ∂ i ( a j T ij ) d 2 x = ( 2 /δ t ) and so is topological : for a simply connected domain only a boundary term for a closed manifold, only contributions from nontrivial windings of a i only h ij = 2 a i , j needs to be single valued TT deformation

Torus L+L’ L’ L 0 torus made by identifying opposite edges of a parallelogram with vertices at ( 0 , L , L ′ , L + L ′ ) in C saddle point is translationally invariant h ∗ ij = δ t ǫ ik ǫ jl � T kl ( 0 ) � = δ t ǫ ik ǫ jl ( 1 / A ) � � L k ∂ L l + L ′ log Z ( t ) k ∂ L ′ l ( A = L ∧ L ′ = area) change in log Z ( t ) is � � T ij ( x ) h ∗ ij [ T ] � c d 2 x = ( δ t ) ǫ ik ǫ jl � � L i ∂ L j + L ′ i ∂ L ′ � T kl ( 0 ) � j TT deformation

Evolution equation for partition function ∂ ∂ tZ ( t ) ( L , L ′ ) = ǫ ik ǫ jl � � � � L k ∂ L l + L ′ L k ∂ L l + L ′ Z ( t ) ( L , L ′ ) k ∂ L ′ ( 1 / A ) k ∂ L ′ l l In terms of Z ( t ) ≡ Z ( t ) / A ∂ t Z = ( ∂ L ∧ ∂ L ′ ) Z simple linear PDE, first order in ∂ t if log Z ∼ − f t A , f 0 ∂ t f t = − f 2 ⇒ f t = t 1 + f 0 t – no new UV divergences in the vacuum energy TT deformation

interpretation as a stochastic process ∂ t Z = ( ∂ L ∧ ∂ L ′ ) Z is of diffusion type, where Z ( t ) is the pdf for a Brownian motion ( L t , L ′ t ) in moduli space with ( L t 1 − L t 2 ) ∧ ( L ′ t 1 − L ′ � � E t 2 ) = | t 1 − t 2 | In particular the mean area E [ L t ∧ L ′ t ] ∼ t as t → + ∞ , with, however, absorbing boundary conditions on L ∧ L ′ = 0 . The relation between the two approaches is � � Z ( t ) ( L 0 , L ′ Z ( 0 ) ( L t , L ′ 0 ) = E t ) TT deformation

Finite-size spectrum L+L’ L’ H P L 0 Tr e − ( R Im τ )ˆ H ( R )+ i ( R Re τ )ˆ Z ( L , L ′ ) P ( R ) = e − ( R Im τ ) E ( t ) � n ( R )+ i ( R Re τ ) P n ( R ) = n where R = | L | , τ = L ′ / L and ˆ H , ˆ P are the energy and momentum operators for the theory defined on a circle of circumference R . PDE for Z ( t ) then leads after some algebra to [Zamolodchikov 2004] ∂ t E ( t ) n ( R ) = − E ( t ) n ( R ) ∂ R E ( t ) n ( R ) − P 2 n / R For P n = 0 this is the inviscid Burgers equation. TT deformation

If T ( 0 ) is a CFT, E ( 0 ) n ( R ) = 2 π ˜ ˜ ∆ n / R where ∆ n = ∆ n − c / 12 Solution is then (with P n = 0 )  �  1 − 8 π ˜ ∆ n t n ( R ) = R E ( t )  1 −  R 2 2 t energies with ˜ ∆ n > 0 become singular at some finite t > 0 energies with ˜ ∆ n < 0 become singular at some finite t < 0 TT deformation

Thermodynamics identifying R ≡ β = 1 / kT , E ( t ) 0 ( β ) = β f t ( β ) , where f t = free energy per unit length for fixed t < 0 there is a transition at finite T ∼ 1 / ( − ct ) 1 / 2 this is of Hagedorn type where the density of states grows exponentially if T ( 0 ) is a free boson, then E ( t ) n ( R , P ) is the spectrum of the Nambu-Goto string [Caselle et al. 2013] which is known to have a Hagedorn transition as a world sheet theory on the other hand for t > 0 the free energy is analytic but the energy density E is finite as T → ∞ , suggesting another branch with negative temperature TT deformation

S -matrix if T ( 0 ) is a massive QFT, the single particle mass spectrum M is not affected by the deformation the 2-particle energies for MR ≫ 1 have the expected form � M 2 + P 2 E = 2 where however P is quantized according to PR + δ ( P ) ∈ 2 π Z , where δ ( t ) ( P ) is the scattering phase shift consistency with the evolution equation then requires δ ( t ) = δ ( 0 ) − t M 2 sinh θ where P = M sinh ( θ/ 2 ) this is equivalent to a CDD factor in the 2-particle S -matrix [Smirnov-Zamolodchikov 2017] S ( t ) ( θ ) = e − itM 2 sinh θ S ( 0 ) ( θ ) TT deformation

if T ( 0 ) is integrable, so is T ( t ) , and applying Thermodynamic Bethe Ansatz to the deformed S -matrix yields the expected form for E ( t ) n ( R ) [Cavaglià et al. 2016] in fact this dressing of the S -matrix works for non-integrable theories as well: [Dubovsky et al. 2012, 2013] a p j a < b ǫ ij p i S ( t ) ( { p } ) = e − i ( t / 8 ) � b S ( 0 ) ( { p } ) it corresponds to the dressing of the original theory by Jackiw-Teitelboim [1985, 1983] (topological) gravity: � √− g ( φ R − Λ) d 2 x S ( t ) = S ( g ij , matter ) + where Λ ∼ t − 1 the torus partition function of this theory has been computed and shown to satisfy the PDE of the TT deformed theory [Dubovsky et al. 2017, 2018] TT deformation

Simply connected domain boundary action is � � � ǫ jl ( a j ∂ k a l ) ds k − 2 ǫ ik a j T ij ds k − λ a k ds k ( 1 / 8 δ t ) a k ds k = 0 � λ = lagrange multiplier enforcing fermion on boundary: coupling to T simplifies with conformal boundary condition T ⊥� = 0 gaussian integration then gives � � G ( s − s ′ ) � T ⊥⊥ ( s ) T ⊥⊥ ( s ′ ) � c ǫ kl ds k ds l δ log Z ∝ δ t | s − s ′ | <ℓ/ 2 2 sgn ( s − s ′ ) − ( s − s ′ ) /ℓ where ℓ = perimeter and G ( s − s ′ ) = 1 TT deformation

for a disk | x | < R , we may decompose into modes n a n e in θ a ⊥ ( θ ) = � the n = 0 mode gives the evolution equation ∂ t Z = ( 1 / 4 π )( ∂ R − 1 / R ) ∂ R Z where R = perimeter / 2 π corresponds to the stochastic (Bessel) process ∂ t R = − 1 η ( t ′ ) η ( t ′′ ) = 1 2 πδ ( t ′ − t ′′ ) 4 π R + η ( t ) , curvature driven dynamics as in 2d coarsening TT deformation