KdV charges in $T\bar{T}$ theories and new models with super-Hagedorn behavior

Two-dimensional CFTs and integrable models have an infinite set of conserved KdV higher spin currents. These currents can be argued to remain conserved under the $T\bar{T}$ deformation and its generalizations. We determine the flow equations the KdV charges obey under the $T\bar{T}$ deformation: they behave as probes"riding the Burgers flow"of the energy eigenvalues. We also study a Lorentz-breaking $T_{s+1}\bar{T}$ deformation built from a KdV current and the stress tensor, and find a super-Hagedorn growth of the density of states.

1 Introduction and summary which we start deforming as the seed theory. 85 It was shown in [4, 5] that the energy spectrum of TT -deformed relativistic theories on 86 the cylinder is governed by the equation 87 ∂ λ E n = −π 2 E n ∂ L E n + P 2 n L , (1.1) where E n and P n are the energy and momentum eigenvalues, L is the circumference of the 88 circle, and λ is the deformation parameter. In this paper we derive that the quantum KdV 89 charges P s n of the eigenstate |n , if present in the seed theory, obey 90 ∂ λ P s n = −π 2 E n ∂ L P s n + P n s P s n L . (1. 2) The allowed values of s are ±1, ±3, . . . . This equation was also obtained using integrability 91 techniques of [5,57]. Our field theory derivation applies more broadly, to the TT deformation 92 of any Lorentz-invariant theory that contains at least one higher spin conserved charge, 93 and hence rules out the possibility that the evolution equation (1.2) is a miracle of some 94 special models. 95 There is a beautiful analogy with hydrodynamics. Equation (1.1) is the forced inviscid 96 Burgers equation where the . . . stand for terms that involve derivatives and lower powers of the stress tensor.
These can be adjusted to remain conserved after deformation, namely To show that this is indeed possible we have to review the methods of [4]. We work in the 149 Hamiltonian formalism. 150 In a CFT the algebra of conserved charges is the universal enveloping algebra of the 151 Virasoro algebra U(Vir), which is formed by the sums of products of L n 's. 1 In fact, a (T s+1 (x)Θ t−1 (z) − Θ s−1 (x)T t+1 (z)) + (reg. terms) , (2.7) 1 There is an antiholomorphic copy as well. These charges are integrals of local holomorphic currents of the form z n T m . These charges in general do not commute with the Hamiltonian H, they are conserved because their noncommutativity with H is compensated by their explicit time dependence. The maximal commuting set of these charges are the KdV charges. 2 In the notations of [4] our operators A s σ are equal to iAσ,s for 0 < σ, s, iBσ,−s for s < 0 < σ, iĀ−σ,−s for σ, s < 0, and iB−σ,s for σ < 0 < s. proof of (2.9), which relaxes the assumption of non-degenerate energy spectrum that was 181 needed in [4], and also allows for the generalization of factorization to the operator: 182 X s 1 ...s k σ 1 ...σ k ≡ k! A s 1 [σ 1 · · · A s k σ k ] reg , n|X s 1 ...s k σ 1 ...σ k |n = k! n|A s 1 [σ 1 |n n|A s 2 σ 2 |n · · · n|A s k σ k ] |n . (2.10) Note that X these operators can also be written as X t ±1 . These deformations by conserved current 220 components correspond to turning on background gauge fields. The condition (2.12) is which then gives δP = 0, δH = dy A t 1 as required. In Section 2 we understood how the KdV charges change under irrelevant deformations.

230
Let us now choose joint eigenstates |n of the commuting charges P s , and denote their 231 eigenvalues by P s n ≡ n|P s |n . We can use the Hellman-Feynman theorem for the 232 infinitesimal deformation δ P s n = n|δP s |n to write: Lorentz-invariance even for u = −t, by assigning a spin to the coupling λ. This spurion 252 analysis is performed in Appendix C. 253 We have analyzed and solved a problem similar to (3.1) for a family of deformations made 254 out of an abelian current and the stress tensor in [56]. There we have also demonstrated 255 that our current tools are inadequate to determine A t s n in general. In the rest of this 256 section we focus on the case X 1,−1 = TT . In Section 4 we analyze a special case of (3.1)

257
where we can make progress, while we discuss perturbative aspects in Section 4 and in 258 Appendix F.  Then transforming to complex coordinates, 4 where in the second line we used (2.3) and that ∂ L P n = −P n /L which follows from 273 momentum quantization, P n ∈ 2πZ L . Similarly, we get T 0 n − Θ −2 n = 2π∂ L P −1 n .

274
This motivates us to compute ∂ L P s . In Appendix E we show which reduces to the equations above for s = ±1. Taking diagonal matrix elements and 276 using that ∂ L P s n = ∂ L P s n (valid in eigenstates of P s ), we find where in the second equality we used (2.6) valid in Lorentz invariant theories. 280 ∂ λ P s n = −π 2 E n ∂ L P s n + P n s P s n L .
(3.9) This is our main result. Setting s = ±1 and using (2.3) we recover the Burgers equation 281 for E n , and the fact that P n remains undeformed:  Let us start by solving the equations in two special case. We drop the expectation value 289 symbols and the n subscript to lighten the notation. If we set P = 0, the equation simply  Burgers equation with this initial data is familiar from the literature: 5 (3.13) Once we know the Burgers flow, we can solve for the KdV charges that probe it. In fact we 296 do not have to know the explicit form of the solution, (3.13), to verify that (3.14) solves (3.9), if we use that P ±1 (λ, L) satisfies (3.10).

A check from integrability and concluding comments 299
Integrable field theories provide a useful testing ground of our results. The TT deformation 300 changes the two particle S-matrix of an integrable field theory by a simple CDD factor: where m is the mass, and θ i the rapidities. Plugging this result into the nonlinear integral 302 equation that determines the spectrum gives the deformed spectrum in terms of the initial 303 one. This computation was done for the energy in [5] and extended to KdV charges and 304 other deformations in [57]. Instead of repeating their derivation, we simply copy their 305 equations (5.27) and (5.30) in our notation in (F.1), and here we specialize to the TT case 306 (corresponding to taking u = 1 in (F.1)). The equation reads (3.16) We recognize that the flow equation is identical to (3.9). This match is a strong check of 308 our results. In Appendix F we discuss in detail their deformations with u = 1.

309
Let us comment on the regimes of validity of the different derivations of (3.9). The    319 We return to the analysis of (3.1): we study deformations such as X u,−1 = T u+1T − Θ u−1Θ 320 (sometimes called T u+1T for short) that break Lorentz invariance. Specifically, we write 321 an evolution equation for the spectrum of zero-momentum states under the X 1,u − X −1,u 322 deformation. This incidentally implies that our main result (3.9) holds for zero-momentum 323 states even without assuming Lorentz invariance. We then explain why our current methods 324 do not allow writing an evolution equation for general states under these deformations. 325 Finally, we solve the evolution of zero-momentum states and find the asymptotic density of 326 states to shows super-Hagedorn growth.  Importantly, this choice is preferred if the seed theory is a CFT, as we show in Appendix C.

331
This makes it nontrivial to compare our results with those of [57]. What we find in 332 Appendix F is that the two papers describe different deformations, even after accounting   For the deformation by X 1,u − X −1,u , (3.1) gives The relation (3.8) A 1 s − A −1 s n = −2π∂ L P s n holds without assuming Lorentz invariance.

343
As always, there is no general way to determine A u s n . For states |n with zero momentum 344 this issue does not show up since P 1 − P −1 n = −P n = 0, and one has 345 ∂ λ P s n = 2π 2 P u n ∂ L P s n if P n = 0 .  We note that (4.2) also describes the deformation JT − JΘ, which is a special case of and (4.2) surprisingly agree for states that have zero momentum and P u n = P −u n (for 356 instance states that are parity-invariant in the seed theory).
Finally, an easy calculation shows that the X 1,u − X −1,u and X 1,v − X −1,v deformations 358 commute (in the zero-momentum sector), since the following result is symmetric in u ↔ v:

359
∂ λu ∂ λv P s n = 2π 2 ∂ λu P v n ∂ L P s n + P v n ∂ L ∂ λu P s n = 4π 4 P u n ∂ L P v n ∂ L P s n + P v n ∂ L P u n ∂ L P s n + P v n P u n ∂ 2 L P s n .   The evolution equation (4.2) transports KdV charges along characteristics determined 380 by P u (to avoid clutter we leave implicit the dependence on |n ), so we can simply adapt 381 results (3.11) from the TT case and get (4.6) As for TT the solution with CFT initial conditions is much more explicit. We use the same 383 logic as around (3.12). First we set s = u, and using the CFT initial conditions (3.12) we 384 find the solution: where f u is the unique solution to the polynomial equation The JT − JΘ deformation (u = 0) has to be treated separately, and the solution of (4.2) is (4.10) Note that we get a divergence for λ = −L/(πQ n ), which is the analog of the branch point  It is particularly interesting to consider the asymptotic behavior of the spectrum. For that 395 we need to solve (4.8) for For p uλ negative enough (see footnote 7) we formally get a complex solution, a familiar 398 behavior from the study of TT .

399
In the CFT, high energy primary states in the zero momentum sector have where the inconvenient alternating sign ultimately follows from the sign in the decomposition To have a real asymptotic spectrum, it follows from 402 the condition p uλ > 0 that λ < 0 for u = ±1, ±5, . . . and λ > 0 for u = ±3, ±7, . . . .

403
Plugging (4.12) into (4.11) for s = ±1 we get (again for p uλ 1) (4.13) 7 One can write a series solution and recast it as a hypergeometric function which takes real values for x > xmin ≡ − |u| |u| /(|u| + 1) |u|+1 and has a branch point at x = xmin. Another way to find this branch point is to compute the discriminant of (4.8), when seen as a polynomial of fu(x).
, which vanishes at xmin, indicating that two solutions collide for this value of x. This is the analogue of the square-root singularity in the usual Burgers equation. 8 To recover this result from the formulas (6.4) of [56], we take A = 0 corresponding to the JT − JΘ deformation, then E = (1/L)(s − C/B) = e/(L + πλ Qn), where we simply set g JT = −gJΘ = 1, and = λ and specialize to zero momentum. The very attentive reader will notice that we absorbed an i in the definition of the deformation compared to [56] to make formulas real. In [56] the special A = 0 case was not analyzed separately, this was first done in [34]. 9 The equation does not have a real solution for x → −∞. (4.14) where we used that E = e L in the CFT. Expressing e with the energies of the deformed 406 theory from (4.13), we obtain for the appropriate sign of λ that depends on the value of u as discussed above. Note that the density of states is now independent of L, in stark contrast to the extensive entropy

413
A generalization is to deform a CFT by a linear combination u λ u (X 1,u − X −1,u ).

414
Similar calculations 10 lead to (4.16) Different choices of λ u appear to accomodate arbitrarily strong (e.g., doubly exponential) 416 super-Hagedorn growth of the density of states. 11 However, since our results only concern 417 zero-momentum states, they are not sufficient to determine when the deformation remains 418 well-defined: there could be divergences in the sum over u for some states.

419
In the case of the JT − JΘ deformation the Cardy growth remains, but the central 420 charge is replaced by a charge dependent expression: (4.17) where we took the full density of states, hence the replacement (c − 1) → c.  10 A convenient shortcut goes as follows. Charges are transported along characteristics, specifically Ps(λ, L) = P (0) s L + 2π 2 u λuPu as in (4.6). High-energy primary states of the CFT obey (4.12) P (0) s ≈ (−1) (|s|+1)/2 (E (0) /2) |s| . This relation is transported along characteristics. Now use the definition e = L E (0) (L ) valid for any L combined with the transport equation to express the initial dimensionless energy e in terms of the deformed energy E: this gives e = L+ u 2π 2 (−1) (|u|+1)/2 λu(E/2) |u| E. Deleting the negligible term L from this expression and plugging into the Cardy growth (4.14) for e gives (4.16).
11 Even though (4.16) formally allows for depletion of the density of states if λu is fine tuned, the formula breaks down for those cases due to a Jacobian factor that we neglected, and we expect descendent states to ruin cancellations either way.
The result (4.15) however provides extra motivation to study the T u+1Tu+1 deformation, 430 as these Lorentz invariant theories may give rise to exotic UV asymptotics, which would 431 manifest itself in a density of states similar to (4.15). A natural guess based on the simple 432 dependence of ρ(E) on λ of (4.15) and dimensional analysis for the density of states in 433 these theories is New ideas will be needed to establish (or rule out) this guess.
On the other hand, [P λ , P τ ] = 0 and the Jacobi identity imply that First, we can deduce a symmetry property. To make the derivation easier to parse, 446 above the equal signs we write the relation we use. We repeatedly transpose neighboring 447 subscripts to find We take a derivative with respect to one of the coordinates only (say, the first), keeping 467 implicit the position dependence of each A s j σ i (z j ) for brevity: (A.8) The notation means that σ i indices (but not ±1) are antisymmetrized in each term. The

469
result is a sum of [P σ i , •] and we shall call it a P σ -commutator. Similarly, derivatives of 470 A s 1 [σ 1 · · · A s k σ k ] with respect to any of the z j orz j are P σ -commutators. In fact, (A.8) also 471 holds with ±1 replaced by any τ , but we will not use that observation. 472 We have just shown that all derivatives ofX s 1 ...s k σ 1 ...σ k are P σ -commutators. Let us use the  In this appendix we show the factorization property of the composite operators X in 502 diagonal matrix elements between energy eigensates. We work in a basis of states |n in 503 which all charges P s are diagonal. We assume that the theory has a non-degenerate spectrum, 504 namely that each joint eigenspace of all the charges P s is one-dimensional. This is a much 505 weaker assumption than the assumption in [1] that the energy spectrum is non-degenerate.

506
(For instance CFTs have a highly degenrate energy spectrum.)

507
Consider basis states |n and |n of equal P σ for some σ. A s σ restricted to fixed P σ is diagonal, X s 1 ...s k σ 1 ...σ k restricted to fixed P σ 1 , . . . , P σ k is diagonal.
(A.16) 12 Relatedly, when point-splitting we only showed that the OPE is regular (up to Pσ-commutators) when antisymmetrizing the σi: antisymmetrizing the si instead may not give a well-defined operator.
Next, insert a complete set of states in a diagonal matrix element ofX s 1 ...s k σ 1 ...σ k : Then consider one of the off-diagonal terms (m = n) and let P τ be one of the charges 515 for which P τ (n) = P τ (m). Such a charge exists by our non-degeneracy assumption.

516
Then we can perform a calculation very similar to (A.8) but using additionally that 517 [P σ , |m m|] = P σ m |m m| − |m m| P σ m = 0. We find  13 For example, consider a theory with flavor symmetry su(2) and consider an irreducible representation R of su(2) inside the Hilbert space. Our reasoning shows the factorization property for eigenstates of iσ3 ∈ su(2), but also by symmetry for eigenstates of any other element of su(2). How can the non-linear property of factorization hold for all these linearly-related states in R at the same time? The key is that T andT and TT commute with su(2) hence are multiples of the identity when acting on R.
Let us first derive a consequence of (B.1). These relations can be stated as sA s ±1 = ±A ±1 s . identity, hence must be zero unless its spin s + t is the same as that of the identity operator.

541
In that case (s + t = 0) the multiple of the identity can be absorbed into the definition of First we work out All terms except the n = 0 ones are manifestly x-derivatives. Let us check the n = 0 terms 557 also are: (B.8) 14 We sometimes denote On(A, B) without specifying the point y when that point is clear from context.

560
To make factors of spin appear, we consider commutators with the spin operator S From the fact that T s+1 and Θ s−1 have spins s ± 1 we learn that We obtain that derivatives of A 1 s − sT s+1 and A −1 s − sΘ s−1 are quite complicated:

567
It is not immediately obvious how to absorb right-hand sides into an improvement of 568 (T s+1 , Θ s−1 ). Because 2πT tt = T 2 + Θ 0 + T 0 + Θ −2 , the sum of these equations simplifies 569 and gives second derivatives and higher: This is precisely as expected because the time component T s+1 + Θ s−1 of a current is 571 shifted by a space derivative upon improvements. For s = 0 the right-hand side of (B.13) 572 is absorbed by using the following improved current (in the main text we drop the hats) 16 Explicitly, 15 We left the point of origin x implicit in our notation for S. S is the charge corresponding to the rotation current jµ(y) ≡ αβ (y − x)αT βµ that is conserved by virtue of the symmetry of the stress tensor. Since the coordinates x, y are not well-defined on the cylinder, the expression we gave for S only makes sense locally, but our calculations are local so doing them on the plane would be equivalent. 16 For s = ±1 the left-hand side of (B.13) vanishes by construction, so the right-hand side must vanish. This is difficult to prove by direct calculations.
(B.16) Applying the Jacobi identity and the conservation equations for T tµ and T s+1 ± Θ s−1 gives The space derivatives ∂ x and ∂ y can be pulled out of the commutators, which can then Equating coefficients of ∂ n x δ(x − y) in (B.16) and (B.18) teaches us that for n ≥ 1

(B.20)
Returning to (B.12) and using T tx = T xt we work out Since if the space derivative of an operator with spin vanishes, it must be the zero operator, 588 (B.15) and (B.22) conclude the proof of (2.6).

590
In this Appendix we collect results about ambiguities that we encountered in our derivation.
where each term in the sum can be rewritten as [P σ i , .
There is a trivial ambiguity in the definition of A s σ , the shift by multiples of the identity: This mixing of charges is a special case of the ambiguities discussed next.
where the shift of A s σ is a particular choice that preserves (2.5). There are other satisfactory KdV charges. This ambiguity is frozen by our choice (2.16).

652
It is still worth contemplating how easy would it be to recognize the evolution considered 653 in this paper, if we were handed the spectrum of the theory with a different choice of rescaling.

654
Since the rescaling acts the same way on each eigenvalue, the ratio of two eigenvalues is 655 unambiguous, and it would readily lead to the identification of the deformation and the 656 rescaling used.

657
Next, consider a relativistic seed theory, but deform it by an arbitrary X tu . The key to 658 using Lorentz-invariance of the original theory is to promote the coupling λ to a background 659 field (also called a spurion) that has spin −t − u, so that the action is deformed by the 660 Lorentz-invariant combination d 2 x λX tu . To illustrate how the spurion helps, note that 661 our minimal prescription for ∂ λ P s is an integral of operators X tu s,±1 of spin s + t + u ± 1, 662 consistent with the spins of the current components ∂ λ T s+1 and ∂ λ Θ s−1 . Using the same 663 idea, the only mixing ambiguities in the X tu deformation of a relativistic seed are for some coefficients α s,k (more generally one should allow in each term any charge of the 665 same spin as P s+k(t+u) ). Without further input these ambiguities cannot be eliminated.

666
If the seed is a CFT then we use dimensional analysis: λ has dimension − |t| − |u| while 667 P s has dimension |s|. Only terms with |s + kt + ku| = |s| + k |t| + k |u| are dimensionally 668 consistent. This condition means (s, kt, ku) have the same sign or are zero. 3) for the X u,−u deformation we find ∂ λ Pu → ∂ λ Pu + πa −u u Pu and ∂ λ P−u → ∂ λ P−u − πa u −u P−u. This means that (2.16) does not fully define a choice of charges Pu and P−u: specifically one could rescale both of them (by the same factor because a −u u = −a u −u ). This caveat does not affect our results: for the TT deformation, a −1 1 = 0 because of (2.5), so (2.16) fully defines all ∂ λ Ps.
For tu > 0 deformations of a CFT (say, t, u > 0), X tu vanishes because it is an 671 antisymmetric combination of holomorphic currents. The deformation thus ought to be 672 trivial, but our general prescription (2.16) turns out to mandate a change of basis among 673 holomorphic currents. Indeed, it sets ∂ λ P s to an integral of operators X tu s,±1 . For s < 0 674 this vanishes because A t s and A u s vanish, as P s is built from a different Virasoro algebra 675 than P t and P u . For s > 0 however, the operator X tu s1 may be non-zero: it is simply a 676 holomorphic conserved current. We see that our general prescription is in this case not a 677 "minimal" choice of how charges are deformed, as one could have taken simply ∂ λ P s = 0.

678
(This minimal choice cannot be generalized to non-CFTs.) The spurion and dimensional 679 analysis above simply teaches us that for s < 0, ∂ λ P s = 0 is not ambiguous, while for s > 0 680 the variation ∂ λ P s has the full ambiguity (C.5). That ambiguity is enough to relate the  to all of the coupling constants. In particular let us discuss the X 1,u − X −1,u deformation 686 of Section 4, taking for definiteness u > 1 (the case u = 1 is TT ). For the case of a CFT 687 seed we will eliminate the whole ambiguity.

688
Assume first that we start from a Lorentz-invariant theory. The couplings λ ± of X ±1,u 689 have different spins −u ∓ 1. A charge P s can thus be mixed with λ k + λ l − P s+k(u+1)+l(u−1) for 690 k, l ≥ 0. We can now reduce to a single coupling λ ± = ±λ and write the ambiguity as  Next we work out the right-hand side of (E.1). We compute (E.8) The term n = 0 is a derivative, like the other terms: up to shifts by multiples of the identity (the only local operator whose ∂ x derivative 736 vanishes). Then We are done showing (E.1), because the right-hand sides of (E.7) and (E.11) agree up to F A comment on an integrability result 740 We show here that the evolution equation (3.16) found in [57] using integrability describes 741 some deformation that is outside the class of operator deformations that we study. Our 742 results cannot be compared. Let us copy their equation for the u-th deformation here in 743 our notations: 744 ∂ λ P k n = π 2 L ∂ L P k n − k θ 0 P k n , L ≡ P u n + P −u n − π 2 (u − 1)λ P u n − P −u n θ 0 , .
where we used the translation I u → −P u , τ → −π 2 λ, R → L and kept their notation for where we simplified a derivative by using that momentum depends on L as −P 1 + P −1 • n ∼ 763 1/L. In an updated version of [57] another momentumP is also defined, and it is found 764 not to depend on λ and hence coincides with the momentum we are using in the main text.

765
The relation betweenP and P in [57]  currents. 768 We would thus have two conserved charges: −P 1 + P −1 , and momentum P . Their because the sum of L −m L m cannot cancel in all states. 793 We conclude that (F.1) cannot describe in general for u = 1 the evolution of local 794 charges under a deformation that respects periodic translation invariance and locality. If 795 (F.1) describes the effect of field-dependent changes of coordinates as proposed in [57], then 796 it is perhaps not surprising that periodicity of the space coordinate is not preserved. It may 797 be the case that the deformation only makes sense on the plane rather than the cylinder.

798
Another possibility may be that the charges P k n appearing in (F.   In fact, dimensional analysis and spurion analysis together rule out such mixing for the αX −1,u +βX 1,−u deformation (u > 0). Since X s,t (s, t > 0) vanish in a CFT, it is not possible to distinguish (at linear order around the CFT) the αX −1,u + βX 1,−u deformation from a sum of this deformation and of any X s,t (s, t > 0). While the couplings of X s,t are invisible in the Hamiltonian at this order, they weaken dimensional and spurion analysis because of their varied dimensions and spins. These couplings allow a large class of mixing ambiguities. We thus move on with the proof without using dimensional and spurion analysis. 21 In fact, this essentially happens in Section 4. To linear order around a CFT the deformation studied there is X −1,u , corresponding to α = 1 and β = 0 here, and we focus there on the zero-momentum sector. In that sector we can check P−1 • n A u k • n = P1 • n A u k • n = (L/2π) X 1u 1k • n + P k • n Pu • n (2π/L). The first term is a shift by the conserved charge of the holomorphic current X 1u 1k . The second is expressed in terms of charges that we have control on. Away from the zero-momentum sector this switch to holomorphic quantities is not possible.  collision, we can define X ab = ( µν J a,µ J b,ν ) reg by point-splitting, modulo total derivatives.

868
Indeed, conservation leads to 869 ∂ z J a,z (z,z)J b,w (w,w) − J a,z (z,z)J b,w (w,w) = (∂ z + ∂ w ) J a,z (z,z)J b,w (w,w) + (∂ z +∂ w ) J a,z (z,z)J b,w (w,w) , (G.7) hence the collision µν J a,µ J b,ν is independent of the offset (z − w,z −w), modulo total 870 derivatives. Amusingly we did not need to assume that the charges Q a and Q b commute.

871
Now deform the action by X ab . The key question is which symmetries Q c can be 872 preserved. As we showed in (2.12), the condition is that [Q c , X ab ] needs to be a total 873 derivative. One can compute A word of warning: the bilinears J a,µ A cb − A ca J b,µ regulated by point splitting have 875 significantly more ambiguities than those we discuss in Appendix A.2 for the case of 876 commuting charges.

877
In order for the deformation to make sense beyond linear order, the symmetries Q a and 878 Q b that define the deformation must themselves be preserved by the deformation. Setting 879 c = a and c = b we see that the above commutator is only a total derivative if [Q a , Q b ] is 880 both proportional to Q a and to Q b , hence is simply zero.

881
We learn that it only makes sense to deform by bilinears X ab of commuting currents.
(G.10) The first two and last two terms combine into x derivatives, while the middle two terms 899 In Appendix C we analyze ambiguities that affect the definition of currents, charges 900 and A ab appearing throughout the paper. In this appendix we worked with the specific 901 fixing of ambiguities and saw that the symmetry algebra remains undeformed. If we were 902 to reintroduce ambiguities, the nonabelian structure would get deformed. Hence, if a 903 nonabelian algebra is preserved, requiring it to remain undeformed is an efficient principle 904 to fix the ambiguities.