Component twist method for higher twists in D1D5 CFT
CComponent twist method for higher twists in D1D5 CFT
Zaq Carson † , Ian T. Jardine † , and Amanda W. Peet †§ † Department of Physics, University of Toronto, Toronto, ON M5S 1A7, Canada § Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada
Abstract
The deformation operator of the D1D5 orbifold CFT, a twist 2 operator, drives theCFT towards the black hole dual and its physics is key to understanding thermalizationin the D1D5 system. To further study this deformation, we extend previous work onthe effect of twist 2 operators to a method that works for higher orders, in the con-tinuum limit. Our component twist method works by building higher twist operatorsout of twist 2 operators together with knowledge of Bogoliubov transformations. Con-sequently, this method sidesteps limitations in Lunin-Mathur technology by avoidinglifts to the covering space. We verify the method by reproducing results obtainablewith Lunin-Mathur technology. Going further, our method upholds a previously con-jectured scaling law in the continuum limit that applies to any generic configurationof twists. We illustrate this with computations for a new configuration of two twist 2operators that twists three copies together. [email protected] [email protected] [email protected] a r X i v : . [ h e p - t h ] A p r Introduction
Holography plays an important role in exploring key questions in quantum gravity such asthe black hole information problem. The first concrete realization of holography was theAdS/CFT correspondence of string theory, invented by taking near-horizon limits of largenumbers of certain types of branes [1]. Since then, other holographic dualities have beenengineered using systems of branes, including classes with less supersymmetry, but this maynot be the most efficient way to discover holographic dualities in general. Another approachis to start by inspecting CFTs and asking what conditions a CFT should satisfy in order forit to have a well defined AdS gravity dual. These include a large central charge and a sparsespectrum of light operators [2]. With these assumptions, useful information can be extractedin the context of quantum gravity to help address questions about black hole physics, forexample universalities arising via 1 /c and h/c expansions [3, 4, 5, 6, 7].Finding CFTs that satisfy holographic restrictions is not easy: they appear to be rare.One fruitful line of investigation is to focus on symmetric orbifold CFTs, which are morelikely to be holographic [8, 9, 10, 11]. They possess the required sparse light spectrum, whilealso giving the desired growth at large energies consistent with existence of black holes inthe dual theory. Accordingly, investigating the general structure of orbifold CFTs and thestates in them is likely to be important to understanding holography more generally.The CFT of the D1D5 brane system is a well known prototype holographic CFT. Atone point in the moduli space it is described by a free ( T ) N /S N orbifold. Of course, thedual at this point is strongly coupled and we need to deform towards the supergravitydescription. The deformation operator that accomplishes this is the superdescendent of thetwist 2 operator. This twisted operator introduces interactions between the copies, whichis dual to thermalization in the CFT. So understanding the action of the twist operator isimportant in investigating how the black hole information problem might be resolved.One physical property known for 1+1 dimensional free orbifold theories is that actingon the vacuum with twist operators will produce a squeezed state. This behavior holds evenwhen one applies an arbitrary number of twists [12, 13]. We can then try to find the matricesdescribing these squeezed states. An existing technique for computing quantities with twistedoperators in free symmetric orbifold CFTs, the Lunin-Mathur method, involves mapping theorbifold CFT into a covering space where all fields are single-valued [14, 15]. These mapsmust later be inverted to return to physical spacetime. Eventually the maps become quinticand higher-order polynomials once sufficiently many twists are applied. Such polynomialscannot be inverted, which severely limits the utility of these techniques for higher numbersof twist operators. Going to the continuum limit, where the radius of the circle on whichD1-branes are wrapped becomes large, enables one to sidestep this problem.The effect of twist 2 operators in the D1D5 CFT has been studied before in the contextof deformations towards a black hole spacetime description [16, 17, 13, 18, 19, 20, 21, 22,23, 24, 25, 26, 27, 28]. We continue the investigations begun there to study general twistedstates, specifically in the continuum limit. Our method, using twist 2 operators to buildup general twist configurations, reproduces previous results obtainable using Lunin-Mathurtechnology in the continuum limit. We also compute a new second-order twist 2 configurationand find that the coefficients defining the squeezed state scale as conjectured in [26]. Ourmethodology gives further evidence that these general scalings should hold in the continuum1imit for arbitrary configurations of twist operators.The remainder of this paper is organized as follows. In Section 2, we present the methodin detail for twist operators in orbifold CFTs. In Section 3 we discuss briefly the particularorbifold CFT arising from the D1D5 black hole construction. We also discuss here thespecific supersymmetry relations that relate the bosonic and fermionic sectors of this theory.In the subsequent two sections, we apply our method to two different twist configurationsin the D1D5 CFT. In Section 4 we tackle a previously-studied configuration and find goodagreement with known results, while in Section 5 we tackle a new twist configuration. Section6 concludes with a few closing remarks. Our work builds heavily on [21], which presentedthe basics of this method entirely in the context of the D1D5 CFT. Given a free orbifold CFT in 1 + 1 dimensions, consider any arbitrary combination of indi-vidual twist operators σ n , which together we will call ˆ σ . This operator takes a free theoryliving on one set of windings to a free theory on another set of windings, with the totalwinding number remaining constant. For the moment, consider a scalar field φ exists onall windings. The allowed excitation modes for this field will differ between the pre-andpost-twist configurations. The result isˆ σ | (cid:105) = C exp (cid:88) m,n> (cid:88) ( j )( j (cid:48) ) γ ( j )( j (cid:48) ) mn a ( j ) † m a ( j (cid:48) ) † n | (cid:48) (cid:105) ≡ | χ (ˆ σ ) (cid:105) ˆ σa ( i ) † m | (cid:105) = (cid:88) n> (cid:88) ( j ) f ( i )( j ) mn a ( j ) † n | χ (ˆ σ ) (cid:105) . (2.1)The new parenthetical indices refer to the various CFT copies of the theory. For clarity, weuse the index ( i ) exclusively for CFT copies before the twist operators and the index ( j )exclusively for copies after the twist operators. The index ( k ) will be used when speakingof an arbitrary copy without specifying one side side of the twist operators (including copystructures between the twist operators). Primes are used for additional copies of the sametype. For calculational convenience, the mode indices m and n indicate the energy of eachexcitation. They can be fraction-valued for multiwound copies. The excitation level ofeach excitation is its mode index times the winding number of the CFT copy on which theexcitation lives.A method for handling the bosonic sector of these twist operators was explored in [21]in the context of the D1D5 CFT, though the need to invert infinite matrices inhibited anyuseful generalization. We present a straightforward workaround for obtaining the requiredinverse matrices. For the remainder of this paper, we will work in the continuum limit ofour orbifold CFT. 2 .1 Bogoliubov approach for arbitrary orbifold CFTs Consider a particular component CFT copy ( i ) with winding number N ( i ) before the twistinsertions. On this copy, the field φ may be expanded as: φ ( x ) = (cid:88) mN ( i ) ∈ Z + (cid:0) h ( i ) ( x ) a ( i ) m + h ( i ) ∗ m ( x ) a ( i ) † m (cid:1) , (2.2)Now consider a particular copy ( j ) with winding number N j after the twist insertions. Here φ may be expanded as: φ ( x ) = (cid:88) mN ( j ) ∈ Z + (cid:0) h ( j ) ( x ) a ( j ) m + h ( j ) ∗ m ( x ) a ( j ) † m (cid:1) . (2.3)As described in [21], these two expansions share a linear relationship: a ( i ) m = (cid:88) nN ( j ) ∈ Z + (cid:0) α ( i )( j ) mn a ( j ) n + β ( i )( j ) mn a ( j ) † n (cid:1) . (2.4)The two matrices appearing here are also related to the coefficients of (2.1). f ( i )( j ) mn = (cid:0) α ( i )( j ) (cid:1) − nm γ ( j )( j (cid:48) ) mn = (cid:88) kN ( i ) ∈ Z + (cid:88) ( i ) (cid:0) α ( i )( j ) (cid:1) − mk β ( i )( j (cid:48) ) kn . (2.5)It will be convenient to occasionally write these relations in their matrix form. f = (cid:0) α − (cid:1) T , γ = α − β = f T β. (2.6)The Bogoliubov matrices, α and β , are straightforward to calculate. They can be ex-pressed in terms of the positive frequency solutions to the wave equation used in the expan-sions (2.2) and (2.3). The functions h form a complete orthonormal basis with respect tothe inner product: ( h, g ) ≡ − i (cid:90) Σ d Σ µ ( f ∂ µ g ∗ − g ∗ ∂ µ f ) , (2.7)where Σ is the Cauchy hypersurface where the domains of f and g overlap. Applying thisorthonormality to (2.2) and (2.3) and comparing to (2.4), one finds: α ( i )( j ) mn = (cid:0) h ( i ) m , h ( j ) n (cid:1) β ( i )( j ) mn = (cid:16)(cid:0) h ( i ) m (cid:1) ∗ , h ( j ) n (cid:17) . (2.8) α − Once one has calculated the Bogoliubov matrices, it is necessary to then invert α . This taskis far from trivial. The Bogoliubov matrices are infinite matrices, so there is no mathematicalguarantee that an inverse even exists. Fortunately α − gives the unique physical transition3atrix f . Rather than inverting α directly, we present a direct method of computing f forarbitrary twist configurations based on the behaviour of the single component twists. Inprincipal one then need calculate only the β matrix to proceed.Let us consider the twist configuration ˆ σ again. While we allowed this configuration tocontain any combination of twists σ n , each such component can itself be written in terms oftwo-twist operators σ . We thus decompose ˆ σ in terms of these two-twist operators.ˆ σ = Q (cid:89) q =1 σ { k q } ( x q ) , (2.9)where each { k q } is an ordered pair of component CFTs. Since two-twist operators do notcommute, it is important to specify that we will choose indices such that the twists act inorder of increasing q .We now act on a state containing only a single excitation of φ . The left hand side isgiven by (2.1). On the right, we will pass each twist through one at a time.ˆ σa ( i ) † m | (cid:105) = (cid:88) n, ( j ) f ( i )( j ) mn a ( j ) † n | χ (ˆ σ ) (cid:105) = Q (cid:89) q =1 σ { k q } ( x q ) a ( i ) † m | (cid:105) = (cid:32) Q (cid:89) q =2 σ { k q } ( x q ) (cid:33) (cid:88) n , ( k ) f ( i )( k ) m,n a ( k ) † n | χ (cid:105) = (cid:32) Q (cid:89) q =3 σ { k q } ( x q ) (cid:33) (cid:88) n , ( k ) (cid:88) n , ( k ) f ( k )( k ) n ,n f ( i )( k ) m,n a ( k ) † n | χ (cid:105) = (cid:32) Q (cid:89) q =4 σ { k q } ( x q ) (cid:33) (cid:88) n , ( k ) (cid:88) n , ( k ) f ( k )( k ) n ,n (cid:88) n , ( k ) f ( k )( k ) n ,n f ( i )( k ) m,n a ( k ) † n | χ (cid:105) . . . = (cid:88) { n i } , { ( k i ) } (cid:16) f ( k Q )( k Q − ) n Q ,n Q − (cid:16) f ( k Q − )( k Q − ) n Q − ,n Q − (cid:0) . . . (cid:0) f ( k ) , ( k ) n ,n f ( k ) , ( i ) n ,m (cid:1)(cid:1)(cid:17)(cid:17) a ( k Q ) † n Q | χ (ˆ σ ) . (2.10)Here each | χ i (cid:105) is the state obtained by acting with the first i twists on the vacuum. Theparentheses in the last line are used to present the correct order for the infinite sums. The factthat this order matters is equivalent to the non-associativity of infinite matrix multiplication.We have however played a little loose with our copy notation. We have used k i to indicateboth a pair of copies defining a twist operator (when in brackets) and an intermediate CFTcopy index (when in parentheses).By construction, the sum over ( k Q ) in the last line of (2.10) is the same as the sum over( j ) the first line. Similarly, the sum over n Q in the last line is the same as the sum over n in4he first line. We thus find: f ( i )( j ) mn = (cid:88) { n i } , { ( k i ) } (cid:16) f ( j )( k N − ) n,n N − (cid:16) f ( k N − )( k N − ) n N − ,n N − (cid:0) . . . (cid:0) f ( k ) , ( k ) n ,n f ( k ) , ( i ) n ,m (cid:1)(cid:1)(cid:17)(cid:17) . (2.11)In matrix form, this gives f (ˆ σ ) = ( f Q ( f Q − ( . . . ( f f )))) , (2.12)where each f i is the transition matrix for the i th two-twist in the decomposition (2.9).Now that we have an expression for the transition matrix, and by extension α − , forarbitrary twist configurations, the generalization of [21] is straightforward. Calculate thetransition matrix and Bogoliubov matrices as above. The matrix α can be used to checkthe transition matrix calculation by verifying αf T = f T α = I . One then finds the matrixcharacterizing the sqeezed state via γ = f T β . To show this method explicitly, we turn tothe D1D5 CFT. The D1D5 CFT has all of the ingredients needed to apply our method. For a comprehensivereview, see [29]. In fact, supersymmetry relations in the theory allow us to access mostfermion modes from the bosonic sector in the continuum limit as well. In the next twosections we will look at two specific twist configurations in this theory, one with a knownexact solution and one with no known exact solution. Here we introduce our notation andreview the relations that allow us to probe the fermionic sector.The D1D5 system is constructed in Type IIB string theory compactified as M , × S × T .The N D5 branes wrap the full compactification, while the N D1 branes wrap S . We take S to be much larger than T so that in the low energy limit only modes along S are excited.This gives a CFT on S with (4 ,
4) supersymmetry. We begin at the orbifold point, wherethe CFT is a free 1 + 1 dimensional sigma model with target space ( T ) N N ) /S N N . Thetwist operator is a part of the blow-up mode that allows us to perturbatively deform awayfrom this orbifold point.There are two SO (4) symmetries in this theory, an exact symmetry from rotations in thefour noncompact spatial directions, labelled SO (4) E , and an approximate symmetry fromrotations in T , labeled SO (4) I . We express each of these symmetries as SU (2) × SU (2),giving four different SU (2) symmetry groups. We use a different index type for each SU (2)charge. SO (4) E → SU (2) L × SU (2) R , α, ˙ αSO (4) I → SU (2) × SU (2) , A, ˙ A. (3.1)The bosonic and fermionic fields, along with their excitation modes, then carry the following SU (2) structure (in addition to their CFT copy index): X ( k ) A ˙ A → a ( k ) † A ˙ A,n , ¯ a ( k ) † A ˙ A, − n ψ ( k ) ,αA → d ( k ) † ,αAn ¯ ψ ( k ) , ˙ α ˙ A → ¯ d ( k ) † , ˙ α ˙ An . (3.2)5ote that for bosons, the chiral primary field is actually ∂X . The left-moving (holomor-phic) and right-moving (anti-holomorphic) sectors largely decouple. In most cases, includingthe properties of bare twists, it is sufficient to work only in the holomorphic sector. Theanti-holomorphic sector behaves analogously. We will therefore work predominately in theholomorphic sector for the remainder of this paper.The theory’s twist operator carries charge under the SO (4) E group, which we write as SU (2) L and SU (2) R charges. This has no effect on the bosons so we will often drop suchindices from our notation. The theory also has a supercharge operator. G α ˙ A = ψ αA ∂X A ˙ A . (3.3)The blow-up mode that deforms the CFT away from its orbifold is an SU (2) L singlet com-bination of the supercharge and the twist ( SU (2) R singlet in the anti-holomorphic sector).It was shown in [13] that the two singlet charge combinations are proportional. Becauseof this, it is traditional to work solely with twists of positive charge. We therefore do notconsider negative-charge twists in this paper.Even restricting to only positive-charge twists, one might still expect a proliferation of SU (2) indices on all of our characteristic coefficients. This is not the case. Aside from thefermion zero modes, there is only a single linearly independent γ and f matrix for each copycombination. In practice, however, it is convenient to write different matrices for bosons andfermions and to also add an SU (2) L index to the fermion transition matrix. One then hasthe particular SU (2) structures: | χ (ˆ σ ) (cid:105) = C exp (cid:104) γ B ( j )( j (cid:48) ) mn (cid:16) − a † ( j )++ ,m a † ( j (cid:48) ) −− ,n + a † ( i )+ − ,m a † ( j ) − + ,n (cid:17)(cid:105) × exp (cid:104) γ F ( j )( j (cid:48) ) mn (cid:0) d † ( j )++ m d † ( j ) −− n − d † ( j )+ − m d † ( j ) − + n (cid:1)(cid:105) | (cid:105) (3.4)The transition matrices are independent of all SU (2) charges except SU (2) L . We thus addan index α to the fermionic piece. f B ( i )( j ) mn , f F α ( i )( j ) mn . (3.5)Our method only allows us to calculate the bosonic coefficients directly. However, the non-trivial relationships between the fermionic and bosonic coefficients were shown in [25], [26].We can thus access the fermionic sector (except zero modes). Our method was derived for canonically normalized modes, but unfortunately for us it hasbecome standard for work in the D1D5 CFT to make use of modes that are not canonicallynormalized. We thus provide a translation between the standard mode normalizations andour canonical normalizations. α ( k ) A ˙ A, − n = √ n a ( k ) † A ˙ A,n ˜ d ( k ) A ˙ A, − n = (cid:112) N ( k ) d ( k ) A ˙ A, − n , (3.6) Our inability to access fermion zero modes means that our method is insensitive to the particular SU (2)structure of the Ramond vacuum. This insensitivity is expected in the continuum limit. γ B ( j )( j (cid:48) ) mn = √ mn γ B ( j )( j (cid:48) ) mn ˜ γ F ( j )( j (cid:48) ) mn = (cid:112) N ( j ) N ( j (cid:48) ) γ F ( j )( j (cid:48) ) mn , (3.7)while for the transition matrix we have˜ f B ( i )( j ) mn = (cid:114) nm f B ( i )( j ) mn ˜ f F α ( i )( j ) mn = (cid:115) N ( j ) N ( i ) f F ( i )( j ) mn . (3.8)We also translate the relationships between the bosonic and fermionic coefficients from [25],[26]. ˜ γ F ( i )( j ) mn = − m √ mn (cid:112) N ( i ) N ( j ) γ B ( i )( j ) mn ˜ f F − , ( i )( j ) mn = (cid:114) nm (cid:115) N ( i ) N ( j ) f B ( i )( j ) mn ˜ f F + , ( i )( j ) mn = (cid:114) mn (cid:115) N ( j ) N ( i ) f B ( i )( j ) mn . (3.9)In all of the above expressions, the traditional forms are written on the the left-hand side. → → Results
We now look at the case of two singly wound copies twisted into one doubly wound copyat w = τ + iσ and then untwisted back into two singly wound copies at w = τ + iσ .For convenience, we will use translation and rotation invariance to select τ = τ = 0 and σ > σ . This means that our Cauchy hypersurface Σ is a collection of σ contours.Since we have two singly-wound CFTs in both the initial and final states, we will placeprimes on the explicit final-state copy indices. So the initial copies are (1) and (2) while thefinal copies are (1 (cid:48) ) and (2 (cid:48) ). We begin by writing the positive frequency solutions to the wave equation. h ( k ) m = 1 √ π √ m e im ( σ − τ ) h ( k )¯ m = 1 √ π √ m e − i ¯ m ( σ + τ ) , (4.1)7igure 1: The layout of the copies as we apply the twists. Copy (1) is the black line, copy (2)is red. Here the A,B ends are identified under the σ = 0 ∼ π identification of the cylinder.We perform all calculations at τ = 0, though we separate the windings here visually forclarity. (a) Copies before twists. (b) Application of σ (12) at σ , twisting the copies together.(c) Application of σ (21) at σ , splitting into two new copies (1’) (top) and (2’) (bottom)where m is an integer. The functional form is the same for all copies since each copy has thesame winding number. However, different copies are valid over different regions. Importantly,copies (1) and (2) have no domain overlap, and similarly for copies (1 (cid:48) ) and (2 (cid:48) ). What weneed now is the domain of overlap for each initial-final copy pair. There is in fact someambiguity here, owing to the freedom to define our copies however we choose. We will followthe conventions of [25]. The intermediate doubly-wound CFT has its interval [0 , σ ] takenfrom copy (1), while the second twist takes the first 2 π interval of this intermediate CFTinto copy (1 (cid:48) ). The initial-final domains of overlap can then be seen in figure 1.(1) , (1 (cid:48) ) or (2) , (2 (cid:48) ) = ⇒ Σ = [0 , σ ] ∪ [ σ , π ](1) , (2 (cid:48) ) or (2) , (1 (cid:48) ) = ⇒ Σ = [ σ , σ ] . (4.2)Since the expansion functions are the same for all copies, each matrix’s (1 , (cid:48) ) and (2 , (cid:48) )components are identical, as are (1 , (cid:48) ) and (2 , (cid:48) ) components. This is expected, as thephysics is invariant under the combination of interchanges 1 ↔ (cid:48) ↔ (cid:48) .We can now calculate our Bogoliubov matrices. Starting with α , we have α (1)(1 (cid:48) ) mn = − i (cid:18)(cid:90) σ + (cid:90) πσ (cid:19) π √ mn (cid:0) e im ( σ − τ ) ine − in ( σ − τ ) − e − in ( σ − τ ) ( − im ) e im ( σ − τ ) (cid:1) dσ = 14 π √ mn (cid:18)(cid:90) σ + (cid:90) πσ (cid:19) ( m + n ) e i ( m − n ) σ dσ. (4.3)The result is piecewise. For m = n , we have α (1)(1 (cid:48) ) mm = 14 π m m ( σ + 2 π − σ )= 1 − σ − σ π = 1 − ∆ w πi . (4.4)8or m (cid:54) = n , we find α (1)(1 (cid:48) ) mn = 14 π √ mn m + ni ( m − n ) (cid:0) e − i ( m − n ) σ − − e − i ( m − n ) σ (cid:1) = 12 π m + n √ mn ( m − n ) e i ( m − n ) σ σ i (cid:16) e i ( m − n ) σ − σ − e − i ( m − n ) σ − σ (cid:17) = 12 π m + n √ mn ( m − n ) e i ( m − n ) σ σ sin (cid:18) ( m − n ) ∆ w i (cid:19) . (4.5)We now use translation invariance to set the σ midpoint to zero. This eliminates the phase. α (1)(1 (cid:48) ) mn = 12 π m + n √ mn ( m − n ) sin (cid:18) ( m − n ) ∆ w i (cid:19) . (4.6)As noted earlier, this is also the purely left-moving (2 , (cid:48) ) portion of the α matrix. Thecalculation for (1 , (cid:48) ) is identical up to a change in integration limits. The result is α (1)(2 (cid:48) ) mn = (cid:40) ∆ w πi m = n − α (1)(1 (cid:48) ) mn m (cid:54) = n, (4.7)which is also the purely left-moving (2 , (cid:48) ) result. As expected by their general decoupling,the purely right-moving results are identical to their purely left-moving counterparts. Thereare also parts of the α matrix with mixed holomorphicity. These are nonphysical, terms withthem vanish, and serve only to illuminate the kernel of f . We have omitted these portionsof the matrix for brevity. We thus have all the physically-relevant parts of α .The calculation for β is analogous. It gives: β (1)(1) mn = 12 π m − n √ mn ( m + n ) sin (cid:18) ( m + n ) ∆ w i (cid:19) β (1)(2) mn = − β (1)(1) mn . (4.8)Conveniently, there is no need for a piecewise expression. Each component simply vanisheson the diagonal. An analytic form of the transition matrix for this twist configuration in the continuum limitwas identified in [26]. Shifting to canonical modes, for 0 < ∆ w ≤ πi one has: f (1)(1 (cid:48) ) mn = f (2)(2 (cid:48) ) mn = − ∆ w i m = n − π ( m − n ) sin (cid:18) ( m − n ) ∆ w i (cid:19) m (cid:54) = nf (1)(2 (cid:48) ) mn = f (2)(1 (cid:48) ) mn = ∆ w i m = n π ( m − n ) sin (cid:18) ( m − n ) ∆ w i (cid:19) m (cid:54) = n . (4.9)9 γ Residuals, Δ w = i π , Λ = Figure 2: Residual vs n for γ (1 (cid:48) )(1 (cid:48) ) n,n at ∆ w = πi .We can multiply this matrix by α to check our calculations. Using a cutoff of Λ = 2000 forthe sums, the result is I to within 0 .
01% for mode indices of order 10 or larger.One can also check the method of building this two-twist transition amplitude out ofthe behaviour of each component twist. Each component transition amplitude also scales as1 / ( m − n ), so the summand behaves as an inverse square. The convergence is fast. Λ = 2000is sufficient for 0 .
2% accuracy for mode indices of order 10 or larger.Like the transition matrix, the matrix γ has the following copy relations: γ (1 (cid:48) )(1 (cid:48) ) mn = γ (2 (cid:48) )(2 (cid:48) ) mn = − γ (1 (cid:48) )(2 (cid:48) ) mn = − γ (2 (cid:48) )(1 (cid:48) ) mn . (4.10)We therefore compute only the (1 (cid:48) ) , (1 (cid:48) ) sector. This computation is more arduous. Thesummand scales with a power of − / γ with rising mode indices wasnot valid for low-lying modes.Figure 2 shows the percent error of our calculation for Λ = 10 for various excitationlevels. These high-cutoff calculations were performed on SciNet [30]. Computations withlower cutoffs were performed on personal computers. Figure 3 shows the cutoffs requiredfor a desired accuracy, specifically for γ (1 (cid:48) )(1 (cid:48) )200 , . To get a feel for the accuracy/computationtradeoff, see Figures 4 and 5. 10
00 000 400 000 600 000 800 000 1 × Λ Diagonal γ Residuals, n = Δ w = i π Figure 3: Residual vs cutoff for γ (1 (cid:48) )(1 (cid:48) )200 , at ∆ w = πi .
50 100 150 200 Time ( s ) Off Diagonal γ Residuals, One Calculation, p = = Δ w = i π Figure 4: Residual vs computation tame for γ (1 (cid:48) )(1 (cid:48) )200 , at ∆ w = πi .11 ( s ) Diagonal γ Residuals, One Calculation, n = Δ w = i π Figure 5: Residual vs computation time for γ (1 (cid:48) )(1 (cid:48) )200 , at ∆ w = πi . → → We now turn to a new twist configuration which has not yet been addressed. Here we beginwith three singly-wound CFT copies and through two two-twist operators join them intoa single triply-wound copy. We can again use rotation and translation invariance to set τ = τ = 0 and σ > σ . The twist at σ will join copies (1) and (2) while the twist at σ joins copies (2) and (3). There is only one final copy so we drop its index. We begin by writing the positive frequency solutions to the wave equation. For each of thethree initial copies, we have: h ( i ) m = 1 √ π √ m e im ( σ − τ ) h ( i )¯ m = 1 √ π √ m e − i ¯ m ( σ + τ ) , (5.1)with integer m . For the final copy, we instead have: h s = 1 √ π √ m e is ( σ − τ ) h ¯ s = 1 √ π √ m e − is ( σ + τ ) , (5.2)where s is a multiple of 1 / σ = 0 ∼ π . Weperform all calculations at τ = 0, though we separate the windings here visually for clarity.(a) Copies before twists. (b) Application of σ (12) at σ , twisting the copies together. (c)Application of σ (23) at σ , which twists all three together.figure 6. (1) = ⇒ Σ = [0 , σ ] ∪ [4 π + σ , π ](2) = ⇒ Σ = [ σ , σ ] ∪ [2 π + σ , π + σ ](3) = ⇒ Σ = [ σ , π + σ ] , (5.3)where each line is the listed copy’s domain overlap with the unique final copy.Before we proceed further, let us pause to identify an important point. Note that all ofour wavefunctions are identical after shifts of σ → σ + 6 π . The domain overlap for copy (1)can thus be rewritten as: (1) = ⇒ Σ = [4 π + σ , π + σ ] (5.4)It is now clear that copies (1) and (3) both have contiguous domain overlaps while copy(2) does not. This will manifest in significant similarity between all of copy (1) and copy(3) matrix components, while the copy (2) components are significantly different. This isphysically expected, as copies (1) and (3) each only see one two-twist operator while copy(2) sees both operators.We can now proceed with the calculation of α and β for each of our three initial copies.The mathematics behaves similarly to the previous case, so we present only the results. Forconvenience, we identify a commonly-occurring factor: µ s ≡ − e πis , (5.5)13hich vanishes for integer s . For α , we again obtain a piecewise result. α (1) ms = √ δ ms s ∈ Z − i π √ sm s + ms − m µ s e i ( s − m ) σ s / ∈ Z α (2) ms = √ δ ms s ∈ Z − i π √ sm s + ms − m (cid:0) µ − s e i ( s − m ) σ − µ s e i ( s − m ) σ (cid:1) s / ∈ Z α (3) ms = √ δ ms s ∈ Z i π √ sm s + ms − q µ − s e i ( s − m ) σ s / ∈ Z . (5.6)The result for the β matrix is: β (1) ms = i π √ sm s − ms + m µ − s e − i ( s + m ) σ β (2) ms = i π √ sm s + ms − m (cid:0) µ s e − i ( s + m ) σ − µ − s e − i ( s + m ) σ (cid:1) β (3) ms = − i π √ sm s + ms − m µ s e − i ( s − m ) σ . (5.7)As expected form our configuration, the results for copies (1) and (3) are identical to thebehaviour of a single two-twist operator, which was solved in general in [21]. It is thebehaviour of copy (2) here which is new. While this twist configuration has not been solved before, the behaviour of copies (1) and(3) is the same as a single two-twist configuration since each copy only interacts with one ofour twists. We can thus identify their transition matrices from [21]. f (1) ms = √ δ ms s ∈ Z − πi √ s − m µ − s e − i ( s − m ) σ s / ∈ Z f (3) ms = √ δ ms s ∈ Z πi √ s − m µ s e − i ( s − m ) σ s / ∈ Z . (5.8)We now turn to f (2) , which is where we must apply our method of combining transitionbehaviours for each individual two-twist operator. Noting that the intermediate state is ona doubly-wound CFT, we have: f (2) ms = (cid:88) q ∈ Z + / f [(1) , (2)] mq ( σ ) f [(1+2) , (3)] qs ( σ ) , (5.9)14here here the copy indices in brackets are used to denote which copies are being twistedtogether. The expression (1 + 2) refers to the intermediate doubly-wound CFT that resultsform the application of the twist at σ . All first-order transition matrices are known fortwo-twist operators , so we simply refer to [21] for their continuum limit approximations. f [(1) , (2)] mq = √ δ mq q ∈ Z πi √ q − m µ q e − i ( q − m ) σ q / ∈ Z f [(1+2) , (3)] qs = (cid:114) δ qs s ∈ Z − πi √ s − q µ s e − i ( s − q ) σ s / ∈ Z . (5.10)We now break our computation into two cases. Case 1: s ∈ Z Here f [(1+2) , (3)] is nonzero only for q = s , which means q ∈ Z . Thus f [(1)(2)] is nonzero onlyfor m = q = s . This leaves only one nonzero term in the sum. (cid:2) f (2) ms (cid:3) s ∈ Z = 1 √ δ ms . (5.11) Case 2: s / ∈ Z Here we cannot have s = q since the allowed values for the two indices only coincide atintegers. We can however still have q = m . We separate out this term, which is the onlynonzero term for integer q . This gives: (cid:2) f (2) qs (cid:3) s/ ∈ Z = µ s π √ is − m e − i ( s − m ) σ − π e imσ e − isσ (cid:88) q ∈ Z + , q> e iq ( σ − σ ) ( s − q )( q − m ) . (5.12)As before, we will tackle this sum by using a large but finite cutoff. A quick check
Since there is no previous result to compare f (2) to, we instead multiply by α (2) and use alarge but finite cutoff to handle the sum. The product quickly reaches the identity thanksto the inverse square scaling of the summand. As with the previous configuration, a cutoffof Λ = 2000 is sufficient for 0 .
01% accuracy.The behaviour of f (2) ms is shown in Figures 7 and 8. As anticipated by [26], the continuumlimit behaviour for s (cid:54) = m scales as (for canonical modes): f (2) ms ∼ m − s ) h f ( m, s, ∆ w ) , (5.13) Of positive SU (2) L charge Δ w / i0.020.040.060.080.10 | f ( ) / | | f ( ) / | , Λ = Figure 7: Transition Amplitude as a function of separation for f (2)10 , / .where h f is a function oscillating in ∆ w with amplitude independent of m and s . This indexfalloff has been shown to hold for all cases of a single two-twist operator as well as for bothsecond-order cases considered so far. We recommend this form of transition matrices as afirst-guess fit for numeric work for any twist configurations in the continuum limit of theD1D5 CFT. We can now combine our transition matrix with β to obtain the matrix γ that characterizesthe squeezed state. All three initial copies contribute. γ ss (cid:48) = (cid:2) f T β (cid:3) ss (cid:48) = (cid:88) m> (cid:88) ( i ) f ( i ) ms β ( i ) ms (cid:48) . (5.14)There are technically two sums at work here, as a sum is required for the intermediatecalculation of f (2) . As usual we terminate each sum at some large cutoff. The resulting γ ss (cid:48) behaviour is shown in Figures 9 and 10. Once again, the anticipated falloff from [26] is agood approximation for the continuum limit. γ ss (cid:48) ∼ s + s (cid:48) h γ ( s, s (cid:48) , ∆ w ) , (5.15)where h γ is a different function oscillating in ∆ w with amplitude independent of s and s (cid:48) .We therefore recommend this falloff form for the squeezed state coefficient as a first-guess fitfor numeric work with any twist configurations in the continuum limit of the D1D5 CFT.16 | f ( ) q , - q + / | | f ( ) q , - q + / | , Λ = Figure 8: Transition Amplitude as a function of its index for f (2) q, − q +1 / at ∆ w = πi . Δ w / i0.0010.0020.0030.004 | γ / / | | γ / / | , Λ = Figure 9: Gamma as a function of separation for γ / , / .17
10 15 20 25 30 n γ nn γ nn , Δ w = i π , Λ = Figure 10: Gamma as a function of its index for γ s,s at ∆ w = πi . Our goal here was to further investigate the effect of twist operators on quantum states inthe D1D5 orbifold CFT, in order to bring us closer to the general goal of understandinghow thermalization works in the microscopic picture of black holes. We wished to findresults in the continuum limit that went beyond the already known twist 2 results foundpreviously. The method we presented here, using component twists, has the advantage ofbeing applicable to higher twist configurations, unlike the Lunin-Mathur method.We used knowledge of Bogoliubov transformations and the fact that twist two operatorsˆ σ can be used to build up higher-twist operators ˆ σ n to find the transition matrices f andsqueezed state coefficient matrices γ associated to states created by ˆ σ n . To show our methodworks, we checked our component twist method by comparing to a previous result calculatedwith Lunin-Mathur technology.With our method verified, we explicitly calculated a new configuration and found thatthe scaling predicted in [26] holds true: we see good agreement with equations (5.13) forthe transition matrices f and (5.15) for the squeezed state coefficients γ . We expect thatthe form of these matrices will hold in the continuum limit for any general twist operatorconfigurations. This conclusion is based on our component twist method, the forms of thetwist 2 transition matrices, and simple power counting.We have seen, using calculations in the D1D5 orbifold CFT, that it is possible to buildup the f matrices from components and to obtain the γ matrices from them. The only partthat made our calculations specific to this CFT was the form of the twist two transitionamplitudes. Twist operators in other orbifold CFTs might have a similar decomposition.18wist operators also show up in other orbifold CFTs that are not themselves holographicbut are useful for computing quantities of physical interest in holographic CFTs. In particu-lar, cyclic orbifolds show up when using the replica trick for computing entanglement entropyvia Renyi entropies in quantum field theories [31, 32, 33, 34, 35]. The entanglement entropyof multiple regions is written as a correlation function of multiple pairs of twist operators.Better understanding the effect of applying twist operators in the prototype D1D5 orbifoldCFT may assist with learning about information theoretic quantities of interest for quantumgravity. Acknowledgements
We would like to thank Samir D. Mathur for useful discussions. This work is supported bya Discovery Grant from the Natural Sciences and Engineering Research Council of Canada.Computations were performed on the GPC supercomputer at the SciNet HPC Consortium.SciNet is funded by: the Canada Foundation for Innovation under the auspices of ComputeCanada; the Government of Ontario; Ontario Research Fund - Research Excellence; and theUniversity of Toronto.
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