Aspects of D1-D5 SCFT near the orbifold point: string...

Context & Motivation D1-D5 Conformal Perturbation Theory Building the Deformation Operator Mixing of Low-Lying String States

Aspects of D1-D5 SCFT near the orbifold point:string state operator mixing

and twist-nontwist correlators

Amanda W. Peet, University of Torontoand Ida G. Zadeh, Brandeis University

Co-author: Ben Burrington, Troy UniversityBased on: 1211.6689, 1211.6699, work in progress

Black Holes in String Theory workshop, University of MichiganSlides: http://amandapeet.ca/talks/mctp1310/

I Special thanks to Finn and Angie and other organizers.

http://amandapeet.ca/talks/mctp1310/


Context and MotivationString theory and the black hole information paradoxD-branes and BH entropy/informationMicrostates and the fuzzball programmeOur sports arena: the D1-D5 systemDeforming away from the orbifold point

D1-D5 Conformal Perturbation TheoryD1-D5 SCFT: fields, generators, chiral primariesSuperconformal algebra and OPEsConformal perturbation theory (3 slides)

Building the Deformation OperatorLunin-Mathur technology for beginnnersPizza diagramsFermionsBosonizationMarginal deformation operator Od

Mixing of low-lying string statesOperator mixing4-point functions and factorization channelsMixing for ∂X∂X ∂X ∂XWork in progressSummary and future directions II handoff to Ida Zadeh

1 / 21


Context and Motivation


String theory and the black hole information paradox

String theory is a powerful approach to studying both gravitation and QFTs.Has Newtonian limit. Its low-E SUGRA limit contains classical black holesolutions. Rich context for investigating BHs. Inherently higher-dimensional.Best feature: large set of technical tools for calculating details of how quantumgravity corrects classical spacetime, rather than waffling about it.Most maligned feature: insisting on solid foundations in the UV makes flowingdown to the experimentally relevant IR messy and hard. Similarly for BHs: wecan do BPS systems best but astrophysical BHs are maximally far from BPS.Hawking (’75) : quanta radiated by BHcarry no behind-horizon information –other than what can be measured atinfinity: M, J,Q. ⇒ BH info paradox.Mathur (’09) : subleading quantum gravitycorrections cannot resolve info paradox.Theorem does not assume what quantumgravity is, only strong subadditivity & thateach Hawking pair created independently.Only O(1) corrections to semiclassical BH expectations around the horizon canrescue unitarity. So we need to look for hair – lots of hair.

2 / 21


D-branes and BH entropy/information

String theory is not just a theory of strings. Also has nonperturbative D-branes,loci where strings end, with tension ∼ 1/gs . They carry R-R charge Polchinski

’95 and gravitate with strength ∝ gsN, which can be parametrically large.Strominger-Vafa ’96 kicked off a revolution in computing SBH from quantumstatistical mechanics of strings and D-branes.Their entropy agreement was underpinned by a SUSYic nonrenormalizationtheorem. This is broken at finite T , or in systems with less SUSY wheredegeneracy of states can jump as parameters vary, c.f. wall crossing.Generically, physical observables will differ in dual descriptions, and not just bya factor of 3/4. But for certain near-BPS configurations, gorgeousmulti-parameter agreement was demonstrated between spectra for emissionfrom spacetime ergoregion and from microscopic CFT physics.Caveat: getting the entropy or the emission spectrum right does not mean youget the entanglement or the quantum states right. Morally speaking, to resolvethe information paradox, you need to know about the wavefunction inside theblack hole horizon as well as outside it.Bonus: string theory has a very rich solution space for black spacetimes,especially in D ≥ 5: holes, branes, rings (’01) , multis, combos, etc. Hair! Yay!

3 / 21


Microstates and the fuzzball programme

If you make me vote on the BH info paradox, I will pick Team Mathur. Why?Mathur’s theorem shows unitarity cannot be saved by perturbative quantumgravity corrections. Fuzzballs finagle traditional (4D) no-hair intuition byhaving hair that is nonperturbative, i.e. not explained by linearizedperturbation theory about semiclassical BH spacetimes.Get parametric enhancements of string effects, by factors of N. Also: phasespace of fuzzballs with macroscopic quantum numbers is exponentially large.You can do the usual desirable duo – calculate and speculate. You stand onUV-solid string foundations by constructing the microstates.Traditional BHs are conjectured to emerge from thermal averaging over stringtheory microstates with same conserved quantum numbers as BH.

Key idea: study gravitational fields of quantum string theory ingredients intheir natural higher-D context. (A “4D” BH may not be 4D everywhere.)

4 / 21


Our sports arena: the D1-D5 system

The D1-D5 system, our focus here, is obtained by wrapping N1 D1-branes onS1 and N5 D5-branes on S1 × T 4.To make the SV BH we would also add momentum P along S1 to makehorizon macroscopic, while keeping R(S1) and Vol(T 4) small to keep BH 5D.IIB is a nice duality frame for 2-charge system: which description of thephysics is valid where is determined only by parameters. Specifically: stringperturbation theory for D1-D5 system when gsN1,5 � 1 (and g 2

s Np � 1 forthe SV BH). SUGRA spacetime is reliable in the opposite limit.

In low-energy limit with R(S1)� 4√Vol(T 4), D1-D5 system is described by

D=1+1 conformal field theory with N = (4, 4) SUSY living on S1.

String theory D1-D5 system has a(20-parameter) moduli space. At onepoint it is best described in terms ofblack spacetime geometry. At another,the CFT is a sigma model withsymmetric product orbifold targetspace (T 4)N/SN where N = N1N5.

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Deforming away from the orbifold point

Mathematical SN orbifold structure is related to physical phenomenon offractionation in ‘long string’ picture: lowest excitation mode has energy1/(N1N5R) rather than the naive 1/R. D1-D5 boundstate is thought of as onelong multiply wound string rather than multiple singly wound strings.It is easy to calculate in the microscopic D1-D5 SCFT at the orbifold pointwhere it is free. But to connect honestly with black hole physics, we need todeform the CFT away from the orbifold point towards the BH. This is thefocus of the two papers and works in progress reported here by me and Zadeh.Other fuzzbally people have focused on the black spacetime side, egBena-Warner, and quantum string corrections away from the black spacetimepoint, eg Giusto-Russo. Technically complicated; huge literature.

Some other recent (’10-’13) related orbifold CFT developments:-

Constructing/categorizing new class of fuzzball geometries, eg

Giusto-Lunin-Mathur-Turton

Absorption/emission of quanta by extremal D1-D5 BH using dual orbifoldCFT, eg Lunin-Mathur

Computation of entanglement Renyi entropies, eg Headrick-Lawrence-Roberts

and Hartman’s ultra-clear talk from yesterday after lunch.6 / 21


D1-D5 CFT andConformal Perturbation Theory


D1-D5 SCFT: fields, generators, chiral primaries

D1-D5 CFT lives in 1 + 1D and has N = (4, 4) supersymmetry. It has a verylarge central charge c ∝ N1N5. c.f. critical superstrings with ctot = 0.Our orbifold is by the symmetric group SN . This is different from Zn cyclicgroup orbifolds sometimes used in superstring model building.Symms? SU(2)L × SU(2)R R-symmetry, SO(4)I = SU(2)1 × SU(2)2 from T 4.Four real bosons, X i i ∈ {1, · · · , 4}. Write as doublets of SU(2)1 and SU(2)2:

X AA = 1√2

X i (σi )AA .

Four real fermions in the left moving and four in the right moving sector.Combine to form complex fermions which are doublets of SU(2)L and SU(2)2:

ψαA, (ψαA)† = −εαβεABψβB .

Generators of superconformal algebra:-

T = 14εAB εAB ∂X AA ∂X BB + 1

2εαβ εAB ψαA∂ψβB ,

GαA =√

2εAB ψαA ∂X BA,

Ja = 14εAB ψ

αA (σ∗a)βγ ψγB .

Chiral primaries: G +A− 1

2

|χ〉 = 0, h = m, correspond to SUGRA modes in bulk.

We want to do better, so we will investigate lowest-lying excited states.7 / 21


Superconformal algebra and OPEs

∂X AA(z1) ∂X BB(z2) ∼ εAB εAB

(z1 − z2)2,

ψαA(z1)ψβB(z2) ∼ −εαβ εAB

z1 − z2

Ja(z1) GαA(z2) ∼ 1

2(σ∗a)αβ

GβA(z2)

(z1 − z2),

T (z1) Ja(z2) ∼ 1Ja(z2)

(z1 − z2)2+

∂Ja(z2)

(z1 − z2),

T (z1) GαA(z2) ∼ 3

2

GαA(z2)

(z1 − z2)2+∂GαA(z2)

(z1 − z2),

Ja(z1) Jb(z2) ∼ c

12

δab

(z1 − z2)2+ i εabc

Jc(z2)

(z1 − z2),

GαA(z1) GβB(z2) ∼ −2

3c

εαβ εAB

(z1 − z2)3

+2εAB εβγ (σ∗a)βγ

[2Ja(z2)

(z1 − z2)2+

∂Ja(z2)

(z1 − z2)

],

−2εαβ εABT (z2)

(z1 − z2)

T (z1) T (z2) ∼ c

2

1

(z1 − z2)4+ 2

T (z2)

(z1 − z2)2+

∂T (z2)

(z1 − z2).

[Jam,G

αAn

]=

1

2(σ∗a)αβ GβA

m+n,

[Lm, Jan ] = −nJa

m+n,[Lm,G

αAn

]=(m

2− n)

GαAm+n,[

Jam, J

bn

]=

c

12m δab δm+n,0 + i εabc Jc

m+n,[GαAm ,GβB

n

]= −c

3(m2 − 1

4)εαβ εABδm+n,0

+2(m − n) εβγ εAB (σ∗a)αγ Jam+n

−2εαβ εAB Lm+n,

[Lm, Ln] =c

12(m3 −m) δm+n,0 + (m − n) Lm+n .

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Conformal Perturbation Theory I

Consider two-point functions (2pf):

〈φi (z1, z1)φj(z2, z2)〉0 = δij z−2hi12 z−2hi

12 ,

where φi , φj are quasi-primary operators transforming tensorially under global

conformal transformations: φ′i (z ′, z ′) = (∂z ′/∂z)−h (∂z ′/∂z)−h φ(z , z).Add a small perturbation to the action of the free SCFT:

δS = λ

∫d2z Od(z , z) + h.c .

Two ways to evaluate change in the two-point function.First: evaluate derivative of 2pf with respect to λ:

〈φi (z1, z1) φj(z2, z2)〉λ = δij z−2hi (λ)12 z

−2hi (λ)12

= δij z−2(hi+λ

∂hi (λ)

∂λ

)12 z

−2(hi+λ

∂hi (λ)

∂λ

)12

To first order in conformal perturbation theory (CPT):∂∂λ 〈φi (z1, z1) φj(z2, z2)〉λ =(

−2∂hi (λ)∂λ ln(z12)− 2∂hi (λ)

∂λ ln(z12))〈φi (z1, z2) φj(z2, z2)〉0.

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Conformal Perturbation Theory II

Second way of evaluating change in 2pf: use path integral formulation:

〈φi (z1, z1)φj(z2, z2)〉λ =

∫d [X , ψ] e−Sfree+λ

∫d2zOd (z,z) φi (z1, z1)φj(z2, z2)∫

d [X , ψ] e−Sfree+λ∫

d2zOd (z,z)

= 〈φi (z1, z1)φj(z2, z2)〉0 + λ∫

d2z〈φi (z1, z1)Od(z , z)φj(z2, z2)〉+ O(λ2).

So to first order in CPT:

∂

∂λ〈φi (z1, z1)φj(z2, z2)〉λ =

∫d2z 〈φi (z1, z1)Od(z , z)φj(z2, z2)〉

Three point function in integrand:

〈φi (z1, z1)Od(z , z)φj(z2, z2)〉 = CiAj ×× 1

(z1−z)hi +1−hj (z−z2)hj +1−hi (z12)hi +hj−1 (z1−z)hi +1−hj (z−z2)hj +1−hi (z12)hi +hj−1.

Regularize integral by putting cutoffs at insertion points of φi , φj :

|z − z1| > ε, |z − z2| > ε.

Assume temporarily that there is no degeneracy in conformal weight of φi :

hi = hj , hi = hj ⇒ φi = φj .

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Conformal Perturbation Theory III

This yields

∂

∂λ〈φi (z1, z1)φj(z2, z2)〉λ =

(2πCiAi ln(z12)+2πCiAi ln(z12)−2πCiAi ln(ε2)

)〈φi (z1, z1)φi (z2, z2)〉0, hi = hj

−2π δsi ,sj CiAj

(εdj−di

dj−di1

z2hj12 z

2hj12

+ εdi−dj

di−dj1

z2hi12 z

2hi12

), hi 6= hj

where si = hi − hi , di ≡ hi + hi .ε-dependent term are absorbed into renormalization of φi .Compare finite result with what obtained earlier. This yields the perturbedanomalous dimension to first order in CPT:

∂hi

∂λ= −πCiAi ,

∂hi

∂λ= −πCiAi .

Kadanoff; Cardy; A. B. Zamolodchikov; Dijkgraaf, Verlinde, Verlinde.

If there are {φk} with weights (hk , hk), need to diagonalize CiAk in the entireblock of fields with same conformal dimension. Find operators by identifyingall φk that mix with φi and iterating.

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Building theDeformation Operator


Lunin-Mathur technology for beginners

What is a symmetric product orbifold MN/SN? Get an orbifold when mod outan ordinary manifold by a finite symmetry group, which is in our case thesymmetric group SN , where N = N1N5. M can be T 4 or K3 in the D1-D5system; we picked the case M = T 4.In modding out by a symmetry, you might think we lose a bunch of d.o.f. fromthe spectrum. But with critical string theory orbifolds we actually gain backjust as many d.o.f. from taking into account the twisted sectors, closed stringanimals which only close up to a symmetry transformation.For our large-c symmetric product orbifold MN/SN , we need to distinguishwhich M is which, using a copy index. Twisted sector states in our SCFT areconfigurations that return to themselves up to a SN transformation.Physically, you can think about the insertion of a twist operator (e.g. σ2) astaking (e.g.) two singly wound strings and twisting them together to make asingle doubly wound string.Like with other orbifolds, it is easier to figure out the correct group-invariantexpressions for physical observables by lifting to the covering space. Morally,this is just like using the method of images.Fractionally moded R-currents Ja

−m/N allow adding excitations inexpensively.

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Pizza diagrams

Contributions to correlators from higher genera in cover are always suppressedby 1/N = 1/(N1N5). So our covering space will be the sphere.All details of untwisting the orbifold boundary conditions – for bosons andfermions – are contained in the magic of the map . Each point where a twistoperator sits in the base gets ramified. (II See Zadeh’s talk for workedexamples.) Interestingly, this map is not unique, and different maps may makedifferent calculations easier. c.f. Pakman-Rastelli-Razamat .In our large-c SCFT, have a conformal anomaly. For gµν = eφgµν in cover,

SL =c

96π

∫d2t√

g [gµν∂µφ∂νφ+ 2R(g)φ]

Fields which are not quasi-primary, eg stress-energy tensor T , pick up animportant Schwarzian derivative term in being uplifted to the cover.Regularizing SL is necessary. LM show how to do this by cutting out smalldiscs of radius ε around twists and operator insertions; ramified points aresmooth in the cover. Regulate infinity by bounding with a large disk of radius1/δ. Work at g = 0 in cover, so can choose to concentrate the curvature in aring, which ensures the R(g) term in SL contributes there.The resulting pictures you draw on the blackboard look like pizza diagrams.

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Fermions

Construct generic states in twist sector of SCFT by putting excitations onnormalized but undressed twist operators σ.Because we have a SCFT, there are bosonic and fermionic excitations flyingaround. Fermions have branch cuts associated with them. Owing to the magicof the map, branch cuts can be cancelled or arise newly because of theJacobian associated with going to the cover, where spin fields are needed.

Bosonizing the fermions allows us to get rid of all the nasty branch cuts andspin fields and replace them with normal ordered exponentials of bosons. Thiscomes at the price of introducing cocycles to ensure the correct(anti)commutation relations among the fundamental X , ψ fields.The degree of technical simplification bosonization affords is very significant.It is totally worth doing.When constructing cocycles it is important to get right the (anti)commutationrelations but also all the OPEs, including OPEs of spin fields with themselvesand others. And the symmetry properties (oy!).

Limitations of LM technology? Best for low numbers of twist operators, andaided by low order of twists.

14 / 21


Bosonization

Schematically, ψ ∼ :ekφ:. Define multiple bosonic fields φ5, φ6, φ5, φ6.Unrelated bosons do not share an OPE and commute as operators.Small problem: fermions obey Pauli exclusion while normal-ordered bosonexponentials do not. Introduce algebraic cocycle operators Ck whose only

physical role is to guarantee anticommutation of Ck e i(k5φ5+k6φ6+k5φ5+k6φ6),

Ck = e iπck , ck ≡ (k5 k6 k5 k6) M

p5

p6

p5

p6

, M =

12

12− 1

212

− 12− 1

212− 1

212− 1

2− 1

2− 1

2

− 12

12

12

12

.

New bit c.f. Polchinski text: diagonal entries in M. Bosonized fermions are:

ψ+1 = e iπc e−iφ6 , ψ+2 = e iπc e iφ5 ,

ψ−2 = −e iφ6 e−iπc , ψ−1 = e−iφ5 e−iπc , e iπc ≡ eiπ2

(p5+p6−p5+p6) .

Cocycles for spin fields:

S+ ≡ e iπc ei2

(φ5−φ6), S− ≡ e−iπc e−i2

(φ5−φ6),

S 1 ≡ e iπ2 e−

i2

(φ5+φ6), S 2 ≡ e−i π2 e

i2

(φ5+φ6).

Fermionic zero modes act on spin fields and map them to other spin fields.They satisfy the gamma matrix algebra {Γi , Γj} = 2δij . 15 / 21


Marginal deformation operator Od

There are four marginal deformation operators in the twist sector of the D1-D5CFT. Conformal weights (h, h) = (1, 1); singlets under SU(2)L × SU(2)R .We want a blow up of the orbifold, which transforms as 3 + 1 of SU(2)1.In SUGRA this corresponds to a1χ+ a2C6789 (the other fellow is b+

ij ).

Construct by applying modes of G−A−1/2 , G −B

−1/2 to CPs of twist-2 sector (σ++2 ).

Od (0, 0) = εAB G−A

− 12

G −B

− 12

σ++2 (0, 0) =

∮0

dz

2πi

∮0

dz

2πiG−A(z) G −B(z)σ++

2 (0, 0)

→∮

0

dt

2πi

(dz

dt

)1− 32∮

0

dt

2πi

(dz

dt

)1− 32

G−A(t) G −B(t)

(1

b18 b

18

S++(0, 0)

)=

1

|b|54

(: e

i2

(−φ6−φ5−φ6−φ5)(∂X 21∂X 22 − ∂X 22∂X 21

):

+ : ei2

(−φ6−φ5+φ6+φ5)(∂X 21∂X 12 − ∂X 22∂X 11

):

+ : ei2

(+φ6+φ5−φ6−φ5)(∂X 11∂X 22 − ∂X 12∂X 21

):

+ : ei2

(+φ6+φ5+φ6+φ5)(∂X 11∂X 12 − ∂X 12∂X 11

):

)(0, 0).

Complete deformation: λOd + λ∗O†d .

O†d = G+A

− 12

(z ′) G +B

− 12

(z ′)σ−−2 = G+A

− 12

(z ′) G +B

− 12

(z ′) J−0 (z) J−0 (z)σ++2 (z0, z0).

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Mixing ofLow-Lying String States


Operator mixing

For degenerate conformal weights there is potential first-order operator mixing,so it is necessary to identify all operators participating in mixing and evaluateCiAj to find the sought-after anomalous dimensions of states not protected bySUSY (i.e., everything above SUGRA modes). Generically, this is obviously anextremely hard problem. We choose to work with low-lying string states to makethe story tractable.We expect only a finite number of operators with the correct conformal weightto mix with a given low-lying string state. Why? For a given operator O, theconformal weight h is the energy eigenvalue of the corresponding state on thecircle. Since the circle has gapped modes, we expect a finite number of modesat or below a given energy, which corresponds to having a finite number ofoperators with h at or below a given cutoff. So the procedure for diagonalizingto find anomalous dimensions will truncate in a finite number of steps. This iswhy we decided to attempt it.We consider non-twist sector operators φi . Their 3-point functions with Od ,〈φi Od φi 〉, vanish because Od ’s twists are uncancelled: φi is not twisty.Our method of attack is based on the idea of using four-point functions andfactorization channels to figure out who mixes with whom and how strongly.Cuts number of independent 3pf we need to calculate. 17 / 21


4-point functions and factorization channels

To investigate first-order operator mixing we evaluate four-point functions

〈φi (z1, z1)Od(z2, z2)Od(z3, z3)φi (z4, z4)〉.Take coincidence limit:z1 → z2, z3 → z4.There are singularities.Find leading singular term andits coefficient.

Singularity signals intermediate quasi-primary ops mixing with φi at first order.Sum over CiAj of these ops gives coefficient of leading singular term.Subtract conformal families from leading order singular limit. Each conformalfamily is composed of an ancestor quasi-primary and its descendants underVirasoro algebra. Find the remaining leading order singularity, mixingcoefficient, and subtract their conformal families. Continue the procedure.We are interested in conformal families whose ancestors φj have sameconformal weight as φi : they contribute to anomalous dimension of φi . Otherquasi-primaries contribute only to wave function renormalization at first order.Computing 4pf and taking coincidence limits is a robust way to find mixingcoefficients of all quasi-primaries which mix with φi at first order.

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Mixing for ∂X∂X ∂X ∂X

Consider a specific low-lying string state belonging to non-twist sector:∂X∂X ∂X ∂X This excitation obviously has (h, h) = (2, 2).In two prior papers we computed the 4pf function of this with Od andcomputed the precise mixing coefficient for this string state. Main result:evidence of operator mixing contributing to anomalous dimensions at 1st order.II For a nice exposition of the technical tools we use to compute results,please proceed to Zadeh’s talk after the coffee break. You are guaranteed lesshand-waving and more equations and diagrams.Data needed for correction of h to 1st order? Find set of all (2, 2) operators φkthat contribute to mixing and evaluate their mixing coefficients. Do thisiteratively until nail down all such (2, 2) operators.Work by identifying CFT states at each twist level n. We figure that there willbe twist level no greater than 8 in the cover.Tentative estimate: we expect roughly of order 30 operators to mix with∂X∂X ∂X ∂X .Technically, it is fortunate that we found mixing at first order in conformalperturbation theory, rather than second order. Was unclear in advance whetherthis would occur.

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Work in progress

Our eventual goal is of course to diagonalize the matrix of structure constants,in order to find how deforming away from the orbifold point changes theanomalous dimensions of low-lying string states. Computing correlationfunctions and mixing coefficients for so many operators is a laborious process.Cocycles are fiddly. Work with {holo}×{anti-holo} tensored together to get allsymmetry properties to turn out right. Just finished them!Newest PhD student Ian Jardine and I are investigating usingmathematica/maple to automate parts of the work involved in computingcorrelation functions.

Also in progress: a related deformed D1-D5 orbifold CFT project with SamirMathur (Ohio State), Ida Zadeh (Brandeis), and Ben Burrington (Troy).More about eigenvectors than eigenvalues. To appear, 13mm.nnnn.

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Summary and future directions

Started programme to compute anomalous dimensions of low-lying stringstates in the D1-D5 orbifold SCFT deformed by a blowup.Focused on factorization channels of 4pf to identify intermediate states.Main result: evidence of operator mixing at first order in deformationparameter in CPT. This implies low-lying string states acquire anomalousdimension; h ↑ as λ ↑ for some, h ↓ as λ ↑ for others.II Go to Ida Zadeh’s talk after the break for a nice explanation of how wegeneralized Lunin-Mathur technology to calculate correlators involving bothtwist and non-twist operators and demonstrated low-lying string state mixing.

To-do listEnumerate all the operators that mix with ∂X∂X ∂X ∂X (etc.), and operatorsthat they mix with, and diagonalize the matrix of structure constants.Compute anomalous dimensions of low-lying twist sector states, c.f. Gava and

Narain . These states contain excitations on both left and right parts.Integrability? Unclear if persists under deformation. See if we can find a spinchain model that reproduces the matrix of anomalous dimension coefficients.Study problems involving infall of quanta into fuzzballs in the context of thedeformed symmetric product orbifold CFT.

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Aspects of D1-D5 SCFT near the orbifold point: string...

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