Renormalization of Twisted Ramond Fields in D1-D5 SCFT$_2$

We explore the Ramond sector of the deformed two-dimensional $\cal N = (4, 4)$ superconformal $(T^4)^N /S_N$ orbifold theory, describing bound states of D1-D5 brane system in type IIB superstring. We derive the large-$N$ limit of the four-point function of two R-charged twisted Ramond fields and two marginal deformation operators at the free orbifold point. Specific short-distance limits of this function provide several structure constants, the OPE fusion rules and the conformal dimensions of a few non-BPS operators. The second order correction (in the deformation parameter) to the two-point function of the Ramond fields, defined as double integrals over this four-point function, turns out to be UV-divergent, requiring an appropriate renormalization of the fields. We calculate the corrections to the conformal dimensions of the twisted Ramond ground states at the large-$N$ limit. The same integral yields the first-order deviation from zero of the structure constant of the three-point function of two Ramond fields and one deformation operator. Similar results concerning the correction to the two-point function of bare twist operators and their renormalization are also obtained.


Introduction
The Ramond ground states of the two-dimensional N = (4, 4) superconformal (T 4 ) N /S N orbifold theory and its marginal moduli deformation are fundamental ingredients for the microscopy of (nearly) extremal black holes (BH). The microstates involved in the original BH entropy computation by Strominger and Vafa [1] are formed by the momentum excitations of such Ramond states, and the coherent superpositions of these states also appear to be an important constitutive part of a variety of fuzzball geometries [2][3][4][5][6][7].
The N = (4, 4) orbifold SCFT 2 is known to be a dual description of certain lowenergy limits of the D1-D5 brane system -a type-IIB supergravity solution with the structure of a two-charge, extremal 1/4-BPS horizonless black hole, resulting from the bound states of N 1 D1-and N 5 D5-branes, see [3,6,8] for reviews. In the near-horizon decoupling limit [9], the asymptotic geometry becomes AdS 3 × S 3 × T 4 , with large Ramond-Ramond charges [10,11], from which one can reconstruct its holographic dual SCFT 2 . There is strong indication that the D1-D5 SCFT 2 flows in the infrared to a free field theory whose sigma model is (T 4 ) N /S N , an orbifold of T 4 by the symmetric group S N , with N = N 1 N 5 . The supergravity description is obtained by moving in moduli space with a deformation away from this 'free orbifold point'. Extensive research has achieved considerable progress in the understanding of the free orbifold and its deformation, as well as in the construction of 'superstratum' geometries corresponding to the microscopic picture . Nevertheless, the description of the dynamics of the deformed SCFT 2 is still not fully understood. One of the open problems concerns the selection rules separating protected states from "lifted" ones, whose conformal data flow in the deformed theory after renormalization [36][37][38][39][40][41].
The present paper investigates the effects on the conformal properties of twisted ground states in the N = (4, 4) orbifold SCFT 2 when the theory is deformed by a marginal scalar modulus operator λO (int) [2] [20, 21,[42][43][44]. The first-order correction, in powers of λ, of the two-point function of a ground state is known to vanish. Our main result is an explicit derivation of the finite part of the second-order correction to two-point functions of n-twisted primary operators O [n] , by eliminating the UV divergences with an appropriate renormalization of the fields. As a consequence, the scaling dimension ∆ O n (0) in the free orbifold point flows with λ according to where 2 < n < N and J O (n) is a regularized integral defined in Sect.6 below. Its value for the particular case of twisted Ramond fields has been recently reported in our short letter [45]. The main ingredient in the calculation of second-order corrections to the two-point function is finding an explicit analytic expression for the four-point function Our main interest will be on the relevant cases when O [n] is a twisted R-charged Ramond ground state R ± [n] , or a bare twist field σ [n] . We present a detailed derivation of the large-N approximation of the corresponding functions, by applying covering surface techniques [12,13] combined with the 'stress-tensor method' [46]. Our computation of the leading term in the 1/N expansion of the connected part of (1.2), takes into account only the terms contributing to genus zero surfaces, i.e. we use the well known map [47,48] between the "base" branched sphere to its genus-zero covering surface [12]. The alliance of the covering surface with the stress-tensor method emphasizes some interesting mathematical properties of the correlation functions, and their relation to Hurwitz theory, as discussed in detail in Refs. [16,17,49].
Corrections to the anomalous dimensions at second-order follow from the integral of (1.2) over the positions of the interaction operators. The analogous integral in the case where there are NS chiral fields at z 1 and z 4 has been computed in [17], and shown to vanish, as expected for protected operators which should not renormalize. For the Ramond ground states and the twist operators, however, the integrand has a more complicated structure, with one more branch cut, and without appropriate regularization the integrals are divergent. In order to define and evaluate their finite parts, we have elaborated a regularization procedure and a specific renormalization scheme for the fields in the deformed SCFT 2 . Our starting point is the observation that the integrals we are interested in can be put in a form studied by Dotsenko and Fateev [50][51][52] in a different context, as integral representation of the conformal blocks of primary fields (curiously; not of their integrals) in the c < 1 series 1 of minimal CFT 2 models. While, in the one hand, they can be formally written as specific contour integrals in the complex plane -with the contours ensuring a series of algebraic properties -on the other hand these integrals can be represented by four 'canonical functions' which are analytic in their parameters, even in cases where the integral itself diverges. Thus, by analytic continuation, the canonical functions give a regularized result for the desired integrals of the four-point functions (1.2). When applied to the parameters of NS chirals, this procedure gives a vanishing result, as expected; but when applied to the Ramond and twisted fields, we find finite, non-vanishing corrections to the conformal dimensions. The analytic expressions for the renormalized conformal dimensions of R ± [n] and σ [n] is one of the most important results of this paper. As a byproduct of the computation of the integrals, we can also present the first-order correction to the structure constants R − [n] (∞)O (int) [2] (1)R + [n] (0) , and σ [n] (∞)O (int) [2] (1)σ [n] (0) , which do not vanish in the deformed SCFT 2 . It is worthwhile to mention the recent use of similar methods to the renormalization of certain composite Ramond fields, for example R + [n] (z)R − [m] (z) [56]. In the composite case, an important consequence of the renormalization procedure is the existence of a condition, namely m+n = N , selecting a class of protected (non-renormalized) states. The remaining states, with n+m < N , are lifted: their renormalized conformal dimensions flow with λ, and are given by the sum of the second-order corrections (1.1) for each one of the constituents, i.e. ∆ R n (λ) + ∆ R m (λ). Another contribution of the present paper is the analysis of short-distance limits of the four-point function (1.2). In the limits where operators coincide, u → 0, 1, ∞, we are able to derive several structure constants, the OPE fusion rules and the conformal dimensions of some non-BPS operators. These OPE data add to the description of the Ramond and twisted sectors of the free-orbifold point. Our results for the non-BPS fields are consistent with what is known about the chiral NS and twisted sectors [17,23]. They are also in agreement with the recently conjectured universality of OPEs of certain chiral fields and the deformation operator in the large-N limit [57,58], and represent an extension of these results for all other sectors of the free orbifold theory. In particular, we find that the OPE algebra of the deformation operator and the Ramond ground states includes a set of R-charged twisted non-BPS operators Y ± m , appearing in the OPEs O (int) [2] (z,z)R ± n (0). Similarly, the algebra of O (int) [2] and σ n includes new twisted operators Y m . We have calculated the dimensions of these operators, as well as the values of structure constants such as R ± n (∞)O (int) 2 (1)Y ± n±1 (0) . Applying the fractional spectral flows of Ref. [58] with ξ = n/(n + 1), we find that our results for the twisted Ramond fields' OPEs are in complete correspondence with those obtained from OPEs in the NS sector resulting in specific non-BPS NS fields.
The structure of the paper is as follows. In Sects.2 and 3, we fix our notations by defining first the free orbifold SCFT 2 , and then its deformation away from the free orbifold point; we also review some key features of conformal perturbation theory used later. In Sect.4, we give a detailed calculation of the four-point functions involving Ramond and bare twist fields, necessary for the second-order correction of the two-point functions. In Sect.5, we investigate certain short-distance limits of the four-point function in order to extract OPE fusion rules, conformal weights and structure constants of several operators in the free-orbifold point. In Sect.6, we return to conformal perturbation theory, with a detailed study of the regularization and the final computation of integrals resulting in the change of the conformal weights of R ± [n] and σ [n] ; we also explain how the renormalization scheme can be extended to a generic primary field O [n] . In Sect.7 we present a compact summary of our results, together with a short discussion of a few open problems and the eventual consequences of the continuous (λ-dependent) conformal dimensions of the renormalized twisted Ramond fields for their geometric bulk counterparts. Some auxiliary topics are left for the appendices.
The unitary representations of the holomorphic N = 4 algebra are characterized by three numbers {h, j 3 , j 3 }, respectively the conformal weight and the semi-integer charges under the R-current J 3 (z) of SU(2) L , and a current J 3 (z) of the global SU(2) 1 . Similar numbers {h,j 3 ,j 3 } characterize the anti-holomorphic sector with SU(2) R and SU(2) 2 groups.
The theory can be realized in terms of free bosons XȦ A (z,z) and free fermions ψ αȦ (z),ψαȦ(z), whereas the stress-tensor, the R-current and the super-current are expressed as with similar expressions for the anti-holomorphic sector. Conventions for SU (2) indices are given in Appendix A. The complex bosons and the complex fermions obey reality conditions (A.3), and can be written in terms of real bosons and fermions X i (z,z), ψ i (z) andψ i (z), i = 1, 2, 3, 4; see (A.2). The fermions can be described in terms of chiral scalar bosons φ r (z) andφ r (z), with r = 1, 2. In the holomorphic sector, Every exponential should be understood to be normal-ordered (and we ignore cocycles). The stress-tensor (2.1a) can be written in the completely bosonic form, 2 Bosons are assumed to be periodic, so e.g. XȦ A (e 2πi z) = XȦ A (z). Fermions can have Neveu-Schwarz or Ramond boundary conditions on C. The Ramond sector has a collection of degenerate vacua with holomorphic dimension h = c 24 = 1 4 , and different charges under the global and R-symmetry SU(2) groups. The set of Ramond vacua can be obtained from the NS vacuum by the action of spin fields, conveniently realized as exponentials, e.g. for the SU(2) doublet S α (z), To construct the orbifold (T 4 ) N /S N , one makes N copies of the free SCFT and identifies them under the action of S N ; more explicitly, we take the N -fold tensor product (⊗ N T 4 )/S N , and label operators O I in each copy by an index I = 1, · · · , N . The energy tensor becomes Normal ordering of two operators A 1 and A 2 is defined by and the total central charge is c orb = N c = 6N .
Permutations of the copies can be realized by the insertion of twist operators σ g (z), g ∈ S N , which give a representation of S N , and act on the other operators by twisting their boundary conditions [59], We are going to consider only single-cycle twists, which are the building blocks of the Hilbert space [60]. So, denoting by (n) a generic cycle of length n, we consider g = (1) N −n (n) ∼ = (n), leaving the trivial one-cycles implicit. The single-cycle twist sector of the Hilbert space is created by the orbit-invariant combination of twists defined as where the representing cycle can be taken to be (n) = (1 · · · n), and the combinatorial factor S n (N ) makes the two-point function normalized, i.e. (2.8) The well-known (total) conformal dimension of a twist σ n (z,z) is [13,59] ∆ σ n = h σ n +h σ n , h σ n = The twisted Ramond sector is generated by twisted spin operators. For the representative permutation (1 · · · n), the charged fields R α n (z) are with a similar construction for the neutral RȦ n (z). From these, we can compose the S N -invariant combinations R ζ [n] (z), ζ = ±,Ȧ using the normalized S N -invariant twists σ [n] , which give the Ramond vacua |∅ ζ R[n] of the (holomorphic) n-twisted sector. For example, the R-charged fields, with which we will be primarily concerned, are written explicitly as where in the exponential we sum over h(I) = {h(1), · · · , h(n)}, the image of the original copy-set I = {1, · · · , n} under the permutation h. The R ± [n] , like the spin fields S ± , form a doublet of the R-symmetry SU(2) L and a singlet of SU(2) 1 , their R-charges being j 3 = ± 1 2 and j 3 = 0. On the other hand, the RȦ [n] (z) form a singlet of R-symmetry and a doublet of SU(2) 1 , with charges j 3 = 0 and j 3 = ± 1 2 . The conformal weights of all holomorphic Ramond ground states is computed from (2.9), which is the correct dimension for a Ramond vacuum in the n-twisted sector. Completely analogous fieldsR ζ [n] (z), with dimensionh R n = h R n , make the anti-holomorphic sector. The normalization factor S n (N ) ensures that the two-point functions are normalized, granted that the non-S N -invariant functions are normalized: The main objective of this paper is to describe how the dimensions h R n are corrected when the free orbifolded SCFT is perturbed by a marginal operator.

Away from the free orbifold
A marginal deformation of the free orbifold turns the theory into an interacting SCFT, with the (Euclidean) action parameterized by a dimensionless deformation parameter λ. In the large-N limit, in which we will be interested, the deformation parameter λ should scale with N in such a way that the 't Hooft coupling λ * ≡ λ/ √ N is held fixed as N → ∞; see [12,49]. The "scalar modulus" interaction operator O (int) [2] is marginal, with total conformal dimension ∆ = h +h = 2. This dimension should not change under renormalization. Also, O (int) [2] must be a singlet of R-symmetry, in order for N = (4, 4) SUSY not to be broken. From the 20 deformation operators, which correspond to the 20 SUGRA moduli (see [42]), we consider the S N -invariant singlet constructed as a descendent of the NS chiral field O (0,0) Let us review a few key results in conformal perturbation theory used in the next sections; see for example [38] for more detail. For a marginal perturbation, the twopoint function of a neutral and hermitian (for simplicity) operator O is still fixed by conformal symmetry, hence the effect of the marginal perturbation has to be a change of its conformal dimension. The λ expansion of the functional integral gives where absence of a λ-index (e.g. in ∆) always indicates evaluation in the free theory, and the objects in the r.h.s. are defined as follows. At first order, C is the structure constant coming from the three-point function At second order, G(u,ū) is the undetermined part of the four-point function in terms of the anharmonic ratio u ≡ (z 12 z 34 )/(z 13 z 24 ), and Λ is a cutoff for the integral The log Λ divergence in the two-point function requires the introduction of an appropriate regularization and a corresponding renormalization of the field O. The logarithmic form of the divergent terms indeed has the effect of changing the exponent of the renormalized two-point function, thus changing ∆. The operators we are interested in have a vanishing three-point function with O (int) [2] , i.e. C = 0. The corrections in (3.4) therefore start at second order in λ, and the renormalized field is We can see that so the Λ-divergence is canceled, and the free-theory dimension ∆ has flowed to a λ-dependent value The integral J also gives the first-order λ-correction to the particular structure constant in (3.5). This can be seen from the functional integral expansion of the corresponding three-point function. For our case where the free-theory constant vanishes, we find To compute the integral (3.9), we need to be able to calculate the four-point function (3.6) in the free orbifold theory. In the next section, we show how to do this.

Four-point functions
Our goal is to compute four-point functions 3 for primary operators O in the n-twisted sectors of the orbifold SCFT 2 . Some aspects of the computation are universal, depending only on the nature of the twists: we start by describing the covering surface appropriate to the twisted structure of (4.1); then we describe the stress-tensor method to compute the simplest four-point function with this structure, containing only bare twists. Finally, we turn to the cases containing interaction operators with O [n] as the charged Ramond ground state, or as a bare twist field.

The covering surface
Twisted correlators such as (4.1) are complicated functions, with specific monodromies of their arguments fixed by their (bare) twist fields constituents. The standard way [12] of implementing the boundary conditions (2.6) for G(u,ū) is to map the 'base sphere' S 2 base = C ∪ ∞ to a ramified 'covering surface' Σ cover , whose ramification points correspond to the position and the order of twists operators. At large N , the leading contribution comes from genus-zero covering surfaces. Denote coordinates on the base by z ∈ S 2 base , and coordinates on the covering sphere by t ∈ S 2 cover , and fix the four punctures on each surface to be 3 A note on convention: in this paper, fields inside correlation functions are to be understood in two-dimensional theory, e.g. σ n (z,z), instead of σ n (z). However, when fixing a point in C 2 we only write one argument for economy of notation. Thus, in (4.1), it should be understood that [2] (1) = O (int) [2] (1,1), etc.
The method for finding z(t) for generic monodromies was pioneered in [12] and generalized in [49]. For the specific monodromies (and topology) above, The monodromies at z = 0 and z = ∞ are evident, but at z = 1 and z = u they are implicit in the derivative z (t), which must vanish at every branching point. Indeed, vanishes at t = 0 with the correct monodromy, while x and t 1 must be the roots of the quadratic expression in brackets. This quadratic equation relates the parameters t 1 , t 0 , t ∞ and x, We are free to choose one of the ratios t 0 /x, t 0 /t 1 , t ∞ /t 1 and t ∞ /x as long as they satisfy the two conditions (4.4), and we choose a rational function.

Four-point functions
The covering map encodes the monodromies of functions like (4.1), with the twist structure into the ramification points of the covering surface. One way of computing g(u,ū), formulated by Lunin and Mathur [12], is to cut circles around the ramification points, replace them with vacua and compute the functional integral directly. An alternative 4 [59] is to use the conformal Ward identity: if one is able to find the residue r(u) of the following function on the base, the Ward identity gives a differential equation which can be solved for the holomorphic part of g(u,ū) = g(u)g(ū). The antiholomorphic partg(ū) =ḡ(ū) is obtained likewise, usingT (z).
In simpler orbifold theories, it is possible to find r(u) by engineering the function with the appropriate poles and monodromies [59]. Here, we can follow Refs. [16,[47][48][49] and use the covering surface as an aid, by computing the correlation functions on S 2 cover , where the monodromies are trivial, and then mapping back: f (z) is a function of the position of the stress tensor which, unlike the twists, is not placed on a branching/ramification point -hence mapping from covering to base is just a conformal transformation. On the covering, because the twists disappear, and when mapping back to base only the anomalous transformation of T does not cancel in the fraction, so The position of the twists appear as parameters implicit in the inverse maps z → t, which encode the twist structure of (4.7). There is a sum over I in Eq.(4.11) because T (z) is a sum over copies (2.5). Around a branching point, there is one inverse map t I (z) for each copy entering the corresponding twist; at z = u, the insertion point of σ 2 , there are two maps, which can be found locally [17,48], as follows. Take the logarithm of the ratio z(t)/z(x), i.e. log(z/u) = n log t x + log t−t 0 x−t 0 − log t−t∞ x−t∞ , and expand both sides, In the first equation, the coefficients are found from the Taylor expansions, The coefficients c k are solved in terms of a k and b k order by order, by inserting the c k power series into the first equation in (4.12). The multiple inverses z → t appear as multiple solutions for the c k . After solving for the c k , we can put the powers series into the r.h.s. of Eq.(4.11), expand to order (z − u) −1 and extract the desired residue. The coefficients c 1 , c 2 and c 3 completely determine the result up to this order, (4.14) As expected, there are two solutions. When the parameters t 0 and t ∞ in a k , b k are written explicitly in terms of x, these coefficients are functions of x alone, thus we find the residue r as a function of x. One can check that r(x) is the same for both choices of the c k . Solving Eq.(4.9) requires expressing r(x) as an explicit function of u, but there are multiple inverses of u(x). It is easier to make a change of variables, and solve instead the differential equation whose solution is The integration constant c σ has to be determined by looking at OPE limits (see App.C). 5 Now, we have found a function parameterized by the pre-image of u under the covering map z(t). For fixed u = u * , there are H different pre-images x a , a = 1, · · · , H, solutions of the equation The degree of the polynomial shows that H = 2n. Note that this is not the number of sheets of the ramified covering (u is the position of a branching point), it is the number of different covering maps with the assumed monodromy conditions; H is a Hurwitz number [16,17,49]. The method has thus yielded H functions g(x a (u)). This was expected, because the S N structure of the composition of cycles in Eq.(4.7) is not completely fixed. Labeling cycles by the position of their twists operators, those entering g(u,ū) must compose to the identity, otherwise the correlator vanishes. There are several collections {(n) ∞ , (2) 1 , (2) u , (n) 0 } of cycles which solve Eq.(4.18), 6 and these collections can be arranged into equivalence classes defined by The existence of different such equivalence classes is the reason for the existence of different functions g(x a (u)); there are precisely H = 2n equivalence classes [49]. Inside each of these classes, let C s (N ) be the number of collections {(n) ∞ , (2) 1 , (2) u , (n) 0 } for which the cycles involve a fixed number s of distinct elements of {1, 2, · · · , N }. Then it can be shown [49] that C s is the same for all classes, and that, for large N , it scales as where n 1 = n = n 4 and n 2 = 2 = n 3 are the order of the q = 4 twists involved in (4.7). But n r − 1 is also the order of the ramification points of the covering surface, s is the number of its sheets, hence its genus is fixed by the Riemann-Hurwitz formula We thus see that C s (N ) ≡ C g (N ) ∼ N −g−1 , therefore the covering surface with g = 0 constructed in §4.1 gives the leading contribution at large N [12]. For our four-point functions, the Riemann-Hurwitz formula gives s = −g + n + 1, hence we see that, for the covering surface to have genus zero, we must have 2 < n < N. When we sum over the orbits of individual cycles to make an S N -invariant correlation function, we get all terms in each of the equivalence classes above, This sum corresponds to different OPE channels resulting from composing the twist permutations, not only for g(u,ū) but for the other functions G(u,ū) which share the same twist structure.

Charged Ramond fields
Let us now turn to the function The Ramond fields R ± [n] (z,z) are lifted to the corresponding spin field S ± (t,t), so we compute and then find the residue H of the function with t(z) one of the maps obtained from Eqs.(4.12) and (4.14). The deformation operator, denoted by O (int) (t,t) -without a twist index since there are no twists on the covering surface -can be expressed on S 2 cover in terms of the basic fields only, because the contour integrals in the super-current modes G αA dzG αA (z) just pick up a residue (see e.g. [18]). The result is a sum of products of bosonic currents, free fermions and spin fields coming from the lifting of the NS chiral field O (0,0) [2] (z,z) → S + (t)S+(t). Writing spin fields as exponentials, The constant a int can be conveniently chosen by a redefinition of the deformation parameter λ. For now, we leave it unspecified. To compute the correlators, the strategy is to show that contractions of T (t) with the fields in the numerator of (4.25) are always proportional to , appearing in the denominator of Eq.(4.25). We can decompose T (t) = T B (t) + T F (t) into bosonic and fermionic parts, respectively As far as bosons are concerned, each term of the product O (int) (t 1 )O (int) (x) has the structure ∂XĊ C (t 1 )∂XĖ E (x) multiplied by "transparent" fermionic or anti-holomorphic factors. Using the conformal Ward identity and the two-point functions (A.5), Hence we can recompose G, and obtain (4.28) For the fermionic part of the calculation, it is very helpful to organize O (int) as : (4.29c) the (a.h)s being combinations of anti-holomorphic fields which can be read from (4.27). This makes it is clear that contractions with O (int) [2] are very simple, and The second line in the r.h.s. can be further simplified because, since the only nonvanishing two-point functions (A.7) are between a field and its conjugate, it follows Putting this back in (4.30), G appears as a common factor canceled in (4.25), Combining (4.28) and (4.32), we get (4.33) Inverting the maps, we find the residue H(x) of F (z) to be The solution of the differential equation where C R is an integration constant.

Bare twists
Let us also consider appearing in the second-order correction of the two-point function of bare twist fields. The computation of goes as before (but is simpler), and we find where C σ is an integration constant. The same function has been computed in App.E of Ref. [17], but using a different parameterization map u(x), in place of (4.6) (hence their function G(x) is different from ours).

OPE limits, fusion rules and structure constants
The short-distance behavior of G(u,ū) in the limits u → 1, 0, ∞ contains the complete conformal data of the operator product expansions of the fields involvedi.e. the OPE fusion rules. Recall that super-conformal invariance fixes the form of the OPE algebra of generic primary holomorphic fields O with structure constants C 12k , and j 3 k = j 3 1 + j 3 2 .

The OPE of two interaction operators
The OPE of two interaction operators appears in the limit u → 1 of G(u,ū). To extract this limit from G(x), we have to find the inverse maps x a (u) which contribute to the singularities near u = 1. For both G R (x) and G σ (x), there are clearly only two contributions, i.e. limits where G(x) becomes singular, namely: 7 x = ∞ and x = 1−n 2 , the former with multiplicity one, and the latter with multiplicity three. We label the two corresponding functions, given in (B.4), as x 1 a (u), with a (gothic) index a = 1, 2, and the superscript indicating that u → 1. Each function gives a channel of the fusion rule, according to Eq.(4.23). Both functions G R (x) and G σ (x) have the same behavior in these limits, as it was necessary for consistency, since both functions should give the [3] ], where the r.h.s. is based on the composition of permutations. We mostly focus on G R (x) in what follows, similar calculations for G σ (x) are listed in Appendix C.

Determining the constants of integration
1 (u) given by Eq. (B.4), we obtain By formula (5.1), since O (int) [2] has weight h int = 1, the leading singular term shows an operator of dimension h = 2 − 2 = 0 -the identity operator. Also, the coefficient of the term ∼ (1 − u) −1 is zero, hence there is no contribution from a field of dimension h = 1, as it was to be expected for a truly marginal deformation.
The function in Eq.(5.2) corresponds to a correlator where the permutations in the twists form one representative element of the equivalence class where the 2-cycles of the interaction operators cancel. This happens when they share both elements. At order N −1 , there must be s = n + 1 elements entering the permutation, c.f. Eq.(4.20), so we can take this representative function to be or any other with a global relabeling of elements in the cycles. We now fix the constant a int in (4.27) so that the non-S N -invariant two-point functions are normalized, O Note that in these functions the two-cycles must share both of their elements, since, as in Eq.(4.18), we must have (2) ∞ (2) 1 = 1. With this definition, the normalized Together with the normalization (2.13), inserting the limit (5.2), back into the fourpoint function (5.3) we find 16n 2 C R = 1. The same reasoning can be applied to the function G σ (x), which has the exact same limit as (5.2) in this channel. Therefore With the functions G(x) completely fixed, we can now look at other OPEs and derive structure constants.
The σ 3 channel In the other channel corresponding to u → 1, we must expand G R (x) around x = 1−n 2 , and insert x 1 2 (u) given by Eq. (B.4), Once again, the coefficient of next-to-leading divergence, ∼ (1 − u) 3/3 , vanishes, showing that there is no dimension-one operator in this conformal family either. The leading singularity shows the presence of an operator of dimension 2 3 = h σ 3 , so we have found σ 3 itself, and the OPE whose structure constant is given in Eq.(C.5), and found independently from G σ (x). Inserting the OPE into the correlation function we find the structure constant This corresponds to an operator of dimension zero, and is in fact the correct expression for the two-point function of Ramond fields, Eq.(2.13). In the channel (5.7) we now find the behavior ∼ (1 − u) −n+ 8 3 , indicating a twist-three operator of holomorphic weight To understand the appearance of σ 3 in a channel of the OPE R − n R + n , let us consider the simpler case of the correlator with bare twists only. Changing the points of Eq.(4.7), we can find the OPE σ n σ n from the limit u → 1 of the function Of course, there are other twists in the r.h.s. but they cannot be found from the four-point function we have began with, because of the condition (4.18). As discussed above, in channel (C.6) the two twists σ 2 in the correlator have inverse cycles, hence it is necessary that the two twists σ n also be the inverse of each other; this gives 1 in the fusion rule. As for the channel (C.7), we have seen that the cycles in σ 2 then only have one overlapping element, say, σ (k ) σ (km) = σ (k m) . Hence for Eq.(4.18) to be satisfied the two σ n operators must compose to σ n σ n = σ (m k) , which is why σ 3 appears.

Non-BPS operators in the OPEs of
We now turn to the limit u → 0, where the interaction operator collides with either the Ramond field R + n (0) or with the bare twisted field σ n (0), depending on the function we analyze, if either G R or G σ . Now one can find all 2n solutions of Eq.(4.17), viz. x = 0 (with multiplicity n − 1) and x = −n (with multiplicity n + 1), all contributing to the OPE limits.
The function G R (u,ū) gives the OPE O (int) [2] (u,ū)R + [n] (0). Using (B.1), Counting powers of u, we find that the OPE O which follows from the same procedure of fixing points used to find (5.13). Since the factor of (1 − u)

4−n 2
does not contribute to the leading term near u = 0, we immediately find the same expansion as before. Now the resulting fields Y − m have the same dimensions (5.20), but opposite R-charge, j 3 = − 1 2 . In summary, we have found the fusion rules where the fields Y ± m have the dimension (5.20). The appearance of m = n ± 1 in the r.h.s. is a basic consequence of permutation composition, see Eq.(C.9). We take the Y ± m to be normalized, so that (by charge conservation) the non-vanishing two-point functions are Inserting the OPE back into the four-point function, the leading short-distance coefficients C a give us information about the product of structure constants (Recall that we must take |G R (x 0 a (u))| 2 .) In the l.h.s. we actually have products of conjugate three-point functions/structure constants, (with the twists in the subscripts) and taking the explicit expressions for C a , found from the expansions (5.18)-(5.19), we get = + n − 2 2 log(n + 1) − n 2 + 4n − 2 2(n + 1) log n. σ n , found in the limit u → 0 of G σ (u,ū). The two channels reveal twisted operators Y m with zero R-charge and dimensions the terms in the r.h.s. corresponding to x 0 1 (u) and x 0 2 (u), respectively. The coefficients calculated from the expansion of G σ give structure constants as before: We have found that the operator algebras of Ramond fields with the deformation operator include non-BPS fields. These fields are consistent with the fractional spectral flow with ξ = n n+1 of twisted non-BPS fields in the NS sector, recently found [58] to be a part of the OPEs of the deformation operator and NS chiral operators. A complete study of the algebras found here requires knowledge of OPEs such as [2] ]. For that, the new fields have to be explicitly constructed. From our discussion of their properties, and in particular from the conformal dimension (5.20), we can infer that This explicit construction should be sufficient for the study of the remaining OPEs by the computation of four-point functions such as (which, incidentally, can be computed with the same covering map used here).

Analytic regularization and field renormalization
We now turn to the calculation of the conformal dimension of Ramond fields in the interacting SCFT 2 . At second-order in perturbation theory, this requires computation of the integral (3.9), using the functions we have found in Sect.4.

Dotsenko-Fateev integrals
We want to compute integrals J = d 2 u G(u,ū), given an analytic expression for , with α 1 + n − 2 = α 4 − n α 2 − n − 2 = α 3 + n (6.1) from which G(u,ū) is obtained by inversion of the map (4.6). Both the Ramond function (4.35) and the bare twist function (4.38) have the form (6.1). We can perform a change of variables from u to x in the integral which, taking the special relation between the exponents into account, becomes We then make the following change of variables [17], such that every term in the new integrand is expressed simply in terms of y, J(n) = 1 2 n(n + 1)C 2 I(n), (6.4) (6.6) We will refer to I(n) as a 'Dotsenko-Fateev (DF) integral', as it has been studied in detail by Dotsenko and Fateev, as a representation of correlation functions in degenerate CFTs [50][51][52]. 9 The properties of I(n) crucially depend on the exponents of its critical points y = {0, w n , 1}. For example, the exponents for G R (x) are thus, for general n, all three critical points are branching points. The integral diverges at 1 and ∞, and vanishes at 0 for all n; at w n , it converges for n ≤ 6 and diverges for n > 6. Clearly, some regularization procedure is needed. Following Ref. [52], we now show that I(n) can be expressed in terms of hypergeometric functions, leading to a regularization by analytic continuation. We do this in two steps: 1. Assume that the parameters a, b, c are such that the DF integral exists.
2. Express the integrals in terms of an analytic function of a, b, c that is well-defined also for values of a, b, c, such as (6.7), for which the original integral diverges.
(Such functions will turn out to be a product of hypergeometric and Gamma functions.) This leads to an extension of the definition of the integrals by their maximal analytic continuation.
As we shall see, the procedure is consistent. Let us write y ∈ C as y = y 1 + iy 2 in (6.5), and perform a rotation of y 2 , such that y 2 → i(1 − 2iε)y 2 , with ε a positive arbitrarily small parameter. Defining v ± = y 1 ± y 2 (where y 2 now refers to the new, rotated coordinate), and expanding the integrand to first order in ε, The double integrals have been factorized into a product of two one-dimensional integrals, because the variable v ± only appears in the v ∓ integral multiplied by the infinitesimal parameter ε. The effect of the ε-terms is to specify how the otherwise real integrals of go around the points 0, w n , 1. To further disentangle the integrals, we split integration over v + at 0, w n , 1, so that ε-terms can be ignored, while the v − integrals go around the contours γ k dictated by the infinitesimal terms ε(v − − v + ) as in Fig.1(a), (6.10) For example, for v + ∈ (0, w n ), hence the contour γ 1 goes above v − = 0, and below v − = w n , 1.
The function f (ζ) has branch cuts, so closing the contours γ k with semi-circles is non-trivial. Here is the point were our regularization procedure effectively starts. Assume that a, b, c are such that the DF integral is convergent. Precisely, assume that If we try to close γ 1 or γ 2 in Fig.1(a), we are deemed to cross branch cuts, and move to another Riemann sheet of f (v − ). One way out of this is to cross the cut on a branching point, where f (v − ) is single-valued. That the integral exists at the branching points is assured by our assumptions (6.11). Thus we choose the branch cuts to align with the Real axis in two different ways: for the integral over γ 1 they extend to −∞, and for γ 2 they extend to +∞; then we close the contours with semicircles as in Fig.1(b). In one case, we cross the real axis at v − = 0, in the other at v − = w n . Next, we deform the contours as in Fig.1(c). Given our assumptions (6.11), as R → ∞ the integral over the (almost closed) circle vanishes, and we have where the contours C i are shown in Fig.1(d). Integration over C i is standard: the effect of coasting the two margins of a branch cut, turning at the branch point is to produce a phase 2i sin(πθ).
Thus we arrive at the following form of (6.10), where s(θ) ≡ sin(πθ) and we have defined four 'canonical integrals': TheĨ 1,2 can actually be written in terms of the I 1,2 with a different arrangement of their arguments: Also, by combining deformed contours such as the ones in Fig.1, it can be shown [50] that I 1,2 (a, b, c, w n ) andĨ 1,2 (a, b, c, w n ), with the same arguments, form a linear system: The four canonical integrals are proportional to the Euler representation of the hypergeometric function [63], valid for |arg(1 − w)| < π, 0 < Re(β) < Re(γ).

(6.16)
With the substitution t = 1/v + in I 1 , and t = v − /w n in I 2 , we find The restrictions (6.16), required for both integrals to be represented by hypergeometrics, translate to a, b, c as (6.19) and also 0 < w n < 1, cf. (6.6). These conditions are consistent with our starting hypothesis (6.11), therefore Eq.(6.12) can be read as a product of hypergeometric and Gamma functions.
The 'canonical functions' (6.17) and (6.18) are analytic functions of each of the parameters a, b, c, on the domain of validity (6.19). This is evident for the Gamma functions, and is also true for the hypergeometrics, see [64, §2.1.6]. Note that in (6.17) and (6.18) what actually appears is the 'regularized hypergeometric function' Hence I(a, b, c; w n ) is analytic in a, b, c separately. Consequently, an analytic continuation of I(a, b, c; w n ) to outside of the domain of definition (6.19) is unique, when it exists. We take this analytic continuation to be the definition of the DF integral (6.5) for arbitrary parameters. Note that it is not precluded that, outside the domain (6.19), I(a, b, c; w n ) might develop a singularity -there may be a barrier to the analytic continuation -it just happens that, for the applications below, the continuation is, indeed, (almost) always well-defined.

The integral for R-charged Ramond fields
Let us apply our results to the Ramond function (4.35). As noted before, the parameters (6.7) do not lie within the domain (6.19), hence we are indeed using the analytic continuation. Eqs.(6.17), (6.18), (6.14) yield F (− 1 2 + n 4 , − 1 2 ; 1 + n 4 ; 1 − w n ) (6.22c) Several observations are in order. The expression (6.22a) does not correspond immediately to the formula (6.17), because here we have Γ(−1) in the denominator. In this case, we must use Eq.(6.21) to find the correct expression for I 1 in (6.22a). Expression (6.22b) can be found immediately from (6.18). The factor s(c) in (6.22b) can be found either from Γ(z)Γ(1 − z) = π/ sin(πz), or from the linear system (6.15), by noting that in the present case we have Eqs.(6.22c) and (6.24d) follow immediately from (6.14), but (6.24d) is only valid when n is odd. For n even there are two cases. When n = 4(k + 1) a pole of the Gamma function in the denominator of (6.24d) requires that we use Eq.(6.21) again, leading tõ I 2 =Ĩ 1 . This can also be found from the linear system (6.15) by noting that, besides (6.23), now s(b R + c R ) = 0. All of the peculiarities above are taken into account if we simply replace the hypergeometrics by the well-behaved regularized hypergeometric, We can now use Eqs.(6.12) and (6.4) to write Before we analyze this result further, let us consider what happens if n = 4k + 2.
The case n = 4k + 2 When n = 4k + 2, a pole of the Gamma function appears in the numerator of (6.24d), so I(n) is infinite. We can isolate the divergence, however. First, we list again the four canonical integrals, now in terms of k = n−2 4 , Here we note that in this case we have s(a R ) = s(c R ) = 0 besides (6.23), and the linear system (6.15) is not valid anymore. This is related to the fact that there is now only one branch point in the canonical integrals, instead of the three branchings of the general case. Eq.(6.12) is, however, still valid. Moreover, we have I 2 (k) = −I 1 (k)/ sin(πk). The sine is cancelled in Eq.(6.12), from which we separate the finite part and the divergence: where ψ(ζ) is the digamma function. Using (6.27) and (6.4), we end up with which is the final regularized expression for J R when n = 4k + 2.

Comments
We present a unified plot of J R (n) for every n in Fig.2; for n = 4k + 2, we plot the regularized function J reg R (k). One can distinguish a peculiar "almost periodicity" of the function, with period 4. We believe that this might be related to some combinatoric relation between the twists of R ± n and the twists of the interaction operators appearing in the four-point function. As can be seen from Fig.2, J R (n) stabilizes around small, negative values for large n. As a reference, for k = 30 we have Note that an analytic form of J R (n) for large n is very hard to find because it involves taking simultaneous limits of the multiple arguments of the hypergeometric function.

The integral for bare twists
The function G σ (x) also has the form (6.1), and where I(n) is a DF integral (6.5) with exponents The canonical integrals I 1 (a σ , b σ , c σ ) = − π(n − 1) 2 2(n + 1) 2 F 3 2 , 1+6n+n 2 4n ; 3; w n (6.37a) ; 3; w n (6.37b) ; 1 − w n (6.37c) are all well-defined and convergent for all values of n ∈ N -the arguments of the Gamma functions are never a negative integer (for n > 1). We plot the values of J σ (n) = − n + 1 32n  to be compared with the corresponding values for J R (n) given above. For n = 4k + 2, J σ (n) grows with n, instead of stabilizing around a small value; note that these values of n are also those for which the Ramond integral J R (n) diverged, and had to be regularized.
The regularization of the divergent integral J described in this section gives welldefined, finite two-point functions in the deformed theory, to second order in λ. Here we have considered the renormalization of bare twists and Ramond ground states, but the method is more general, and can be applied to all sectors of the SCFT 2 . Our procedure relied on the fact that J can be reduced to a Dotsenko-Fateev integral for the functions G R (x) and G σ (x). This, in turn, relied on the structure of these functions, which had the form (6.1). It is not hard to check that, for any primary twisted O [n] that we insert in the general correlation function (4.1), the corresponding G(x) always has the form (6.1), including the specific relations between the pairs of exponents α 1 , α 3 and α 2 , α 4 .
To see that this is true, one can reverse-engineer the reasoning developed in Sect.5. Given an operator O n consider the correlator This function must be singular in the short-distance limit u → 0, and consistent with the OPE rule (5.1). The associated function G(x) must therefore be singular when x goes to one of the values x 0 a for u → 0, or x ∞ a for u → ∞, where x 0 a , x ∞ a are the channels in the limits u → 0 and u → ∞, respectively; see App.B. This fixes the numerator of while the denominator is fixed similarly by the channels in u → 1. But (6.46) is just another way to write (6.1). Note that this argument only makes use of the properties of the function u(x) and its inverses, i.e. only on the structure of the twists in the correlator -not on the specifics of O n nor, even, on the properties of O . Thus O [n] can be, say, a primary NS field, an R-charged or R-neutral Ramond ground state, or a bare twist field; also, we can replace O (int) [2] by, say, the simplest chiral NS primaries O (p,q) [2] (defined e.g. in [6]).
Having proved that G(x) must have the structure (6.46), it remains for us to show that the exponents satisfy the two relations in (6.1). This is also a consequence of the OPEs. Take the channel x 0 1 in the limit u → 0. We have the OPE O (int) 2 O n ∼ X n−1 for some operator of twist n − 1, whose dimension is fixed by the power of u appearing in G(x 0 1 (u)). Since G(x 0 (6.47) Now, in the limit u → ∞, we will have the OPE O n−1 , and the dimension of X † n−1 is to ∞. But X m and X † m have the same dimension, so subtracting Eqs.(6.47) and (6.48) we find gives the first relation in (6.1). The second relation, between α 1 and α 3 , is found similarly in the channels x 0 2 and x ∞ 2 , completing the proof that (6.1) holds in general.
Thus we have shown that, for any primary twisted filed O [n] , we can always reduce J O to a Dotsenko-Fateev integral, for some set of parameters a, b, c. Then, we can apply our regularization procedure and subsequent renormalization of the two-point function -if that is necessary. A very important example of fields for which there is no renormalization is the class of BPS-protected NS chiral twisted fields. Explicit computation of their non-renormalization was given in [17], for O as expected.
As a final remark, let us point out that our regularization and renormalization procedure is even more general. It can be extended almost intactly for the analysis of two-point functions of operators with a more complicated twist structure. In Ref. [56] we have studied the double-cycle composite Ramond fields R ± [n] R ± [m] (z,z). In this case, the covering map is more complicated, and, correspondingly, so is the form of G(x) which generalizes (6.1); but just as explained above, there are relations between exponents which allow a transformation of J(n, m) into a Dotsenko-Fateev integral, and then everything follows as in here.

Discussion
The investigation of the twisted Ramond sector of marginally-deformed D1-D5 SCFT 2 presented in this paper is based on the explicit construction of the large-N limit of the four-point function (4.24) of two R-charged Ramond fields and two scalar modulus operators O (int) [2] (z,z). In fact, this function provides dynamical information about both theories: the "free-orbifold point" SCFT 2 , and its marginal deformation at second order in λ. In what follows, we will briefly address a some open problems whose solutions can eventually be reached by adapting the methods developed in the present paper.
More on the properties of non-BPS fields. The four-point functions that we have calculated can be used not only for accessing the deformed SCFT 2 , but also to give a more complete description of the free orbifold itself. For example, the OPE data we have extracted from short-distance limits reveal important features of the Ramond sector of the free SCFT 2 , such as the conformal weights, R-charges and a few structure constants of the non-BPS twisted Ramond operators Y ± n±1 given by Eq.(5.31). Their four-point functions with the deformation operator, [2] R ± n , which are known to describe specific features of the effective fuzzball geometries, including (among other options) BTZ black holes with specific conic singularities, or eventually singular black rings [14,67].
R-neutral Ramond ground states. Here we have focused on the R-charged SU(2) L,R doublets R α [n] . The set of Ramond ground states RȦ [n] , which are neutral under Rsymmetry, and form a doublet of SU(2) 1 , have only been mentioned in passing. These fields have a very important role in counting black hole entropy, because "typical" microstates, which dominate the ensemble from which the entropy is derived, have zero R-charge [6,14]. The renormalization of single-cycle 10 neutral fields RȦ [n] in the deformed SCFT 2 can be studied with the same methods of the present paper. One must compute their four-point function with O (int) [2] by lifting to covering space with the corresponding R-neutral spin fields SȦ, etc. In practice, the actual computation of the four-point function is slightly more complicated then the one presented in §4.2, because some simplifying cancelations only occur for the R-charged fields. Once the function is found, however, all the methodology developed here for exploration of operator algebras via short-distance limits, as well as the renormalization scheme, can be applied.
Lifted vs. protected states. As the conformal field theory flows away from the free orbifold in the deep IR, one looks for protected fields whose moduli-independent properties extrapolate to the strong coupling SCFT 2 related to the semi-classical supergravity limit, where a weakly-coupled description of black holes is possible.
We have demonstrated that the twisted Ramond ground states in deformed SCFT are not protected. The only exception are the Ramond fields of minimal and maximal twists, n = 2 and n = N , which remain protected at the leading order in the 1/N approximation, since their four-point functions must be computed with a genus-one covering surface. The renormalization of Ramond fields, starting at second order in perturbation theory, means, at first sight, that the mass of the associated supersymmetric black hole is slightly corrected, while its charge is preserved. Then the black hole becomes nearly-extremal, with M > Q, and less supersymmetric. Under certain restrictions on the parameters describing the model, the holographic three-dimensional bulk images of the renormalized Ramond states could be certain BTZ-like heavy objects with conic singularities, related to the values of λ [14,67]. Indeed, for a large range of the parameters, the most expected consequence of the presence of such lifted twisted Ramond states lies beyond the low-energy, weak-supergravity picture. They are expected to represent specific massive states of the effective superstring sigma model for AdS 3 × S 3 × T 4 with finite R-R fluxes and large T 4 volume (the decompactification limit) [11], that could be related to the little or/and long strings considered in [32].
In the standard holographic description of the deformed SCFT 2 as the dual of a black hole in the near-horizon decoupling limit, where the geometry becomes effectively AdS 3 ×S 3 ×T 4 , the construction of fuzzballs uses specific compositions of twisted Ramond states. Here it is important to note that the fact that single-cycle Ramond fields undergo renormalization does not mean that more complicated composite Ramond fields cannot be protected. The simplest of such fields, the double-cycle operators R ± [n] R ± [m] (z,z) in the twisted sector with conjugacy class g = (n)(m)(1) N −n−m ∈ S N , were discussed in Ref. [56]. There, it was shown that, for generic values of the twists such that 2 < m + m < N , the four-point function has a 'partially disconnected' part -a factorization into a four-point function of noncomposite operators -leading to a flow of their conformal dimensions as a corollary of the results of the present paper. However, operators with m + n = N do constitute a protected family of ground states. Similar phenomena are expected for more general composite states, including higher "powers" of R ± [n] and R-neutral fields. Note that these protected operators are the ones with weight h H = 1 24 c orb ; also, the singlecycle operators with h H are those with n = N for which our renormalization results do not apply. A series of works [24,25,31,[68][69][70] has shown that it is possible to obtain bulk geometries dual to the free-orbifold point by considering 'heavy-heavylight-light' (HHLL) four-point functions involving two such 'heavy' Ramond states, with h H = 1 24 c orb = 1 4 N 1, and two 'light' operators with a fixed conformal weight h L in the large-N limit (e.g. h L = 1 2 ). In general, heavy states are expected to dominate the microstate ensemble dual to black holes [14].
Despite the existence of families of protected operators, one cannot avoid the question of what are (if any) the bulk holographic images of generic renormalized composite Ramond ground states, with their continuous, λ-dependent conformal dimensions. The answer remains to be discovered, and there are indications that tools necessary for this end include the description of the symmetry algebra of the deformed SCFT 2 , and the representations of its unitary ground states. the kind hospitality of the Federal University of Espírito Santo, Vitória, Brazil, where part of his work was done.
The SCFT can be realized in terms of four real bosons X i (z,z), four real holomorphic fermions ψ i (z) and four real anti-holomorphic fermionsψ i (z), with i = 1, · · · , 4. They are related to the complex fields XȦ A (z,z), ψ αȦ (z) andψαȦ(z) by There are analogous constructions for the right-moving sector. The Levi-Civita symbol always has the structure 12 = +1. Pauli matrices are defined such that σ 3 = Diag(1, −1). The "Pauli vector" σ i = (σ 1 , σ 2 , σ 3 , σ 4 ) and its conjugateσ i have components (we work in Euclidean space) σ a = −σ a σ 4 = i1 2×2 =σ 4 . The reality condition of X i and ψ i implies that where the last equation is for the bosonized fermions (2.2). The non-vanishing bosonic two-point functions are between a current ∂XȦ A and its complex conjugate; explicitly, as can be checked from (A.4) using the reality conditions (A.3).

C. OPEs with bare twists and structure constants
In this appendix, we examine the OPE limits of the functions G σ (x) and g(x). We derive several structure constants, some of which are known in the literature, thus checking our expressions for g(x) and G σ (x).