Renormalization of Twisted Ramond Fields in D1-D5 SCFT₂

In the near-horizon decoupling limit Maldacena:1997re of the type-IIB supergravity description of bound states of $N_{1}$ D1- and $N_{5}$ D5-branes, the asymptotic geometry becomes $AdS_{3}\times S^{3}\times T^{4}$, with large Ramond-Ramond charges Maldacena:1998bw ; Seiberg:1999xz , from which one can reconstruct its holographic dual SCFT₂; see David:2002wn ; Mathur:2005zp ; Skenderis:2008qn for reviews. There is strong indication that this D1-D5 SCFT₂ 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 $N=N_{1}N_{5}$, while the supergravity description is obtained by moving in moduli space with a deformation away from this ‘free orbifold point’. Gravitational solutions, which include the Strominger-Vafa black hole Strominger:1996sh and fuzzball geometries Lunin:2001jy ; Mathur:2005zp ; Kanitscheider:2007wq ; Kanitscheider:2006zf ; Skenderis:2008qn ; Mathur:2018tib , are all dual to states in the Ramond sector of the SCFT₂, and one can find correspondences between the geometries and Ramond states in many cases where the latter are BPS-protected from renormalization. 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 Lunin:2000yv ; Lunin:2001pw ; Balasubramanian:2005qu ; Avery:2009tu ; Pakman:2009ab ; Pakman:2009mi ; Burrington:2012yq ; Bena:2013dka ; Carson:2014ena ; Carson:2015ohj ; Fitzpatrick:2016ive ; Burrington:2017jhh ; Galliani:2017jlg ; Bombini:2017sge ; Tormo:2018fnt ; Eberhardt:2018ouy ; Bena:2019azk ; Dei:2019osr ; Eberhardt:2019ywk ; Giusto:2018ovt ; Martinec:2019wzw ; Hampton:2019csz ; Warner:2019jll ; Dei:2019iym ; Galliani:2016cai ; Bombini:2019vnc ; Giusto:2020mup . Nevertheless, the description of the dynamics of the deformed SCFT₂ 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 Guo:2019pzk ; Keller:2019suk ; Keller:2019yrr ; Guo:2019ady ; Belin:2019rba ; Guo:2020gxm .

The present paper investigates the effects on the conformal properties of twisted ground states in the $\mathcal{N}=(4,4)$ orbifold SCFT₂ when the theory is deformed by a marginal scalar modulus operator $\lambda O^{(\mathrm{int})}_{[2]}$ Avery:2010er ; Avery:2010hs ; Carson:2014ena ; Carson:2015ohj ; Carson:2016uwf . The first-order correction, in powers of $\lambda$, 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 $\mathscr{O}_{[n]}$, by eliminating the UV divergences with an appropriate renormalization of the fields. As a consequence, the scaling dimension $\Delta^{\mathscr{O}}_{n}(0)$ in the free orbifold point flows with $\lambda$ according to

where $J_{\mathscr{O}}(n)$ is a regularized integral defined in Sect.6 below. We will be mostly interested in two specific operators: the bare twist field $\sigma_{[n]}$, and, more importantly, the twisted R-charged Ramond fields $R^{\pm}_{[n]}$, with bare (holomorphic) conformal weight $h=\frac{n}{4}$. In both cases, our main result — the lifting of the conformal dimensions — holds for $n<N$. Ramond fields with $n=N$ and $n=2$, i.e. with maximal and minimal twist, are protected at leading order in the large-$N$ approximation. Bare twist fields with $n=N$ are also protected at leading order. For the particular case of twisted Ramond fields, this has been recently reported in our short letter Lima:2020boh .

The main ingredient in the calculation of second-order corrections to the two-point functions is finding an explicit analytic expression for the four-point function

for the fields $\mathscr{O}_{[n]}$ we are interested in. We will present a detailed derivation of the large-$N$ approximation of the corresponding functions, by applying covering surface techniques Lunin:2000yv ; Lunin:2001pw combined with the ‘stress-tensor method’ Dixon:1985jw . Our computation of the leading term in the $1/N$ expansion of the connected part of (2), takes into account only the terms contributing to genus zero surfaces, i.e. we use the well known map Arutyunov:1997gi ; Arutyunov:1997gt between the “base” branched sphere to its genus-zero covering surface Lunin:2000yv . 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 Pakman:2009ab ; Pakman:2009zz ; Pakman:2009mi .

Corrections to the anomalous dimensions at second-order follow from the integral of (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 Pakman:2009mi , and shown to vanish, as expected for protected operators which should not renormalize. For the twisted Ramond fields 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 SCFT₂. Our starting point is the observation that the integrals we are interested in can be put in a form studied by Dotsenko and Fateev Dotsenko:1984nm ; Dotsenko:1984ad ; dotsenko1988lectures in a different context, as integral representation of the conformal blocks of primary fields (curiously; not of their integrals) in the $c<1$ series²²2See Refs.Mussardo:1987eq ; Mussardo:1987ab ; Mussardo:1988av for the extension of the Dotsenko-Fateev integral representation to the Ramond and twisted sectors of the $\mathcal{N}=1$ and $\mathcal{N}=2$ supersymmetric minimal models. of minimal CFT₂ 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 (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 $R^{\pm}_{[n]}$ and $\sigma_{[n]}$ 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 $\langle R_{[n]}^{-}(\infty)O^{(\mathrm{int})}_{[2]}(1)R_{[n]}^{+}(0)\rangle$, and $\langle\sigma_{[n]}(\infty)O^{(\mathrm{int})}_{[2]}(1)\sigma_{[n]}(0)\rangle$, which do not vanish in the deformed SCFT₂. It is worthwhile to mention the recent use of similar methods to the renormalization of certain composite Ramond fields, for example $R_{[n]}^{+}(z)R^{-}_{[m]}(z)$ Lima:2020nnx . 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 $\lambda$, and are given by the sum of the second-order corrections (1) for each one of the constituents, i.e. $\Delta_{n}^{R}(\lambda)+\Delta_{m}^{R}(\lambda)$. What distinguishes the protected Ramond fields from the lifted ones is that the former have conformal weight $h=\frac{1}{24}c_{orb}=\frac{1}{4}N$, i.e. they are Ramond ground states of the full orbifold theory; in contrast, the fields with $h=\frac{1}{4}n$, $n<N$ are Ramond ground states only of the $n$-twisted sector, or of the ‘$n$-wound component string’ in the familiar description of the D1-D5 SCFT in terms of effective stings.

Another contribution of the present paper is the analysis of short-distance limits of the four-point function (2). In the limits where operators coincide, $u\to 0,1,\infty$, 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 Pakman:2009mi ; Burrington:2017jhh . 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 Burrington:2018upk ; deBeer:2019ioe , 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 fields includes a set of R-charged twisted non-BPS operators $Y^{\pm}_{m}$, appearing in the OPEs $O^{(\mathrm{int})}_{[2]}(z,\bar{z})R^{\pm}_{n}(0)$. Similarly, the algebra of $O^{(\mathrm{int})}_{[2]}$ and $\sigma_{n}$ includes new twisted operators $\mathcal{Y}_{m}$. We have calculated the dimensions of these operators, as well as the values of structure constants such as $\langle R^{\pm}_{n}(\infty)O^{(\mathrm{int})}_{2}(1)Y^{\pm}_{n\pm 1}(0)\rangle$. Applying the fractional spectral flows of Ref.deBeer:2019ioe with $\xi=n/(n+1)$, 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 SCFT₂, 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 $R^{\pm}_{[n]}$ and $\sigma_{[n]}$; we also explain how the renormalization scheme can be extended to a generic primary field $\mathscr{O}_{[n]}$; we also comment on the spectral flow between the fields $R^{\pm}_{[n]}$ and NS chiral operators in the free theory, and how it is “broken” after the deformation. 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 ($\lambda$-dependent) conformal dimensions of the renormalized twisted Ramond fields for their geometric bulk counterparts. Some auxiliary topics are left for the appendices.

The ‘free orbifold point’ of the D1-D5 system is the SCFT₂ with central charge $(c_{orb},\tilde{c}_{orb})=(6N,6N)$, obtained by taking $N$ copies of the free $\mathcal{N}=(4,4)$ SCFT₂, identified under the symmetric group $S_{N}$, with target space $(T^{4})^{N}/S_{N}$. The $\mathcal{N}=(4,4)$ superconformal algebra of the ‘seed theory’ has central charge $(c,\tilde{c})=(6,6)$, R-symmetry group $\mathrm{SU(2)}_{L}\times\mathrm{SU(2)}_{R}$, and ‘internal’ group $\mathrm{SO(4)_{I}}=\mathrm{SU(2)}_{1}\times\mathrm{SU(2)}_{2}$ corresponding to the torus $T^{4}$ of target space. We work on the complex plane, with coordinates $z,\bar{z}$.

The unitary representations of the holomorphic $\mathcal{N}=4$ algebra are characterized by three numbers $\{h,j^{3},\mathfrak{j}^{3}$}, respectively the conformal weight and the semi-integer charges under the R-current $J^{3}(z)$ of SU(2)_L, and a current $\mathfrak{J}^{3}(z)$ of the global SU(2)₁. Similar numbers $\{\tilde{h},\tilde{j}^{3},\tilde{\mathfrak{j}}^{3}\}$ characterize the anti-holomorphic sector with SU(2)_R and SU(2)₂ groups.

The theory can be realized in terms of free bosons $X^{\dot{A}A}(z,\bar{z})$ and free fermions $\psi^{\alpha\dot{A}}(z)$, $\tilde{\psi}^{\dot{\alpha}\dot{A}}(\bar{z})$, 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 (158), and can be written in terms of real bosons and fermions $X_{i}(z,\bar{z})$, $\psi_{i}(z)$ and $\tilde{\psi}_{i}(\bar{z})$, $i=1,2,3,4$; see (157). The fermions can be described in terms of chiral scalar bosons $\phi_{r}(z)$ and $\tilde{\phi}_{r}(\bar{z})$, with $r=1,2$. In the holomorphic sector,

Every exponential should be understood to be normal-ordered (and we ignore cocycles). The stress-tensor (3a) can be written in the completely bosonic form,³³3Normal ordering of two operators $\mathscr{A}_{1}$ and $\mathscr{A}_{2}$ is defined by $$\,\colon\!\mathscr{A}_{1}(z)\mathscr{A}_{2}(z)\!\colon\,\equiv\lim_{\varepsilon\to 0}\Big{[}\mathscr{A}_{1}(z+\varepsilon)\mathscr{A}_{2}(z)-\langle\mathscr{A}_{1}(z+\varepsilon)\mathscr{A}_{2}(z)\rangle\Big{]}.$$

Bosons are assumed to be periodic, so e.g. $X^{\dot{A}A}(e^{2\pi i}z)=X^{\dot{A}A}(z)$. Fermions can have Neveu-Schwarz or Ramond boundary conditions on $\mathbb{C}$. The Ramond sector has a collection of degenerate vacua with holomorphic dimension $h=\frac{c}{24}=\frac{1}{4}$, 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 $S^{\alpha}(z)$,

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 $(\otimes^{N}T^{4})/S_{N}$, and label operators $\mathscr{O}_{I}$ in each copy by an index $I=1,\cdots,N$. The energy tensor becomes

and the total central charge is $c_{\mathrm{orb}}=Nc=6N$.

Permutations of the copies can be realized by the insertion of twist operators $\sigma_{g}(z)$, $g\in S_{N}$, which give a representation of $S_{N}$, and act on the other operators by twisting their boundary conditions Dixon:1986qv ,

We are going to consider only cyclic twists, which form the building blocks of the Hilbert space of the orbifold theory Dijkgraaf:1996xw . So, denoting by $(n)$ a generic cycle of length $n$, we consider $g=(1)^{N-n}(n)\cong(n)$, leaving the trivial cycles implicit. We denote by $\sigma_{n}$ the twist operator for a generic cycle of length $n$; they cyclically permute the $n$ copies of the fields appearing in the cycle $(n)$, while leaving the remaining copies invariant. The conjugacy class is represented by the orbit-invariant combination

where the representing cycle can be taken to be $(n)=(1\cdots n)$, and the combinatorial factor $\mathscr{S}_{n}(N)$ makes the two-point function normalized, i.e.

The well-known (total) conformal dimension of a twist $\sigma_{n}(z,\bar{z})$ is Lunin:2001pw ; Dixon:1986qv

The $n$-twisted Ramond sector is generated by twisted spin operators with the appropriate $\frac{1}{n}$ rescaling of their weights. For the representative permutation $(1\cdots n)$, the $n$-twisted R-charged fields $R^{\alpha}_{n}(z)$ are

with a similar construction for the neutral $R^{\dot{A}}_{n}(z)$. From these, we can compose $S_{N}$-invariant combinations $R^{\zeta}_{[n]}(z)$, $\zeta=\pm,\dot{A}$ by summing over orbits, as we did for the normalized $S_{N}$-invariant twists $\sigma_{[n]}$. For example, the R-charged fields, with which we will be primarily concerned, are written explicitly as

where in the exponential we sum over $h(I)=\{h(1),\cdots,h(n)\}$, the image of the original copy set $I=\{1,\cdots,n\}$ under the permutation $h$. The $R^{\pm}_{[n]}$, like the spin fields $S^{\pm}$, form a doublet of SU(2)_L and a singlet of SU(2)₁, with charges $j^{3}=\pm\tfrac{1}{2}$ and $\mathfrak{j}^{3}=0$. On the other hand, the $R^{\dot{A}}_{[n]}(z)$ form a singlet of R-symmetry and a doublet of SU(2)₁, with charges $j^{3}=0$ and $\mathfrak{j}^{3}=\pm\frac{1}{2}$. The conformal weight of the $R^{(\zeta)}_{n}$ is

obtained from the combined weights of the exponential and the twist. Completely analogous fields $\tilde{R}^{\zeta}_{[n]}(\bar{z})$, with dimension $\tilde{h}^{R}_{n}=h^{R}_{n}$, make the anti-holomorphic sector. The normalization factor $\mathscr{S}_{n}(N)$ ensures that the two-point functions are normalized, granted that the non-$S_{N}$-invariant functions are normalized:

Let us examine these fields a little further. The Hilbert space of the orbifold theory, $\mathcal{H}_{orb}=\oplus_{[g]}\mathcal{H}_{g}$, is a direct sum of sectors invariant under the conjugacy classes of $S_{N}$ Dijkgraaf:1996xw . The latter are given by the irreducible decomposition of $g\in S_{N}$ into disjoint cycles, $[g]=\prod_{k\in\mathbb{N}}(k)^{q_{k}}$, with $\sum_{k}kq_{k}=N$, and we are interested in the simplest sector, corresponding to $[g]=(1)^{N-n}(n)$. This Hilbert space, $\mathcal{H}_{(n)}$, is invariant under the centralizer subgroup $S_{N-n}\times\mathbb{Z}_{n}$, where $S_{N-n}$ permutes the $N-n$ trivial cycles $(1)$, and $\mathbb{Z}_{n}$ acts on the elements permuted by $(n)$. It can be further decomposed as Dijkgraaf:1996xw

where $\mathcal{S}^{(N-n)}$ is the symmetric tensor product of copies entering the trivial cycles, and where $\mathcal{H}^{\mathbb{Z}_{n}}_{(n)}$ corresponds to the copies permuted by $(n)\in\mathbb{Z}_{n}$. States in $\mathcal{H}_{(n)}$ can be interpreted as a string with winding number $n$. We can think of the construction of the operators $R^{(\zeta)}_{n}$ as exciting the $n$-wound copies to the Ramond ground state, while leaving the unwound copies in the NS vacuum. Thus the operators $R^{(\zeta)}_{n}$ correspond to states $\ket{R^{(\zeta)}_{n}}=\ket{\varnothing}^{\mathrm{NS}}_{N-1}\otimes\ket{\varnothing,\zeta}^{\mathrm{R}}_{(n)}$ where $\ket{\varnothing}^{\mathrm{NS}}_{N-1}\in\mathcal{S}^{N-1}$ is the NS vacuum of the non-twisted copies and $\ket{\varnothing,\zeta}^{\mathrm{R}}_{(n)}$ are Ramond ground states of a CFT defined on $\mathcal{H}^{\mathbb{Z}_{n}}_{(n)}$. Since the latter CFT involves $n$ copies of the $\mathcal{N}=4$ SCFT, it has central charge $c=6n$; its Ramond ground states have the conformal weight $\frac{c}{24}=\frac{n}{4}$ in Eq.(14). We will sometimes refer to $R^{\alpha}_{[n]}$ as ‘Ramond ground states’, but it should be kept in mind that this is an abuse of nomenclature, as $R^{\alpha}_{n}$ are ground states only of the $n$-wound string; the true Ramond ground states of the orbifold theory have conformal weight $h=\frac{1}{24}c_{orb}=\frac{1}{4}N$, and are made either by $R^{\alpha}_{[N]}$, i.e. by single-cycle fields with maximal twist $n=N$, or, more generally, by “composite fields” $\prod_{k}(R^{\alpha_{k}}_{n_{k}})^{k}$ with $\sum_{k}kn_{k}=N$.

The main objective of this paper is to describe how the dimensions $h^{R}_{n}=\frac{1}{4}n$ are corrected when the free orbifolded SCFT is perturbed by a marginal operator.

A marginal deformation of the free orbifold turns the theory into an interacting SCFT, with the action

parameterized by a dimensionless deformation parameter $\lambda$. In the large-$N$ limit, in which we will be interested, the deformation parameter $\lambda$ should scale with $N$ in such a way that the ’t Hooft coupling $\lambda_{*}\equiv\lambda/\sqrt{N}$ is held fixed as $N\to\infty$; see Lunin:2000yv ; Pakman:2009zz .

The “scalar modulus” interaction operator $O^{(\mathrm{int})}_{[2]}$ is marginal, with total conformal dimension $\Delta=h+\tilde{h}=2$. This dimension should not change under renormalization. Also, $O^{(\mathrm{int})}_{[2]}$ must be a singlet of R-symmetry, in order for $\mathcal{N}=(4,4)$ SUSY not to be broken. From the 20 deformation operators, which correspond to the 20 SUGRA moduli (see Avery:2010er ), we consider the $S_{N}$-invariant singlet

constructed as a descendent of the NS chiral field $O^{(0,0)}_{[2]}(z)$ with $h=\frac{1}{2}=j^{3}$.

Let us review a few key results in conformal perturbation theory used in the next sections; see for example Keller:2019yrr for more detail. For a marginal perturbation, the two-point function

of a neutral and hermitian (for simplicity) operator $\mathscr{O}$ is still fixed by conformal symmetry, hence the effect of the marginal perturbation has to be a change of its conformal dimension. The $\lambda$ expansion of the functional integral gives

where absence of a $\lambda$-index (e.g. in $\Delta$) 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, $G(u,\bar{u})$ is the undetermined part of the four-point function in terms of the anharmonic ratio $u\equiv(z_{12}z_{34})/(z_{13}z_{24})$,

and $\Lambda$ is a cutoff for the integral

The $\log\Lambda$ divergence in the two-point function requires the introduction of an appropriate regularization and a corresponding renormalization of the field $\mathscr{O}$. The logarithmic form of the divergent terms indeed has the effect of changing the exponent of the renormalized two-point function, thus changing $\Delta$. The operators we are interested in have a vanishing three-point function with $O^{(\mathrm{int})}_{[2]}$, i.e. $C=0$. The corrections in (19) therefore start at second order in $\lambda$, and the renormalized field is

We can see that

so the $\Lambda$-divergence is canceled, and the free-theory dimension $\Delta$ has flowed to a $\lambda$-dependent value

The integral $J$ also gives the first-order $\lambda$-correction to the particular structure constant in (20). 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 (24), we need to be able to calculate the four-point function (21) in the free orbifold theory. In the next section, we show how to do this.

Our goal is to compute four-point functions⁴⁴4A note on convention: in this paper, fields inside correlation functions are to be understood in two-dimensional theory, e.g. $\sigma_{n}(z,\bar{z})$, instead of $\sigma_{n}(z)$. However, when fixing a point in ${\mathbb{C}}^{2}$ we only write one argument for economy of notation. Thus, in (27), it should be understood that $\mathscr{O}_{[n]}(0)=\mathscr{O}_{[n]}(0,\bar{0})$, $O^{(\mathrm{int})}_{[2]}(1)=O^{(\mathrm{int})}_{[2]}(1,\bar{1})$, etc.

for primary operators $\mathscr{O}$ in the $n$-twisted sectors of the orbifold SCFT₂. 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 (27); 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 $\mathscr{O}_{[n]}$ as the charged Ramond ground state, or as a bare twist field.