# Screening stringy horizons

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## Abstract

It has been argued recently that string theory effects qualitatively modify the effective black hole geometry experienced by modes with radial momentum of order \(1/\sqrt{\alpha '}\). At tree level, these \({\alpha '}\) effects can be explicitly worked out in two-dimensional string theory and have a natural explanation in the T-dual description as coming from the integration of the zero mode of the linear dilaton, which yields a contribution that affects the scattering phase shift in a peculiar manner. It has also been argued that the phase-shift modification has its origin in a region of the moduli space that does not belong to the exterior black hole geometry, leading to the conclusion that at high energy the physics of the problem is better described by the dual model. Here, we elaborate on this argument. We consider the contribution of world-sheet instantons in the two-dimensional Euclidean black hole \(\sigma \)-model and study its influence on the phase shift at high energy.

## Keywords

Black Hole Vertex Operator Black Hole Background Tachyon Condensate Black Hole Geometry## 1 Introduction

In the recent paper [1], it has been argued that string theory effects qualitatively modify the effective black hole geometry experienced by modes with radial momentum of order \(1/\sqrt{\alpha ^{\prime }}\). At tree level, these \(\alpha ^{\prime }\) effects can be explicitly worked out in two-dimensional string theory, where the black hole background admits an exact world-sheet description in terms of the gauged \(SL(2,\mathbb {R})/U(1)\) Wess–Zumino–Witten (WZW) model. In addition, this model is well known to have a dual description (in a sense similar to T-duality) that involves a two-dimensional flat tachyonic linear dilaton background, known as Fateev–Zamolodchikov–Zamolodchikov (FZZ) dual [2]. In the FZZ dual model, the \(\alpha ^{\prime }\) effects studied in [1] have a natural explanation as coming from the integration over the zero mode of the linear dilaton, which yields a contribution that affects the scattering phase shift in a peculiar manner. Such contributions and, consequently, the phase-shift modification they produce seem to come from a region of the moduli space that does not belong to the exterior black hole geometry. This led the authors of [1] to conclude that the high-energy physics is better described by the FZZ dual model, rather than by the two-dimensional (2D) black hole \(\sigma \)-model. Here, we elaborate on this argument. We consider world-sheet instanton contributions and study their influence to the phase shift at high energy. The 2D black hole \(\sigma \)-model contains an operator that controls the world-sheet instanton contributions. This is given by a *second* screening operator in the world-sheet CFT, and the integration over the zero mode of the linear dilaton, once such a screening operator is considered, yields an \(\alpha '\)-dependent function of the string states momenta. Therefore, it is natural to ask to what extent such an \(\alpha '\)-dependent function accounts for the high-energy phase shift discussed in [1]. We will study this in Sect. 5. Before, in Sect. 2, we review the 2D Euclidean black hole \(\sigma \)-model. In Sect. 3, we discuss string scattering amplitudes on the black hole background and the high-energy modification of the phase shift. In Sect. 4, we review how such a modification is gathered in the FZZ dual theory. In Sect. 5, we first show how the second screening is associated to world-sheet instantons, and then we study their contribution in relation to the stringy phase shift.

## 2 String 2D black hole

*r*is large, namely far from the horizon. The metric above describes a semi-infinite cigar-like geometry of the \( \mathbb {R}^{2}\) topology. The compact coordinate \(\theta \) represents the compactified Euclidean time. The horizon of the Lorentzian black hole gets mapped to the tip of the cigar, which is located at \(r=0\). In the large

*r*region, the Euclidean metric approaches a cylinder of radius

*L*.

*r*, one can identify \(\phi \sim \sqrt{k}r\), and this corresponds to a linear dilaton background. The constant

*M*appearing in (3) can be shown to be related to the two-dimensional black hole mass. Its precise value is not relevant for perturbative physics, as it can be set to 1 by simply rescaling \(\phi \). The relevant parameter is actually

*L*. The classical limit thus corresponds to

*k*going to infinity.

*B*–

*C*ghost system [6, 7, 8]. This amounts to improve the stress tensor (7) with an extra piece, namely

^{1}

*k*limit.

*U*(1) of the coset model is

*j*,

*m*, and \(\bar{m}\) are isospin variables that label the \(SL(2, \mathbb {R)\times }SL(2,\mathbb {R)}\) representations. These variables represent the momenta associated to the radial and Euclidean time coordinates and the winding number along the latter. More precisely, we have the radial momentum

States with \(p_{\phi }\in \mathbb {R}\) and \(m-\bar{m}\in \mathbb {Z}\) correspond to vectors of the continuous series representation of \(SL(2, \mathbb {R})\). Discrete representations have \(j\in \mathbb {R}\), \(m-\bar{m }\in \mathbb {Z}\) and \(m+\bar{m}-j\in \mathbb {Z}\). Here, we will be involved with the former.

## 3 High-energy scattering

*M*in (14) refers to the fact that expectation value is defined by interacting action (3). More precisely, we can write

*k*-dependent factor in the definition of the path integral, one finds

*s*additional integrated vertices \(M\int \beta \bar{\beta }\mathrm{e}^{-\sqrt{ \frac{2}{k-2}}\tilde{\phi }}\) in (20) comes from the factor \( ( \tilde{S}_{\text {I}}) ^{s}\) in (18). Recall we have

*resonant*correlators. The poles \(s\in \mathbb {Z}_{> 0}\) admit a physical interpretation similar to that proposed in Ref. [9] for the analogous poles in Liouville field theory.

Formula (19) can also be interpreted within the context of the Coulomb gas realization of 2D CFT correlation functions, where the insertion of *s* operators \(\tilde{S}_{\text {I}}\) correspond to the inclusion of screening charges needed to satisfy the charge condition imposed by the presence of the background charge \(Q=-1/\sqrt{k-2}\) at infinity.

The black hole mass parameter *M* in the amplitudes (19) plays the role of string coupling constant: Its power depends on the genus of the surface goes like \(M^{s+1-g}\) and, as mentioned above, its absolute value is determined by the zero mode of the dilaton, i.e. \(M\sim \mathrm{e}^{\Phi _{0}}\).

When integrating over the \(\beta \)–\(\gamma \) system, and because \(\beta \) is a 1-differential, the Riemann–Noch theorem, once combined with (21), yields exactly the same conservation law as obtained from the integration over the zero mode of the field *X*, namely \(\sum _{i=1}^{N}(m_{i}+\bar{m}_{i})=0\). The condition on the total winding number \(\sum _{i=1}^{N}(m_{i}-\bar{m}_{i})\), on the other hand, is more subtle [10, 11].

The prefactor \(\Gamma (-s)\) in (19) can be alternatively obtained by virtually integrating over the imaginary part of \(\phi _{0}\). This produces a \(\delta \)-function that selects a precise amount of operators \(\tilde{S} _{\text {I}}\) from the series expansion of \(\mathrm{e}^{-S_{\text {I}}}\). More precisely, the \(\delta \)-function selects the \(s{\text {th}}\) term \(\frac{ (-1)}{s!}\left( S_{\text {I}}\right) ^{s}\) of the series, with \( s=\sum _{i=1}^{N}j_{i}+1\). Then the infrared divergence \(\delta (0)\sim \Gamma (0)\) can be combined with the multiplicity factor \((-1)/s!\) in order to produce the factor \(\Gamma (-s )\) by recalling the formula \( \lim _{\varepsilon \rightarrow 0}\Gamma (-s+\varepsilon )/\Gamma (\varepsilon )=(-1)^{s}/\Gamma (s+1)\) for \(s\in \mathbb {Z}_{\ge 0}\).

*M*can be adjusted to absorb the

*k*-dependent functions in the first factor of (22), the only relevant

*k*-dependent piece in the two-point function is given by the prefactor

*z*| one has \( \Gamma (z)\simeq \sqrt{{2\pi }/{z}}\left( {z}/{e}\right) ^{z}\ \left( 1+ \mathcal {O}(1/z)\right) \). Applying this to (23), one finds that in the regime \(p_{\phi } \gg \sqrt{k-2}\), one finds

## 4 The dual theory

*X*is compact, while \(\varphi \) takes values on the real line. That is to say, unlike the 2D Euclidean black hole \(\sigma \)-model, the model defined by (25) has the topology \(\mathbb {R\times }S^{1}\). Thought of as a string \(\sigma \)-model action, sine-Liouville theory (25) represents a flat linear dilaton background in the presence of a non-homogeneous tachyon condensate.

The duality between (3) and (25) has been conjectured by Fateev, Zamolodchikov, and Zamolodchikov (FZZ) in an unpublished work [2], and it has been reviewed and elaborated by Kazakov, Kostov, and Kutasov in Ref. [12]. It represents a kind of T-duality. In fact, the supersymmetric version of the FZZ duality actually corresponds to mirror symmetry [13], which relates the \(\mathcal {N}=2\) version of Liouville theory with the Kazama–Susuki \(SL(2,\mathbb {R})/U(1)\)-model. In [14, 15], Maldacena explained how the bosonic FZZ duality emerges as a consequence of the supersymmetric extension. More recently, a proof of the FZZ conjectured duality was given by Hikida and Shomerus in Ref. [16]; see also [17].

*N*-point function can be violated up to \(N-2\) units [2]. The winding number preserving correlation functions correspond to the particular cases \(s_{+}=s_{-}=(\sum _{i=1}^{N}j_{i}+1)/(k-2)\). On the other hand, correlators with \(\sum _{i=1}^{N}\omega _{i}\ne 0\) correspond to correlators computed with \( s_{+}-s_{-}\ne 0\), so that the quantity \((\sum _{i=1}^{N}j_{i}+1)/(k-2)\) in the latter case is not necessarily an integer number. In the case of the two-point function (\(N=2\)) the winding number is preserved, and thus \(s_-=s_+\). This implies that one can describe the sine-Liouville 2-point correlation functions by inserting \(s_+ + s_-\) operators \(\mathrm{e}^{-\sqrt{\frac{k-2}{2}}\varphi }\cos ( \sqrt{k/2} \tilde{X})\). It can be shown [12] that, in this case, the integration over the zero mode of \(\varphi \) in the sine-Liouville two-point function yields a factor

^{2}

^{3}(24). In addition, it was observed there that the integral over the zero mode of \(\varphi \), which in the case of sine-Liouville theory produces the \(\Gamma \)-function (28), receives dominant contributions coming from a region of the moduli space that, in the 2D Euclidean black hole \(\sigma \)-model side, has no representative. This suggests that sine-Liouville description is the appropriate one to describe these finite-\(\alpha '\) effects. To complete the argument, in the next section we will address the following two questions: First, whether (and how) finite-\(\alpha '\) effects (finite-

*k*effects) can be gathered in the 2D Euclidean black hole \(\sigma \)-model by the integration over \(\phi _0 \). Secondly, whether (or to what extent) such \(\alpha '\) effects recover the phase shift (24).

## 5 World-sheet instantons

*s*and \(\tilde{s}\) are the amount of operators of the type \(S_{ \text {I}}\) and of the type \(S_{\text {II}}\), respectively.

*s*screening operators of the type (17) and no operators of the type (29) exactly agree with the correlation functions computed by inserting \(\tilde{s}=s/(k-2)\) operators of the type (29) and no operators of the type (17) provided the couplings

*M*and \(\tilde{M}\) are related by

*N*-point function, yields the pole condition

*N*-point correlation functions. One confirms from (37) that such poles actually appear. In [11], these poles were interpreted as world-sheet instantons, corresponding to classical string configurations of momentum \(j\sim k\) that can extend to the large \(\phi \) region with no cost of energy. These configurations correspond to holomorphic maps \(\gamma =\gamma (z)\), associated to classical solutions, which extend in the \(\phi \) direction with no potential preventing them from going to \(\phi =\infty \). When integrating over auxiliary fields \(\beta \) and \(\bar{\beta }\), one produces an effective potential \(\int \mathrm{d}^{2}z\partial \bar{\gamma }\bar{ \partial }\gamma \mathrm{e}^{\sqrt{\frac{2}{k-2}}\phi }\), which vanishes for configurations with \(\bar{\partial }\gamma =0\) (as operators (17) and (29) do). These classical configurations are closely related to the long strings discussed in [19].

*k*-dependent function of the momenta that only accounts for

*one half*of the leading order modification that the phase shift suffers at high energy. In turn, one concludes, with the authors of [1], that at high energy the physics of the problem is better described by the FZZ dual model, even if the operator that controls the world-sheet instantons are considered in the 2D Euclidean black hole \(\sigma \)-model CFT.

## Footnotes

- 1.
Hereafter we omit the diffeomorphism

*b*–*c*ghost system contributions. - 2.
Here, the symbol \(\simeq \) means that this has to be understood as valid in the limit \(p_{\phi } >>\sqrt{k-2}\).

- 3.
Although the coefficient of the term in \(\delta \) that is linear in \(p_{\phi } \) differs from that in (24).

## Notes

### Acknowledgments

Work partially funded by FNRS-Belgium (convention FRFC PDR T.1025.14 and convention IISN 4.4503.15), by the Communauté Française de Belgique through the ARC program and by a donation from the Solvay family. The support of CONICET, FNRS\(+\)MINCyT, FONDECyT and UBA through grants PIP 0595/13, BE 13/03, Fondecyt 1140155 and UBACyT 20020120100154BA, respectively, is greatly acknowledged. The Centro de Estudios Científicos (CECs) is funded by the Chilean Government through the Centers of Excellence Base Financing Program of CONICYT-Chile.

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