${\bf AdS}_{\bf 3}$’s with and without BTZ’s

There are two ways to address this issue, which we consider in turn. The first IR regulator we will discuss is to separate the fivebranes slightly in their transverse space ${\mathbb{R}}^{4}$, so that the harmonic function ${\mathsf{H}}$ becomes

$${\mathsf{H}}=\sum_{\ell=1}^{n_{5}}\frac{\alpha^{\prime}}{|{\bf y}-{\bf y}_{\ell}|^{2}}~{}.$$

(3.3)

Naively, this doesn’t improve the situation, simply replacing one coincident brane singularity by several individual ones. However, the level $\kappa_{\rm tot}={n_{5}}$ of the supersymmetric ${SU(2)}$ WZW model describing the angular space in (3.1) gets a contribution $\kappa={n_{5}}\!-\!2$ from the bosonic ${SU(2)}$ WZW model and $\kappa\!=\!2$ from its fermionic superpartners. Thus there is no unitary worldsheet description of local string dynamics in the throat of a single NS5-brane.⁴⁴4Recently, a worldsheet theory has been proposed to describe a single fivebrane in the presence of a macroscopic number of wound fundamental strings Eberhardt:2018ouy . We are not aware of an extension of this description where there is a fivebrane but no wound strings; and furthermore, in this description the wound strings have a single allowed radial wavefunction, whose radial profile is essentially constant, so there is still no local string dynamics in the single fivebrane throat. The way the worldsheet theory accommodates this fact is to not allow fundamental strings to penetrate into the throat of an isolated fivebrane at low energy Giveon:1999px ; Giveon:1999tq . This property has been studied in detail for the particular configuration where the fivebranes are separated in a circular array in say the $y^{1}$-$y^{2}$ plane (see figure 2(a))

$$y_{\ell}^{1}+iy_{\ell}^{2}=a\,e^{2\pi i\ell/{n_{5}}}~{}.$$

(3.4)

This configuration is special in that it still admits an exactly solvable worldsheet description Giveon:1999px ; Giveon:1999tq .

The version of this exact worldsheet theory that will be useful for us is a presentation in terms of a gauged WZW model Israel:2004ir ; Martinec:2017ztd

$$\frac{\mathcal{G}}{\mathcal{H}}=\frac{SL(2,{\mathbb{R}})\times{SU(2)}}{U(1)_{L}\times U(1)_{R}}$$

(3.5)

where $U(1)_{L}\times U(1)_{R}$ is generated by the left- and right-moving null currents

$$\mathcal{J}=J^{3}_{\text{sl}}+J^{3}_{\text{su}}~{}~{},~{}~{}~{}~{}\bar{\mathcal{J}}=\bar{J}^{3}_{\text{sl}}-\bar{J}^{3}_{\text{su}}~{}.$$

(3.6)

Gauging these currents and integrating out the gauge field leads to a geometry which when written in terms of the natural Euler angle coordinates on the $SL(2,{\mathbb{R}})\times{SU(2)}$ group manifold⁵⁵5The metric and NS $B$-field in Euler angles on the angular three-sphere ${SU(2)}$ is given in (3.1). For $SL(2,{\mathbb{R}})$, substitute $\theta\to i\rho$, $\phi\to\sigma$, $\psi\to\tau$; it is convenient to fix $\tau\!=\!\sigma\!=\!0$ as a gauge choice. See Martinec:2017ztd ; Martinec:2018nco ; Martinec:2019wzw ; Martinec:2020gkv for details. takes the form

	$\displaystyle ds^{2}$	$\displaystyle=\Bigl{(}-dt^{2}+dx^{2}+ds_{\scriptscriptstyle\mathbf{T}^{4}}^{2}\Bigr{)}+n_{5}\Bigl{[}d\rho^{2}+d\theta^{2}+\frac{1}{\Sigma_{0}}\Bigl{(}{\cosh}^{2}\!\rho\sin^{2}\!\theta\,d\phi^{2}+{\sinh}^{2}\!\rho\cos^{2}\!\theta\,d\psi^{2}\Bigr{)}\Bigr{]},$
	$\displaystyle B$	$\displaystyle=\frac{n_{5}\cos^{2}\theta\cosh^{2}\rho}{\Sigma_{0}}\,d\phi\wedge d\psi\,,\qquad e^{-2\Phi}=\frac{a^{2}\Sigma_{0}}{{n_{5}}}\,,\qquad\Sigma_{0}\;\equiv\;{\sinh^{2}\!\rho+\cos^{2}\!\theta}\,.$		(3.7)

This geometry is just what one would get from smearing the Coulomb branch NS5 sources over the circle along which they are uniformly arrayed Sfetsos:1998xd , with the coordinate map

$$y^{1}+iy^{2}=a\,\cosh\rho\,\sin\theta\,e^{i\phi}~{}~{},~{}~{}~{}~{}y^{3}+iy^{4}=a\,\sinh\rho\,\cos\theta\,e^{i\psi}~{}.$$

(3.8)

From the point of view of the effective supergravity theory, the geometry is still singular along the circle $\rho=0,\theta=\pi/2$ parametrized by $\phi$, as one sees from the the vanishing of $\Sigma_{0}$ in (3.1); however, the worldsheet CFT is completely nonsingular – its correlation functions are simply a projection of the nonsingular correlation functions of the parent WZW model onto the subset that are neutral under the currents (3.6). Thus the apparent singularity is a fake, at least from the point of view of low-energy strings – it is effectively resolved by $\alpha^{\prime}$ corrections. The effective coupling in the theory is set by the value of the dilaton away from this source ring but in the cap of the geometry, say at the center of the source ring at $\rho=\theta=0$. Note that the effective supergravity geometry (3.1) does not resolve the individual fivebrane locations; rather the source appears to be smeared all along the circle. Nevertheless, the exact worldsheet theory keeps track of the precise location of the fivebrane sources.

A physical picture of what is happening is provided by the FZZ dual description of the near-source region Giveon:1999px ; Giveon:1999tq ; Martinec:2020gkv . This duality initially arose in the present context as a quantum equivalence between the supersymmetric $\frac{SL(2,{\mathbb{R}})}{U(1)}$ gauged WZW model, and $N=2$ supersymmetric Liouville field theory Giveon:1999px . Ultimately, though, it derives from a certain quantum equivalence of $SL(2,{\mathbb{R}})$ current algebra representation theory, as explained in Maldacena:2000hw ; Chang:2014jta ; Giveon:2015cma ; Martinec:2020gkv .

In this quantum equivalence of 2d CFT’s, the IR cap in the geometry (3.1) is seen to be equivalent to a Liouville-like wall, described by the $N=2$ supersymmetric worldsheet superpotential

$$\mathcal{W}={\mathsf{Z}}^{n_{5}}-\lambda_{0}\,e^{-{n_{5}}{\mathsf{X}}}=\prod_{\ell=1}^{n_{5}}\Big{(}{\mathsf{Z}}-\mu_{\ell}\,e^{-{\mathsf{X}}}\Big{)}~{}~{},~{}~{}~{}~{}\mu_{\ell}=\lambda_{0}^{1/{n_{5}}}\,e^{2\pi i\ell/{n_{5}}}~{}.$$

(3.9)

The $N=2$ Liouville field ${\mathsf{X}}$ has a linear dilaton for its real part $\rho=\Re({\mathsf{X}})$, which represents the radial direction away from the fivebrane source; the exponential wall keeps strings from pushing into the strong coupling region. The circular array of fivebranes is coded in the zeros of this superpotential; the coupling $\lambda_{0}$ thus sets a scale related to the parameter $a$ in the geometrical description. This worldsheet dual description captures information that is non-perturbative in $\alpha^{\prime}$ relative to the sigma model description (3.1), such as the unsmeared locations of the fivebrane sources.

Thus the throats of isolated fivebranes cannot be probed by fundamental strings at low energy, because a wall prevents strings from reaching the region of strong coupling near such a fivebrane. However, pushing the branes back together by sending $a\to 0$ and hence $\lambda_{0}\to 0$ makes the effective wall recede and strings are no longer kept away from the region of strong coupling near the fivebranes.

Note also that the parameters $\mu_{\ell}$ are marginal couplings in the superpotential (3.9), related to the positions of the individual fivebranes ${\bf y}_{\ell}$ in the $y^{1}$-$y^{2}$ plane. One can push any pair $\ell,\ell^{\prime}$ of fivebranes together by sending $\mu_{\ell}\to\mu_{\ell^{\prime}}$, in which case the superpotential develops a flat direction and once again, along that flat direction the dilaton grows and perturbative strings can explore it, because the throat of two fivebranes has a unitary worldsheet description as outlined above.

Separating the fivebranes onto their Coulomb branch moduli space thus regulates the strong coupling IR dynamics of the nonabelian little string theory that lives inside coincident NS5-branes.

A second infrared regulator results from dropping into the throat ${n_{1}}$ fundamental strings which wrap ${\mathbb{S}}^{1}_{x}$. Inside the fivebrane charge radius $r_{\rm F1}^{~{}}$, the linear dilaton throat of the NS5-branes rolls over into an infrared $AdS_{3}$ geometry, and the dilaton saturates to a constant $e^{2\Phi}\approx V_{{\mathbb{T}}^{4}}^{~{}}n_{5}/n_{1}$; see figure 3. The theory is weakly coupled in the IR if the F1 winding charge ${n_{1}}$ is sufficiently large.

A worldsheet description of this transition between linear dilaton asymptotics and $AdS_{3}$ in the IR can be achieved via a suitable modification of the gauged WZW formalism described above Martinec:2017ztd . One simply generalizes the embedding of the gauge group to the $U(1)_{L}\times U(1)_{R}$ generated by

$$\mathcal{J}=J^{3}_{\text{sl}}+J^{3}_{\text{su}}-R_{x}(\partial t+\partial x)~{}~{},~{}~{}~{}~{}\bar{\mathcal{J}}=\bar{J}^{3}_{\text{sl}}-\bar{J}^{3}_{\text{su}}-R_{x}(\bar{\partial}t-\bar{\partial}x)~{}.$$

(3.10)

Gauging these currents in the WZW model leads to a somewhat more complicated geometry

	$\displaystyle ds^{2}$	$\displaystyle=\Bigl{(}-du\,dv+ds_{\scriptscriptstyle\mathbf{T}^{4}}^{2}\Bigr{)}+{n_{5}}\Bigl{[}d\rho^{2}+d\theta^{2}+\frac{1}{\Sigma}\Bigl{(}{\cosh}^{2}\!\rho\sin^{2}\!\theta\,d\phi^{2}+{\sinh}^{2}\!\rho\cos^{2}\!\theta\,d\psi^{2}\Bigr{)}\Bigr{]}$
		$\displaystyle\hskip 17.07182pt+\frac{2R_{x}}{\Sigma}\Bigl{(}{\sin^{2}\!\theta\,dt\,d\phi+\cos^{2}\!\theta\,dx\,d\psi}\Bigr{)}+\frac{R_{x}^{2}}{{n_{5}}\Sigma}\Bigl{[}{n_{5}}\sin^{2}\!\theta\,d\phi^{2}+{n_{5}}\cos^{2}\!\theta\,d\psi^{2}+du\>\!dv\Bigr{]},$
	$\displaystyle B$	$\displaystyle=\frac{\cos^{2}\!\theta(R_{x}^{2}+{n_{5}}\cosh^{2}\!\rho)}{\Sigma}{d\phi\wedge d\psi-\frac{R_{x}^{2}}{n_{5}\Sigma}\,dt\wedge dx}$
		$\displaystyle\hskip 17.07182pt{}+\frac{R_{x}\cos^{2}\!\theta}{\Sigma}dt\wedge d\psi+\frac{R_{x}\sin^{2}\!\theta}{\Sigma}dx\wedge d\phi~{},$
	$\displaystyle e^{-2\Phi}$	$\displaystyle=\frac{n_{1}\Sigma}{R_{x}^{2}\,V_{4}}~{},\qquad~{}~{}~{}\Sigma=\frac{R_{x}^{2}}{{n_{5}}}+\Sigma_{0}\ ,$		(3.11)

where $u,v=t\pm x$.

This background has the advertised feature, that it has a linear dilaton structure for $e^{2\rho}\gg R_{x}^{2}/{n_{5}}$, and an $AdS_{3}$ structure for $e^{2\rho}\ll R_{x}^{2}/{n_{5}}$. One way to see these asymptotics is to note that the $SL(2,{\mathbb{R}})$ current $J^{3}_{\text{sl}}$ in the null current $\mathcal{J}$ behaves as $J^{3}_{\text{sl}}\sim{\textstyle\frac{1}{2}}e^{2\rho}\partial(\tau+\sigma)$. Therefore at large $\rho$ the gauge orbits lie mostly along $SL(2,{\mathbb{R}})$, the gauging is approximately that of the Coulomb branch fivebranes, and $t,x$ parametrize an asymptotic cylinder; on the other hand, for small $\rho$, the gauge orbit is mostly along ${\mathbb{R}}_{t}\times{\mathbb{S}}^{1}_{x}$, which are gauged away leaving $AdS_{3}$ largely intact. Note also that the $B$ field now contains, in addition to the magnetic flux represented by the first term in the first line, an electric flux represented by the second term in the first line. This arises because the $H_{3}$ flux in the $SL(2,{\mathbb{R}})$ factor is no longer unphysical after tilting the orientation of the gauge orbits to partly lie along ${\mathbb{R}}_{t}\times{\mathbb{S}}^{1}_{x}$.

The $AdS_{3}$ decoupling limit of this background takes $R_{x}\to\infty$. Because spacetime fermions have periodic boundary conditions around ${\mathbb{S}}^{1}_{x}$, the resulting background describes a state in the Ramond sector of the dual spacetime CFT; in fact one obtains the Ramond sector state of maximum R-charge, which is the spectral flow of the $SL(2,{\mathbb{R}})$ invariant state in the NS sector Maldacena:2000dr ; Lunin:2001fv . This property explains why the geometry looks like (a large diffeomorphism of) global $AdS_{3}$ in the IR.⁶⁶6A suitable generalization of the gauge group embedding allows the realization of conical defect geometries $AdS_{3}\times{\mathbb{S}}^{3})/{\mathbb{Z}}_{q}$, with R-charge a faction $1/q$ of the maximal value Martinec:2017ztd .

In fact, this second IR regulator has a kinship with the Coulomb branch regulator above Martinec:2017ztd . T-duality along $x$ converts the F1 charge to momentum charge P, and the fivebrane source configuration in fact still has the ${n_{5}}$ fivebranes uniformly spread around a circle, whose radius is now dynamically determined by the tension of the fivebranes balanced against the angular momentum they carry; see figure 2(b) (this balancing determines the value of the dilaton in the NS5-F1 frame). Thus, the source fivebranes are indeed separated along this T-dual circle $\widetilde{\mathbb{S}}^{1}_{x}$. This uniformly spiraling source generates a particularly symmetric geometry, which is amenable to exact solution on the worldsheet.

The maximal R-charge Ramond ground state is of course one of a large ensemble of $\frac{1}{2}$-BPS states. The geometry sourced by these states can be given in terms of Green’s function integrals for the coefficients in the supergravity fields; for intance, if only NS-NS fields are sourced, one has (see Mathur:2005zp ; Skenderis:2008qn for reviews)

$\displaystyle\begin{aligned} ds^{2}&\;=\;{\mathsf{K}}^{-1}\bigl{[}-(d\tau+{\mathsf{A}})^{2}+(d\sigma+{\mathsf{B}})^{2}\bigr{]}+{\mathsf{H}}\,d{\bf y}\!\cdot\!d{\bf y}+ds^{2}_{\mathcal{M}}\,,\\[5.69046pt] B&\;=\;{\mathsf{K}}^{-1}\bigl{(}d\tau+{\mathsf{A}}\bigr{)}\wedge\bigl{(}d\sigma+{\mathsf{B}}\bigr{)}+{\mathsf{C}}_{ij}\,dy^{i}\wedge dy^{j}\,,\\[5.69046pt] e^{2\Phi}&\;=\;\frac{{\mathsf{H}}}{{\mathsf{K}}}\,,\qquad~{}~{}~{}d{\mathsf{C}}=*_{\scriptscriptstyle\perp}d{\mathsf{H}}\;,\qquad~{}~{}d{\mathsf{B}}=*_{\scriptscriptstyle\perp}d{\mathsf{A}}\,,\end{aligned}$

(3.12)

where again ${\bf y}$ are Cartesian coordinates on the transverse space to the fivebranes. The harmonic forms and functions appearing in this solution can be written in terms of a Green’s function representation, which in the $AdS_{3}$ decoupling limit takes the form

$\displaystyle\begin{aligned} {\mathsf{H}}\,=\,\frac{1}{2\pi}\sum_{\ell=1}^{n_{5}}\int\limits_{0}^{2\pi}\frac{d\tilde{v}}{|{\bf x}-{\mathsf{F}}_{\ell}(\tilde{v})|^{2}}~{},&\qquad~{}{\mathsf{K}}\,=\,1+\frac{1}{2\pi}\sum_{\ell=1}^{n_{5}}\int\limits_{0}^{2\pi}\frac{d\tilde{v}\,\dot{\mathsf{F}}_{\ell}\!\cdot\!\dot{\mathsf{F}}_{\ell}}{|{\bf x}-{\mathsf{F}}_{\ell}(\tilde{v})|^{2}}\;,\\[5.69046pt] {\mathsf{A}}\,=\,{\mathsf{A}}_{i}dy^{i}\,,\qquad{\mathsf{A}}_{i}\,=\,&\frac{1}{2\pi}\sum_{\ell=1}^{n_{5}}\int\limits_{0}^{2\pi}\frac{d\tilde{v}\,\dot{\mathsf{F}}^{\ell}_{i}(\tilde{v})}{|{\bf x}-{\mathsf{F}}_{\ell}(\tilde{v})|^{2}}\;,\end{aligned}$

(3.13)

involving source profile functions ${\mathsf{F}}_{\ell}(\tilde{v})$, $\ell=1,2,\dots,{n_{5}}$, that describe the locations of the fivebranes in their transverse space (overdots denote derivatives with respect to $\tilde{v}$). It is convenient to choose twisted boundary conditions ${\mathsf{F}}_{\ell}(\tilde{v}+2\pi)={\mathsf{F}}_{\scriptscriptstyle\ell+1}(\tilde{v})$, so that all the fivebranes are bound together into one long fivebrane, see figure 4. For example, the choice

$${\mathsf{F}}_{\scriptscriptstyle\ell}(\tilde{v})=a\,{\rm exp}\Big{[}\frac{i\tilde{v}}{{n_{5}}R_{y}}+\frac{2\pi i\ell}{{n_{5}}}\Big{]}$$

(3.14)

is the source profile that generates the round “supertube” geometry of equation (3.2) and figure 2(b).

One now has a fancier version of the Coulomb branch story. For profiles which only excite wiggles of the fivebranes in the $y^{1}$-$y^{2}$ plane, there is a proposed FZZ dual description Martinec:2020gkv in terms of an $N=2$ superpotential

$$\mathcal{W}=\prod_{\ell=1}^{n_{5}}\Bigl{(}{\mathsf{Z}}\,e^{i{\mathsf{v}}/{n_{5}}}-\mu_{\ell}({\mathsf{v}})\,e^{-{\mathsf{X}}}\Bigr{)}$$

(3.15)

where the $\mu_{\ell}$ are simply related to the source profiles ${\mathsf{F}}_{\ell}(\tilde{v})$. When the $\mu_{\ell}$ are the $n_{5}^{\rm th}$ roots of unity (a common overall scale can be absorbed in a shift of ${\mathsf{X}}$), one has a spiral structure of zeros due to the adiabatically rotating phase multiplying ${\mathsf{Z}}$; allowing more general $\mu_{\ell}({\mathsf{v}})$ with the twisted boundary condition $\mu_{\ell}({\mathsf{v}}+2\pi)=\mu_{\ell+1}({\mathsf{v}})$ reproduces the wiggly profile of the source in the geometrical construction of Lunin-Mathur, and exhibits the unsmeared source in the worldsheet theory.

Singularities arise when the profile self-intersects. In the geometry (3.12), the source profile ${\mathsf{F}}(\tilde{v})$ specifies the location of a KK monopole core transverse to the source contour.⁷⁷7The tilted source profile of figure 2(b) has a component transverse to the circle ${\mathbb{S}}^{1}_{x}$ that is being T-dualized to go from NS5-P to NS5-F1, and this component of NS5 charge then T-dualizes into KKM charge. This is a dipole charge because its orientation rotates as one travels around the $\phi$ circle at $\rho=0,\theta=\frac{\pi}{2}$, such that the net KKM charge vanishes. When the profile approaches a self-intersection, there is a minimal size holomogical ${\mathbb{S}}^{2}$ in the geometry whose polar direction is the interval between the two points of closest approach of the contour, and whose azimuthal direction is the fibered circle of the KK monopole. The size of this cycle vanishes in the limit where the profile actually self-intersects, see figure 5.

In the FZZ dual description, this self-intersection corresponds to a point along ${\mathsf{v}}$ where two zeros of the superpotential (3.15) collide, for instance $\mu_{\ell}({\mathsf{v}})$ and $\mu_{\ell^{\prime}}({\mathsf{v}})$; again the superpotential degenerates as it did on the Coulomb branch, a flat direction opens up, along which strings are no longer walled off from strong coupling in this direction in field space.

All of this structure reinforces the picture of a brane gas underlying the density of states. Below the black hole threshold, we can enumerate the $\frac{1}{2}$-BPS spectrum as the phase space of a wiggly fivebrane source carrying KK-dipole charge. When we put energy into the system, it starts to explore this phase space, until it finds a self-intersection of the wiggly fivebrane. The formation of little strings will trap the intersecting fivebranes and start the black hole formation process. A deep throat opens up in this wiggly geometry at the intersection. In the black hole phase that is emerging, the phase space is dominated by that of nonabelian little string excitations. The manifestation of these nonabelian little strings in perturbative string theory below the transition, and their emergence as light excitations as we approach the black hole threshold, is the subject we turn to next.

As argued in Giveon:2005mi and elaborated in Balthazar:2021xeh , the $SL(2,{\mathbb{R}})$ level $k=1$ marks the dividing line of the string/black hole correspondence principle Horowitz:1996nw , in that for $k>1$ the high-energy density of states is that of black fivebranes, equation (2); and in the $AdS_{3}$ limit $R_{x}\to\infty$, the high-energy density of states is that of BTZ black holes (2), where in both these equations we substitute ${n_{5}}\to k$. In the critical dimension theory on ${\mathbb{T}}^{4}$, one has $k=\kappa={n_{5}}$; more generally, the density of states is governed by $k$, which is related to the ${SU(2)}$ levels, and hence any magnetic $H_{3}$ fluxes present, through the central charge constraint (5.4). In particular, in the $r=1$ models we consider, it is natural to continue to interpret ${n_{5}}=\kappa$, which differs from $k$ by (5).

One way to see that the black hole spectrum is governed by $k$ rather than $\kappa$ is to examine the linear dilaton regime of the throat. The black hole entropy in the fivebrane decoupling limit is governed by the Euclidean black NS5 geometry

$$ds^{2}=k\big{(}\tanh^{2}\!\rho\,dt^{2}+d\rho^{2}\big{)}+\kappa\,d\Omega_{{\mathbb{S}}^{3}_{\flat}}^{2}+ds^{2}_{{\mathbb{S}}^{1}_{x}}+ds^{2}_{{\mathbb{T}}^{d}}\quad,\qquad e^{-2\Phi}=\mu\,\cosh^{2}\!\rho~{}.$$

(5.6)

where $\mu=\sqrt{k}\,\ell_{\textit{s}}M$. The metric in the $\rho$-$t$ plane is that of the $\frac{SL(2,{\mathbb{R}})}{U(1)}$ “cigar” coset and this exact solution of the model gives us confidence in the results even when $k\sim 1$. Feeding this geometry into black hole thermodynamics yields a black hole inverse temperature $\beta_{H}=\sqrt{k}\,\ell_{\textit{s}}$, and Hagedorn entropy

$$S=2\pi\beta_{H}M=2\pi\sqrt{k}\,\ell_{\textit{s}}\,M~{}.$$

(5.7)

One can extend these considerations to Euclidean black fivebranes carrying F1 winding charge by modifying the coset to $\frac{SL(2,{\mathbb{R}})\times U(1)_{x}}{U(1)}$ where again the embedding of the gauge group lies partly in each numerator factor; see Giveon:2005mi .

FZZ duality again plays a crucial role. The coset geometry is FZZ dual to $N=2$ Liouville theory, as outlined above. This means that the near-horizon region at the tip of the Euclidean “cigar” geometry comes inextricably linked with a winding string condensate represented by the worldsheet superpotential

$$\mathcal{W}=\tilde{\mu}\,e^{-k(\rho-i\tilde{x})}~{},$$

(5.8)

($\tilde{x}$ is the coordinate T-dual to $x$). This winding condensate decays exponentially away from the tip of the cigar, and can be interpreted as the Euclidean manifestation of a thermal string atmosphere outside the horizon, in equilibrium with the black hole.

The linearized deformation of the geometry from an asymptotically cylindrical form with linear dilaton, $\delta(ds^{2})=e^{-2\rho}dx^{2}$, is normalizable in the full geometry. It corresponds to a worldsheet dimension (1,1) operator in the the $\frac{SL(2,{\mathbb{R}})}{U(1)}$ coset model.⁸⁸8This deformation among other things shifts the value of the dilaton at the tip of the cigar, so one can think of it as the dilaton zero mode. But this normalizable deformation of the geometry comes together with the winding condensate (5.8), and the norm of this string winding component of the deformation

$$\int\!d\rho\,dx\sqrt{G}e^{-2\Phi}|\mathcal{W}|^{2}$$

(5.9)

is finite for $k>1$ and blows up for $k<1$. The dimension (1,1) operator includes this stringy contribution and is thus non-normalizable for $k<1$. The conclusion of Giveon:2005mi is that because the black hole background itself has such a condensate, it ceases to have a finite action for $k<1$, due to stringy effects, and therefore does not contribute to the thermodynamics.

So what governs the asymptotic density of states in noncritical little string theory for $k<1$? For both the regimes with linear dilaton asymptotics and with $AdS_{3}$ asymptotics, the perturbative string spectrum persists up to arbitrarily high energy and saturates the leading order entropy Giveon:2005mi . One can compute the spectrum of perturbative strings in the linear dilaton region, with the result

$$S=2\pi\ell_{\textit{s}}E\sqrt{2-\frac{1}{k}}~{}.$$

(5.10)

When $k<1$ this entropy is lower than the black fivebrane entropy one would obtain from geometry, equation (5.7), but due to the non-normalizable contribution of the winding condensate the naive geometric entropy formula doesn’t apply. Note however that the two expressions agree at $k=1$. At the same point that black holes leave the spectrum, the high energy density of states can be accounted for in perturbative string theory. The coefficient $\beta_{H,{\rm eff}}$ in the density of states is plotted in figure 9.

The regime $k<1$ is on the stringy side of the string/black hole correspondence transition, where black holes leave the normalizable spectrum and instead the spectrum consists of a gas of strings and D-branes whose thermodynamics is well-characterized by perturbative string theory.

What is true of the full little string theory is also true of its CFT limit. Here one has a spacetime Virasoro algebra, realized on the worldsheet through the string vertex operator amplitude realization of the spacetime CFT correlation functions Kutasov:1999xu ; Giveon:2001up . The central extension of this algebra arises as the usual leading singularity in the stress tensor OPE

$$\mathcal{T}(z_{1})\mathcal{T}(z_{2})\sim\frac{3k\mathcal{I}}{(z_{1}-z_{2})^{4}}+\frac{2\mathcal{T}(z_{2})}{(z_{1}-z_{2})^{2}}+\frac{\partial\mathcal{T}(z_{2})}{z_{1}-z_{2}}~{},$$

(5.11)

but in the worldsheet realization of this algebra, $\mathcal{I}$ is a nontrivial vertex operator. This vertex operator has two components, again due to FZZ duality. One is the zero mode of the dilaton, which again is a normalizable contribution. The other component is again associated to a winding string condensate; here it represents the ${n_{1}}$ fundamental strings winding ${\mathbb{S}}^{1}_{x}$ dissolved in the fivebranes.⁹⁹9More precisely, the parameter $\tilde{\mu}$ in (5.8) is the chemical potential conjugate to the winding number ${n_{1}}$ Porrati:2015eha . Just as in the linear dilaton case, the geometrical deformation is always normalizable, but the winding component is non-normalizable for $k<1$. As a result, there is no normalizable $SL(2,{\mathbb{R}})$ invariant state associated to the identity operator, a situation sometimes encountered in non-compact CFT’s Seiberg:1992bj . But in the torus partition function of the spacetime CFT, the $SL(2,{\mathbb{R}})$ invariant state is related by modular invariance to the asymptotic density of states Cardy:1986ie , which in the holographic context is the BTZ black hole spectrum. The conclusion of Giveon:2005mi is that the black hole states also cease to be normalizable.

A more refined computation of the asymptotic density of states in conformal field theories with no $SL(2,{\mathbb{R}})$ invariant vacuum yields Kutasov:1990sv

$$S=2\pi\sqrt{c_{\rm eff}(\varepsilon+p)/2}+2\pi\sqrt{c_{\rm eff}(\varepsilon-p)/2}\quad,\qquad c_{\rm eff}=c-24h_{\rm min}$$

(5.12)

where $h_{\rm min}=\bar{h}_{\rm min}$ is the energy eigenvalue of the lowest energy state in the theory.

There is now also a proposal for the spacetime CFT in the $k<1$ regime Balthazar:2021xeh , namely a deformation of the symmetric product

$$({\mathbb{R}}_{\rho}\times{\mathbb{S}}^{3}_{\flat})^{n_{1}}/S_{n_{1}}$$

(5.13)

where ${\mathbb{R}}_{\rho}$ has a linear dilaton, ${\mathbb{S}}^{3}_{\flat}$ is the same squashed ${SU(2)}$ that appeared in the worldsheet construction (5.3), and the deformation is by a ${\mathbb{Z}}_{2}$ twist operator dressed by an exponential of $\rho$ to make it marginal. This exponential vanishes as $\rho\to+\infty$ where the symmetric product structure reproduces the Fock space of free strings winding ${\mathbb{S}}^{1}_{x}$, and grows as $\rho\to-\infty$ where it erects a wall to keep the strings out of strong coupling. It was shown in Balthazar:2021xeh that the spectrum of this CFT agrees with that of the worldsheet theory on the $d=0$ geometry (5.2) for states whose wavefunctions are concentrated in the asymptotic weak-coupling region.

Fundamental strings in purely NS $AdS_{3}$ and linear dilaton backgrounds have a continuum of winding string states (so-called long strings) above a gap

$$h_{\rm min}(w)=\frac{(k-1)^{2}}{4kw}+\frac{k}{4}\Big{(}w-\frac{1}{w}\Big{)}\quad,\qquad w=1,2,...$$

(5.14)

The symmetric product reproduces this long string continuum. This continuum thus has a lowest dimension state where all the strings have $w=1$, leading to a perturbative approximation

$$c_{\rm eff}\approx 6{n_{1}}\Big{(}2-\frac{1}{k}\Big{)}~{}.$$

(5.15)

The resulting $k<1$ entropy (5.12) joins smoothly onto the BTZ entropy with $c=6k{n_{1}}$ at $k=1$.

As argued in Balthazar:2021xeh , this approximation is not expected to have large (i.e. leading order in ${n_{1}}$) corrections. One can imagine assembling a state by adding strings one at a time from the continuum; when the strings are out at large $\rho$, the free string approximation accurately calculates their energies in the asymptotically free spacetime CFT elaborated in that work. The only way this approximation could fail is if there were a bound state spectrum well below the continuum (i.e. by an amount of order ${n_{1}}$). Given that as the energy increases, the wavefunctions spill out to larger and larger $\rho$ where the coupling is weaker and weaker, and thus the perturbative approximation better and better, this possibility seems unlikely.

The three classes of models we chose in order to explore the range of $k$ – the critical theory on $AdS_{3}\times{\mathbb{S}}^{3}\times{\mathbb{T}}^{4}$ with $k\geq 2$, the $d=2$ models on $AdS_{3}\times{\mathbb{S}}^{3}_{\flat}\times{\mathbb{T}}^{2}$ with $1\leq k<2$, and the models on $AdS_{3}\times{\mathbb{S}}^{3}_{\flat}$ with $k<1$ – have a similar structure on the worldsheet. In each case, a worldsheet CFT consisting of a gauged WZW model describes a background that interpolates between a linear dilaton UV and the $AdS_{3}$ background in the IR. The ${\mathbb{T}}^{4}$ and ${\mathbb{T}}^{2}$ cases involve $SL(2,{\mathbb{R}})$ WZW models with $k\geq 1$, and so have an asymptotic BTZ spectrum; the last class has $k<1$ and so the asymptotic spectrum has no BTZ spectrum.

The D-brane avatars of the little string have transverse oscillations in the first two cases but not the last, suggesting that when the fivebranes are pushed together and nonabelian little string physics arises in the deep IR, black holes associated to the resulting Hagedorn little string gas will form in the first two cases but not in the last case, because in the last case the little strings have no Hagedorn spectrum. This qualitative distinction of D-brane spectra between the three classes of models thus agrees with the general considerations above for the asymptotic spectrum as a function of $k$. While the little string naively has a tension lighter than the fundamental string, and hence would be expected to dominate the entropy when it is deconfined, for $k<1$ the little string wrapping ${\mathbb{S}}^{1}_{x}$ has no transverse oscillations while for $k>1$ it does. The absence of a black hole spectrum kicks in precisely when the phase space for little string oscillations disappears.

The asymptotic spectrum of the $k<1$ models is accounted for by the perturbative string spectrum of the bulk description, extrapolated from low energies and isolated strings to high energies and strings on top of one another. The energies in question are of order ${n_{1}}$ or more, in which case the $1/{n_{1}}$ expansion is suspect. But the fact that perturbative analysis appears to get the correct asymptotics gives us confidence that it is a reliable guide, at least qualitatively. Running with this idea, let us piece together a picture of the dynamics in generic high-energy states.

The worldsheet theory has both a discrete spectrum of bound states as well as a continuum of scattering states. The bound states are seen as poles in the S-matrix of the scattering states of wound strings that descend into the middle of $AdS_{3}$ and bounce back out. At leading order in $1/{n_{1}}$, the amplitude is diagonal in the winding number $w$ of the string, and there are bound states for every $w$, whose energies are given by the pole structure in the reflection coefficient of the single-string wavefunction. A multi-string bound state is just a Fock space of such bound strings. The generic (non-BPS) bound state is not protected by any symmetry,¹⁰¹⁰10The BPS bound state spectrum is of course protected by supersymmetry. and can merge with the continuum, i.e. decay. The pole locations gain small imaginary parts, and the bound states become resonances which decay by emission of long strings that escape to infinity.

Now, it could be that string perturbation theory is completely breaking down at large energies, and the generic state gains a decay width of order its mass, in which case it no longer makes much sense to talk about a bound state spectrum. However, we expect the size of the bound state wavefunction to grow with energy and extend further out into the weak-coupling region of the asymptotically free spacetime CFT, in which case string perturbation theory is getting better and better at describing the dominant support of the wavefunction, suggesting that the decay width remains much smaller than the mass (perhaps down by $1/{n_{1}}$). The resonant states would then be long-lived, like an ordinary black hole that traps incoming energy and releases it slowly.

Note that as an aside, as the quasi-bound spectrum decays, the late-time state consists of a number of strings heading out toward the boundary $\rho\to\infty$. But then by time-reversal symmetry the generic state came from the boundary in the far past, by assembling it from an in-state of scattering strings. To have a steady state in Lorentz signature, it would seem that there would have to be strings all along the $\rho$ direction, both incoming and outgoing, and thus ${n_{1}}\to\infty$. One needs to keep feeding strings in from infinity to replenish the ones that are leaking away.¹¹¹¹11This feature is reminiscent of the $c=1$ matrix model, where the underlying sea of D0-branes (the eigenvalues of the dual matrix model) must be continually fed in, otherwise the sea drains away.

Once we have dispensed with the radial center-of-mass motion of the long strings by putting them into a quasi-bound state, or by putting the system in a bath of scattering strings, the remaining structure of the Hilbert space seems not all that different from the picture of the ${\mathbb{T}}^{4}$ symmetric product that describes the NS5-F1 system in the critical dimension in a far corner of its moduli space. That corner has an effective description in terms of which there is only one fivebrane, and the oscillator spectrum of the symmetric product is essentially identical. Of course, with a single fivebrane, we are again at or near the correspondence transition; nevertheless, the long string spectrum is often taken to be a useful model of black hole dynamics. To go toward the regimes where an effective supergravity description approximates the bulk dual to the spacetime CFT in the critical dimension, one has to turn on a marginal ${\mathbb{Z}}_{2}$ twist operator in the symmetric orbifold, just as the $k<1$ deformed symmetric product CFT has such a ${\mathbb{Z}}_{2}$ twist operator turned on at finite $\rho$. Thus we see that there are many points of commonality between the two situations.

Another set of interesting questions surrounds the degree to which the evolution of the generic highly excited state in the spectrum approximates that of a black hole, particularly in the $k\to 1$ limit, both in its formation and in its decay. Consider a situation where most of the CFT is in some low-lying state, for instance one of the BPS ground states discussed in appendix D of Balthazar:2021xeh , with a probe long string travelling in from large $\rho$ with some radial momentum. The incoming scattering state in the deformed symmetric product travels toward strong coupling. Since the wall that protects the strings from strong coupling is a ${\mathbb{Z}}_{2}$ twist operator that joins and splits long strings, the way that a string reflects off the wall is to interact with the condensate of strings already sitting in the IR ensemble.

At low momentum, the worldsheet two-point function predicts that the incident string will reflect off the wall, and return in the same state with a phase shift. As one dials up the incident $\rho$ momentum, the probe string will climb further up the wall before it reflects, thus it probes further into the strong coupling region Giveon:2015cma . The likelihood that it returns unscathed decreases; instead it can transfer some of its energy to long strings in the IR ensemble. Ideally, the result will match the known winding non-conserving structure of the perturbative string S-matrix (see Dei:2021xgh ; Dei:2021yom for a recent discussion and further references).

At some point, for instance at momenta/energies well above the central charge $c$, the probe string scattering off the IR ensemble will cease to be coherent; instead, the probe string will thermalize into the IR ensemble as it rapidly undergoes many incoherent scatterings deep in the wall. These scatterings should mediate fast scrambling of the incoming state into the full ensemble of available high-energy states at the incident energy. Rapid joining/splitting interactions reshuffle and thermalize segments of long string that make up the state.

Here, the picture is not all that different from that of thermalization in the symmetric product of ${\mathbb{T}}^{4}$ in the critical dimension, except that in the present case the string wavefunction can spread out in $\rho$ to a place where the interactions become weak, whereas in the critical dimension symmetric product the interaction strength is roughly uniform over a compact target space; thus the string wavefunction cannot disperse to a place of weak coupling.

Let us suppose then that the system is thermalized into a generic excited bound state. At the surface of such a state, long strings can radiate into the continuum of outgoing modes. Many aspects of this picture mimic the picture of near-extremal black holes Callan:1996dv ; Das:1996wn ; Gubser:1997se ; Klebanov:1997cx that foreshadowed the development of gauge/gravity duality.

The $AdS_{3}$ models discussed here provide a convenient realization of both the center of $AdS_{3}$ and a radiation sector, all within a broader CFT or its linear dilaton extension. The quanta that are capable of leaving the system are in the continuum of wound fundamental strings, with the same sign winding as the background; this will serve as the “radiation bath” that the part of the CFT describing the center of $AdS_{3}$ is coupled to.¹²¹²12Unwound strings lie in bound states rather than in radiation modes. When one backs away from the decoupling limit, their decaying profiles match onto outgoing spherical waves in the asymptotically flat part of the geometry, which are the usually considered Hawking radiation modes. But in the strict decoupling limit, the only modes that radiate energy to spatial infinity are the winding strings, assuming that we are working in the locus in moduli space where these strings are not bound but have a continuum of ingoing/outgoing modes. The models under discussion sit in this locus Seiberg:1999xz ; Larsen:1999uk . The effective field theory Hawking process at the horizon of BTZ black holes in the $k>1$ regime involves the creation of wound-antiwound F1 pairs in the continuum. The fact that F1 charge is being radiated away can be compensated (if desired) by feeding in unexcited wound strings at an appropriate rate.

The investigation of the thermal radiation from the hot “lump of string” that comprises the generic highly excited state of the $k<1$ symmetric product, in particular in the correspondence limit $k\to 1$ where it might start to approximate the Hawking radiation of wound strings, is likely to be a fascinating endeavor.

In Martinec:2020gkv , the condensation of normalizable $\frac{1}{2}$-BPS vertex operators had the effect of deforming the Lunin-Mathur profile function ${\mathsf{F}}(\tilde{v})$. The effect is not to change the theory, but to change the state in the theory from the $SL(2,{\mathbb{R}})$ invariant state to some other $\frac{1}{2}$-BPS state.¹³¹³13Note that because the UV theory has $k>1$, the $SL(2,{\mathbb{R}})$ invariant state exists in the Hilbert space. From the worldsheet point of view, the low-lying modes of deformation correspond to holographic RG flows that deform the background away from the $AdS_{3}$ vacuum. One can apply the same logic to non-critical little string models.

In the class of models (5.2), we can consider situations in which we engineer a holographic RG flow between $k>1$ and $k<1$ backgrounds Harvey:2001wm ; Balthazar:2021xeh . For instance, we can consider the class of models (5.2) with $d=0,r=2$. The associated Landau-Ginsburg potential is

$$\mathcal{W}_{0}={\mathsf{Z}}_{1}^{\kappa_{1}}+{\mathsf{Z}}_{2}^{\kappa_{2}}-\lambda_{0}\,e^{-\kappa_{1}\kappa_{2}{\mathsf{X}}}$$

(6.1)

and the $SL(2,{\mathbb{R}})$ level is

$$k=\frac{2\kappa_{1}\kappa_{2}}{2\kappa_{1}+2\kappa_{2}+\kappa_{1}\kappa_{2}}$$

(6.2)

which lies in the range $\frac{2}{3}\leq k<2$. Starting from a theory with both $\kappa_{1},\kappa_{2}$ large, one can condense an operator whose worldsheet effect is to flow the superpotential to e.g. $\kappa_{2}=2$, in which case the IR theory has $k=\frac{\kappa_{1}}{\kappa_{1}+1}<1$.

The associated gauged WZW model of the theory before adding the deformation is Brennan:2020bju

$$\frac{\mathcal{G}}{\mathcal{H}}=\frac{{\mathbb{R}}_{t}\times{\mathbb{S}}^{1}_{x}\times SL(2,{\mathbb{R}})\times{SU(2)}_{\kappa_{1}}\times{\widetilde{SU}(2)}_{\kappa_{2}}}{U(1)_{L}\times U(1)_{R}\times U(1)_{S}}$$

(6.3)

where the gauge currents $U(1)_{L,R,S}$ are taken to be

	$\displaystyle\mathcal{J}_{L}$	$\displaystyle=J^{3}_{\text{sl}}+\sqrt{\frac{k}{2\kappa_{1}}}\,J^{3,\flat}_{\text{su}}+\sqrt{\frac{k}{2\kappa_{2}}}\,J^{3,\flat}_{\widetilde{\text{su}}}-R_{X}\big{(}i\partial t+i\partial X\big{)}~{}~{},~{}~{}~{}~{}\mathcal{J}_{S}=+J^{3,\flat}_{\text{su}}-J^{3,\flat}_{\widetilde{\text{su}}}$
	$\displaystyle\bar{\mathcal{J}}_{R}$	$\displaystyle=\bar{J}^{3}_{\text{sl}}-\sqrt{\frac{k}{2\kappa_{1}}}\,\bar{J}^{3,\flat}_{\text{su}}-\sqrt{\frac{k}{2\kappa_{2}}}\,\bar{J}^{3,\flat}_{\widetilde{\text{su}}}-R_{X}\big{(}i\bar{\partial}t-i\bar{\partial}X\big{)}~{}~{},~{}~{}~{}~{}\bar{\mathcal{J}}_{S}=-\bar{J}^{3,\flat}_{\text{su}}+\bar{J}^{3,\flat}_{\widetilde{\text{su}}}$		(6.4)

The first two currents $\mathcal{J}_{L},\bar{\mathcal{J}}_{R}$ are null, while $\mathcal{J}_{S},\bar{\mathcal{J}}_{S}$ is a standard (spacelike) axial gauging. The squashing factors of ${SU(2)},{\widetilde{SU}(2)}$ are respectively

$$R_{i}=\sqrt{\frac{\kappa_{i}}{2k}}~{}~{},~{}~{}~{}~{}i=1,2~{},$$

(6.5)

in order that the background is $N=(2,2)$ supersymmetric. The normalizable $\frac{1}{2}$-BPS vertex operators include

$\displaystyle{\widetilde{\mathcal{W}}}_{j_{1}^{\prime},j_{2}^{\prime}}^{(-1)}=e^{-\varphi-\bar{\varphi}}\,\,e^{i\frac{(Y+\bar{Y})}{\sqrt{2k}}(j-k/2)}\,\Lambda^{\left(0,0\right)}_{j_{1}^{\prime};j_{1}^{\prime},j_{1}^{\prime}}\,\widetilde{\Lambda}^{\left(0,0\right)}_{j_{2}^{\prime};j_{2}^{\prime},j_{2}^{\prime}}\,\Phi^{(-1)}_{j;j,j}\,~{}.$

(6.6)

The notation is as follows:

• •

  $Y$ here parametrizes the physical combination of the scalars that bosonize $J^{3}_{\text{su}},J^{3}_{\widetilde{\text{su}}}$ left over after eliminating $U(1)_{S}$;

• •

  $\Lambda^{(\alpha,\bar{\alpha})}_{j_{1}^{\prime};m_{1}^{\prime},{\bar{m}}_{1}^{\prime}}$ are operators in the $\frac{{SU(2)}}{U(1)}$ superparafermion theory in the sector $(\alpha,\bar{\alpha})$ of R-symmetry spectral flow (similarly for $\widetilde{\Lambda}^{(\alpha,\bar{\alpha})}_{j_{2}^{\prime};m_{2}^{\prime},{\bar{m}}_{2}^{\prime}}$ and $\frac{{\widetilde{SU}(2)}}{U(1)}$);

• •

  $\Phi^{(w)}_{j;m,{\bar{m}}}$ are primaries of winding $w$ and spin $j$ in the bosonic $SL(2,{\mathbb{R}})$ WZW model.

See Brennan:2020bju ; Balthazar:2021xeh for further details; our conventions follow Balthazar:2021xeh . The ${SU(2)}$ spins take the values $j_{i}^{\prime}=0,\frac{1}{2},...,\frac{1}{2}\kappa_{i}-1$, and

$$j=k\Big{(}\frac{1}{2}-\frac{j_{1}^{\prime}}{\kappa_{1}}-\frac{j_{2}^{\prime}}{\kappa_{2}}\Big{)}~{}~{},~{}~{}~{}~{}p_{Y}=\frac{j-k/2}{\sqrt{2k}}~{}~{},~{}~{}~{}~{}p_{X}=-\bar{p}_{X}=0~{}.$$

(6.7)

For $j_{1}^{\prime},j_{2}^{\prime}$ such that

$$\frac{\kappa_{1}\kappa_{2}-2j_{1}^{\prime}\kappa_{2}-2j_{2}^{\prime}\kappa_{1}}{\kappa_{1}\kappa_{2}-2\kappa_{1}-2\kappa_{2}}>\frac{1}{2}~{}.$$

(6.8)

these vertex operators lie in the range of the unitary discrete series of normalizable $SL(2,{\mathbb{R}})$ operators

$$\frac{1}{2}<j<\frac{k+1}{2}~{},$$

(6.9)

which correspond to normalizable string states in the $AdS_{3}$ cap. Let $\mathcal{N}_{1}$ bet the set of such normalizable operators with $w=-1$. There are additional normalizable $\frac{1}{2}$-BPS (NS,NS) sector vertex operators in higher winding sectors $w<-1$ Argurio:2000tb ; Martinec:2020gkv ; Balthazar:2019rnh , but they are explicitly $t$-dependent while the operators above are $t$-independent. One can also consider the non-normalizable operators either inside or outside the range (6.8). For operators inside the normalizable range, this corresponds to turning on a chemical potential for the corresponding states; operators outside the range deform the theory rather than the state in a given theory.