--- title: Deep Hole Problem in Lattices & Black Holes url: https://www.emergentmind.com/topics/deep-hole-problem type: topic --- # Deep Hole Problem in Lattices & Black Holes The phrase **Deep Hole Problem** has distinct technical meanings in different research areas. In lattice theory, it concerns points farthest from a lattice and, in the planar case, can be reformulated through a canonical transformation that sends a lattice \(L\) to an affiliated **deep hole lattice** \(H(L)\) generated by a shortest vector and a distinguished deep hole. In supersymmetric black-hole microstate geometry, the same phrase is used for the problem of constructing **smooth horizonless supergravity solutions** that have the same asymptotic conserved charges as a macroscopic black hole while lying **deep inside the black-hole regime**, meaning both that the classical black hole exists and that the microstate reproduces its throat over a parametrically large region before capping off smoothly [2310.14091] [1607.03908]. ## 1. Terminological scope and problem statements In the planar lattice setting, a **deep hole** of a lattice \(L\subset \mathbb R^2\) is a point of \(\mathbb R^2\) that is farthest from the lattice, and the common distance from a deep hole to its nearest lattice points is the **covering radius** \(\mu\). The relevant problem is not merely to identify deep holes, but to use a distinguished one to define a new lattice \[ H(L):=\operatorname{span}_{\mathbb Z}\{\mathbf x_1,\mathbf z\}, \] where \(\mathbf x_1\) is a shortest vector and \(\mathbf z\) is the **fundamental deep hole**, the unique deep hole contained in the triangle with vertices \(\mathbf 0,\mathbf x_1,\mathbf x_2\) for shortest basis vectors \(\mathbf x_1,\mathbf x_2\). This converts the geometric notion of a deep hole into an explicit operation on similarity classes of planar lattices [2310.14091]. In the black-hole setting, the problem is formulated for supersymmetric rotating D1-D5-P black holes in five dimensions. The central question is whether one can construct **smooth, horizonless supergravity solutions** that have the **same asymptotic conserved charges** as a macroscopic supersymmetric black hole and lie **deep inside the black-hole regime**, namely a region of charge and angular-momentum space where the corresponding classical black hole has a large regular horizon and the microstate geometry reproduces its throat over a parametrically large region. The relevant classical regime is characterized by the cosmic censorship inequality \[ n_1 n_5 n_P - j^2 > 0 \,. \] In this context, “deep inside” also has a geometric meaning: the solution should be well approximated by the black-hole geometry over a long throat region and differ only very near where the black-hole singularity or horizon would otherwise appear [1607.03908]. A common misconception is that the phrase denotes a single standard problem across disciplines. The cited literature instead uses it in two sharply different senses: one geometric-arithmetic and discrete, the other supergravity-based and holographic. ## 2. Planar lattices, deep holes, and the affiliated lattice Let \(L\subset \mathbb R^2\) be a lattice with successive minima \(\lambda_1\le \lambda_2\) and corresponding minimal basis vectors \(\mathbf x_1,\mathbf x_2\). By changing signs if necessary, one can ensure that the angle \(\theta\) between \(\mathbf x_1,\mathbf x_2\) lies in \[ [\pi/3,\pi/2]. \] This angle is an invariant of the lattice and is called the **angle** of \(L\). The lattice is **well-rounded (WR)** if \(\lambda_1=\lambda_2\), and **semi-stable** if \(\lambda_1\ge \det(L)^{1/2}\). For planar lattices, WR implies semi-stable. Two lattices \(L_1,L_2\) are **similar**, written \(L_1\sim L_2\), if \[ L_2=\alpha U L_1 \] for some \(\alpha\in \mathbb R_{>0}\) and \(U\in O_2(\mathbb R)\) [2310.14091]. Every planar lattice is similar to a unique lattice \[ \Lambda_\tau:=\begin{pmatrix}1 & a\\ 0 & b\end{pmatrix}\mathbb Z^2, \qquad \tau=a+bi, \] with \[ \tau\in \mathcal F^+:=\{\tau=a+bi\in \mathbb H:\ 0\le a\le 1/2,\ |\tau|\ge 1\}. \] Thus \(\mathcal F^+\) parameterizes similarity classes of planar lattices. In this model, WR similarity classes correspond to \[ \{\tau\in \mathcal F^+ : |\tau|=1\}, \] and semi-stable similarity classes correspond to \[ \{\tau=a+bi\in \mathcal F^+ : b\le 1\}. \] This gives a moduli-theoretic description of the planar problem in terms of the upper half-plane [2310.14091]. The planar theory is tractable because the distinguished deep hole is geometrically explicit. If \(T\) is the triangle with vertices \[ \mathbf 0,\quad \mathbf x_1,\quad \mathbf x_2, \] then there is a unique deep hole \(\mathbf z\) of \(L\) contained in \(T\), and \(\mathbf z\) is the center of the circumcircle of \(T\). Accordingly, \[ \|\mathbf z\|=\|\mathbf x_1-\mathbf z\|=\|\mathbf x_2-\mathbf z\|=\mu. \] The covering radius is \[ \mu = \frac{\sqrt{\lambda_1^2 + \lambda_2^2 - 2\lambda_1\lambda_2\cos \theta}}{2 \sin \theta}. \] If \(A=(\mathbf x_1\ \mathbf x_2)\in GL_2(K)\) is the basis matrix of \(L\), then \[ \mathbf z = \frac{1}{2}(A^\top)^{-1} \begin{pmatrix} \lambda_1^2\\ \lambda_2^2 \end{pmatrix}. \] For \(L=\Lambda_\tau\) with \(\tau=a+bi\), \(\mathbf x_1=(1,0)^t\), \(\mathbf x_2=(a,b)^t\), the fundamental deep hole is \[ \mathbf z_1= \begin{pmatrix} 1/2\\ (a^2+b^2-a)/2b \end{pmatrix}, \] and hence \[ H(\Lambda_\tau)=\Lambda_{\tau_1},\qquad \tau_1=\frac12+\frac{a^2+b^2-a}{2b}i. \] This formula is the basic recursion underlying the deep-hole sequence [2310.14091]. ## 3. Dynamics of the planar deep-hole operator The deep-hole operator has a precise geometric structure in dimension \(2\). If \(L\) is a lattice in the plane with angle \(\theta\in[\pi/3,\pi/2]\) and successive minima \(\lambda_1\), \(\lambda_2=\alpha \lambda_1\), \(\alpha\ge 1\), then the first principal theorem gives four basic properties: a criterion for \(H(L)\) to be WR, the fact that semi-stability implies that one deep-hole step yields a WR lattice, the fact that WR lattices are fixed up to similarity, and preservation of the field of definition [2310.14091]. The exact sufficient criterion is \[ \alpha \leq 2 \sin (\theta + \pi/6), \] which implies that \(H(L)\) is WR. This arises from the angle \(\nu\) between \(\mathbf z\) and \(\mathbf x_1-\mathbf z\). Since \(\mathbf z\) is the circumcenter of \(T\), the triangle with sides \(\mathbf z,\mathbf x_1,\mathbf x_1-\mathbf z\) is isosceles, and one gets \[ \sin(\nu/2) = \frac{\sin \theta}{\sqrt{1+\alpha^2-2 \alpha \cos \theta}}. \] If \(\nu\in[\pi/3,\pi/2]\), then \(H(L)\) is well-rounded, because \(\mathbf z\) and \(\mathbf x_1-\mathbf z\) have equal length and form a minimal basis. After rearrangement, this leads exactly to the inequality above [2310.14091]. A particularly strong specialization is the semi-stable case. If \(L\sim \Lambda_\tau\) with \[ \tau=\alpha e^{i\theta}=\alpha\cos\theta+\alpha i\sin\theta\in \mathcal F^+, \] and \(L\) is semi-stable, then \[ \alpha=|\tau|\le \sqrt{5}/2<\sqrt 3 = \min\{2\sin(\theta+\pi/6):\theta\in[\pi/3,\pi/2]\}, \] so the criterion automatically holds. Hence, if \(L\) is semi-stable, then \(H(L)\) is WR [2310.14091]. The operator also has a fixed-point description. If \(L\) is WR, then \(\alpha=1\), and the proof shows that the angle of \(H(L)\) agrees with the angle of \(L\); therefore \(H(L)\sim L\). In the moduli space \(\mathcal F^+\), WR similarity classes are precisely the fixed points of the deep-hole operation modulo similarity. This suggests that, in the planar setting, the deep-hole problem admits a canonical terminal locus rather than merely a set of extremal points [2310.14091]. ## 4. Iteration, elliptic curves, CM, and counting The second major innovation is to iterate the construction. Given \[ \tau_0=a_0+b_0 i\in \mathcal F^+, \qquad a_0,b_0\in K, \] one constructs a finite sequence \[ \tau_1,\dots,\tau_n,\qquad \tau_k=a_k+b_k i, \] with \[ \Lambda_{\tau_k}=H(\Lambda_{\tau_{k-1}}). \] The first step already sends any \(\tau=a+bi\) to \[ \tau_1=\frac12+\frac{a^2+b^2-a}{2b}i, \] so after one deep-hole step all subsequent representatives lie on the vertical line \(\Re(\tau)=1/2\). The proof gives the crucial estimate \[ b_k<\frac{b_{k-1}}{2}, \qquad\text{hence}\qquad b_k<\frac{b_0}{2^k}. \] The stopping criterion is \(|\tau_n|\le 1\), and since \(a_n=1/2\), this occurs once \[ b_n<\frac{\sqrt 3}{2}. \] Therefore, if \[ n\ge \log_2\left(\frac{2b_0}{\sqrt3}\right), \] then \(|\tau_n|<1\), so \(\tau_n\notin\mathcal F^+\), and at that point \(\Lambda_{\tau_n}\) is WR. The theorem states \[ n \le \log_2\left(\frac{2b_0}{\sqrt3}\right). \] Thus every planar similarity class reaches the WR locus after finitely many deep-hole steps [2310.14091]. The same paper gives an elliptic-curve interpretation. A point \(\tau\in \mathbb H\) corresponds to the elliptic curve \[ E_\tau=\mathbb C/\Lambda_\tau, \qquad \Lambda_\tau=\operatorname{span}_{\mathbb Z}\{1,\tau\}\subset \mathbb C. \] The full standard modular domain \[ \mathcal F=\{\tau=a+bi\in \mathbb H:\ -1/20}: da_k,\ d^2b_k^2\in \mathbb Z\}. \] The same work also studies the inverse problem of counting planar lattices with a prescribed deep hole lattice. For \[ \tau_0=\frac12+it\in \mathcal F^+ \] lying over a number field \(K\), the set \[ S_{K,\tau_0} = \left\{ \tau\in \mathcal F^+: \tau \text{ is defined over }K \text{ and } H(\Lambda_\tau)=\Lambda_{\tau_0} \right\} \] is infinite, and the preimages lie on the circular arc \[ S_{\tau_0} := \left\{ a+bi: 0\le a\le \frac12,\ b>t,\ \left(a-\frac12\right)^2+(b-t)^2=\frac14+t^2 \right\}. \] For primitive height \(\mathfrak H^p\), the height-bounded set \[ S_{K,\tau_0}(T) = \{\tau\in S_{K,\tau_0}:\ \mathfrak H^p(\tau)\le T\} \] satisfies \[ |S_{K,\tau_0}(T)| \le \left( \frac{4^{r_1}\pi^{2r_2}}{8\zeta(2n)\left(2t+\sqrt{4t^2+1}\right)|\Delta_K|} \right)T^{2n} + O(T^{2n-1}), \qquad T\to\infty. \] This gives a quantitative asymptotic upper estimate for planar similarity classes over \(K\) with prescribed deep-hole image [2310.14091]. ## 5. Deep-hole problem for D1-D5-P black holes In the supersymmetric black-hole literature, the deep-hole problem is posed for Type IIB on \[ \mathbb{R}^{4,1} \times S^1 \times \mathcal{M}, \] with \(\mathcal{M}=T^4\) or \(K3\), microscopic \(\mathcal{M}\), and macroscopic \(S^1\) of radius \(R_y\). The charges are carried by \(n_1\) D1-branes on \(S^1\), \(n_5\) D5-branes on \(S^1\times \mathcal{M}\), and momentum \(P\) along \(y\). The supersymmetric rotating black hole compared to is the five-dimensional D1-D5-P BMPV black hole, or equivalently its six-dimensional black string lift [1607.03908]. The cited construction gives the first family of **supersymmetric six-dimensional D1-D5-P supergravity solutions** obtained by adding specific momentum-carrying deformations to a two-charge maximally rotating D1-D5 supertube seed using **superstratum technology** and **coiffuring**. The fluctuations are labeled by integers \[ k\in \mathbb{Z}_{>0},\qquad m,n\in \mathbb{Z}_{\ge 0}, \] with mode amplitude \(b_{k,m,n}\). The metric is written as \[ ds_6^2 = -\frac{2}{\sqrt{\mathcal{P}}}\,(dv+\beta)\bigl(du + \omega + \tfrac{1}{2}\, \mathcal{F} \, (dv+\beta)\bigr) + \sqrt{\mathcal{P}} \, ds_4^2, \] where the flat \(\mathbb R^4\) base is \[ ds_4^2 = \frac{\Sigma\, dr^2}{r^2 + a^2} + \Sigma\, d \theta^2 + (r^2 + a^2) \sin^2 \theta \, d\phi^2 + r^2 \cos^2 \theta \, d\psi^2, \] with \[ \Sigma \equiv r^2 + a^2 \cos^2 \theta,\qquad \mathcal{P} \equiv Z_1 Z_2 - Z_4^2. \] The \(v\)-fibration is \[ \beta = 2^{-1/2}\, a^2 R_y \, \Sigma^{-1} \, ( \sin^2 \theta \, d\phi- \cos^2 \theta \, d\psi ) . \] Momentum is added via deformations with phase \[ \hat{v}_{k,m,n} \equiv \sqrt{2}\, R_y^{-1}\, (m+n) \,v+ (k-m)\phi - m\psi . \] The fluctuating ansatz takes \[ Z_1 = \frac{Q_1}{\Sigma} + \frac{R_y^2}{2 Q_5} b_{k,m,n}^2 \frac{\Delta_{2k,2m,2n}}{\Sigma} \cos \hat{v}_{2k,2m,2n}, \] \[ Z_2 = \frac{Q_5}{\Sigma}, \qquad Z_4 = b_{k,m,n} R_y\frac{\Delta_{k,m,n}}{\Sigma} \cos \hat{v}_{k,m,n}, \] with profile \[ \Delta_{k,m,n} \equiv a^k \, r^n (r^2+a^2)^{-(k+n)/2} \cos^{m}\theta \, \sin^{k-m}\theta . \] Coiffuring is used so that, although the tensor fields depend on \(\hat v_{k,m,n}\), the metric does not [1607.03908]. These solutions have the same conserved quantum numbers as **general supersymmetric rotating D1-D5-P black holes in five dimensions**. The explicit expressions are \[ j = \frac{\mathcal{N}}{2} \left(a^2 + \frac{m}{k} \, b^2\right), \qquad \tilde{j} = \frac{\mathcal{N}}{2} a^2, \qquad n_P = \frac{\mathcal{N}}{2} \frac{m+n}{k} \, b^2, \] where \[ \mathcal{N} \equiv \frac{n_1 n_5 R_y^2}{Q_1 Q_5}, \qquad b^2 = x_{k,m,n}\, b_{k,m,n}^2, \qquad x_{k,m,n}^{-1}\equiv {k \choose m}{k+n-1 \choose n}. \] A key feature is that the family permits **arbitrarily small finite angular momenta**. For the subclass with \(m=0\), \[ j = \tilde{j} = \frac{\mathcal{N}}{2}a^2,\qquad n_P = \frac{\mathcal{N}}{2}\frac{n}{k}b^2. \] Thus taking \(a\to 0\) with \(b\) adjusted appropriately sends both angular momenta to zero while retaining momentum charge, producing the first microstate geometries of the non-rotating D1-D5-P Strominger-Vafa black hole [1607.03908]. ## 6. Throat geometry, holography, and limitations The black-hole comparison geometry is the supersymmetric rotating D1-D5-P black hole, whose six-dimensional near-horizon geometry is \(S^3\) fibered over extremal BTZ with metric \[ ds^2_{\rm BTZ}=\ell^2\Bigl[\rho^2(-dt^2+dy^2)+\frac{d\rho^2}{\rho^2}+\rho_*^2(dt+dy)^2\Bigr] . \] The parameters obey \[ \rho = \frac{r}{\sqrt{Q_1Q_5}},\qquad \ell^2=\sqrt{Q_1Q_5}, \] and \[ \rho_*^2=\frac{Q_P}{Q_1Q_5}. \] The microstate geometries are **well approximated by the corresponding black-hole solution**, and in particular they exhibit the **same near-horizon throat**. When \[ a^2 \ll Q_P, \] the cap lies deep in the AdS\(_2\) region. For \(r\gg a\), the leading terms in the metric coincide with those of the corresponding black hole; for \(r\ll a\), the geometry caps off smoothly. The proper length of the throat is \[ L_{\rm throat}\sim \log\!\left(\frac{Q_P}{a^2}\right), \] so \(a\to 0\) gives an arbitrarily deep throat. The family lies within the cosmic censorship bound whenever \[ \frac{b^2}{a^2}> \frac{k}{\,n+\sqrt{(k-m+n)(m+n)}\,}. \] This is the explicit criterion showing that smooth horizonless geometries exist in the same parameter region where a large BMPV black hole exists [1607.03908]. Regularity is controlled by the constraints \[ \frac{Q_1Q_5}{R_y^2}=a^2 + \frac{b^2}{2},\qquad b^2 = x_{k,m,n} \, b_{k,m,n}^2, \] together with the condition that \(\mu_{k,m,n}\) vanish at \[ r=0,\qquad \theta=0. \] To ensure \(Z_1>0\), the construction uses the bound \[ \frac{\Delta_{2k,2m,2n}}{x_{k,m,n}} \le \frac{a^2 }{r^2 +a^2} \le 1, \] which implies \(b^2_{k,m,n} \Delta_{2k,2m,2n} < b^2\), and hence \(Z_1>0\). The explicit \(k=1,m=0\) family is stated to be easy to verify as regular and CTC-free everywhere [1607.03908]. The holographic dual is the \({\cal N}=(4,4)\) D1-D5 orbifold CFT with central charge \[ c=6 n_1 n_5 \equiv 6N. \] The states are supersymmetric Ramond-sector states at the symmetric-orbifold point \({\cal M}^N/S_N\). In strand language, momentum is carried on \(|00\rangle_k\) strands by commuting operators \[ J^+_{-1}, \qquad (L_{-1}-J^3_{-1}), \] and the identified coherent superposition is \[ (|\!+\!+\rangle_1)^{N_1 } \biggl(\frac{(J^+_{-1})^{m}}{m!} \frac{(L_{-1}- J^3_{-1})^n}{n!} |00\rangle_k\biggr)^{N_{k,m,n}}, \] with strand budget \[ N_1+ k N_{k,m,n} = N. \] The average strand numbers \[ \langle N_{|\!+\!+\rangle_1}\rangle = \mathcal{N} a^2, \qquad \langle N_{|00\rangle_k\text{ excit.}\rangle = \frac{\mathcal{N} b^2}{2k} \] reproduce the supergravity charges \(j,\tilde j,n_P\) exactly. At the same time, the work is explicit that these are **not generic typical black-hole states**. They are protected, supersymmetric, Ramond-sector states counted by the elliptic genus, but they are highly special coherent states built from a restricted set of generators. The paper therefore **partially resolves and substantially advances** the deep-hole problem: it establishes that smooth horizonless microstates can exist deep inside the classical black-hole regime and with arbitrarily small angular momenta, but it does not prove that generic black-hole microstates are smooth horizonless geometries, nor does it address non-extremal or non-supersymmetric cases [1607.03908]. A plausible implication is that the two uses of the term share a structural theme despite their very different mathematical content. In both cases, the “deep-hole” perspective turns an extremal geometric notion into a dynamical or constructive framework: in the planar lattice setting, repeated passage to \(H(L)\) drives every similarity class to the well-rounded locus, while in the D1-D5-P setting, the problem is to realize microstates that sit deep down the classical throat yet terminate in a smooth cap rather than a horizon. Source: https://www.emergentmind.com/topics/deep-hole-problem