Superstrata and Multimode Families

Superstrata are smooth supersymmetric D1–D5–P solutions whose momentum waves depend nontrivially on several coordinates. Their BPS equations are layered: the first layer is linear, while quadratic sources in the second layer impose “coiffuring” relations among Fourier coefficients. We develop one exact multimode family and accompany it with a reproducible consistency benchmark whose narrow scope is explicit.

Required background. Gauge Theory and String-Theory Preparation supplies supergravity fields, D-brane charges, and Fourier modes. Quantum Information supplies the coherent-state and Hilbert-space distinctions used when interpreting a classical family.

Helpful background. Quantum Gravity frames the controlled supergravity limit, while Black-Hole Information supplies the broader microstate question.

Six-dimensional BPS sector

The construction uses six-dimensional $\mathcal N=(1,0)$ supergravity coupled to two anti-self-dual tensor multiplets, in the D1–D5–P BPS ansatz. The metric is a null fibration over a four-dimensional base, and the first-layer fields $(Z_I,\Theta_I)$ obey linear equations. The second-layer fields $(\mathcal F,\omega)$ are sourced quadratically by first-layer modes. A generic superposition therefore does not remain regular merely because its individual modes are regular.

For the $(1,0,n)$ multimode subsector, define

$$\xi=\frac{r}{\sqrt{r^2+a^2}} \exp\!\left(\frac{i\sqrt2\,v}{R_y}\right), \qquad F_0(\xi)=\sum_{n\geq1}b_n\xi^n, \qquad F_1=0.$$

Mayerson, Walker, and Warner show that this family lies in a consistent $U(1)^2$ truncation and give its six-dimensional uplift in Mayerson, Walker, and Warner 2020, §4.1, pp. 20–21, Eqs. (4.1)–(4.5), and Appendix D, pp. 30–38. “Consistent” means every solution of the retained lower-dimensional equations uplifts to a solution of the stated six-dimensional theory; it does not mean the truncation contains every D1–D5–P microstate.

Coiffuring, regularity, and charges

Holomorphy packages the coiffured fields so that dangerous purely holomorphic quadratic sources cancel or become fixed by sesquilinear data. With real $b_n$, the asymptotic Fourier norm fixes

$$c^2=\sum_{n\geq1}b_n^2, \qquad \frac{Q_1Q_5}{R_y^2}=a^2+\frac{c^2}{2}.$$

The second relation removes a cap singularity and makes the asymptotic gauge transformation periodic. The conserved charges for this subsector are

$$Q_P=\frac12\sum_{n\geq1}n\,b_n^2, \qquad J_L=J_R=\frac{R_y a^2}{2}.$$

These formulae and the regularity/closed-timelike-curve analysis are derived in Mayerson, Walker, and Warner 2020, Appendix D.3–D.5, pp. 35–38, Eqs. (D.27)–(D.49). For the $(1,0,n)$ family the no-CTC result uses analyticity of $F_0$; a finite grid is a diagnostic, not a replacement for that proof.

The explicit benchmark retains two modes,

$$(k,m,n)=(1,0,1),(1,0,3), \qquad (b_1,b_3)=\left(\frac65,\frac45\right),$$

and chooses $(Q_1,Q_5,R_y)=(9,16,4)$. Exact algebra then gives

$$c^2=\frac{52}{25},\quad a^2=\frac{199}{25},\quad Q_P=\frac{42}{25},\quad J_L=J_R=\frac{398}{25}.$$

The benchmark reports numerical values after choosing one reference length. Restoring that length $\ell_0$ gives $a,R_y,b_n$ units of $\ell_0$, $Q_1,Q_5,Q_P,c^2$ units of $\ell_0^2$, and the geometric angular-momentum parameters $J_L,J_R$ units of $\ell_0^3$. These last parameters are not the dimensionless quantized CFT spins; their conversion also involves the gravitational normalization. In particular, the cap equation equates quantities with dimension length squared.

The analytic benchmark is the exact rational evaluation of the coiffuring relation and all four conserved quantities above. It is independent of the angular finite-difference calculation: the symbolic branch computes the sums over declared modes directly, while the grid branch samples the phase dependence. Agreement therefore tests data plumbing and normalization as well as numerical convergence; it is not merely the finest-grid value compared with itself.

Reproducible benchmark

Download the superstrata benchmark program and its pinned machine-readable JSON result. Together they expose the benchmark calculation and its expected output for independent inspection.

The reference environment is Python 3.11.4, SymPy 1.13.3, and mpmath 1.3.0. Exact positive rational inputs are retained symbolically; numerical grids use 80 decimal digits, $\rho=r/a\in[0,12]$, and uniform phase points. Save the downloaded program as superstrata-multimode-benchmark.py, then run:

python3.11 superstrata-multimode-benchmark.py --output superstrata-multimode-benchmark.json

The script refuses a version mismatch unless —allow-version-drift is supplied for a separately labeled diagnostic. It exits nonzero if an exact constraint fails, the phase-derivative residual does not decrease, the finest residual exceeds $3\times10^{-3}$, or the sampled scalar determinant falls below $0.9$.

The grid evaluates the holomorphy identity

$$\partial_\theta F_0 =i\sum_n n b_n\xi^n$$

with a centered finite difference. It also checks Parseval’s relation for $c^2$ and the $U(1)^2$ scalar determinant. From Eqs. (4.20)–(4.22),

$$\sqrt{\det m}=1-S_1^2-S_2^2, \qquad \Omega^2=\frac{2}{R_y^2g_0^4} \left(1-S_1^2-S_2^2\right).$$

Positivity of both is a necessary causal and regularity check over the sampled domain.

The boundary check includes both radial endpoints $\rho=0$ and $\rho=12$ and the periodic phase seam, so the centered derivative never silently changes its state choice at the cap or outer sampling boundary. The associated state regularity check verifies the exact cap relation before sampling and requires the scalar determinant and warp factor to remain positive at every stored point. These tests do not replace the analytic no-CTC proof, but they catch inconsistent inputs and boundary indexing errors.

Three refinements and benchmark result

Radial × phase grid	Maximum holomorphy residual	Minimum $\sqrt{\det m}$	Minimum $\Omega^2$ factor
$33\times64$	$3.61109464382\times10^{-2}$	$0.962847407534$	$17.3312533356$
$65\times128$	$9.05576406139\times10^{-3}$	$0.962417580287$	$17.3235164452$
$129\times256$	$2.26569610758\times10^{-3}$	$0.962379549921$	$17.3228318986$

The derivative residual decreases by approximately four under each doubling, the expected second-order behavior. The exact coiffuring/regularity residual is zero, the discrete Parseval residual is zero at 80-digit precision, and the causal factors remain comfortably positive. The slight drift of the sampled minimum reflects locating the minimum more accurately, not a loss of convergence.

Deliberate failures and scope

Three failing inputs are serialized with the result. Shifting $c^2$ by $1/20$ while holding the other data fixed gives regularity residual $-1/20$. Replacing $(b_1,b_3)$ by $(4,3)$ yields $a^2=-7/2$, so the cap-size parameter is unphysical. Using only four phase points aliases a highest mode $n=3$ and invalidates the quadratic Fourier check. Each failure targets a different vulnerability: missing coiffuring, over-rotation, and inadequate resolution.

This benchmark does not solve the full six-dimensional PDE system numerically, scan all metric components for every causal curve, quantize the family, or establish entropy coverage. It checks exact identities and a bounded grid diagnostic for a family already proved analytically. Shigemori finds that presently counted superstrata have parametrically smaller entropy than the black hole in Shigemori 2019, §§3–5, pp. 8–21, so a large functional family must not be confused with a complete state count.

Exercises

For the benchmark modes, recompute $c^2$, $a^2$, and $Q_P$. Then explain why changing $b_3$ to $3$ while keeping $b_1=4$, $Q_1Q_5/R_y^2=9$ forces failure before any grid is run.

Solution

The exact sums give $c^2=(6/5)^2+(4/5)^2=52/25$, so $a^2=9-c^2/2=199/25$. Momentum is $Q_P=\tfrac12[(6/5)^2+3(4/5)^2]=42/25$. For amplitudes $(4,3)$, $c^2=16+9=25$ and the regularity equation demands $a^2=9-25/2=-7/2$. Since $a^2$ sets a squared cap scale, negativity already violates the admissible input regime; refining a grid cannot repair an algebraic obstruction. Therefore the correct diagnostic order is to check exact constraints first, then boundary and state regularity, and only afterward interpret a convergence table. A decreasing derivative residual would be irrelevant for data that fail the exact cap equation.

References

• Mayerson, Daniel R., Robert A. Walker, and Nicholas P. Warner. “Microstate Geometries from Gauged Supergravity in Three Dimensions.” Journal of High Energy Physics 10 (2020): 030. DOI. Open manuscript.
• Shigemori, Masaki. “Counting Superstrata.” Journal of High Energy Physics 10 (2019): 017. DOI. Open manuscript.