TASI lectures on complex structures

TL;DR

Denef’s TASI notes synthesize complex-system methods across physics: (i) replica-based analyses of disordered models yield the Parisi order parameter and ultrametric state organization, connecting to mean-field spin glasses via the SK model; (ii) supersymmetric quantum mechanics provides exact ground-state counting, instanton-induced lifting, and links to stochastic dynamics and Morse theory; (iii) wrapped D-brane systems generate vast, holographically tractable energy landscapes whose ground states and degeneracies can be computed through SUSY QM and topological indices, with concrete ties to black hole entropy via attractor flows and multicentered bound states. The work emphasizes how hierarchical, ultrametric structures and topological indices organize complexity—from spin glasses to D4-D0/D2 bound states and black hole duals—while highlighting the utility of holography and modular properties in counting states and comparing micro- and macro-entropy in string theory. Core results include Parisi’s RSB solution in the SK model, the Witten index framework for SUSY QM, and the construction/counting of supersymmetric D-brane ground states and their black-hole duals, illustrating deep structural parallels across seemingly disparate systems.

Abstract

These lecture notes give an introduction to a number of ideas and methods that have been useful in the study of complex systems ranging from spin glasses to D-branes on Calabi-Yau manifolds. Topics include the replica formalism, Parisi's solution of the Sherrington-Kirkpatrick model, overlap order parameters, supersymmetric quantum mechanics, D-brane landscapes and their black hole duals.

Figures (16)

• Figure 1: An ultrametric tree. Distances between the points in the top layer are set by the vertical distance to the first common ancestor node.
• Figure 2: AABDG Dendrogram plots and overlap matrices for the SK model with $N=800$ spins. We used parallel tempering Monte Carlo Marinari1998Earl2005 to reach thermal equilibrium, with 50 replicas at equally spaced temperatures between $T=0.1$ and $T=1.2$. Based on overlaps of 100 configurations sampled with separation of 100 sweeps, clustered using Mathematica. Results are shown for $T/T_c=1.2, 0.86, 0.55, 0.12$. Red = maximal positive overlap $q=+1$ (i.e. minimal distance), white = overlap $q=0$, dark blue = maximal negative overlap $q=-1$. Although the data is too limited to draw firm conclusions, the plots are suggestive of hierarchical clustering.
• Figure 3: From Crisanti2002a. Disorder-averaged overlap distribution $\overline{P(q)}$ for the SK model at various temperatures, from $T = 0.95$ on right to $T = 0.30$ on left in steps of 0.05 ($T_c = 1$). The dotted lines represent delta-functions, localized at $q_{\alpha\alpha}=q_{EA}$.
• Figure 4: Tree and matrix representation of a Parisi matrix $Q_{ab}$. Different colors and different tree connection heights correspond to different values of $Q_{ab}$, with darker red representing larger values. In this example, $n=90$, $K=3$, $\{m_0,m_1,m_2,m_3,m_4\}=\{1,3,15,45,90\}$, $\{u,q_0,q_1,q_2,q_3,q_4\} = \{ 1, 0.9, 0.6, 0.4, 0.1,0 \}$.
• Figure 5: The cluster size function $m(q)$ for the $K=3$ example of fig. \ref{['ParisiMatrix']}, and the corresponding tree. The arrow indicates the direction of the RG flow.