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A generalization of the Virasoro algebra to arbitrary dimensions

Razvan Gurau

TL;DR

The work generalizes colored tensor models to arbitrary interactions and derives Schwinger-Dyson equations in the large $N$ limit, showing that leading-order constraints form a Lie algebra indexed by colored rooted $D$-ary trees, a higher-dimensional analogue of the Virasoro algebra. Melonic graphs are shown to be in bijection with these trees, enabling closed SDEs and a tractable description of leading critical behavior via the tree-based algebra. The analysis extends to an infinity of couplings, translating tensor-network bubble observables into a constraint system for the partition function $Z$ and its leading free energy $F_ ext{∞}$. The results establish a framework for multi-critical phenomena in higher-dimensional random geometries and highlight avenues for exploring representations, central extensions, and continuum symmetries.

Abstract

Colored tensor models generalize matrix models in higher dimensions. They admit a 1/N expansion dominated by spherical topologies and exhibit a critical behavior strongly reminiscent of matrix models. In this paper we generalize the colored tensor models to colored models with generic interaction, derive the Schwinger Dyson equations in the large N limit and analyze the associated algebra of constraints satisfied at leading order by the partition function. We show that the constraints form a Lie algebra (indexed by trees) yielding a generalization of the Virasoro algebra in arbitrary dimensions.

A generalization of the Virasoro algebra to arbitrary dimensions

TL;DR

The work generalizes colored tensor models to arbitrary interactions and derives Schwinger-Dyson equations in the large limit, showing that leading-order constraints form a Lie algebra indexed by colored rooted -ary trees, a higher-dimensional analogue of the Virasoro algebra. Melonic graphs are shown to be in bijection with these trees, enabling closed SDEs and a tractable description of leading critical behavior via the tree-based algebra. The analysis extends to an infinity of couplings, translating tensor-network bubble observables into a constraint system for the partition function and its leading free energy . The results establish a framework for multi-critical phenomena in higher-dimensional random geometries and highlight avenues for exploring representations, central extensions, and continuum symmetries.

Abstract

Colored tensor models generalize matrix models in higher dimensions. They admit a 1/N expansion dominated by spherical topologies and exhibit a critical behavior strongly reminiscent of matrix models. In this paper we generalize the colored tensor models to colored models with generic interaction, derive the Schwinger Dyson equations in the large N limit and analyze the associated algebra of constraints satisfied at leading order by the partition function. We show that the constraints form a Lie algebra (indexed by trees) yielding a generalization of the Virasoro algebra in arbitrary dimensions.

Paper Structure

This paper contains 12 sections, 8 theorems, 68 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Matrix Models and the Virasoro algebra
  3. A Lie algebra indexed by colored, rooted, D-ary trees
  4. Colored rooted D-ary trees: Definitions
  5. Colored rooted D-ary trees: Properties
  6. The Lie algebra indexed by colored rooted D-ary trees
  7. Schwinger Dyson equations in the large N limit of Colored Tensor Models
  8. The i.i.d. colored tensor models with one coupling
  9. Leading order graphs
  10. From one to an infinity of coupling constants
  11. Schwinger Dyson equations
  12. Conclusions

Key Result

Lemma 1

If $(V) =(k,U) \in {\cal T}$ then $({\cal T}\star_V {\cal T}_1)^k ={\cal T}^k \star_U {\cal T}_1$, and, for $i\neq k$, $({\cal T}\star_{V} {\cal T}_1)^i = {\cal T}^i$.

Figures (7)

  • Figure 1: A colored rooted $D$-ary tree.
  • Figure 2: Gluing of two trees at a vertex ${\cal T} \star_{(2)} {\cal T}_1$.
  • Figure 3: Two equivalent trees ${\cal T} \sim {\cal T}'$, with ${\cal T}' = \Bigl( {\cal T}^2 \star_{(22)} \{(\;),(2)\} \Bigr) \star_{(222)} \tilde{{\cal T}}^2$.
  • Figure 4: Eliminating a $D$-bubble with two vertices.
  • Figure 5: First order.
  • ...and 2 more figures

Theorems & Definitions (10)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Theorem 1
  • Definition 1
  • Definition 2
  • Lemma 6
  • Lemma 7