[Paper Review] On 'categories' of quantum field theories

Yuji Tachikawa|arXiv (Cornell University)|Dec 26, 2017

Algebraic structures and combinatorial models26 references3 citations

TL;DR

This paper proposes a categorical framework for quantum field theories (QFTs) structured by spacetime geometry, enabling a unified language to express deep mathematical conjectures from physics. It introduces categories of QFTs for different structures (e.g., Riemannian, conformal), defines point operators and correlation functions, and establishes connections to hyperkähler cones and vertex operator algebras via functors like $ \mathcal{M}_{\text{Higgs}}$ and $\mathbb{VOA}$, leading to precise conjectures on $X_G$ and $\mathbb{V}_G$ for $G = \mathrm{SU}(3N), \mathrm{SU}(4N), \mathrm{SU}(6N)$, with explicit character formulas and links to $E_6, E_7, E_8$ W-algebras.

ABSTRACT

We give a rough description of the 'categories' formed by quantum field theories. A few recent mathematical conjectures derived from quantum field theories, some of which are now proven theorems, will be presented in this language.

Motivation & Objective

• To formalize quantum field theories (QFTs) not as isolated objects but as a category, capturing their interrelationships and structural properties.
• To provide a mathematical language that clarifies the origin of recent physical conjectures in QFT, especially those involving dualities and duality cascades.
• To establish precise conjectural relationships between QFTs, hyperkähler cones $X_G$, and vertex operator algebras $\mathbb{V}_G$ via functors $\mathcal{M}_{\text{Higgs}}$ and $\mathbb{VOA}$.
• To connect the cohomology of instanton moduli spaces to W-algebras of simply-laced Lie types, using a new functor $Z_{\text{Nek}}$.
• To present explicit character formulas for $X_G$ and $\mathbb{V}_G$ using Hall-Littlewood polynomials and Macdonald theory.

Proposed method

• Define a category $\mathcal{Q}^d_\mathcal{S}$ of QFTs on $\mathcal{S}$-structured manifolds, where $\mathcal{S}$ includes Riemannian, conformal, or spin structures.
• Assign to each QFT $Q$ a space of point operators $\mathcal{V}_Q$, with group actions (e.g., $\mathrm{SO}(d)$, $\mathrm{Spin}(d)$, $\mathrm{SO}(d+1,1)$) and invariant subspaces $\mathcal{V}_Q^{\text{inv}}$.
• Construct correlation functions $Z_Q(M; \varphi_1(x_1)\cdots\varphi_n(x_n))$ as complex-valued amplitudes for point operators on manifolds $M$.
• Use the state-operator correspondence $\mathcal{V}_Q \simeq \mathcal{H}_Q(S^{d-1})$ in conformal QFTs to relate operators to states on spheres.
• Define functors $\mathcal{M}_{\text{Higgs}}: \mathcal{Q}^d_\mathcal{S} \to \text{hyperkähler cones}$ and $\mathbb{VOA}: \mathcal{Q}^d_\mathcal{S} \to \text{VOAs}$ to map QFTs to geometric and algebraic objects.
• Derive character formulas for $X_G = \mathcal{M}_{\text{Higgs}}(T_G)$ and $\mathbb{V}_G = \mathbb{VOA}(T_G)$ using Hall-Littlewood polynomials and Macdonald measure integrals.

Experimental results

Research questions

• RQ1 How can the interrelationships among quantum field theories be formalized in a categorical framework?
• RQ2 What is the precise mathematical structure of the space of point operators $\mathcal{V}_Q$ and its invariants under rotation or conformal groups?
• RQ3 How do the functors $\mathcal{M}_{\text{Higgs}}$ and $\mathbb{VOA}$ map QFTs to hyperkähler cones and vertex operator algebras, respectively?
• RQ4 What is the explicit character formula for the hyperkähler cone $X_G$ and vertex operator algebra $\mathbb{V}_G$ associated to $T_G = S_{G,0,3}$?
• RQ5 Can the cohomology of instanton moduli spaces $\mathcal{M}_{G,n}^{\text{inst}}$ be shown to carry an action of the W-algebra of type $G$ via a new functor $Z_{\text{Nek}}$?

Key findings

• The hyperkähler cone $X_G = \mathcal{M}_{\text{Higgs}}(T_G)$ for $G = \mathrm{SU}(3N)$ satisfies $X_{\mathrm{SU}(3N)}\wr[N^3],[N^3],[N^3] = \mathcal{M}^{\text{inst}}_{E_6,N}$, and similar identities hold for $E_7$ and $E_8$.
• The vertex operator algebra $\mathbb{V}_G = \mathbb{VOA}(T_G)$ satisfies $\mathbb{V}_{\mathrm{SU}(3)} = (\hat{\mathfrak{e}_6})_{-6}$, $\mathbb{V}_{\mathrm{SU}(4)}\wr[2,2],[1^4],[1^4] = (\hat{\mathfrak{e}_7})_{-8}$, and $\mathbb{V}_{\mathrm{SU}(6)}\wr[3,3],[2,2,2],[1^6] = (\hat{\mathfrak{e}_8})_{-12}$.
• The character of $X_G$ is given by $\mathop{\mathrm{ch}} X_G(z_1,z_2,z_3) = \sum_{\lambda} \frac{\prod_{i=1}^3 K_0(z_i)\underline{H}_\lambda(z_i)}{K_{e_{\text{prin}}} \underline{H}_\lambda(q^\rho)}$, where $\underline{H}_\lambda$ is the orthonormalized Hall-Littlewood polynomial.
• The character of $\mathbb{V}_G$ is $\mathop{\mathrm{ch}} \mathbb{V}_G(z_1,z_2,z_3) = \sum_{\lambda} \frac{\prod_{i=1}^3 K_0(z_i)\chi_\lambda(z_i)}{K_{e_{\text{prin}}} \chi_\lambda(q^\rho)}$, with $\chi_\lambda$ the irreducible character of $G$.
• The construction of $X_G$ as a hyperkähler cone with $G^4$ and $S_4$ symmetry is realized via $(X_G \times X_G)/\!/\!/G \simeq (T_G \times T_G)\not{\!\!\!\!\!-} G$, confirming the $S_4$ symmetry.
• The W-algebra action on $\mathcal{H}_G = \oplus_n H_G^*(\mathcal{M}_{G,n}^{\text{inst}})$ is conjectured via the functor $Z_{\text{Nek}}$, which sends QFTs to sections of $\mathcal{H}_G$-bundles, and is now proven via geometric representation theory.