[Paper Review] An introduction to decomposition

Eric Sharpe|arXiv (Cornell University)|Apr 19, 2022

Black Holes and Theoretical Physics4 citations

TL;DR

This paper introduces decomposition in quantum field theories (QFTs) with (d−1)-form symmetries, showing that such theories are equivalent to disjoint unions of simpler QFTs called 'universes.' The framework explains physical phenomena like instanton selection rules via multiverse interference and applies to 2D gauge theories and orbifolds, resolving anomalies via the Wang-Wen-Witten mechanism.

ABSTRACT

We review work on `decomposition,' a property of two-dimensional theories with 1-form symmetries and, more generally, d-dimensional theories with (d-1)-form symmetries. Decomposition is the observation that such quantum field theories are equivalent to (`decompose into') disjoint unions of other QFTs, known in this context as "universes." Examples include two-dimensional gauge theories and orbifolds with matter invariant under a subgroup of the gauge group. Decomposition explains and relates several physical properties of these theories -- for example, restrictions on allowed instantons arise as a "multiverse interference effect" between contributions from constituent universes. First worked out in 2006 as part of efforts to resolve technical questions in string propagation on stacks, decomposition has been the driver of a number of developments since. We give a general overview of decomposition, describe features of decomposition arising in gauge theories, then dive into specifics for orbifolds. We conclude with a discussion of the recent application to anomaly resolution of Wang-Wen-Witten in two-dimensional orbifolds. This is a contribution to the proceedings of the conference Two-dimensional supersymmetric theories and related topics (Matrix Institute, Australia, January 2022), giving an overview of a talk given there and elsewhere.

Motivation & Objective

• To clarify the concept of decomposition in QFTs with (d−1)-form symmetries, particularly in two-dimensional theories.
• To resolve physical inconsistencies in string compactifications on stacks and gerbes by introducing decomposition as a solution.
• To demonstrate how decomposition explains restrictions on allowed instantons through multiverse interference effects.
• To apply decomposition to the anomaly-resolution mechanism of Wang-Wen-Witten in 2D orbifolds.
• To distinguish decomposition from superselection sectors by emphasizing the absence of continuous field-space paths between universes.

Proposed method

• Use of mutually commuting topological projection operators $\Pi_i$ that satisfy $\Pi_i\Pi_j = \delta_{ij}\Pi_j$ and $\sum_i \Pi_i = 1$, enabling simultaneous diagonalization of the Fock space into universe eigenstates.
• Leveraging the partition function identity $Z = \sum_i Z_i$ to show that correlation functions in the full theory are sums of correlation functions in the constituent universes.
• Applying the framework to 2D gauge theories with non-minimal charges, showing equivalence to sums of $U(1)$ theories with minimal charges.
• Analyzing orbifolds where a subgroup of the gauge group acts trivially, leading to decomposition into orbifolds by subgroups.
• Using the Wang-Wen-Witten anomaly resolution procedure, where the choice of $B$-field in the orbifold $[X/\Gamma]_B$ determines which anomaly is resolved, and the resulting theory is equivalent to a disjoint union of non-anomalous orbifolds.
• Constructing tables of quantum symmetries and corresponding resolved anomalies, showing that $d_2(B)$ maps the quantum symmetry to the anomaly that can be canceled.

Experimental results

Research questions

• RQ1 How does decomposition explain the selection rules for instantons in 2D gauge theories with 1-form symmetries?
• RQ2 In what way does decomposition resolve apparent physical inconsistencies in string compactifications on stacks?
• RQ3 How does the decomposition of orbifolds with trivially-acting subgroups differ from standard orbifolds in terms of their physical content?
• RQ4 What is the role of the $B$-field in the Wang-Wen-Witten anomaly resolution procedure when applied to decomposed theories?
• RQ5 How does decomposition differ from spontaneous symmetry breaking and superselection sectors in terms of field-space connectivity between universes?

Key findings

• The partition function of a decomposed QFT is the sum of the partition functions of its constituent universes: $Z = \sum_i Z_i$.
• Correlation functions in the full theory decompose as $\langle \mathcal{O}_1 \cdots \mathcal{O}_m \rangle = \sum_i \langle \tilde{\mathcal{O}}_1 \cdots \tilde{\mathcal{O}}_m \rangle_i$, where $\tilde{\mathcal{O}}_i$ are projections into each universe.
• For $\Gamma = \mathbb{Z}_2 \times \mathbb{Z}_4$, choosing $B$ such that $d_2(B)$ matches the anomaly leads to physical theories like $[X/\langle a \rangle]$ or $[X/\langle b \rangle]$, which are non-anomalous and equivalent to disjoint unions.
• In the case $\Gamma = \mathbb{Z}_2 \times \mathbb{H}$, the resolved theory becomes a $\mathbb{Z}_2$-product of non-anomalous orbifolds, such as $\prod_2 [X/\langle b \rangle]$, depending on $B(a)$ and $B(b)$.
• The first row of Table 3 corresponds to no quantum symmetry, while the next three rows show that anomalies can be resolved by selecting $B$ such that $d_2(B)$ maps to the anomalous subgroup.
• The pattern holds across different groups: when $d_2(B)$ describes the anomaly, the resulting $[X/\Gamma]_B$ is equivalent to an orbifold by a subgroup not containing the anomalous one, confirming the Wang-Wen-Witten mechanism.