On the Algebra of the Infrared with Twisted Masses

Ahsan Z. Khan; Gregory W. Moore

RECORD

On the Algebra of the Infrared with Twisted Masses

Ahsan Z. Khan, Gregory W. Moore

High Energy Physics - Theory · Submitted 15 Aug 2024

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We show that even in this simple setting there are interesting new phenomenon including the emergence of Fock spaces of closed solitons and a natural appearance of Koszul dual algebras.

how it is stated · asserted 1.00 read the record →

Comment (0) Read on arXiv ↗

Cite ▾

Claim paper

The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Open the compiled record

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,508 input tokens.

What the abstract itself states, as probabilities: claims first 0.24 · states limitation 0.58 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.81 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The Algebra of the Infrared \cite{Gaiotto:2015aoa} is a framework to construct local observables, interfaces, and categories of supersymmetric boundary conditions of massive $\mathcal{N}=(2,2)$ theories in two dimensions by using information only about the BPS sector. The resulting framework is known as the ``web-based formalism.'' In this paper we initiate the generalization of the web-based formalism to include a much wider class of $\mathcal{N}=(2,2)$ quantum field theories than was discussed in \cite{Gaiotto:2015aoa}: theories with non-trivial twisted masses. The essential new ingredient is the presence of BPS particles within a fixed vacuum sector. In this paper we work out the web-based formalism for the simplest class of theories that allow for such BPS particles: theories with a single vacuum and a single twisted mass. We show that even in this simple setting there are interesting new phenomenon including the emergence of Fock spaces of closed solitons and a natural appearance of Koszul dual algebras. Mathematically, studying theories with twisted masses includes studying the Fukaya-Seidel category of A-type boundary conditions for Landau-Ginzburg models defined by a closed holomorphic one-form. This paper sketches a web-based construction for the category of A-type boundary conditions for one-forms with a single Morse zero and a single non-trivial period. We demonstrate our formalism explicitly in a particularly instructive example.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The Algebra of the Infrared \cite{Gaiotto:2015aoa} is a framework to construct local observables, interfaces, and categories of supersymmetric boundary conditions of massive $\mathcal{N}=(2,2)$ theories in two dimensions by using information only about the BPS sector.	background 1.00	0.05	0.06	asserted 1.00
s1	The resulting framework is known as the ``web-based formalism.'' In this paper we initiate the generalization of the web-based formalism to include a much wider class of $\mathcal{N}=(2,2)$ quantum field theories than was discussed in \cite{Gaiotto:2015aoa}: theories with non-trivial twisted masses.	contribution 0.87	0.59	0.09	asserted 0.99
s2	The essential new ingredient is the presence of BPS particles within a fixed vacuum sector.	contribution 0.75	0.18	0.06	asserted 1.00
s3	In this paper we work out the web-based formalism for the simplest class of theories that allow for such BPS particles: theories with a single vacuum and a single twisted mass.	contribution 0.87	0.91	0.07	asserted 0.96
s4	We show that even in this simple setting there are interesting new phenomenon including the emergence of Fock spaces of closed solitons and a natural appearance of Koszul dual algebras.	result 1.00	0.94	0.43	asserted 1.00
s5	Mathematically, studying theories with twisted masses includes studying the Fukaya-Seidel category of A-type boundary conditions for Landau-Ginzburg models defined by a closed holomorphic one-form.	interpretation 0.60	0.17	0.08	asserted 0.98
s6	This paper sketches a web-based construction for the category of A-type boundary conditions for one-forms with a single Morse zero and a single non-trivial period.	contribution 0.92	0.85	0.10	asserted 0.81
s7	We demonstrate our formalism explicitly in a particularly instructive example.	contribution 0.79	0.91	0.05	asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.81 · evidence from proof or derivation 0.98.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original abstract

The Algebra of the Infrared \cite{Gaiotto:2015aoa} is a framework to construct local observables, interfaces, and categories of supersymmetric boundary conditions of massive $\mathcal{N}=(2,2)$ theories in two dimensions by using information only about the BPS sector. The resulting framework is known as the ``web-based formalism.'' In this paper we initiate the generalization of the web-based formalism to include a much wider class of $\mathcal{N}=(2,2)$ quantum field theories than was discussed in \cite{Gaiotto:2015aoa}: theories with non-trivial twisted masses. The essential new ingredient is the presence of BPS particles within a fixed vacuum sector. In this paper we work out the web-based formalism for the simplest class of theories that allow for such BPS particles: theories with a single vacuum and a single twisted mass. We show that even in this simple setting there are interesting new phenomenon including the emergence of Fock spaces of closed solitons and a natural appearance of Koszul dual algebras. Mathematically, studying theories with twisted masses includes studying the Fukaya-Seidel category of A-type boundary conditions for Landau-Ginzburg models defined by a closed holomorphic one-form. This paper sketches a web-based construction for the category of A-type boundary conditions for one-forms with a single Morse zero and a single non-trivial period. We demonstrate our formalism explicitly in a particularly instructive example.

Human review

How Pith checks papers → · Full audit of this paper →

How the literature uses it

2 arXiv papers cite this in 2 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 1 sentence without a confident label

Cites it as background 1 paper

Surely, our framework has a huge linage of cousins in the modern literature: algebras of supersymmetric interfaces [ref] , monopole algebras of 3d theories [ref] , vertex operator algebras of class S theories [ref] and many more others.

Tunnels Under Geometries (or Instantons Know Their Algebras) · Introduction · 0.98

What it cites · 47 sentences without a confident label

Uses the method or tool of 1 paper

forms a “cyclic fan of solitons.” Using the method of [ref] one computes the total degree is two and therefore the fermion degrees of these spaces satisfy

Categorical Wall-Crossing in Landau-Ginzburg Models · Another Rank One Example with Two Vacua · 1.00

Compares with 2 papers

In the introduction we mentioned that although the bulk observables are expected to form an $E_2$ algebra the paper [ref] only constructed an $L_∞$ algebra as we have just described.

Algebra of the Infrared: String Field Theoretic Structures in Massive ${\cal N}=(2,2)$ Field Theory In Two Dimensions · Bulk Observables · 0.86
The example of the $ℂℙ^1$ sigma model gives a particularly interesting example with intricate wall-crossing formally analogous to that of 4d SU(2) N=2 supersymmetric Yang-Mills theory [ref] .

The BPS Spectra of Two-Dimensional Supersymmetric Gauge Theories with Twisted Mass Terms · Conclusions And Future Directions · 0.96

Cites as support 1 paper

[For some simple superpotentials one can even exhibit exact solutions to the $ζ$-instanton equation with fan boundary conditions [ref] . ]

An Exact Solution of BPS Domain Wall Junction · Example: Quartic Landau-Ginzburg Model · 0.96

Cites as background 22 papers

We assume that the vacuum weights are in general position (and also away from “exceptional walls” as defined in [ref] ).

Algebra of the Infrared: String Field Theoretic Structures in Massive ${\cal N}=(2,2)$ Field Theory In Two Dimensions · Infrared Data and Bulk Local Operators · 0.81
For the homological origins of the cobar complex and some worked examples see the relevant section in [ref] , which applies this notion to the problem of categorifying pentagon relations related to wall-crossing in four-dimensional $𝒩=2$ theories.

Categorical Pentagon Relations and Koszul Duality · Koszul Duality · 0.89
That the BPS state counting indices can be used to construct numerical “wall-crossing invariants" containing non-trivial information about supersymmetric observables is a relatively old idea that is explored in many papers (see [ref] for a small sample).

On Classification of N=2 Supersymmetric Theories, (e-mail uncorrupted version) · Introduction · 0.97

Discussion

Know something about this paper? Leave a comment, or highlight any passage above to comment on it.