We study non-invertible topological symmetry operators in massive quantum field theories in (1+1) dimensions. In phases where this symmetry is spontaneously broken we show that the particle spectrum often has degeneracies dictated by the non-invertible symmetry and we deduce a procedure to determine the allowed multiplets. These degeneracies are robust predictions and do not require integrability or other special features of renormalization group flows. We exhibit these conclusions in examples where the spectrum is known, recovering soliton and particle degeneracies. For instance, the Tricritical Ising model deformed by the subleading Z2 odd operator flows to a gapped phase with two degenerate vacua. This flow enjoys a Fibonacci fusion category symmetry which implies a threefold degeneracy of its particle states, relating the mass of solitons interpolating between vacua and particles supported in a single vacuum.

https://cmsa.fas.harvard.edu/event/gcs24_cordova/

I will start with a brief overview of gauge theory phases in 3+1 dimensions through the lens of higher symmetries — in particular the realization of 1-form symmetries acting on loop order parameters. I will then review recent progress in refining this characterization using invertible field theories, or equivalently symmetry protected topological phases (SPTs). This refinement leads to new results in gauge theories with fundamental matter, such as quantum chromodynamics (QCD), which do not possess 1-form symmetries. I will explain why these theories must sometimes undergo a phase transition between their confining and Higgs regimes, despite the fact that classic results and standard lore say they should be continuously connected.

https://cmsa.fas.harvard.edu/event/gcs24_dumitrescu/

Hilbert’s Nullstellensatz says that the complex numbers C satisfy a universal property among all R-algebras: every not-too-large nonzero commutative R-algebra maps to C. Deligne proved a similar statement in categorical dimension 1: every not-too-large symmetric monoidal category over R maps to the category sVec of super vector spaces. In other words, sVec (and not Vec!) is “algebraically closed”. These statements help explain why quantum field theory requires imaginary numbers and fermions. I will describe the universal symmetric monoidal higher category that extends the sequence C, sVec, …. This is joint work in progress with David Reutter, and builds on closely-related work by GCS collaborators Freed, Scheimbauer, and Teleman and Schlank et al.

https://cmsa.fas.harvard.edu/event/gcs24_johnson-freyd/

In recent years, a remarkable paradigm shift has occurred in understanding quantum observables in particle physics and cosmology, revealing their emergence from underlying novel mathematical objects known as positive geometries. The conference will center on the amplituhedron—the first and major example of a positive geometry. Building on the work of Lusztig and Postnikov on the positive Grassmannian, the physicists Arkani-Hamed and Trnka introduced the amplituhedron in 2013 as a geometric object that “explains” the so-called BCFW recurrence for scattering amplitudes in N = 4 super Yang Mills theory (SYM). Simultaneously, cluster algebras, originally introduced by Fomin and Zelevinsky to study total positivity, have been revealed to have a crucial role in describing singularities of N = 4 SYM scattering amplitudes. Thus, one can use ideas from quantum field theory (QFT) to connect cluster algebras to positive geometries, and in particular to the amplituhedron. Additionally, QFT can also be used to discover new examples of positive geometries. The conference will bring together a wide range of mathematicians and physicists both to draw new connections within algebraic combinatorics and geometry and to advance our physical understanding of scattering amplitudes and QFT.

The conference features: Introductory Lectures, an Open Problems Forum, Emerging Scholars Talks, and talks by experts in the fields.

The conference aims to actively engage and empower junior researchers and women, ensuring their integral presence and impactful contributions.

Directions to CMSA

Register Online for in-person talks

Register for Zoom Meeting

Confirmed Speakers:

• Evgeniya Akhmedova, Weizmann Institute of Science
• Nima Arkani-Hamed, IAS
• Paolo Benincasa, MPI
• Nick Early, Weizmann Institute of Science
• Carolina Figueiredo, Princeton University
• Yu-tin Huang, National Taiwan University
• Dani Kaufman, University of Copenhagen
• Chia-Kai Kuo, National Taiwan University
• Thomas Lam, University of Michigan
• Yelena Mandelshtam, UC Berkeley
• Shruti Paranjape, UC Davis
• Elizabeth Pratt, UC Berkeley
• Lecheng Ren, Brown University
• Sebastian Seemann, KU Leuven
• Melissa Sherman-Bennett, MIT & UC Davis
• Marcus Spradlin, Brown University
• Ran Tessler, Weizmann Institute of Science
• Hugh Thomas, Université du Québec à Montréal
• Jaroslav Trnka, UC Davis
• Anastasia Volovich, Brown University

Organizers:

• Matteo Parisi, Harvard CMSA
• Lauren Williams, Harvard Mathematics

This event will be co-funded by the National Science Foundation.

Limited funding to help defray travel expenses is available for graduate students and recent PhDs. If you are a graduate student or postdoc and would like to apply for support, please register above and send an email to amplituhedra@cmsa.fas.harvard.edu no later than Friday, April 19, 2024.

Please include your name, address, current status, university affiliation, citizenship, and area of study. F1 visa holders are eligible to apply for support. If you are a graduate student, please send a brief letter of recommendation from a faculty member to explain the relevance of the conference to your studies or research. If you are a postdoc, please include a copy of your CV.

In recent years, a remarkable paradigm shift has occurred in understanding quantum observables in particle physics and cosmology, revealing their emergence from underlying novel mathematical objects known as positive geometries. The conference will center on the amplituhedron—the first and major example of a positive geometry. Building on the work of Lusztig and Postnikov on the positive Grassmannian, the physicists Arkani-Hamed and Trnka introduced the amplituhedron in 2013 as a geometric object that “explains” the so-called BCFW recurrence for scattering amplitudes in N = 4 super Yang Mills theory (SYM). Simultaneously, cluster algebras, originally introduced by Fomin and Zelevinsky to study total positivity, have been revealed to have a crucial role in describing singularities of N = 4 SYM scattering amplitudes. Thus, one can use ideas from quantum field theory (QFT) to connect cluster algebras to positive geometries, and in particular to the amplituhedron. Additionally, QFT can also be used to discover new examples of positive geometries. The conference will bring together a wide range of mathematicians and physicists both to draw new connections within algebraic combinatorics and geometry and to advance our physical understanding of scattering amplitudes and QFT.

The conference features: Introductory Lectures, an Open Problems Forum, Emerging Scholars Talks, and talks by experts in the fields.

The conference aims to actively engage and empower junior researchers and women, ensuring their integral presence and impactful contributions.

Directions to CMSA

Register Online for in-person talks

Register for Zoom Meeting

Confirmed Speakers:

• Evgeniya Akhmedova, Weizmann Institute of Science
• Nima Arkani-Hamed, IAS
• Paolo Benincasa, MPI
• Nick Early, Weizmann Institute of Science
• Carolina Figueiredo, Princeton University
• Yu-tin Huang, National Taiwan University
• Dani Kaufman, University of Copenhagen
• Chia-Kai Kuo, National Taiwan University
• Thomas Lam, University of Michigan
• Yelena Mandelshtam, UC Berkeley
• Shruti Paranjape, UC Davis
• Elizabeth Pratt, UC Berkeley
• Lecheng Ren, Brown University
• Sebastian Seemann, KU Leuven
• Melissa Sherman-Bennett, MIT & UC Davis
• Marcus Spradlin, Brown University
• Ran Tessler, Weizmann Institute of Science
• Hugh Thomas, Université du Québec à Montréal
• Jaroslav Trnka, UC Davis
• Anastasia Volovich, Brown University

Organizers:

• Matteo Parisi, Harvard CMSA
• Lauren Williams, Harvard Mathematics

This event will be co-funded by the National Science Foundation.

Limited funding to help defray travel expenses is available for graduate students and recent PhDs. If you are a graduate student or postdoc and would like to apply for support, please register above and send an email to amplituhedra@cmsa.fas.harvard.edu no later than Friday, April 19, 2024.

Please include your name, address, current status, university affiliation, citizenship, and area of study. F1 visa holders are eligible to apply for support. If you are a graduate student, please send a brief letter of recommendation from a faculty member to explain the relevance of the conference to your studies or research. If you are a postdoc, please include a copy of your CV.

In recent years, a remarkable paradigm shift has occurred in understanding quantum observables in particle physics and cosmology, revealing their emergence from underlying novel mathematical objects known as positive geometries. The conference will center on the amplituhedron—the first and major example of a positive geometry. Building on the work of Lusztig and Postnikov on the positive Grassmannian, the physicists Arkani-Hamed and Trnka introduced the amplituhedron in 2013 as a geometric object that “explains” the so-called BCFW recurrence for scattering amplitudes in N = 4 super Yang Mills theory (SYM). Simultaneously, cluster algebras, originally introduced by Fomin and Zelevinsky to study total positivity, have been revealed to have a crucial role in describing singularities of N = 4 SYM scattering amplitudes. Thus, one can use ideas from quantum field theory (QFT) to connect cluster algebras to positive geometries, and in particular to the amplituhedron. Additionally, QFT can also be used to discover new examples of positive geometries. The conference will bring together a wide range of mathematicians and physicists both to draw new connections within algebraic combinatorics and geometry and to advance our physical understanding of scattering amplitudes and QFT.

The conference features: Introductory Lectures, an Open Problems Forum, Emerging Scholars Talks, and talks by experts in the fields.

The conference aims to actively engage and empower junior researchers and women, ensuring their integral presence and impactful contributions.

Directions to CMSA

Register Online for in-person talks

Register for Zoom Meeting

Confirmed Speakers:

• Evgeniya Akhmedova, Weizmann Institute of Science
• Nima Arkani-Hamed, IAS
• Paolo Benincasa, MPI
• Nick Early, Weizmann Institute of Science
• Carolina Figueiredo, Princeton University
• Yu-tin Huang, National Taiwan University
• Dani Kaufman, University of Copenhagen
• Chia-Kai Kuo, National Taiwan University
• Thomas Lam, University of Michigan
• Yelena Mandelshtam, UC Berkeley
• Shruti Paranjape, UC Davis
• Elizabeth Pratt, UC Berkeley
• Lecheng Ren, Brown University
• Sebastian Seemann, KU Leuven
• Melissa Sherman-Bennett, MIT & UC Davis
• Marcus Spradlin, Brown University
• Ran Tessler, Weizmann Institute of Science
• Hugh Thomas, Université du Québec à Montréal
• Jaroslav Trnka, UC Davis
• Anastasia Volovich, Brown University

Organizers:

• Matteo Parisi, Harvard CMSA
• Lauren Williams, Harvard Mathematics

This event will be co-funded by the National Science Foundation.

Limited funding to help defray travel expenses is available for graduate students and recent PhDs. If you are a graduate student or postdoc and would like to apply for support, please register above and send an email to amplituhedra@cmsa.fas.harvard.edu no later than Friday, April 19, 2024.

Please include your name, address, current status, university affiliation, citizenship, and area of study. F1 visa holders are eligible to apply for support. If you are a graduate student, please send a brief letter of recommendation from a faculty member to explain the relevance of the conference to your studies or research. If you are a postdoc, please include a copy of your CV.