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Thomas Bartsch

Publications and source records attributed to Thomas Bartsch.

At least 19 recordsLinked to original sources
arXiv2026

Beyond Wigner: Non-Invertible Symmetries Preserve Probabilities

In recent years, the traditional notion of symmetry in quantum theory was expanded to so-called generalised or categorical symmetries, which, unlike ordinary group symmetries, may be non-invertible. This appears to be at odds with Wigner's theorem, which requires quantum symmetries to be implemented by (anti)unitary -- and hence invertible -- operators in order to preserve probabilities. We resolve this puzzle for (higher) fusion category symmetries $\mathcal{C}$ by proposing that, instead of acting by unitary operators on a fixed Hilbert space, symmetry defects in $\mathcal{C}$ act as isometries between distinct Hilbert spaces constructed from twisted sectors. As a result, we find that non-invertible symmetries naturally act as trace-preserving quantum channels. Crucially, our construction relies on the symmetry category $\mathcal{C}$ being unitary. We illustrate our proposal through several examples that include Tambara-Yamagami, Fibonacci, and Yang-Lee as well as higher categorical symmetries.

quant-ph↗
arXiv2025

Unitary Categorical Symmetries

Global invertible symmetries act unitarily on local observables or states of a quantum system. In this note, we aim to generalise this statement to non-invertible symmetries by considering unitary actions of higher fusion category symmetries $\mathcal{C}$ on twisted sector local operators. We propose that the latter transform in $\ast$-representations of the tube algebra associated to $\mathcal{C}$, which we introduce and classify using the notion of higher $S$-matrices of higher braided fusion categories.

hep-th↗
arXiv2024

On Unitary 2-Group Symmetries

Global internal symmetries act unitarily on local observables or states of a quantum system. In this note, we aim to generalise this statement to extended observables by considering unitary actions of finite global 2-group symmetries $\mathcal{G}$ on line operators. We propose that the latter transform in unitary 2-representations of $\mathcal{G}$, which we classify up to unitary equivalence. Our results recover the known classification of ordinary 2-representations of finite 2-groups, but provide additional data interpreted as a type of reflection anomaly for $\mathcal{G}$.

math-ph↗
arXiv2024

Blow-up solutions for the steady state of the Keller-Segel system on Riemann surfaces

We study the following Neumann boundary problem related to the stationary solutions of the Keller-Segel system, a basic model of chemotaxis phenomena: \[ \left\{\begin{array}{ll} -\Delta_g u +\beta u =\lambda\left(\frac{Ve^u}{\int_{\Sigma} Ve^u d v_g}-1\right), &\text { in } \mathring\Sigma\\ \partial_{ \nu_g} u=0, &\text { on } \partial \Sigma \end{array} \right.,\] on a compact Riemann surface $(\Sigma, g)$ of unit area, with interior $\mathring\Sigma$ and smooth boundary $\partial \Sigma$. Here, $\Delta_g$ denote the Laplace-Beltrami operator, $dv_g$ the area element of $(\Sigma, g)$, and $\nu_g$ the unit outward normal to $\partial \Sigma$ and $\lambda$ and $\beta$ are non-negative parameters, $V$ is non-negative with finite zero set. For any integers $m>0$ and $k,l\geq 0$ with $m=2k+l$, we establish a sufficient condition on $V$ for the existence of a sequence of blow-up solutions as $\lambda$ approaches the critical values $4\pi m$, which blows up at $k$ points in the interior and $l$ points on the boundary. Moreover, the study expands to the corresponding singular problem.

math.AP↗
arXiv2024

Partial Blow-up Phenomena in the $SU(3)$ Toda System on Riemann Surfaces

This work studies the partial blow-up phenomena for the $SU(3)$ Toda system on compact Riemann surfaces with smooth boundary. We consider the following coupled Liouville system with Neumann boundary conditions: $$ -\Delta_g u_1 = 2\rho_1\left( \frac{V_1 e^{u_1}}{\int_{\Sigma} V_1 e^{u_1} \, dv_g} - \frac 1 {|\Sigma|_g}\right) - \rho_2\left( \frac{V_2 e^{u_2}}{\int_{\Sigma} V_2 e^{u_2} \, dv_g} - \frac{1}{|\Sigma|_g}\right) \text{in} \,\mathring\Sigma$$ and $$ -\Delta_g u_2 = 2\rho_2\left( \frac{V_2 e^{u_2}}{\int_{\Sigma} V_2 e^{u_2} \, dv_g} - \frac{1}{|\Sigma|_g}\right) - \rho_1\left( \frac{V_1 e^{u_1}}{\int_{\Sigma} V_1 e^{u_1} \, dv_g} - \frac{1}{|\Sigma|_g}\right) \text{in} \,\mathring\Sigma$$ with boundary conditions $ \partial_{\nu_g} u_1 = \partial_{\nu_g} u_2 = 0 \text{ on} \, \partial \Sigma,$ where $(\Sigma, g)$ is a compact Riemann surface with the interior $\mathring\Sigma$ and smooth boundary $\partial\Sigma$, $\rho_i$ is a non-negative parameter and $V_i$ is a smooth positive function for $i=1,2$. We construct a family of blow-up solutions via the Lyapunov-Schmidt reduction and variational methods, wherein one component remains uniformly bounded from above, while the other exhibits partial blow-ups at a prescribed number of points, both in the interior and on the boundary. This construction is based on the existence of a non-degeneracy solution of a so-called shadow system. Moreover, we establish the existence of partial blow-up solutions in three cases: (i) for any $\rho_2>0$ sufficiently small; (ii) for generic $V_1, V_2$ and any $\rho_2\in (0,2\pi)$; (iii) for generic $V_1, V_2$, the Euler characteristic $\chi(\Sigma)<1$ and any $\rho_2\in (2\pi,+\infty)\setminus 2\pi \mathbb{N}_+$.

math.AP↗
arXiv2024

Blow-up solutions for mean field equations with Neumann boundary conditions on Riemann surfaces

On a compact Riemann surface $(\Sigma, g)$ with a smooth boundary $\partial \Sigma$, we consider the following mean field equations with Neumann boundary conditions: $$ -\Delta_g u = \lambda \left(\frac{Ve^u}{\int_{\Sigma} Ve^u \, dv_g} - \frac{1}{|\Sigma|_g}\right) \text{ in } \Sigma \text{ with } \partial_{\nu_g} u = 0 \text{ on } \partial \Sigma, $$ We find conditions on the potential function $V: \Sigma \to \mathbb{R}^+$ such that solutions exist for the parameter $\lambda$ when it is in a small right (or left) neighborhood of a critical value $4\pi(m+k)$ for $k \leq m \in \mathbb{N}_+$ and blow up as $\lambda$ approaches the critical parameter. The blow-up occurs exactly at $k$ points in the interior of $\Sigma$ and $(m-k)$ points on the boundary $\partial \Sigma$.

math.AP↗
arXiv2024

The Morse property of limit functions appearing in mean field equations on surfaces with boundary

In this paper we study the Morse property for functions related to limit functions of mean field equations on a smooth, compact surface $\Sigma$ with boundary $\partial\Sigma$. Given a Riemannian metric $g$ on $\Sigma$ we consider functions of the form \[ f_g(x) := \sum_{i=1}^m\sigma_i^2R^g(x_i)+\sum_{i,j=1\\i\ne j}^m\sigma_i\sigma_jG^g(x_i,x_j)+h(x_1,\ldots,x_m), \] where $\sigma_i \neq 0$ for $i=1,\ldots,m$, $G^g$ is the Green function of the Laplace-Beltrami operator on $(\Sigma,g)$ with Neumann boundary conditions, $R^g$ is the corresponding Robin function, and $h \in \mathcal{C}^{2}(\Sigma^m,\mathbb{R})$ is arbitrary. We prove that for any Riemannian metric $g$, there exists a metric $\widetilde g$ which is arbitrarily close to $g$ and in the conformal class of $g$ such that $f_{\widetilde g}$ is a Morse function. Furthermore we show that, if all $\sigma_i>0$, then the set of Riemannian metrics for which $f_g$ is a Morse function is open and dense in the set of all Riemannian metrics.

math.DG↗
arXiv2023

Normalized solutions to Sch\"odinger equations with potential and inhomogeneous nonlinearities on large convex domains

The paper addresses an open problem raised in [Bartsch, Molle, Rizzi, Verzini: Normalized solutions of mass supercritical Schr\"odinger equations with potential, Comm. Part. Diff. Equ. 46 (2021), 1729-1756] on the existence of normalized solutions to Schr\"odinger equations with potentials and inhomogeneous nonlinearities. We consider the problem \[ -\Delta u+V(x)u+\lambda u = |u|^{q-2}u+\beta |u|^{p-2}u, \quad \|u\|^2_2=\int|u|^2dx = \alpha, \] both on $\mathbb{R}^N$ as well as on domains $r\Omega$ where $\Omega\subset\mathbb{R}^N$ is an open bounded convex domain and $r>0$ is large. The exponents satisfy $2<p<2+\frac4N<q<2^*=\frac{2N}{N-2}$, so that the nonlinearity is a combination of a mass subcritical and a mass supercritical term. Due to the presence of the potential a by now standard approach based on the Pohozaev identity cannot be used. We develop a robust method to study the existence of normalized solutions of nonlinear Schr\"odinger equations with potential and find conditions on $V$ so that normalized solutions exist. Our results are new even in the case $\beta=0$.

math.AP↗
arXiv2023

Representation theory for categorical symmetries

This paper addresses the question of how categorical symmetries act on extended operators in quantum field theory. Building on recent results in two dimensions, we introduce higher tube categories and algebras associated to higher fusion category symmetries. We show that twisted sector extended operators transform in higher representations of higher tube algebras and interpret this result from the perspective of the sandwich construction of finite symmetries via the Drinfeld center. Focusing on three dimensions, we discuss a variety of examples to illustrate the general constructions. In the case of invertible symmetries, we show that higher tube algebras are higher analogues of twisted Drinfeld doubles of finite groups, generalising known constructions in two dimensions. Building on this foundation, we discuss non-invertible Ising-like symmetry categories obtained by gauging finite subgroups. We also consider non-invertible topological symmetry lines described by braided fusion categories and discuss connections to the M\"uger center and braided module categories.

hep-th↗
arXiv2023

Higher representations for extended operators

It is known that local operators in quantum field theory transform in representations of ordinary global symmetry groups. The purpose of this paper is to generalise this statement to extended operators such as line and surface defects. We explain that $(n-1)$-dimensional operators transform in $n$-representations of a finite invertible or group-like symmetry and thoroughly explore this statement for $n = 1,2,3$. We therefore propose higher representation theory as the natural framework to describe the action of symmetries on the extended operator content in quantum field theory.

hep-th↗
arXiv2023

Equilibria of vortex type Hamiltonians on closed surfaces

We prove the existence of critical points of vortex type Hamiltonians \[ H(p_1,\ldots, p_N) = \sum_{{i,j=1},{i\ne j}}^N Γ_iΓ_jG(p_i,p_j)+ψ(p_1,\dots,p_N) \] on a closed Riemannian surface $(Σ,g)$ which is not homeomorphic to the sphere or the projective plane. Here $G$ denotes the Green function of the Laplace-Beltrami operator in $Σ$, $ψ:Σ^N\to\mathbb{R}$ may be any function of class $C^1$, and $Γ_1,\dots,Γ_N\in\mathbb{R}\setminus\{0\}$ are the vorticities. The Kirchhoff-Routh Hamiltonian from fluid dynamics corresponds to $ψ= -\sum_{i=1}^N Γ_i^2h(p_i,p_i)$ where $h:Σ\timesΣ\to\mathbb{R}$ is the regular part of the Laplace-Beltrami operator. We obtain critical points $p=(p_1,\dots,p_N)$ for arbitrary $N$ and vorticities $(Γ_1,\dots,Γ_N)$ in $\mathbb{R}^N\setminus V$ where $V$ is an explicitly given algebraic variety of codimension 1.

math.AP↗
arXiv2023

Curvature effect in the spinorial Yamabe problem on product manifolds

Let $(M_1,\textit{g}^{(1)})$, $(M_2,\textit{g}^{(2)})$ be closed Riemannian spin manifolds. We study the existence of solutions of the spinorial Yamabe problem on the product $M_1\times M_2$ equipped with a family of metrics $\varepsilon^{-2}\textit{g}^{(1)}\oplus\textit{g}^{(2)}$, $\varepsilon>0$. Via variational methods and blow-up techniques, we prove the existence of solutions which depend only on the factor $M_1$, and which exhibit a spike layer as $\varepsilon\to0$. Moreover, we locate the asymptotic position of the peak points of the solutions in terms of the curvature tensor on $(M_1,\textit{g}^{(1)})$.

math.DG↗
arXiv2023

Nodal bubble tower solutions to slightly subcritical elliptic problems with Hardy terms

We study the possible blow-up behavior of solutions to the slightly subcritical elliptic problem with Hardy term \[ \left\{ \begin{aligned} -Δu-μ\frac{u}{|x|^2} &= |u|^{2^{\ast}-2-\varepsilon}u &&\quad \text{in } Ω, \\\ u &= 0&&\quad \text{on } \partialΩ, \end{aligned} \right. \] in a bounded domain $Ω\subset\mathbb{R}^N (N\ge7)$ with $0\inΩ$, as $μ,\varepsilon\to 0^+$. In \cite{BarGuo-ANS}, we obtained the existence of nodal solutions that blow up positively at the origin and negatively at a different point as $μ=O(ε^α)$ with $α>\frac{N-4}{N-2}$, $\varepsilon\to 0^+$. Here we prove the existence of nodal bubble tower solutions, i.e.\ superpositions of bubbles of different signs, all blowing up at the origin but with different blow-up order, as $μ=O(\varepsilon)$, $\varepsilon\to0^+$.

math.AP↗
arXiv2023

Multi-bubble nodal solutions to slightly subcritical elliptic problems with Hardy terms in symmetric domains

We consider the slightly subcritical elliptic problem with Hardy term $$ \left\{ \begin{aligned} -Δu-μ\frac{u}{|x|^2} &= |u|^{2^{\ast}-2-ε}u &&\quad \text{in } Ω\subset\mathbb{R}^N, \\\ u &= 0&&\quad \text{on } \partial Ω, \end{aligned} \right. $$ where $0\inΩ$ and $Ω$ is invariant under the subgroup $SO(2)\times\{\pm E_{N-2}\}\subset O(N)$; here $E_n$ denots the $n\times n$ identity matrix. If $μ=μ_0ε^α$ with $μ_0>0$ fixed and $α>\frac{N-4}{N-2}$ the existence of nodal solutions that blow up, as $ε\to0^+$, positively at the origin and negatively at a different point in a general bounded domain has been proved in \cite{BarGuo-ANS}. Solutions with more than two blow-up points have not been found so far. In the present paper we obtain the existence of nodal solutions with a positive blow-up point at the origin and $k=2$ or $k=3$ negative blow-up points placed symmetrically in $Ω\cap(\mathbb{R}^2\times\{0\})$ around the origin provided a certain function $f_k:\mathbb{R}^+\times\mathbb{R}^+\times I\to\mathbb{R}$ has stable critical points; here $I=\{t>0:(t,0,\dots,0)\inΩ\}$. If $Ω=B(0,1)\subset\mathbb{R}^N$ is the unit ball centered at the origin we obtain two solutions for $k=2$ and $N\ge7$, or $k=3$ and $N$ large. The result is optimal in the sense that for $Ω=B(0,1)$ there cannot exist solutions with a positive blow-up point at the origin and four negative blow-up points placed on the vertices of a square centered at the origin. Surprisingly there do exist solutions on $Ω=B(0,1)$ with a positive blow-up point at the origin and four blow-up points on the vertices of a square with alternating positive and negative signs. The results of our paper show that the structure of the set of blow-up solutions of the above problem offers fascinating features and is not well understood.

math.AP↗
arXiv2022

Non-invertible Symmetries and Higher Representation Theory II

In this paper we continue our investigation of the global categorical symmetries that arise when gauging finite higher groups and their higher subgroups with discrete torsion. The motivation is to provide a common perspective on the construction of non-invertible global symmetries in higher dimensions and a precise description of the associated symmetry categories. We propose that the symmetry categories obtained by gauging higher subgroups may be defined as higher group-theoretical fusion categories, which are built from the projective higher representations of higher groups. As concrete applications we provide a unified description of the symmetry categories of gauge theories in three and four dimensions based on the Lie algebra $\mathfrak{so}(N)$, and a fully categorical description of non-invertible symmetries obtained by gauging a 1-form symmetry with a mixed 't Hooft anomaly. We also discuss the effect of discrete torsion on symmetry categories, based a series of obstructions determined by spectral sequence arguments.

hep-th↗
arXiv2022

Non-invertible Symmetries and Higher Representation Theory I

The purpose of this paper is to investigate the global categorical symmetries that arise when gauging finite higher groups in three or more dimensions. The motivation is to provide a common perspective on constructions of non-invertible global symmetries in higher dimensions and a precise description of the associated symmetry categories. This paper focusses on gauging finite groups and split 2-groups in three dimensions. In addition to topological Wilson lines, we show that this generates a rich spectrum of topological surface defects labelled by 2-representations and explain their connection to condensation defects for Wilson lines. We derive various properties of the topological defects and show that the associated symmetry category is the fusion 2-category of 2-representations. This allows us to determine the full symmetry categories of certain gauge theories with disconnected gauge groups. A subsequent paper will examine gauging more general higher groups in higher dimensions.

hep-th↗
arXiv2022

Existence and asymptotic behavior of normalized ground states for Sobolev critical Schrödinger systems

The paper is concerned with the existence and asymptotic properties of normalized ground states of the following nonlinear Schrödinger system with critical exponent: \begin{equation*} \left\{\begin{aligned} &-δu+λ_1 u=|u|^{2^*-2}u+{να} |u|^{α-2}|v|^βu,\quad \text{in }\mathbb{R}^N, &-δv+λ_2 v=|v|^{2^*-2}v+{νβ} |u|^α|v|^{β-2}v,\quad \text{in }\mathbb{R}^N, &\int u^2=a^2,\;\;\; \int v^2=b^2, \end{aligned} \right. \end{equation*} where $N=3,4$, $α,β>1$, $2<α+β<2^*=\frac{2N}{N-2}$. We prove that a normalized ground state does not exist for $ν<0$. When $ν>0$ and $α+β\le 2+\frac{4}{N}$, we show that the system has a normalized ground state solution for $0<ν<ν_0$, the constant $ν_0$ will be explicitly given. In the case $α+β>2+\frac{4}{N}$ we prove the existence of a threshold $ν_1\ge 0$ such that a normalized ground state solution exists for $ν>ν_1$, and does not exist for $ν<ν_1$. We also give conditions for $ν_1=0$. Finally we obtain the asymptotic behavior of the minimizers as $ν\to0^+$ or $ν\to+\infty$.

math.AP↗
arXiv2020

Normalized solutions of mass supercritical Schrödinger equations with potential

This paper is concerned with the existence of normalized solutions of the nonlinear Schrödinger equation \[ -Δu+V(x)u+λu = |u|^{p-2}u \qquad\text{in $\mathbb{R}^N$} \] in the mass supercritical and Sobolev subcritical case $2+\frac{4}{N}<p<2^*$. We prove the existence of a solution $(u,λ)\in H^1(\mathbb{R}^N)\times\mathbb{R}^+$ with prescribed $L^2$-norm $\|u\|_2=ρ$ under various conditions on the potential $V:\mathbb{R}^N\to\mathbb{R}$, positive and vanishing at infinity, including potentials with singularities. The proof is based on a new min-max argument.

math.AP↗