Speaker: Guglielmo Lockhart (University of Bonn).Abstract: General principles suggest that the spectrum of excitations of quantum gravity theories includes not just pointlike particles, but also higher-dimensional membranes. In this talk I will focus on a specific class of one-dimensional membranes, the faithful strings, whose properties can be described using the language of vertex operator algebras. I will discuss how this mathematical structure encodes symmetries of quantum gravity theories, and makes it possible to determine refined information like gauge group topology. Moreover, I will show that modular invariance of the faithful string torus partition function is a powerful tool to detect inconsistencies in a candidate quantum gravity theory, or conversely to identify apparently consistent quantum gravities and study their properties. This provides an exciting opportunity to study exotic quantum gravity models that do not admit any known weakly coupled realization within string theory, in supersymmetric as well as non-supersymmetric settings.
Speaker: Sanjay Moudgalya (Tata Institute of Fundamental Research)Symmetries in quantum many-body systems are typically characterized by their algebraic structure. In this talk, I will show that for continuous symmetries such as U(1) and SU(2), there are underlying quantum geometric structures that play central roles in many-body dynamics. We demonstrate this in noisy Brownian models of unitary quantum dynamics, focusing on entanglement entropies and correlation functions. By mapping the disorder-averaged late-time dynamics to the low-energy physics of effective replica Hamiltonians, we show that the dynamics is governed by the quantum geometry of their ground-state manifolds. This geometry is, in turn, directly related to that of the k-commutants—the symmetry algebras acting on k replicas of the system—and is independent of microscopic details of the noisy evolution. This perspective enables simple geometric derivations, based on the time-dependent variational principle (TDVP), of several characteristic dynamical phenomena, including sub-ballistic Rényi entanglement growth and the anomalous decay of non-hydrodynamic correlators in interacting systems with continuous symmetries. We compare these behaviors across interacting systems with Abelian and non-Abelian continuous symmetries, as well as free-fermion systems, whose distinct dynamics can be traced to differences in the geometry of their k-commutants. More broadly, these results establish a geometric framework for systematically understanding observables and universal dynamical behavior in noisy quantum systems with continuous symmetries.
Speaker: Anna-Laura Sattelberger (Max Planck Institute for Mathematics in the Sciences)Abstract:Holonomic functions are omnipresent in the sciences. These are functions that are solutions to sufficiently large systems of linear partial differential equations with polynomial coefficients. As such, they can be tackled by algebraic analysis: holonomic functions can be encoded by left ideals in the Weyl algebra, encoding annihilating differential operators. Examples include A-hypergeometric functions, Feynman integrals as well as some probability distributions, just to name a few. In this talk, I explain how to modify and read fundamental properties of holonomic functions in terms of algebro-geometric methods.The second part of the talk is based on joint work with Carlos Rodriguez. Motivated from the theory of Hilbert schemes of points, we transfer the concept of border bases from zero-dimensional ideals in polynomial rings to the rational Weyl algebra in order to encode holonomic ideals. As an application, we visit differential equations behind a Feynman integral in dimensional regularization.
Speaker: Ludwig Zweng (Technische Universität München)Recent works have developed a unifying framework for classifying ground state phases of matter with a wide range of discrete symmetries—both group-like and non-invertible symmetries—via a holographic correspondence between symmetric operators in d dimensions and topological operators in a d+1 dimensional system often known as the SymTO or SymTFT. Its lattice formulation naturally yields fixed-point commuting Hamiltonians that realize these phases in their ground state. In this work, we extend this framework to disordered and non-equilibrium settings by analyzing excited states of the disordered versions of these fixed-point models with non-Abelian group symmetries. Characterizing symmetry breaking patterns in the excited states requires new tools and diagnostic methods, which we develop in the language of bond algebras and their commutants, and which resolve whether the non-commutativity obstructing eigenstate order is local or merely global. We find that the disordered phase structure can differ dramatically from its clean ground state counterpart: distinct clean ground state phases may merge, a single one may split, and orders with no ground state analogue can appear. We illustrate these phenomena by constructing lattice models for S₃, D₈, and S₄, classifying the distinct excited state phases in each case, and showing that the answer depends on the local Hilbert space and not on the symmetry group alone. Finally, we argue that the SymTFTs that capture ground state phases do not fully capture the intricacies of the excited states.
Speaker Minghuan Hu (Centre for Quantum Mathematics)Abstract: Given a toric surface, we count rational nodal curves passing through fixed points on the toric boundary, which can be realized as a log Gromov-Witten invariant. Following the idea of Yau-Zaslow, this count coincides with the Euler number of the moduli space of a family of compactified Jacobians. Using homological mirror symmetry, we identify this space with the moduli space of constructible sheaves microsupported in a Legendrian link. We then show that this moduli space admits a stratification called ``ruling decomposition'', and thereby computes its Euler number using combinatorial data. We conjecture that the ruling decomposition also recovers higher-genus invariants. This is a joint work with Tom Graber and Eric Zaslow.
Speaker: Jonah Baerman (University of Hamburg).Integrable models are omnipresent in theoretical physics, and can often be related to the study of supersymmetric gauge theories, where they can provide unprecedented access to nonperturbative information. For example, the low-energy behavior of certain 4d N=2 supersymmetric gauge theories is known to be described by so-called Hitchin systems, associated to a choice of Riemann surface Σ. By gauge theory/conformal field theory arguments, such models are expected to admit a dual description in terms of certain differential operators on Σ. We demonstrate this correspondence for the case of the twice-punctured sphere (the Toda chain), and find a geometric description of its quantization conditions. This allows us to apply integrability techniques, such as TBA equations, to the study of these differential operators, and vice versa. Lastly, we interpret our results as a real variant of the Analytic Langlands Correspondence. Based on past and ongoing work with Alba Grassi, Giovanni Ravazzini, and Jörg Teschner.
The Niels Bohr Quantum Summer School is a national initiative under the Danish National Quantum Strategy, bringing together Danish and international PhD students for an intensive two-week programme in quantum science and technology.This year’s edition focuses specifically on Quantum Algorithms and Quantum Software, discussing the quantum computational model, quantum algorithm design, and quantum software compilation and optimization, including the principles from quantum mathematics and computer science that underpin these topics.Venue: DIAS Auditorium
Speaker: Kasper Green Larsen (Aarhus University).Abstract: Achieving a provable exponential quantum speedup for an important machine learning task has been a central research goal since the seminal HHL quantum algorithm for solving linear systems and the subsequent quantum recommender systems algorithm by Kerenidis and Prakash. These algorithms were initially believed to be strong candidates for exponential speedups, but a lower bound ruling out similar classical improvements remained absent. In breakthrough work by Tang, it was demonstrated that this lack of progress in classical lower bounds was for good reasons. Concretely, she gave a classical counterpart of the quantum recommender systems algorithm, reducing the quantum advantage to a mere polynomial. Her approach is quite general and was named quantum-inspired classical algorithms. Since then, almost all the initially exponential quantum machine learning speedups have been reduced to polynomial via new quantum-inspired classical algorithms. From the current state-of-affairs, it is unclear whether we can hope for exponential quantum speedups for any natural machine learning task. In this work, we present the first such provable exponential separation between quantum and quantum-inspired classical algorithms. We prove the separation for the basic problem of solving a linear system when the input matrix is well-conditioned and has sparse rows and columns.
Speaker: Patrick Kinnear (University of Hamburg).Abstract: In the study of topological symmetry, defects are embedded submanifolds labelled by algebraic data; the evaluation of defect TQFTs (i.e. TQFTs which make sense for bordisms supporting defects) implements topological symmetry. The (2+1) TQFT of Reshetikhin and Turaev was enhanced to a defect TQFT in work of Carqueville--Runkel--Schaumann. This enhancement is a generalized state sum construction, and requires that the defects are labelled by semisimple data; the state spaces for surfaces are given by a projector from the state space of the original RT theory. On the other hand, given a 3-manifold whose boundary is a surface with defects, the usual definitions of skein theory can be extended to define a defect skein module. In this talk I will report on joint work with Ingo Runkel, where we define the defect skein module and prove that it is isomorphic to the defect RT state space of its boundary. This generalizes the well-known fact that the state spaces of RT theory are skein modules. Our work points to non-semisimple generalizations of defect RT theories using skein methods, while in the other direction it gives a state sum construction for defect skein modules labelled by semisimple data.
Speaker: Gerard Valenti-Rojas (Atom Computing).Abstract: The accumulation of errors in quantum computers limits the execution of powerful algorithms. As a result, current devices require Quantum Error Correction (QEC) to operate reliably in the presence of noise — a paradigm known as Fault-Tolerant Quantum Computation (FTQC).In this talk, I will introduce the principles of QEC, explain why it is essential, and illustrate how it can be implemented in practice on a neutral-atom quantum computing platform. This is meant to give the audience an intuitive sense of what it means to run a fault-tolerant quantum algorithm on real hardware. If time permits, I will discuss some promising quantum low-density parity-check (qLDPC) code constructions and the new opportunities they may offer for scalable fault tolerance.By the end of the talk, participants should be familiar with key concepts such as encoding circuits, logical gate implementation, syndrome extraction, as well as the practical challenges involved in realising these techniques experimentally.
Speaker: Steffen Schmidt (University of Southern Denmark).Abstract: Symmetry serves as a guiding principle in mathematics and physics. The language of symmetry groups and their actionsorganizes a wide range of phenomena. In this setting, Dirac operators form a natural bridge between quantum mechanics, geometry, and representation theory. In representation theory, Dirac operators provide conceptual and computational tools for addressing basic problems concerning a (semisimple) Lie group: they furnish explicit constructions, give effective criteria for unitarity, contribute to classification results, and relate the topology of locally symmetric spaces to representation-theoretic data. A foundational instance is Parthasarathy's use of a representation-theoretic Dirac operator in the study of discrete series representations. Interest in this circle of ideas intensified in the late 1990s, following Kostant's introduction of the cubic Dirac operator and the subsequent development of \emph{Dirac cohomology}, including Huang–Pandžić’s proofof Vogan's conjecture. In this talk, I present a modern construction of (algebraic) Dirac operators and discuss applications of Dirac operators and Dirac cohomology to representations of Lie superalgebras, including localization principles, relations with Duflo–Serganova cohomology, and character formulas.
Speaker: Ben Brown (IBM).Abstract: In order to scale a quantum computer to solve difficult problems, not only do we need to produce a quantum memory that protect quantum information from noise, but we also need to perform logical operations (logic gates) to manipulate the protected states. Like the memory, these logic gates must also protect their encoded quantum information as gates are performed. A number of the major breakthroughs in the field of error-corrected logic gates have been made through building connections between quantum error-correcting codes and topological phases of matter. Indeed, the surface code is central to this connection, and this model has captured the imagination of the entire error-correction community. In this talk I will explain how error-corrected logic is performed with the surface code by introducing lattice dislocations with behaviour that is analogous to anyonic quasiparticle excitations known as Majorana fermions. I will also explain how we can use transformations between different topological phases of matter via gauging operations to create magic states; an essential component in a logic gate set. I will mention experimental progress towards the realisations of these gates, and I will look ahead to describe how these ideas are being adapted for quantum computing architectures with long-range interactions, that can potentially minimise the high resource cost of building a practical quantum computer.
Speaker: Olga Solodovnikova (University of Southern Denmark).Abstract: In this research seminar, I will discuss techniques for simulating continuous-variable circuits, which typically fall into two categories: phase-space simulation (Gaussian), and Fock-space simulation. I will present the methodology behind a CV circuit simulation library, which I developed during my PhD, which extends the Gaussian representation to systems that can be written as linear combinations of Gaussians. By approximating Fock states as superpositions of coherent states, any finite-energy CV state can be represented in this formalism. Furthermore, the complexity of this algorithm scales with the stellar rank of the non-Gaussian circuit elements. In the two manuscripts, I showcase the library by simulating and optimising GKP state preparation circuits in the presence of loss. Having said that, the formalism is applicable to any continuous-variable system. [1] Solodovnikova et al. “Fast Simulations of Continuous-Variable Circuits Using the Coherent State Decomposition.” arXiv:2508.06175 [quant-ph][2] Solodovnikova et al. “The Loss Tolerance of Cat Breeding for Fault-Tolerant Grid State Generation.” arXiv:2508.06193 [quant-ph]
Speaker: Olga Solodovnikova (University of Southern Denmark).Abstract: In this seminar, I will introduce continuous quantum variables [1], which are used to describe systems of oscillators, that is, systems with infinite Hilbert spaces such as photonic systems or the vibrational motion of trapped ions. To build a universal and fault-tolerant quantum computer with a continuous-variable system, it is necessary to encode a qubit in an oscillator, perform gates from the universal gate set, and measure stabilisers to implement error correction. In the second part of the introductory seminar, I will introduce the measurement-based quantum computing framework for optical systems, in which Gottesman-Kitaev-Preskill (GKP) states are injected into a cluster state, and homodyne/heterodyne detectors are used to i) apply gates via teleportation and ii) measure stabilisers [2]. Lastly, I will discuss the experimental progress on building a measurement-based quantum computer [3], where, in particular, the generation of GKP states [4] is a major bottleneck due to photon loss. [1] Braunstein and van Loock. “Quantum Information with Continuous Variables.” Rev. Mod. Phys. 77, no. 2 (2005).[2] Gottesman, et al. “Encoding a Qubit in an Oscillator.” Physical Review A 64, no. 1 (2001): 012310. [3] Aghaee Rad, H., T. Ainsworth, R. N. Alexander, et al. “Scaling and Networking a Modular Photonic Quantum Computer.” Nature 638, no. 8052 (2025): 912–19. [4] Larsen, M. V., J. E. Bourassa, S. Kocsis, et al. “Integrated Photonic Source of Gottesman–Kitaev–Preskill Qubits.” Nature 642, no. 8068 (2025): 587–91.
Speaker: Matthew Yu (University of Oxford)Abstract: I will explain a comprehensive framework for characterizing the infrared (IR) phases of a fermionic QFTs in (3+1)d, based on their quantum anomalies associated with a finite symmetry. We uncover a fundamental dichotomy among these anomalies: the first class of anomalies can always be realized by symmetric gapped states, while the second class can never be realized by gapped states without breaking the given symmetry, establishing the phenomenon of symmetry-enforced gaplessness in these settings. Using the construction of symmetry extension afforded to us by new developments in fusion 2-categories, we construct the candidate gapped states that theories with the first class of anomalies can flow to in the IR. As an application, I will provide examples of concrete predictions for the candidate IR phases of (3+1)d gauge theories based on our results.
Speaker: Sara Maggio (University of Bonn).Abstract: Feynman integrals associated with geometries beyond the Riemann sphere, such as elliptic and Calabi–Yau geometries, are playing an increasingly important role in modern precision calculations. They appear not only in collider cross sections, but also in gravitational-wave scattering. A powerful way to compute these integrals is through systems of differential equations, particularly when these can be brought into canonical form, making their singularity structure manifest. In this talk, I will present a method to construct canonical differential equations for multi-loop Feynman integrals beyond the polylogarithmic setting, with particular emphasis on Calabi–Yau examples. After introducing the method through the two-loop sunrise integral, I will show how it applies to the cutting-edge case of the three-loop multi-parameter banana integrals.
Speaker: Ryan Cullinan (University of York).Abstract: Integrable systems provide a pragmatic point in the landscape of physical theories. Such systems are characterised by their solvability, which is usually underpinned by the possession of an infinite-dimensional symmetry group, the existence of solitonic solutions, or admittance of an algebro-geometric description of the system.Given the abundance of models deemed ‘integrable’ in one of the aforementioned senses, it becomes essential to ask what universal characteristics these models share that enable their integrability. A systematic understanding of Lax integrability for integrable field theories in two dimensions was provided by the introduction of a topological holomorphic theory in four dimensions called four-dimensional Chern-Simons theory, which illuminates many of the superficially mysterious aspects of a system's integrability by describing the data of a two-dimensional integrable field theory in the language of four-dimensional gauge theory.In this talk, we will describe the refinements required to build on this paradigm by studying a topological-holomorphic theory in five dimensions called five-dimensional 2-Chern-Simons. The aforementioned theory is a 2-categorical generalisation of its four-dimensional counterpart, describing Lax integrable field theories in three dimensions via the medium of higher categorical gauge theory. Bringing together results from homotopical and homological algebra, Hodge theory, and the BV formalism, we will outline the subtleties in defining the theory and in attempting to chart this new landscape of 'higher Lax integrable' theories.
Speaker: Boris Kjær (University of Copenhagen).Abstract:In recent years, the analysis of topologically ordered ground states of 2d quantum lattice systems in the thermodynamic limit has been developed to provide a rigorous invariant of gapped phases. The methodology is based on the approach of Doplicher, Haag, and Roberts in the context of algebraic quantum field theory, and derives a braided C*-tensor category which represents the anyonic excitations above the ground state and the fusion and braiding among them. We present the computation of this category for the class of Levin-Wen models, which yields the Drinfeld center Z(C) of an arbitrary unitary fusion category C up to braided monoidal equivalence. Previous methods, applied to quantum double models based on finite groups, relied on defining string operators on the quasi-local observable algebra implementing an action of Z(Vec_G). But there are obstructions in K-theory that limit this approach to group-like models with integer quantum dimensions. Our approach requires detailed understanding of the local ground state spaces of the model in terms of topological quantum field theory based on skein modules. Based on work joint with Alex Bols.
Speaker: Birgit Kaufmann (Purdue University).Abstract: Quantum annealers represent an interesting approach to quantum computing, which make use of the adiabatic theorem to efficiently find the ground state of a physically realizable Hamiltonian. Such devices are currently commercially available and have been most successfully applied to several combinatorial and discrete optimization problems. However, the application of quantum annealers to problems in chemistry remains a relatively sparse area of research due to the difficulty in mapping molecular systems to the Ising Hamiltonian representing the architecture of the DWave quantum computer. In this talk, we review two different methods for finding the ground state of molecular Hamiltonians using Ising-model-based quantum annealers. In addition, we compare the relative effectiveness of each method by calculating the binding energies, bond lengths, and bond angles of the H3+ and H2O molecules and mapping their potential energy curves. We also assess the resource requirements of each method by determining the number of qubits and computation time required to simulate each molecule using various parameter values. While each of these methods is capable of accurately predicting the ground state properties of small molecules, we find that they are still outperformed by modern classical algorithms and that the scaling of the resource requirements remains a challenge.
Speaker: Frederik Nathan (University of Copenhagen).Abstract: The current main paradigm for quantum computing — active quantum error correction (AEC) with overhead qubits — faces two challenges: Realizing long-lived logical qubits at reasonable resource cost, and achieving universal fault-tolerant quantum gates. At the root of these challenges lie two no-go theorems: The Braviy-Terhal theorems, which forbids passively-stabilized, or self-correcting, qubits in <4 dimensions, and the Eastin-Knill theorem which dictates that fault-tolerant non-Clifford gates must be generated with costly magic distillation or cultivation routines. Together, these theorems imply that even simple quantum information processing operations require continuous readout, data processing, and feedback control of very large numbers of overhead qubits. Here I illustrate how the no-go-theorems above are not as far-reaching as one could fear, but can be circumvented via an alternative paradigm for quantum computing currently gaining experimental maturity: bosonic codes (BC), which redundantly encodes a qubit in the infinite-dimensional Fock state of a single continuous variable. In particular, I will demonstrate how tuning the impedance of a simple driven-dissipative superconducting resonator to a “magic” value given by a constant of nature can lead to a self-correcting Gottesman-Kitaev-Preskill qubit that supports exponentially-robust Clifford and non-Clifford gates [1,2]. [1] FN, L. O’Brien, K. Noh, M.H. Matheny, A.L. Grimsmo, L. Jiang, G. RefaelPRX Quantum 6 (3), 030352 (2025)[2] L. O'Brien, G. Refael, FN arXiv:2507.19713 (2025)
Speaker: Bangxin Wang (University of Hamburg)Abstract: We introduce the notion of .-crossed ribbon categories, motivated by constructions in TQFT. There are two well-known non-semisimple 3d TQFTs: BCGP and DGGPR. The BCGP theory assigns invariants to triples (M, T, w), where M is a 3-manifold, T c M is an embedded ribbon graph, and w e H1(M\T;G). The DGGPR construction produces invariants for pairs (M, T). A natural question is whether one can enrich the DGGPR construction by including cohomological data. This leads to the theory of homotopy QFT whose algebraic inputs are G-crossed ribbon categories, expected to recover the BCGP theory. This is an ongoing project with A. Gainutdinov, N. Geer, B. Patureau, and I. Runkel. We generalize the notion of a G-crossed ribbon category by allowing the grading group G and the action group H to be distinct. To make sense of a crossed braiding, it turns out that the pair (G,H) should form a crossed module x. We present two constructions of such categories and prove several classification results. This part is joint work with A. Gainutdinov and I. Runkel.
Speaker: Ingo Runkel (University of Hamburg)Abstract:Given a suitable ribbon category C and a so-called orbifold datum A in C, one can obtain a new ribbon category C_A. This construction subsumes several well-known examples, such as the Drinfeld centre of a spherical fusion category, passing to the category of local modules of a commutative algebra in C, and equivariantisation of a G-crossed ribbon category. In a 3d TFT Z which has C as category of line defects, A can be interpreted as a "gaugeable (possibly non-invertible) symmetry", and C_A as the line defects in the gauged theory Z/A. The state spaces of Z/A can be described in terms of commuting projector Hamiltonians acting on state spaces of Z, giving a variant of the Levin-Wen construction.This is joint work with N. Carqueville, V. Mulevicius, G. Schaumann, D. Scherl, and T. Voß.
Speaker: Adela Zhang (University of Copenhagen)Abstract:With Barkan and Steinebrunner, we show that any open field theory extends canonically to an open-closed field theory whose value at the circle is the Hochschild homology object of the E_1-Frobenius algebras (i.e. E_1-Calabi-Yau objects) A associated to F. In particular, we show that the open-closed bordism category is obtained by formally adjoining iterated Hochchild homology to the open bordism category. As a corollary, we determine that the moduli spaces of surfaces is the space of universal formal operations on the Hochschild homology object of E_1-Frobenius algebras. This provides a space level refinement of previous work of Costello (over Q) and Wahl (over Z).Time permitting, I will also explain work with Andrea Bianchi on canonical local to global extensions of graph TFTs associated with E_\infty-Frobenius objects, i.e. in dimension infinity. In this case, we show that the graph cobordism category is obtained by formally adjoining to the \infty-category of graphs the factorization homology of the universal E_\infty-Frobenius algebra over any sapce X.
Speaker: Albert H. WernerUniversity of CopenhagenAbstract:Our ability to benchmark, verify, and control quantum devices ultimately rests on how well we understand their underlying dynamics. These dynamics are shaped not only by ideal Hamiltonian evolution, but also by time dependent effects and noise that arise in realistic settings. This talk will survey ideas for extracting meaningful dynamical descriptions of many qubit systems using simple, local measurements. Emphasis will be placed on how fundamental notions of locality limit the spread of information and make large scale quantum dynamics accessible through local data. The goal is to outline a set of ideas and methods that contribute to ongoing efforts to understand and characterize the dynamics of complex quantum devices.Location: D-IAS Aud. (V24-501a-0), Danish Institute for Advanced Study - DIASThe event is open to allCrémant and petit fours will be served after the talk
Speaker: Guillaume Laplante-Anfossi (University of Southern Denmark)Abstract:The goal of this talk is to describe a strategy to compute algorithmically unstable homotopy groups of spheres modulo a prime number, using the theory of A-infinity algebras. This approach, which is based on Brown's theory of twisting cochains and its generalisation to A-infinity algebras by Kadeishvili, was proposed by Alain Prouté in 1986. One central ingredient that was missing at the time is a formula for the tensor product of A-infinity algebras and A-infinity morphisms, which are now available and were studied recently in combinatorics. In fact, these techniques can be used more generally to compute the homology of fibered spaces, and could potentially be used to develop a theory of Sullivan minimal models in characteristic p.
Speaker: Alex Takeda (Uppsala University)Abstract:In rational homotopy theory, there are notions of formality and coformality of a topological space, related to how “simple” the higher structures on their minimal models are. In particular, for simply connected spaces, the notion of coformality, which measures how simple is the Quillen model for its rational homotopy groups, can be related to the formality of A-infinity algebras, which is something that can be stated entirely within the homotopy theory of operadic algebras. In this talk I will explain a generalization of this latter notion of formality that extends this notion to a pair (X,[X]) of a space with a fundamental class giving local Poincaré duality. This is phrased in terms of algebras over a certain dioperad Y, which one can show is Koszul using a certain geometric construction. Part of this talk is about joint work with C. Emprin.
Speaker: Jakob Ulmer(Sorbonne Paris Norht University)Abstract: I will recall the work of Costello, Caldararu-Tu on how to extract 'enumerative‘ invariants out of a Calabi-Yau category C, conjecturally matching GW invariants of M symplectic if C=Fuk(M). Their work is an algebraic incarnation of closed string field theory (SFT); the invariants are assembled in the partition function of the closed SFT associated to C. Next I will report on work in progress (by U., Amorim-Tu and using results of Voronov et. al) how to define the partition function of large N open SFT on a stack of N branes of an object x of C, conjecturally related to open GW invariants. Finally I explain how to define the partition function of closed SFT of C, backreacted in the direction of an object x. This is directly inspired by constructions in the ’Twisted Holography’ proposal of Costello-Gaiotto.
Speaker: Erik KjellgrenDepartment of Physics, Chemistry, and PharmacyAbstract: In this talk, we explore how noisy intermediate-scale quantum (NISQ) hardware can be utilized to accurately compute molecular properties. By combining the variational quantum eigensolver (VQE) for wave function circuit preparation with quantum linear response theory, we can extract molecular properties. However, the inherent noise in NISQ hardware presents a significant challenge, impacting the accuracy and reliability of the results. We will discuss how error mitigation strategies can help address these challenges and enable the extraction of meaningful computed results.Location: D-IAS AUD (V24-501a-0)The event is open to all. Crémant and petits fours will be served.
Speaker: Hao Xu(Georg-August University of Göttingen) Abstract: Given a finite symmetry group $G$ with an anomaly $\pi \in \mathrm{H}^4(G,U(1))$, one can construct a twisted 4D gauge theory called a Dijkgraaf-Witten model. These models are exactly solvable and play important roles in both higher energy physics and topological orders of quantum phases of matter. People have previously studied (codim > 1) topological defects (such as generalizations of Wilson lines) within them, which together form a **braided fusion 2-category**, $\mathscr{Z}(\mathbf{2Vect}^\pi_G)$, the Drinfeld center of $\pi$-twisted $G$-crossed finite semisimple linear categories.Anyons (i.e. topological line defects) can condense in 3D quantum systems to produce new ones, with gapped 2D domain walls sitting between these 3D systems. It has been known for a decade that such a physical process can be described mathematically considering connected étale algebras (which in physical terms correspond to a condensable set of anyons), their ordinary and local modules within modular tensor categories. In particular, the after-condensation phase is trivial if and only if the condensable set of anyons is maximal, and hence the gapped 2D domain wall becomes a 2D topological boundary of the original phase. My doctoral project is to develop a mathematical framework for the same picture but in one higher dimension. This goal has been achieved with Décoppet in a series of past papers, where we define **étale algebras** and their **local modules** in braided fusion 2-categories and study their properties. In a recent independent work, I applied the general machinery to $\mathscr{Z}(\mathbf{2Vect}^\pi_G)$, where étale algebras mean $\pi$-twisted $G$-crossed braided multifusion 1-categories. Built upon essential tools developed by Drinfeld et al., I obtain a **classification of connected étale algebras** in $\mathscr{Z}(\mathbf{2Vect}^\pi_G)$. In particular, Lagrangian algebras in $\mathscr{Z}(\mathbf{2Vect}^\pi_G)$ correspond to 3D topological boundaries of 4D Dijkgraaf-Witten Models. Décoppet has shown that the Drinfeld center of an arbitrary fusion 2-category is either equivalent to a 4D Dijkgraaf-Witten model $\mathscr{Z}(\mathbf{2Vect}^\pi_G)$ or a more complicated fermionic version. If time allows, I would introduce my classification result for all **fusion 2-categories** based on studying Lagrangian algebras in 4D Dijkgraaf-Witten models.
Speaker: Benjamin Sung(University of California)Abstract: The theory of Bridgeland stability conditions assigns a complex manifold to the derived category of coherent sheaves on a smooth projective variety. The structure of this complex manifold is central for applications to algebraic geometry, but describing even a connected component is often a difficult, open problem. In this talk, I will describe the stability manifold of the product of three isomorphic elliptic curves without complex multiplication. This gives the first description for a smooth projective threefold of non-minimal Picard rank, and confirms a conjecture of Kontsevich in dimension 3. Based on 2410.08028, joint with Fabian Haiden.
Speaker: Chandan Singh(University of Melbourn)Abstract:In this talk, we show that a groupoid model based on the operad ofparenthesized ribbons admits a cyclic structure. This extends the action ofGT from braids to ribbons, thus providing a new characterization of GT asan automorphism group of the prounipotent cyclic operad of parenthesizedribbons. As a consequence, we exploit this GT action to provide a simpleproof of the formality of the cyclic-framed little disk operad. Finally, thisaction extends to the category of parenthesized tangles, which verifies theconjecture of Kessel-Turaev about Galois actions on framed tangles.The work discussed here lays foundation to exploring GT actions on abroader class of knotted objects such as B-tangles, virtual tangles andwelded tangles, which will be addressed in our future work. This is a jointwork with Marcy Robertson.
Speaker: Linhao Li(Ghent University)Abstract:: In this talk, I will introduce our recent work on how to characterize the 't Hooft anomalies in open quantum systems [1]. Owing to nontrivial couplings to the environment, symmetries in such systems manifest as either strong or weak type. The total symmetry \Gamma can fit into the symmetry extension: 1→K→\Gamma→G→1 with K the strong symmetry and G the weak symmetry. By representing their symmetry transformation through superoperators, we incorporate them in a unified framework that enables a direct calculation of their anomalies. We find that anomalies of bosonic systems are classified by $H^{d+2}(\Gamma,U(1))/H^{d+2}(G,U(1))$ in d spatial dimensions. We also generally prove that anomaly must lead to nontrivial mixed-state quantum phases as long as the weak symmetry is imposed, which can guarantee nontrivial steady states and long-time dynamics for open quantum systems governed by Lindbladians. Finally, we explore the generalization of the ``anomaly inflow" mechanism in open quantum systems.Reference:[1] Zijian Wang, Linhao Li, arXiv: 2403.14533
Speaker: Jonte Gödicke (University of Hamburg) Abstract:In the study of 3-dimensional Topological Quantum Field Theories (short TQFTs), certain monoidal categories called multi-fusion categories are of fundamental importance. Well-known examples of these arise from finite groups through a linearization construction. However, it is less well-known that the same construction can produce more interesting examples of monoidal structures from any 2-Segal space. These include so-called Hall monoidal structures, which are anticipated to have interesting connections to quantum topology and TQFTs.In this talk, I will classify those 2-Segal spaces that induce multi-fusion categories. For this, I will introduce a 2-categorical characterization of multi-fusion categories to translate questions about these monoidal structures to questions about homotopy coherent algebra in span categories. If time permits, I will discuss extensions of this to the context of derived categories and derived TQFTs.
Speaker: Zhi-Zhen Wang(Trinity College Dublin)Abstract:Eberhardt-Mizera (2023) evaluated the one-loop bosonic and Type I open string amplitude via Rademacher method. In this talk, we will introduce two other different strategies to treat the modular integrals appearing in one-loop bosonic and Type I amplitudes of both open and closed strings, and in particular their analytic continuation based on a string theoretic analog of the iε-prescription of quantum field theory. We will identify the two strategies and evaluate various zero- and two-point amplitudes. Beyond the perfect agreement to that obtained from the Rademacher method, our approach further provided an exact expression as a combination of three partition functions in terms of a temperature variable. While each individual term depends on this variable, the sum does not. In particular, the imaginary part of the one-loop amplitude, which accounts for the decay widths, has a compact analytical form. Its relation to the Regge limit will also be discussed. This talk is based on our recent work 2411.02517.
Speaker: Meer Ashwinkumar (University of Bern) Abstract:In this talk, I will present recent developments in the study of 5d non-commutative Chern-Simons theory, which Costello employed to describe twisted M-theory in a Taub-NUT background. This theory is connected to affine Yangians and W-algebras through the structure of its operators. I will first discuss my recent work, where I derived Miura operators (which can be used to realize W-algebras) from the intersection of line and surface defects in 5d Chern-Simons theory. These Miura operators can be shown to have an interpretation as R-matrices. Next, I will explore a holographic duality for 5d Chern-Simons theory, formulated in terms of a novel 4d chiral WZW model. Finally, I will outline how twisted M-theory on more general conifold backgrounds can be described as 5d Chern-Simons-matter theories.
Speaker: Christoph Nega (Technical University of Munich)Abstract: In my talk, I will discuss Feynman Integrals which are the cornerstone for precision computations in perturbative physics ranging from quantum field theory to classical general relativity. The two main examples will be self-energies in quantum electrodynamics and the scattering of black holes in a perturbative treatment of general relativity. I will introduce the relevant concepts in the context of Feynman Integrals, as integration by parts relations, master integrals, canonical integrals, iterated integrals and others. Moreover, I will show that Feynman Integrals are related to period integrals on geometries, particularly on Calabi-Yau varieties. This knowledge can be used to compute them. At the end, I will discuss methods to numerically evaluate the Feynman Integrals appearing in my examples.
Speaker: Guillaume Laplante-Anfossi (University of Southern Denmark) Abstract:Given a curved A-infinity algebra and a bounding cochain (a.k.a Maurer—Cartan element), one can twist the former by the latter to get a new A-infinity algebra where the curvature is zero. This uncurving procedure plays a central rôle in Floer theory, where Fukaya categories are naturally curved objects. The goal of this talk will be to give a universal characterization of the procedure, using the theory of operads: we will show that the operad cMC, whose algebras are curved A-infinity algebras endowed with a Maurer—Cartan element, together with its natural morphism to the A-infinity operad, is terminal in a certain comma category. This category does not contain the Maurer—Cartan equation, thus the theorem provides a way to « rediscover » it as the answer to a universal (algebraic) problem. This is joint work with Adrian Petr and Vivek Shende.
Speaker: Apoorv Tiwari (University of Southern Denmark) Abstract: It has recently been understood that finite (internal) global symmetries in quantum field theory are describe fusion higher categories, sometimes referred to as global categorical symmetries. In this talk, I will introduce a framework based on Symmetry Topological Field Theory (SymTFT) for characterizing and classifying both gapped and gapless phases, as well as phase transitions in quantum systems with global categorical symmetries. This characterization will be expressed in terms of sets of condensed, confined, and deconfined generalized symmetry charges, evoking concepts familiar from superconductivity. Furthermore, I will demonstrate how the SymTFT data can be transformed into an ultraviolet (UV) lattice model—such as an anyon chain model in 1+1 dimensions or a fusion surface model in 2+1 dimensions—that realizes these phases and transitions in the infrared (IR) limit.
Speaker: Shaukat Ali HoD, Research Professor, Chief Research ScientistSimula Research Lab, Norway Abstract:Quantum software will enable fascinating future applications. However, creating dependable quantum software that executes on quantum computers and delivers these applications requires new software engineering methods. To this end, a novel field of Quantum Software Engineering (QSE) is emerging. Additionally, there is great potential to apply quantum computing-enhanced AI techniques to solve classical software engineering problems.This talk will focus on engineering dependable quantum and classical software using two groundbreaking technologies: quantum computing and AI. It will discuss the current challenges in using these technologies to engineer quantum and classical software. Next, the talk will provide an overview of the current state of QSE, followed by an introduction to some techniques developed in our group. Moreover, it will introduce our work on using quantum AI techniques to solve classical software engineering problems (e.g., testing). Finally, the talk will conclude by examining open research questions at the intersection of quantum computing, AI, and software engineering and discussing the future outlook.Location: O100 Aud. (Ø5-112-1) The event is open to all.
Speaker: Franziska Porkert (University of Bonn)Abstract:Feynman integrals are fundamental to multi-loop scattering amplitudes and exhibit interesting connections to various areas of mathematics. For instance, beyond one loop, many Feynman integrals are associated with (hyper)-elliptic curves or Calabi-Yau surfaces. In this talk, I will provide a brief overview of (hyper)-elliptic Feynman integrals, illustrated with specific examples, specifically highlighting ongoing work on solving a family of Feynman integrals related to a genus-two curve.
Speaker: Nadia Ott (University of Southern Denmark) Abstract: An essential ingredient in perturbative string theory is a certain measure on the moduli space Mg of curves. This measure is defined in terms of the Mumford isomorphism, which relates the canonical line bundle on Mg to the determinant of cohomology of the pushforward of the relative canonical line bundle on the universal curve. This pushforward, and thus also its determinant, has a natural hermitian metric given by integration. This metric can be expressed in terms of the period map.For superstring theory, this generalizes to a measure on the supermoduli space Mg. The super Mumford isomorphism relates the canonical bundle on Mg to the fifth power of the Berezinian of the pushforward of the relative canonical bundle on the universal supercurve. However, in the super case, there is no Hermitian metric given by integration. Instead, the metric is defined in terms of the period map. Furthermore, in contrast to the classical case, the super period map is non-holomorphic and develops a pole along the bad locus. Deligne recently proved that the supermeasure extends smoothly over the bad locus.In joint work with Ron Donagi, we define a measure on Mg,0,2r, the supermoduli space with Ramond punctures, using the super Mumford isomorphism and super period map, adapted to the case of Ramond punctures. We show that in Mg,0,2r, the analogous bad locus has codimension 2 or higher for r > 1, allowing us to extend the measure using a Hartog-like argument.
Speaker: Oscar Bendix Harr (University of Copenhagen) Abstract: Not much is known about the cohomology of the moduli space of handlebodies (equivalently, the cohomology of handlebody mapping class groups) except what it stabilizes to, by Hatcher's analogue of the Madsen--Weiss theorem. Hatcher and Wahl have shown that the degree d cohomology of the moduli space of genus g handlebodies is stable for d less than roughly g/2. I will discuss some work from my PhD thesis (in preparation) which expands this range to g roughly less than 2g/3, using the methods of Galatius, Kupers, and Randal-Williams. By a computation of Morita, this is the optimal range for homological stability for moduli spaces of handlebody groups. A similar argument shows that the cohomology of automorphism groups of free groups Aut(F_n) is stable in degrees roughly less than 2n/3, which improves the best known range for integer coefficients. Time provided, I will also say something about ongoing work on the cohomology outside the stable range.
Speaker: Chris Grossack (UC Riverside) Abstract:Often in math it's easier to solve a problem "locally", and then one studies how to glue these local solutions into a global solution of the whole problem. In recent years there's been a lot of progress in our understanding of skein algebras and how they glue, culminating in the work of Ben-Zvi--Brochier--Jordan and Cooke on Skein Categories, which can be computed with Factorization Homology. In a separate line of development, there has been progress in the gluing of (topological) Fukaya Categories and Hall Algebras, using the Dyckerhoff--Kapranov machinery of 2-Segal Spaces. In the two halves of this talk we'll discuss Factorization Homology and 2-Segal Spaces. In any remaining time we'll speculate on how these techniques might be used to globalize certain results in Skein Theory.
Speaker: Peter Samuelson (UC Riverside) Abstract:The Homflypt skein relations were originally used in the 90's to define polynomial invariants of knots in S^3. In this talk we describe several other ways they arise in algebra and geometry: intertwiners for certain quantum group representations, commutator relations in quantum groups, deformation quantization of trace functions on character varieties, commutator relations in Hall algebras of Fukaya categories of surfaces, and higher dimensional Heegaard-Floer homology of configuration spaces of points in cotangent bundles to surfaces. This will be a survey style talk; in particular, no significant prior knowledge of these topics will be assumed, and the precision of the statements will decrease monotonically as the talk progresses. Our main goal is to describe a nonobvious common theme in these topics.
Speaker: Frederic Fauvet (IRMA, University of Strasbourg) Abstract:Paralogarithms constitute a family of resurgent functions which have been introduced by J. Ecalle, in order to solve, in a very general and systematic way, inverse problems of Riemann--Hilbert type that are expressed in the formalism of alien derivations. I will present "paralogarithmic resurgence monomials" and describe how they are implemented to yield differential equations with given sets of Stokes data, focussing on a simple non--linear example.
Speaker: Sergey I. Bozhevolnyi Center for Nano Optics University of Southern Denmark Abstract:Manipulation of single-photon emission from quantum emitters (QEs) has attracted a considerable attention in recent years due to its importance for quantum information technologies in quantum communication, computation, sensing and metrology. Here, recent progress in on-chip manipulation of the polarization, directionality and phase distribution in single-photon emission by making use of planar holographic QE-coupled metasurfaces is presented and discussed. The underlying idea is related to the concept of meta-atom, in which a QE is efficiently and non-radiatively coupled to surface modes, such as surface plasmon polaritons (SPPs), that are subsequently outcoupled into free propagating waves. An innovative metasurface design approach, vectorial scattering (computer-generated) holography, is introduced for the purpose of designing hybrid SPP-QE coupled metasurfaces suitable for generation of well-collimated beams of single photons with desirable polarization characteristics propagating along given directions. Latest results include its extension for realizing single-photon sources with radiation channels that exhibit diverse (including vectorial with spin and orbital angular momenta) wavefronts and polarization characteristics, opening thereby a way to generating quantum structured light in high dimensions.Location: D-IAS Aud. (V24-501a-0), Danish Institute for Advanced Study - DIAS. The event is open to all.
Speaker: Mykola Dedushenko (Simons Center for Geometry and Physics, Stony Brook) Abstract:I will review my recent work with Nekrasov on the aspects of relation between supersymmetric gauge theories and quantum integrable models. An important role is played by the supersymmetric defects (especially interfaces) in gauge theories, which provide a QFT realization of the infinite-dimensional quantum groups underlying the integrability structures. These defects realize, and in many ways are inspired by, the constructions of Maulik with Okounkov, and Aganagic with Okounkov, of stable envelopes and chamber R-matrices in geometry.
Speaker: Thibault Langlais (University of Oxford) Abstract:The Lie group G2 is one of the two exceptional cases in Berger's list of possible Riemannian holonomy groups. I will start by introducing the basics of G2-geometry, emphasizing its relation to Calabi-Yau geometry in (real) dimension 4 and 6. The moduli spaces of G2-manifolds are smooth and carry a natural Riemannian metric, analogous to the Weil-Petersson metric, whose properties are poorly understood. I will show that certain singular G2-manifolds correspond to finite-distance limits in the moduli spaces, which proves that G2-moduli spaces are generally not complete. Time permitting, I will also say a few words about the computation of curvatures.
Speaker: Manuel Meyer Department of Physics, Chemistry and PharmacyUniversity of Southern Denmark Abstract:The nature of dark matter continues to elude us, even after almost one century after first evidence for this non-luminous substance appeared. A leading hypothesis is that dark matter, which makes up more than 80% of all matter in the universe, consists of yet undiscovered fundamental particles. Such dark matter particles might interact feebly with known particles of the Standard Model, making them potentially detectable in the laboratory. The predicted low interaction rates require extremely sensitive detectors and ultra-low background levels. In this talk, I will discuss our recent progress in characterizing and improving the performance of transition edge sensors (TESs). Such TESs are planned to be used in the Any Light Particle (ALPS) II experiment, which searches for axions and axion-like particles. I will also review how we can use our TES to search for weakly interacting massive particles. With our ongoing research, we hope to achieve a new record for background suppression employing both optical filtering inside a cryostat as well as machine learning techniques while maintaining a quantum efficiency close to unity. Location: D-IAS Aud. (V24-501a-0), Danish Institute for Advanced Study - DIAS. The event is open to all.
Speaker: Ron Donagi (University of Pennsylvania) Abstract:After a brief review of the background on super geometry and of Super Riemann surfaces and their punctures, I will discuss the proof (with Witten) that the supermoduli space of super Riemann surfaces is not projected for g ≥5, and the more recent proof, with Ott, that the supermoduli space Mg,0,2r of super Riemann Surfaces with Ramond punctures is also not projected, for all g≥5r+1 and r≥6.
Speaker: Nadia Ott (University of Southern Denmark) Abstract:D’Hoker and Phong’s calculation of the genus g = 2 superstring amplitude uses, in a crucial way, a projection from genus g = 2 supermoduli space to its underlying reduced space. They define this projection using a formula for the genus g = 2 super period matrix. Witten generalized their formula for the super period matrix to higher genus g and found that the super period matrix may develop a pole along a particular divisor in supermoduli space if g ≥ 11. This divisor is commonly called the bad divisor. Witten also considered super period matrices on super Riemann surfaces with a nonzero number of Ramond punctures (note: the word puncture is a bit of a misnomer). He found that in the presence of Ramond punctures, a closed one form has, in addition to the usual 2g ”even” periods (defined by integrals over one cycle in homology), 2r fermionic periods. The fermionic periods of one form w are certain constants appearing in the restriction of w to the Ramond divisor. In joint work with Ron Donagi, we identify the 2r fermionic periods of w with the residues of a particular global section of the twisted spin structure on the underlying curve. As in the unpunctured case, the super period matrix with Ramond punctures may develop a singularities as we vary over supermoduli space. Using this identification of the fermionic periods in terms of residues, we explicitly describe this bad locus in supermoduli space.
Speaker: Nadia Ott (University of Southern Denmark) Abstract:I will give an introductory talk on supermanifolds, super Riemann surfaces, and supermoduli space, starting with the definitions of these objects. I will then introduce a current area of research in supergeometry, super period matrices, and state two famous results in the field: Donagi and Witten’s proof that supermoduli space is not split for genus g ≥ 5, and D’Hoker and Phong’s calculation of the genus g = 2 superstring amplitude.
Speaker: Erik Donovan Hedegård Department of Physics, Chemistry and Pharmacy University of Southern Denmark Abstract:Transition metals in biological systems pose a formidable challenge in modern quantum chemistry. Unfortunately, these metals are close to everywhere in biological systems. For instance, about one-third of all enzymes contain a transition metal. The main issues when dealing with transition metals are (i) the metal typically demands so-called complete active space (CAS) methods with billions of electron configurations included. This quickly becomes computationally costly. (ii) Relativistic effects can be sizable. Addressing these effects also becomes computationally demanding. (iii) The relevant chemistry usually occurs in a surrounding solvent or within a protein environment that also needs to be taken into account, both in terms of the nuclear dynamics as well as the electronic interactions between the metal and the environment.In this talk, we demonstrate how challenges (i)–(iii) can be tackled efficiently and accurately. We discuss how this allows us to understand a part of the global carbon cycle, enzymes that can boost the production of biofuel, and new drugs against cancer.Location: D-IAS Aud. (V24-501a-0), Danish Institute for Advanced Study - DIAS. The event is open to all.
Speaker: Ying-Hsuan Lin (Harvard University) Abstract:We numerically studied an anyon chain based on the Haagerup fusion category, and found evidence that it leads in the long-distance limit to a conformal field theory (CFT) whose central charge is ~2. Our findings were in agreement with concurrent research by Vanhove et al., who explored a closely related statistical lattice model. Subsequent analyses of the operator product expansion by Liu, Zou, Ryu, however, revealed a deeper complexity than our data had initially suggested. In this talk, I will discuss these developments and contextualize them within the broader program of classifying CFTs.
Danish Quantum Community invites to participate in our annual Scientific Quantum Conference 2024 This year’s conference is hosted by Centre for Quantum Mathematics (QM) with support from the Novo Nordisk Foundation. Join us for a two-day deep dive into the Danish quantum research community - across universities and across disciplines. About the event:The conference will present recent results from Danish quantum research with a special focus on younger researchers at Danish universities. Serving as a meeting point for the research community, the conference aims to strengthen the Danish quantum ecosystem and promote industry-academia collaborations.Across two days, the conference will provide a deep dive into four key research tracks: quantum hardware, quantum software/algorithms, components for quantum systems, and early quantum applications in chemistry and finance.The conference is hosted by Centre for Quantum Mathematics (QM) at SDU in Odense with support from the Novo Nordisk Foundation.Participation is free of charges, but requires registration.
Speaker: Merlin Christ (Institut de mathématiques de Jussieu – Paris Rive Gauche) Abstract:A complex of stable infinity-categories is a categorification of a chain complex, meaning a sequence of stable infinity-categories together with a differential that squares to the zero functor. To produce examples of such categorical complexes we introduce a categorification of the totalization construction, which produces a categorical complex from a categorical multi-complex, such as a commuting cube of stable infinity-categories. We will then explain how categorified perverse sheaves, also known as perverse schobers, on C^n (with a certain stratification) can be described in terms of categorical cubes and categorical complexes of spherical functors. We will also see what categorical totalization means in this case geometrically. This talk is based on joint work with T. Dyckerhoff and T. Walde.
Speaker: Edmund Heng (Institut des Hautes Études Scientifiques) Abstract:Classically, finite symmetries are captured by the action of a finite group. Moving to the quantum world, one has to allow for (possibly non-invertible) quantum symmetries — these are instead captured by the action of a more general algebraic structure, known as a fusion category. Such quantum symmetries are actually ubiquitous in mathematics; for example, given a category with an action of a finite group G (e.g. rep(Q), Coh(X) etc.), its G-equivariant category has instead the action of the category of representations rep(G), where rep(G) has the structure of a fusion category (and is not just a group when G is non-abelian). The aim of this talk is to introduce fusion categories and discuss their role as “quantum symmetries” in relation to Bridgeland’s stability conditions. We first introduce a generalised notion replacing “G-invariant stability conditions” in the setting of a triangulated category equipped an action of a fusion category C, which we will “C-equivariant stability conditions”. The first result is that these stability conditions form a closed submanifold of the stability manifold, just as the G-invariant stability conditions do. Moreover, given a triangulated D with a G-action, so that its G-equivariant category D^G has a rep(G)-action, we will see the following (Morita) duality result for stability conditions: the complex manifold of G-invariant stability conditions (associated to D) is homeomorphic to the complex manifold of rep(G)-equivariant stability conditions (associated to D^G).If time allows, I will discuss other more “exotic” actions of fusion categories on triangulated categories, and possibly its relation to Coxeter theory.This is part of joint work with Hannah Dell and Anthony Licata.
DIAS lecture by Professor Francesco Sannino, DIAS Chair of Physics, Head of the Quantum Field Theory Center, Founder of the Centre for Cosmology and Particle Physics Phenomenology (CP³-Origins) at SDU. The lecture takes place in the DIAS Auditorium and is open to all.Abstract We live in an era marked by the LIGO Laboratory discovery of gravitational waves, emitted when two black holes coalesce, and the Event Horizon Telescope imaging of black holes. While these amazing discoveries crystallise Einstein’s theory of general relativity they also beg for a fresh look at the problem of unifying gravity and quantum field theory. These two theories constitute our current understanding of Nature but are notoriously at odds with each other. I will first review basic facts about black holes and then argue in favour of a recent approach that aims at shortcutting the problem of the absence of a theory of quantum gravity. This will be achieved by introducing a quantum gravity model independent approach focussed on effective metric descriptions of quantum black holes. We believe that our findings will herald novel ways to explore quantum corrections to black hole dynamics with impact for our understanding of the fabric of space time.About Francesco SanninoFrancesco Sannino is the Head of the Quantum Field Theory Center, the Founder of the Centre for Cosmology and Particle Physics Phenomenology (CP³-Origins) at SDU. He is also one of the Founders of the Danish IAS and Professor of Theoretical Physics at the Federico II University in Italy.Professor Francesco Sannino is widely recognized for having pioneered the analytical and numerical investigations of the conformal structure of gauge theories of fundamental interactions, for the construction of minimal composite extensions of the standard model, and for the recent discovery of four-dimensional asymptotically safe theories.Recently he has also applied and developed mathematical tools stemming from theoretical physics to describe the evolution of infectious diseases at human and viral level.His work crosses several realms of particle physics and cosmology from bright and dark extensions of the standard model and inflationary cosmology to the mathematical underpinning of theories of fundamental interactions.
The seminar is cancelled.Speaker: Astrid Eichhorn Department of Physics, Chemistry and Pharmacy University of Southern Denmark Abstract:I will present a tentative answer to this question, consisting in a theory of quantum spacetime and matter, to which I will give a pedagogical introduction. I will argue that the main challenge for such a theory is the question of how to probe it experimentally, given that we expect its effects to manifest at tiny distances below the Planck length of 10^-35 m. I will challenge this expectation and show that not all effects are untestably small. Concretely, I will show how consequences of the microscopic physics leave their imprints on macroscopic scales in particle physics and black holes.Location: D-IAS Aud. (V24-501a-0), Danish Institute for Advanced Study - DIAS. The event is open to all.
Speaker: Apoorv Tiwari (Niels Bohr International Academy, University of Copenhagen) Abstract:In this talk, I'll describe how finite symmetries of quantum systems are naturally encoded in fusion higher categories, presenting a universal and systematic construction of quantum systems with such higher categorical symmetries starting from quantum systems with conventional group-like symmetries. I'll discuss the Symmetry Topological Field Theory (SymTFT), a conceptual tool that facilitates the study of the generalized charges and gapped phases realized in quantum systems with categorical symmetries. Utilizing the SymTFT, I'll demonstrate a systematic construction of lattice models realizing categorical symmetry-protected gapped phases and phase transitions between them. Finally, I'll outline ongoing efforts toward realizing these novel phenomena in experimental cold-atom platforms.
Speaker: Kazuki Ikeda (Stony Brook University) Abstract:In this presentation, I will discuss the crucial role of quantum mathematics in understanding quantum many-body systems such as nuclear physics and condensed matter physics. I will also introduce the frontiers of quantum software development and system integration necessary to realize efficient quantum simulations. In particular, I will discuss quantum language development and the category theoretic construction of quantum compilers, and highlight the important and specific role that quantum mathematics plays in this field. Moreover I will introduce new interactions between algebraic geometry, quantum physics and quantum computation. This includes the novel construction of band theory on general Riemann surfaces, topological quantum computation, quantum error correction. Through my presentation, the audience will grasp the diverse applications and progress in Quantum Mathematics to solve various problems in modern QIST.
Speaker: Nate Bottman (Max Planck Institute for Mathematics) Abstract:I will present the 2-associahedra, which I constructed in 2017 in the context of symplectic geometry, and explain some current developments, focusing on symplectic aspects. First, I will explain how 2-associahedra form the right operadic structure for endowing the Fukaya category with functoriality properties. Specifically, the 2-associahedra lead to the notion of the symplectic (A-infinity,2)-category Symp, which is the natural setting for building functors, associated to Lagrangian correspondences, between Fukaya categories. Second, I will describe a related family of posets called constrainahedra, which Daria Poliakova and I constructed in 2022. The constrainahedra also translate into an algebraic structure in symplectic geometry: in ongoing work with Mohammed Abouzaid and Yunpeng Niu, we aim to show that Fukaya categories of Lagrangian torus fibrations are monoidal A-infinity categories, where the latter notion is constructed using of constrainahedra. This will fulfill a longstanding expectation in mirror symmetry.
Speaker: Kaiwen Sun (Uppsala University) Abstract:Nahm sums, also called fermionic sums are some special q-series which are closely related to 2d CFTs and SCFTs. For example, the renowned Rogers-Ramanujan functions are Nahm sums and also the characters of Lee-Yang model M(5,2). It is an open question to classify the Nahm sums with modularity. In rank 1, it was proved there only exist 7 Nahm sums. In rank 2 and 3, a conjectural list has been given by Don Zagier. I will discuss some recent progress on Nahm sums and generalized Nahm sums. This is based on a joint work with Haowu Wang.
Speaker: Matthias Wilhelm (Niels Bohr Institute) Abstract:In this talk, I will give an overview of the numbers and functions that occur in Quantum Field Theory, which are relevant for the calculation of precision predictions for collider and gravitational-wave experiments. In the simplest cases, the functions that occur are multiple polylogarithms, generalizations of logarithms. They are by now very well understood and allow us to reach precision that would be impossible using any other method; in one case up to the eighth subleading order in perturbation theory! In the next cases, we encounter elliptic integrals, where much progress was made in recent years. Beyond these, we encounter an infinite zoo of further functions involving integrals over Calabi-Yau manifolds, which are currently being explored.
Speaker: N. Asger Mortensen POLIMA – Center for Polariton-driven Light-Matter Interactions Danish Institute for Advanced Study University of Southern Denmark Abstract:Polaritons are hybrid particles – quasiparticles – that result from the strong interaction between light (photons) and matter (such as electrons, excitons or phonons). These quasiparticles exhibit mixed characteristics, combining features of both photons and elementary particles like electrons. Colloquially, they constitute half-light-half-matter excitations. The interaction between light and matter in polaritons leads to unique physical properties and behaviors that are now also being explored in sheets of atomically thin two-dimensional materials and in artificial nanostructures. The electrodynamics of matter and optical phenomena are commonly explored within the framework of classical electrodynamics and semiclassical models for the interactions of light with matter. Materials are commonly assumed homogenous, and light-matter interactions are treated in an intuitive local manner. The plasmonic response of metal nanostructures is one such example, where the understanding of mesoscopic electrodynamics at metal surfaces is, however, becoming increasingly important for both fundamental developments in quantum plasmonics and potential applications in emerging light-based quantum technologies. The seminar will discuss recent examples of quantum nonlocal effects that emerge in surface-polaritonic systems, including metal surfaces, 2D materials, and combinations thereof. Biography: Mortensen is a full professor in the Center for Polariton-driven Light-Matter Interactions (POLIMA) and a Chair of Physics in the Danish Institute for Advanced Study, both at the University of Southern Denmark. Previously, he held a full professorship (faculty since 2004) at the Technical University of Denmark (DTU). In addition to his MSc (1998) and PhD (2001) degrees from DTU, he holds higher doctoral degrees from University of Copenhagen (Dr. Scient., 2021) and DTU (Dr. Tech., 2006). He is a fellow of APS, OSA, SPIE, IOP, and European Academy of Sciences. Currently, he serves the AAAS as an associate editor for Science Advances.
Speaker: Jesse Cohen (University of Hamburg) Abstract:Bordered Floer homology, due to Lipshitz, Ozsváth, and Thurston [LOT], is a generalization of Heegaard Floer homology to 3-manifolds with parametrized boundary. The simplest incarnation of this invariant can be regarded as a differential module CFD(Y) and a pairing theorem of [LOT] tells us that the complex of module homomorphisms between two such modules is homotopy equivalent to the Heegaard Floer complex of the 3-manifold obtained by gluing. We will discuss a topological interpretation of composition of module homomorphisms in this context, and applications thereof, including forthcoming work on deformations of arc algebras and a spectral sequence for links in S^1xS^2.
Speaker: Jian Qiu (Uppsala University) Abstract:I want to explain in this talk the framework of quantization using branes, as developed by Gukov-Witten and later refined by Gaiotto-Witten. This approach embeds the symplectic manifold to be quantized into a bigger space (of double the dimension) as a brane, and the open strings ending on the brane play the role of the operators acting on the Hilbert space.The open strings are expected to be quantized via the A-model, while in practice the open string wave functions are identifiable as holomorphic functions on the larger space and therefore many algebraic approach can be applied for its quantization and bypass the A-model.I will apply Kontsevich's deformation quantization to deform the algebra of operators. Contrary to what one expects from deformation quantisation, it is very computable, thanks to some recently developed technical results due to Barmeier and Wang.I will run you through some examples, the simplest being the Kleinian singularity. In the A_1 case, the quantization gives the Verma module of Usu(2) including the integrality condition for hbar, which Is unusual for deformation quantization.
Speaker: Shuhan Jiang (Max Planck Institute, Leipzig) Abstract:In this talk, I will introduce a geometric framework for classical cohomological field theories (CohFTs) using G^{\star} algebras and variational bicomplexes. This framework enables a mathematical interpretation of scalar and vector supersymmetries, along with a systematic classification of solutions to the descent equations for CohFTs. Furthermore, I will provide a BV-BFV extension of this framework, illustrating its application through the (fully extended) Donaldson-Witten theory.
Speaker: Ce Ji (Peking University) Abstract:Over decades of development of the Witten conjecture, many enumerative geometries are related to integrable hierarchies. On the other hand, many of such theories also admit mirror symmetry in the sense of the remodeling conjecture with an underlying object called the spectral curve. In this talk, we propose a generalization of the Witten conjecture from spectral curve, which produce descendent potential functions related to certain reductions of (multi-component) KP integrable hierarchy. Proof for genus zero spectral curve with one boundary will be sketched, which can be applied to deduce the rKdV integrability of deformed negative r-spin theory, conjectured by Chidambaram--Garcia-Falide--Giacchetto. This talk is based on the joint work with Shuai Guo and Qingsheng Zhang.
Speaker: Cris Kuo (University of Southern California) Abstract:For complex manifolds, the Riemann-Hilbert correspondence generalizes the classical correspondence between finite dimensional local systems and vector bundles with connections to perverse sheaves and regular holonomic D-modules. These later objects are in fact microlocal in nature that they can be regarded as living on the cotangent bundles, and the correspondence admits a microlocalization as well. Continuing from an earlier joint work with Côté, Nadler, and Shende, we will globalize this correspondence to general complex contact manifolds in an upcoming work.
Speaker: Alexander Stottmeister (University of Hannover) Abstract:I will discuss a generalized notion of inductive systems of Banach spaces, coined soft inductive systems, that can serve as a flexible tool to describe the limits of physical theories, e.g., in the context of the thermodynamic limit and phase transitions or renormalization of quantum field theories.General criteria for the convergence of dynamics can be obtained in this setting, which I will illustrate using the example of renormalization of free fermion theories on a one-dimensional spatial lattice. Specifically, this example can be used to prove the conformal invariance of various correlation functions of the classical Ising model using operator algebraic methods.This talk is based on work with L. van Luijk, T.J. Osborne, and R.F. Werner.
Speaker: Guillem Cassasus (QM, SDU) Abstract:We will talk about algebraic structures arising in Lagrangian Floer homology in the presence of a Hamiltonian action of a compact Lie group. First, we will show how the Lagrangian Floer complex can be equipped with an A-infinity module structure over the Morse complex of the group, and how this action permits to define equivariant versions of Floer homology. We will then explain how this structure interacts with the structure of the Fukaya category: both can be packaged into (our version of) an A-infinity bialgebra action, giving an alternative answer to a conjecture of Teleman. This should enable one to build an extended topological field theory corresponding to Donaldson-Floer theory. This is based on two joint work in progress, one with Paul Kirk, Mike Miller-Eismeier and Wai-Kit Yeung, and another with Alex Hock and Thibaut Mazuir.
Speaker: Ingmar Saberi (Ludwig-Maximilians-Universität München) Lectures: Tuesday, Wednesday and Thursday 15:00-16:00. Q&A session: Friday 15:00-16:00.Abstract: In this mini-course, I will discuss some approaches to the geometry of supersymmetric field theory that offer promising new perspectives on issues related to M-theory, in particular on eleven-dimensional supergravity and six-dimensional theories with (2,0) supersymmetry. One key ingredient will be a structural analogy between superspaces and almost-complex manifolds, which allows for a unified formulation of supersymmetric theories and their twists. Another will be the notion of factorization current algebra and the factorization Noether theorem of Costello and Gwilliam. By the end, I hope to have reviewed both some recent progress and some outstanding questions related to twisted eleven-dimensional supergravity.
Speaker: David O'Connell (University of Okinawa) Abstract:Topology change is typically modelled by a Lorentz cobordism, which can be seen as a maximal spacelike hypersurface that globally splits into multiple pieces. In this talk we will take the absolute opposite approach, and consider models of topology change in which individual points are duplicated and then allowed to propagate into an eventual cobordism. As we will see, such models are non-Hausdorff, with the Hausdorff-violation occurring along the future nullcones of the duplicated points. Using recent developments in non-Hausdorff differential geometry, we will evaluate the Einstein-Hilbert action on topology changing spacetimes in two dimensions. Interestingly, even in the case of seemingly-flat geometries there will be non-zero curvature contributions coming from Hausdorff-violating submanifolds that sit inside the spacetime. These observations suggest that such spacetimes will generally be suppressed in a path integral that sums over topologies, with more elaborate branching receiving a stronger suppression.
Speaker: Guglielmo Lockhart (Universität Bonn) Abstract:String theory compactification on elliptic Calabi-Yau manifolds provides a bridge between enumerative geometry on one side, and vertex operator algebras and a modular or Jacobi forms of various kinds on the other. This comes about because compactification leads to supersymmetric quantum field or quantum gravity theories that include two-dimensional objects, the noncritical strings, among their degrees of freedom. Exploiting this connection has led to a deeper understanding both of the spectrum of supersymmetric theories and of the modular properties of generating functions of enumerative invariants. In the first part of the talk I will give a broad overview of this subject, while in the second part I will focus on a recent work with Michele Del Zotto, where we find a link between Donaldson-Thomas invariants of higher rank, vector-valued Jacobi forms, and a two-dimensional analogue of Kronheimer-Nakajima quivers.
Speaker: Muyang Liu (QM, SDU) Abstract:F-theory is remarked by its powerful model building potential due to feasible geometric descriptions of string compactifications. It translates physics concepts in the theory concerned to mathematical objects extracted from the beautiful structures of elliptic fibrations. In particular, the resolved Calabi-Yau manifolds are determined within the framework of toric geometry. This gives a combinatorically simple description of the geometry and can be algorithmized with many computer algebra programs. In this talk, I will explore the searching of explicit models in the language of F-theory geometry, ranging from those admitting exact 4D Minimal Supersymmetric Standard Model (MSSM) matter spectra to novel 6D (1,0) heterotic theories and their T-dual network.
Speaker: Paolo Ghiggini (Université Grenoble Alpes) Abstract:Vincent Colin, Ko Honda, Michael Hutchings and I defined embedded contact homology groups for sutured three-manifolds as a slight modification of Hutching's embedded contact homology for closed three manifolds. I will sketch a strategy to prove that those groups are isomorphic to Juhász's sutured Floer homology, and therefore are topological invariants. The strategy is to extend the knot complement to a larger closed manifold, and then apply the isomorphism between Heegaard Floer homology and embedded contact homology to the closed manifold. In the talk I will focus on the effect of that extension on embedded contact homology, and therefore no knowledge of suitured Floer homology will be necessary beyond the fact that it exists and is interesting. On the other hand the definition of embedded contact homology for sutured three-manifolds will be sketched. This is a joint work in progress with Vincent Colin and Ko Honda.
Speaker: Saebyeok Jeong (Department of Theoretical Physics, CERN) Abstract:The Coulomb branch of four-dimensional N=2 supersymmetric gauge theory of class S is well-known to be identical to the moduli space of Hitchin's equations. I will consider the case of genus-0 with regular marked points, where the associated integrable system is the Gaudin model. I'll explain how the quantization of the Gaudin model can be formulated in the N=2 supersymmetric gauge theory with the help of half-BPS surface defects. We consider two types of surface defects: the "canonical" surface defect and the "regular monodromy" surface defect, inserted on top of each other. The action of the former on the latter is diagonal due to cluster decomposition, interpreted as the Hecke eigensheaf property with the eigenvalue given by the oper. In this way, we formulate the geometric Langlands correspondence in the N=2 gauge theory setting. I will also explain that the separation of variables and the bispectral duality of the quantum Hitchin integrable system can be accounted for by surface defect transitions.
Speaker: Greg Stevenson (AU) Abstract:Differential graded algebras, which consist of a graded algebra together with a square-zero derivation, are an algebraic way of modelling multiplication up to homotopy. They are well adapted to, and arise naturally from, the study of derived categories in algebraic geometry. In general, the presence of the derivation makes these very complicated objects which are not necessarily amenable to usual techniques from ring theory.In my talk I'll explain some recent results, by my student Isambard Goodbody and others, in the setting where the underlying algebra is finite dimensional over some field. In this case, a surprising number of results from the theory of finite dimensional algebras can be generalized to the differential graded setting.
Location D-IAS Auditorium, Fioniavej 34, 5230 OdenseProgramme: 09:00 Welcome by DIREC Managing Director Thomas Riisgaard Hansen09:10 Topological Quantum Computing, Prof. & QM Centre Director Jørgen Ellegaard Andersen10:20 Coffee break10:35 Gaussian Boson Sampling, Assistant Professor Shan Shan11:15 Coffee break11:30 Measurement-based Quantum Computing, PhD Student Santiago Q. RiosAll are welcomeCo-organised by DIREC Note: This event will be followed by the SDU Quantum Hub Opening Event – same location from 13:00.
Speaker: Greyson Potter (SDU) Abstract:We develop a computational approach to analyzing the generalized volume conjecture, which relates topological recursion and Chern-Simons theory for hyperbolic 3-manifolds with torus boundary. The conjecture states that there is an equality between two series in a formal parameter hbar: the non-perturbative wave function from topological recursion and a state-integral for SL(2,C) Chern-Simons theory. Both series are expressible as sums over graphs. We develop algorithms for efficiently computing the graph sum for the non-perturbative wave function via quantum Airy structures and a software package, called qairy, which implements our algorithms. We further develop tools and techniques which are widely applicable to the calculation and analysis of the non-perturbative wave functions associated to genus one spectral curves. Using these tools, we are able to verify the generalized volume conjecture in several cases up to much higher order in hbar than was previously accessible as well as analyze the arithmetic aspects of the conjecture.
Speaker: Changlong Zhong (State University of New York at Albany) Abstract:Equivariant elliptic cohomology of symplectic resolutions was recently studied by Okounkov and his collaborators. For example, elliptic stable envelop is defined and it is closely related to geometric representation theory, mathematical physics and 3d mirror symmetry. For the cotangent bundle T^*G/B, it basically says that the restriction to torus fixed points of elliptic stable envelops are related with that for the Langlands dual. This is proved by Rimanyi-Weber. In this talk, I will focus on the elliptic Demazure-Lusztig operators that generate the elliptic classes corresponding to elliptic stable envelop. The (sheaf of) modules spanned by these classes are called the periodic module. The main result of this talk will show that the elliptic Demazure-Lusztig operators can be assembled together to obtain a canonical isomorphism between the periodic module and that for the Langlands dual system. In particular, it recovers Rimanyi-Weber's result. This is joint work with C. Lenart and G. Zhao.
Speaker: Andres Ibanez Nunes (University of Oxford) Abstract:In work of Haiden-Katzarkov-Konsevich-Pandit (HKKP), a canonical filtration, labeled by sequences of real numbers, of a semistable quiver representation or vector bundle on a curve is defined. The HKKP filtration is a purely algebraic object, yet it governs the asymptotic behaviour of a natural gradient flow on the space of metrics of the object.In this talk, we show that the HKKP filtration can be recovered from the stack of semistable objects and a so called norm on graded points. This approach gives a generalisation of the HKKP filtration to other moduli problems of non-linear origin.
Speaker: Guillaume Laplante-Anfossi (University of Melbourne) Abstract:On the one hand, simplex equations are higher dimensional generalizations of the Yang—Baxter (aka triangle) equation. The tetrahedron equation was introduced by Zamolodchikov in 1980, and the equations for higher dimensional simplices can be defined via higher Bruhat orders, a family of posets introduced by Manin and Schechtman in 1989. On the other hand, Steenrod squares are operations acting on the mod 2 cohomology of a topological space, introduced by Steenrod in 1947. They arise from correcting homotopically the lack of cocommutativity of the Alexander—Whitney diagonal at the cochain level. Together with Nicholas Williams, we found a surprising bijection between the terms of these two constructions: each side of a higher Yang—Baxter equation defines a Steenrod cup-i coproduct, and vice-versa. A conceptual explanation for, as well as consequences of this result remain unknown to us, and I hope to stimulate discussions about what could grow out of this new dictionary between pieces of mathematical physics and classical algebraic topology.
Speaker: Matthias Christandl (University of Copenhagen) Abstract:Molecular biology and biochemistry interpret microscopic processes in the living world in terms of molecular structures and their interactions, which are quantum mechanical by their very nature. Whereas the theoretical foundations of these interactions are very well established, the computational solution of the relevant quantum mechanical equations is very hard. However, much of molecular function in biology can be understood in terms of classical mechanics, where the interactions of electrons and nuclei have been mapped onto effective classical surrogate potentials that model the interaction of atoms or even larger entities. The simple mathematical structure of these potentials offers huge computational advantages; however, this comes at the cost that all quantum correlations and the rigorous many-particle nature of the interactions are omitted. In this work, we discuss how quantum computation may advance the practical usefulness of the quantum foundations of molecular biology by offering computational advantages for simulations of biomolecules. We not only discuss typical quantum mechanical problems of the electronic structure of biomolecules in this context, but also consider the dominating classical problems (such as protein folding and drug design) as well as data-driven approaches of bioinformatics and the degree to which they might become amenable to quantum simulation and quantum computation.
Speaker: Robin Kaarsgaard Sales (University of Southern Denmark) Abstract:We provide a universal construction of the category offinite-dimensional C*-algebras and completely positivetrace-nonincreasing maps from the rig category offinite-dimensional Hilbert spaces and unitaries. Thisconstruction, which can be applied to any dagger rig category, isdescribed in three steps, each associated with their ownuniversal property, and draws on results from dilation theory infinite dimension. In this way, we explicitly construct thecategory that captures hybrid quantum/classical computation withpossible nontermination from the category of its reversiblefoundations. We discuss how this construction can be used in thedesign and semantics of quantum programming languages.This talk is based on joint work with Pablo Andrés-Martínez (Quantinuum) and Chris Heunen (University of Edinburgh).
Speaker: Noémie Legout (University of Uppsala) Abstract:To a Legendrian submanifold in the contactisation of a Liouville domain, one can associate a differential graded algebra (DGA) called the Chekanov-Eliashberg algebra (C-E algebra), and from which many Legendrian isotopy invariants are defined. Given any DGA, it is a hard question to know if it can be the C-E algebra of a Legendrian or not. In this talk, we will give a characterization of the C-E algebra, namely we show that for a displaceable Legendrian sphere this algebra is a Calabi-Yau DGA. This means that there is a quasi-isomorphism of DG-bimodules between the diagonal bimodule and the inverse dualizing bimodule. In order to prove this, we define another complex associated to a Legendrian, the Rabinowitz complex, and show that the acyclicity of this complex implies the existence of a Calabi-Yau structure. If time permits we will briefly talk about a work in progress with Georgios Dimitroglou-Rizell for the case when the Legendrian is not necessarily a sphere. In this case, and under certain hypothesis, the C-E algebra admits a relative Calabi-Yau structure.
Speaker: William Elbæk Mistegård (University of Southern Denmark) Abstract:In his seminal work on quantum Chern-Simons field theory and the Jones polynomial, Witten envisioned a 2+1 dimensional TQFT. Further, he presented two procedures for constructing the underlying modular functor, namely by conformal field theory (CFT) and by geometric Kähler quantization of the moduli spaces of flat principal bundles on two-manifolds. Subsequently, Reshetikhin and Turaev constructed a TQFT using surgery of three-manifolds and representation theory of quantum groups, and this is now known as the WRT-TQFT. The CFT approach was initially developed by Tsuchiya, Ueno and Yamada, and then further refined to a full modular functor by Andersen and Ueno, who also established that this is equivalent to the WRT-TQFT. The geometric Kähler quantization approach was initialized by Axelrod, Witten and Pietra, and Hitchin. The quantization approach results in a projective representation of the mapping class group of a two-manifold, which is known due to Laszlo to be projectively equivalent to the representation of the WRT-TQFT. However, the full TQFT is not yet described from the point of view of quantization. In particular, the so-called TQFT factorization axiom has not been proved using only geometric quantization with respect to Kähler polarizations. However, Andersen have understood factorization from the point of view of geometric quantization with respect to reducible non-negative polarizations. Jeffrey and Weitsman introduced a different quantization procedure. Given a trinion decomposition of a closed two-manifold S, one can obtain a system of functions on the moduli space of flat bundles on the two-manifold S. This is given by taking the trace of the holonomy representations of the boundary circles of the trinions. This results in a fibration of the moduli space, the fibres of this map being Lagrangian tori. The vertical distribution of this fibration is a polarization in the sense of geometric quantization. Jeffrey and Weitsman considered the so-called Bohr-Sommerfeld quantization of this system (i.e. this polarization), and proved that it results in a finite dimensional Hilbert space, which we denote by V(S). Further, they showed that V(S) has dimension equal to the dimension of the Hilbert space of the WRT-TQFT (this dimension is the famous Verlinde formula). In this talk, we present aspects of an ongoing project on quantization and TQFT, which is complementary to the factorization construction of Andersen. We consider the two-manifold S' obtained by cutting along a circle of the trinion decomposition. This two-manifold have two boundaries and an induced trinion decomposition. Following Jeffrey and Weitsman, we consider for each element j in the gauge group, the associated Lagrangian torus fibration of the moduli space of flat bundles on S' with boundary holonomy conjugate to j. We consider the Bohr-Sommerfeld quantization of this system, which we denote by V(S',j). We show that in accordance with the TQFT factorization axiom, the Hilbert space of the closed surface, denoted above by V(S), splits as a direct sum indexed by elements of the WRT-TQFT label set, the summand indexed by j being equal to V(S',j). Further, we show that this direct sum decomposition is conjugate to factorization isomorphism of the WRT-TQFT. Thus these results gives a geometric illumination of the factorization axiom of the WRT-TQFT by means of the Bohr-Sommerfeld quantization of Chern-Simons theory.
Speaker: William Elbæk Mistegård (University of Southern Denmark) Abstract:It was envisioned by Witten that quantum Chern-Simons field theory is a 2+1 dimensional topological quantum field theory (TQFT) in the sense of Atiyah. Further, Witten argued that the link invariant known as the Jones polynomial, can be computed in this TQFT as the expectation value of Wilson loop operators. This gave an intrinsic definition of the Jones polynomial, as opposed to the known combinatorial definition, and thus solved an important problem in topology. Subsequently, a full TQFT was constructed mathematically rigorously by Reshetikhin and Turaev, using surgery presentations of three-manifolds and modular tensor categories. This is now known as the Witten-Reshetikhin-Turaev TQFT. To a closed oriented three-manifold M, the WRT-TQFT associates a topological invariant in the form of a complex number. This is called the quantum invariant of M. These invariants are essentially the building blocks of the WRT-TQFT. The goals of this introductory talk are to introduce the quantum invariants and to present the axioms of TQFT (or more precisely, the axioms of a modular functor).
Speaker: Tobias Dyckerhoff (University of Hamburg) Abstract:Derived categories have come to play a decisive role in a wide range of topics. Several recent developments, in particular in the context of topological Fukaya categories, arouse the desire to study not just single categories, but rather complexes of categories. In this talk, we will discuss examples of such complexes in algebra, topology, algebraic geometry, and symplectic geometry, along with some results and conjectures involving them.
Speaker: Robin Kaarsgaard Sales (University of Southern Denmark) Abstract:We provide a universal construction of the category offinite-dimensional C*-algebras and completely positivetrace-nonincreasing maps from the rig category offinite-dimensional Hilbert spaces and unitaries. Thisconstruction, which can be applied to any dagger rig category, isdescribed in three steps, each associated with their ownuniversal property, and draws on results from dilation theory infinite dimension. In this way, we explicitly construct thecategory that captures hybrid quantum/classical computation withpossible nontermination from the category of its reversiblefoundations. We discuss how this construction can be used in thedesign and semantics of quantum programming languages.This talk is based on joint work with Pablo Andrés-Martínez (Quantinuum) and Chris Heunen (University of Edinburgh).
Seminar by Mikhail Gorsky (University of Vienna) Abstract: A braid variety is a certain affine algebraic variety associated with a simple algebraic group G and a positive braid of the corresponding type. These varieties generalise open Richardson varieties and appear in the context of symplectic topology and in the study of link invariants such as HOMFLY-PT polynomials and Khovanov-Rozansky homology. In this talk, I will give a proof of the existence of cluster A-structures and cluster Poisson structures on any braid variety. I will sketch an explicit construction of cluster seeds involving the diagrammatic calculus of weaves and explain certain properties of these cluster algebras such as local acyclicity. The talk is based on joint work with Roger Casals, Eugene Gorsky, Ian Le, Linhui Shen, and José Simental (arXiv:2207.11607).
Seminar by Côme Dattin (Uppsala University) Abstract: Starting with a Riemannian manifold M, its unit bundle UM is a contact manifold, whose Reeb vector field lifts the geodesic flow. Given a submanifold N, its unit conormal is a Legendrian submanifold in UM. Thus this construction takes us from the smooth world to the contact world, and one may use Legendrian invariants to study smooth (sub)manifolds.In this talk we will show that, if the unit conormals of two surface braids are Legendrian isotopic (relatively to their boundary), then the braids are equivalent. The main tool will be an invariant of Legendrians with boundary, called the stopped sutured homology. Using its associated exact sequence, we recover a classical invariant of braids, namely an automorphism of the punctured surface.
Seminar by Ingmar Saberi (University of Munich) Abstract:In recent years, there has been a great deal of progress on ideas related to twisted supergravity, building on the definition given by Costello and Li. Much of what is explicitly known about these theories comes from the topological B- model, whose string field theory conjecturally produces the holomorphic twist of type IIB supergravity. Progress on eleven-dimensional supergravity has been hindered, in part, by the lack of such a worldsheet approach. I will discuss a rigorous computation of the twist of the free eleven-dimensional supergravity multiplet, as well as an interacting BV theory with this field content that passes a large number of consistency checks. Surprisingly, the resulting holomorphic theory on flat space is closely related to the infinite-dimensional exceptional simple Lie superalgebra E(5|10), and another such exceptional simple algebra - E(3|6) - appears in the context of M5-branes. This is joint work with Surya Raghavendran and Brian Williams.
Seminar by Bingyu Zhang (SDU, QM) Abstract: To bounded open sets in a cotangent bundle, we can associate a pair of sheaves, the microlocal kernels, following the idea of Tamarkin and Chiu. Using microlocal kernels for the open set, we can define the Chiu-Tamarkin complex, which is a cohomology invariant associated with the open set. In this talk, I would like to explain the construction of these cohomology theories and some computations. Then I would like to explain how to construct a sequence of capacities based on the Chiu-Tamarkin complex.
By H2 we denote the Lie algebra of polynomial hamiltonian vector fields on the plane. We consider the moduli space of stable twisted Higgs bundles on an algebraic curve of given coprime rank and degree. De Cataldo, Hausel and Migliorini proved in the case of rank 2 and conjectured in arbitrary rank that two natural filtrations on the cohomology of the moduli space coincide. One is the weight filtration W coming from the Betti realization, and the other one is the perverse filtration P induced by the Hitchin map. Motivated by computations of the Khovanov-Rozansky homology of links by Gorsky, Hogancamp and myself, we look for an action of H2 on the cohomology of the moduli space. We find it in the algebra generated by two kinds of natural operations: on the one hand we have the operations of cup product by tautological classes, and on the other hand we have the Hecke operators acting via certain correspondences. We then show that both P and W coincide with the filtration canonically associated to the sl2 subalgebra of H2. Based on joint work in progress with Hausel, Minets and Schiffmann.Read more about Assistant Professor Anton Mellit from University of Vienna.