### Quantum field theory and topology

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### Frobenius algebras and 2D topological quantum field theories

Williams, I will explain a new version of the construction by Rietsch of a mirror for some varieties with a homogeneous Lie group action. The varieties we study include quadrics and Lagrangian Grassmannians i. The mirror takes the shape of a rational function, the superpotential, defined on a Langlands dual homogeneous variety. I will also explain how these properties lead to new relations in the quantum cohomology, and a conjectural formula expressing solutions of the quantum differential equation in terms of the superpotential.

If time allows, I will also explain how these results should extend to a larger family of homogeneous spaces called cominuscule homogeneous spaces. Mirror Symmetry provides a link between different suites of data in geometry. On one hand, one has a lot of enumerative data that is associated to curve counts, telling you about important intersection theory in an interesting moduli problem. On the other, one has a variation of Hodge structure, that is, complex algebro-geometric structure given by computing special integrals.

While typically one has focussed on the case where we study the enumerative data for a symplectic manifold, we here will instead study the enumerative geometry of a Landau-Ginzburg model. This is joint work in preparation with Mark Gross and Ran Tessler. Subvarieties of Grassmannians and especially Fano varieties obtained from sections of homogeneous vector bundles are far from being classified.

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This talk will be mainly devoted to the construction of some new examples of such varieties. This is a work in progress with Giovanni Mongardi. R-matrices are solutions to the Yang-Baxter equation, which was introduced as a consistency equation in statistical mechanics, but has since then appeared in many other research areas, for example integrable quantum field theory, knot theory, the study of Hopf algebras and quantum information theory.

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Twisted K-theory on the other hand is a variant of topological K-theory that allows local coefficient systems called twists. Twists over Lie groups gained increasing importance in the subject due to a result by Freed, Hopkins and Teleman that relates the twisted equivariant K-theory of the group to the Verlinde ring of the associated loop group. In this talk I will discuss how involutive R-matrices give rise to a natural generalisation of the twist appearing in this theorem via exponential functors.

The relation between conformal and dilation covariance is a controversial problem in QFT. Although many models which are dilation covariant are indeed conformal covariant a complete understanding of this implication in the algebraic approach to QFT is missing. Additional properties are either a certain condition on modular covariance, or a variant of strong additivity.

Time permitting we further discuss counterexamples. I will discuss recent progress connecting the physics of certain large classes of 4D superconformal field theories with logarithmic conformal field theories. I will then use this connection to discuss a bridge between the physics of these 4D theories and certain more familiar 2D rational conformal field theories. Quantum computers are designed to use quantum mechanics to outperform any standard, "classical" computer based only on the laws of classical physics. Following many years of experimental and theoretical developments, it is anticipated that quantum computers will soon be built that cannot be simulated by today's most powerful supercomputers.

In this talk, I will begin by introducing the quantum computational model, and describing the famous quantum algorithm due to Grover that solves unstructured search problems in approximately the square root of the time required classically. I will then go on to describe more recent work on a quantum algorithm to speed up classical search algorithms based on the technique known as backtracking "trial and error" , and very recent work on calculating the level of quantum speedup anticipated when applying this algorithm to practically relevant problems.

The talk will aim to give a flavour of the mathematics involved in quantum algorithm design, rather than going into the full details. Quantum walk speedup of backtracking algorithms, Theory of Computing to appear ; arXiv Recently two approaches to twisting of the real structure of spectral triples were introduced. In this talk we present and compare these two approaches. The Yang-Baxter equation YBE is a cubic matrix equation which plays a prominent in several fields such as quantum groups, braid groups, knot theory, quantum field theory, and statistical mechanics.

Its invertible normal solutions "R-matrices" define representations and extremal characters of the infinite braid group. These characters define a natural equivalence relation on the family of all R-matrices, and I will describe a research programme aiming at classifying all solutions of the YBE up to this equivalence. Instead they are busy trying to pass to pass their course.

And let many opportunities pass by. The notion of the instanton or more generally the soliton -- these are the things our partition functions are counting. Individual solutions to partial differential equations may be too difficult to evaluate point-wise but we can still get qualitative information about solutions to PDE. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Why is Quantum Field Theory so topological? Ask Question. Asked 2 years, 4 months ago.

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Active 1 year, 8 months ago. Viewed 4k times. Abdelmalek Abdesselam A Physical newbie A Physical newbie 4 4 silver badges 3 3 bronze badges. The most successful quantum field theories rely on the mathematical background of group theory because they're all gauge theories and the properties of groups are inherently important to gauge theory.

If you imagine a world where calculus wasn't invented to solve physics problems, you might be able to ask the same question along the lines of "why is calculus so heavily utilized in mechanics? Why nature is this way, that's probably a philosophical question. Topological ideas in fluid dynamics go back to 19th century, where vortices, vortex filaments and vorticity tubes, and their topological features were a topic of study by Helmholtz, Lord Kelvin etc. Ofcourse modern fluid dynamicists have taken up those topological ideas and developed them in many different directions.

Aaron Bergman Aaron Bergman 2, 1 1 gold badge 17 17 silver badges 28 28 bronze badges. Jaffe's review referenced by Robert Israel below seems like a good place to start. See my answer to this post for some pointers.

## Differential Topology and Quantum Field Theory by Charles Nash

Here is a stab at answering these two very different questions. Question A: This is moot since it is based on a false premise. To get an idea of what is going on in the field, have a look at the reports for the two recent Oberwolfach meetings: "Recent Mathematical Developments in Quantum Field Theory" "The Renormalization Group". Another recent meeting around the new developments by Hairer and others in the strongly related field of stochastic quantization is: "Rough Paths, Regularity Structures and Related Topics".

Question B: This one is not moot. Abdelmalek Abdesselam Abdelmalek Abdesselam Robert Israel Robert Israel So basically it has nothing to do with the OP's question. There is a lot to disagree with, but perhaps that's the strength of his argument. This is kind of misleading since I don't think we have understood the geometric content of E and M we don't understand the geometry of solutions to Maxwell's equations e.

### Head of Group

Francois Ziegler Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Instantons Pages Schwarz, Albert S. Groups Pages Schwarz, Albert S. Gluings Pages Schwarz, Albert S. Show next xx. Read this book on SpringerLink. Services for this Book Download Product Flyer. Recommended for you. Schwarz Translated by Yankowsky, E. PAGE 1.