Flat Connections in Physics and Geometry

From July 20 to 22, 2017, the Mathematical Physics group in the Mathematisches Institut at Universität Heidelberg will host a three-day workshop on recent topics at the intersection of Physics and Geometry. The workshop is intended to serve as an unofficial satellite workshop of String Math 2017, held in Hamburg July 24–29.

Confirmed speakers

• Matthew Bullimore (Oxford)
• Mykola Dedushenko (Pasadena)
• Ori Ganor (Berkeley)
• Owen Gwilliam (Bonn)
• Justin Hilburn (Philadelphia)
• Kentaro Hori (Tokyo)
• Si Li (Beijing)
• Hyenho Lho (Zurich)
• Mauricio Romo (Princeton)
• Piotr Sułkowski (Warsaw)
• Brian Williams (Evanston)
• Piljin Yi (Seoul)
• Michele del Zotto (Stony Brook)

Participants

Many of the equations that govern the mathematical applications of quantum field theory and string theory arise as flatness equations, either directly from the nilpotency of super-variations, or indirectly during a gauge-fixing procedure. In fact, dressed up with moduli and upon quantization, flat connections have been claimed to encompass our entire subject. An elementary but rich example are the moduli \( M_{\rm flat} \) of flat connections over Riemann surfaces, which occur as phase space of Chern-Simons theory, and whose quantization leads to the computation of dimensions of spaces of sections of certain bundles over \( M_{\rm flat}\). In the process, the theorem of Narasimhan-Seshadri identifies \( M_{\rm flat} \) with the moduli space of holomorphic vector bundles, which in turns allows one to describe them much more explicitly and concretely. When \( G_{\mathbb C} \) is a complex Lie group and \(G\) its compact real form, \( M_{\rm flat}(G_{\mathbb C})\) can be identified with the moduli space of \(G\)-Higgs bungles, connecting \(M_{\rm flat}\) to Hitchin's integrable systems, another flatness landmark.

This workshop views \( M_{\rm flat} \) and related ideas as the backdrop to many recent developments, a catalyst for progress, and a synecdoche for the many surprising connections between gauge theory, string theory, geometry and topology. Our idea is to bring together researchers in whose work \( M_{\rm flat} \) makes unusual or surprising appearances: people who have used or developed new or unusual techniques to study \( M_{\rm flat} \), who have proposed analogues of \( M_{\rm flat}\), or who may not even think of themselves as studying flat connections at all. We hope that people with different viewpoints on these connections will be able to benefit from one another's perspectives this Summer.

The workshop program will highlight recent applications of flat connections in analyzing BPS spectra, D-brane transport and stability, illustrate the power of homological algebra and derived algebraic geometry to address real physical problems, and showcase recent trends emerging from arithmetic questions, including flux quantization in string theory and various analogies between Galois symmetries in number theory and global and local symmetries in gauge theory.

We hope the conference will be fairly unique in bringing together mathematicians working on factorization algebras in QFT and physicists working on categorification, knot homologies, and defects in supersymmetric QFTs.

Thursday, July 20

Friday, July 21

Saturday, July 22

Venue

All talks will be held in the 5th floor Conference Room (5/104) in the Southern wing of the MATHEMATIKON.

Morning and afternoon coffee/tea and lunches will be served in the 5th floor common room.

Accommodation and Provisioning

The Heidelberger Hof hotel is conveniently located midway between the Neuenheimer Feld and the main attractions of old town Heidelberg. For a fast bus connection to Campus, board route 31 at Bismarckplatz, headed to Kopfklinik (not Universitätsplatz!) and get off 5 stops later at Bunsengymnasium. The main entrance of the MATHEMATIKON is across the street. Shops and restaurants are in the middle and Northern wing.

The pleasant walk from Bismarckplatz across the bridge, down the Neckar river, and up Berliner Strasse takes about 25 minutes.

Registration and Workshop Dinner

There is no registration fee, however we ask that all those interested in attending the talks send us an Email ahead of time, so we may have an idea of the number of prospective participants. (We reserve the right to raise a guard in the unlikely event of a general stampede.) In your message, please indicate whether or not you are interested in joining us for the workshop dinner on Friday night. You will be asked for a € 20 contribution for the dinner; further information to follow.

Richard Eager, Ingmar Saberi, Johannes Walcher