Workshop Report

Focus Week: New Invariants and Wall Crossing

Hirosi Ooguri
Principal Investigator

Yukinobu Toda
IPMU Associate Professor

Domenico Orlando
IPMU Postdoctoral Fellow

Susanne Reffert
IPMU Postdoctoral Fellow

This year marks the 80th anniversary of quantum
field theory. Quantum field theory is a programand physicists, the research opportunity in quantum
field theory has not escaped our attention.

to apply the principles of quantum mechanics toThough physicists use integrals over infinite

fields such as the electromagnetic field, and it isdimensional spaces of field configurations to

the fundamental language of elementary particle

perform computations in quantum field theory,physics. However, it is incomplete as a mathematicalsuch integrals are not well-defined. Fortunately, fortheory. The mathematical formulation of the Yang-certain quantities in supersymmetric field theories,

Mills theory (an important example of quantum field theory)the infinite dimensional integrals reduce to finite

has been posed as one of the 7 Millennium Problems

dimensional integrals, for which precise mathematicalby the Clay Mathematics Institute along with theapproaches are possible. This Focus Week has

Poincare Conjecture, the Riemann Hypothesis andgathered leading mathematicians and physicists

the P versus NP Problem.

working on quantum invariants defined by such

Although quantum field theory has not yet

integrals. The wall crossing in the title of the Focus

become a part of mathematics, some approximate

Week refers to a phenomenon where the values ofresults by physicists have been made into precise

quantum invariants change by some kind of phasemathematical conjectures, and they have inspiredtransition. It has important consequences both indevelopments in modern mathematics. For example,mathematics and in physics. In mathematics, it has

roughly 40% of Fields Medalists in mathematics

provided new insights into the classification of higher

since 1990 have worked in areas closely related todimensional geometry. In physics, it has played

quantum theory. In return, such developments havefundamental roles in deriving low energy effective

provided powerful new tools to study quantum

theories from superstring theory, in analyzing

field theory and superstring theory. Since one of thequantum states of black holes, and in studying

stated goals of IPMU is to generate new mathematics

strongly coupled effects in gauge theories.

by enhancing collaborations between mathematicians16

In response to the rapid progress in the field,

IPMU News No. 6 June 2009

Conference in the Time of Influenza

this Focus Week was quickly put together in threeinterdisciplinary meeting, crossing walls between

months by the organizational committee consisting

mathematics and physics.

of 1 mathematician and 3 physicists. It gathered

Since the Focus Week was held in the midst of

11 invited lecturers from all over the world, among

the H1N1 Influenza pandemic, every possible step

whom 7 were mathematicians and 4 were physicists.

was taken to ensure the health and well-being of

The organizers made various arrangements tothe participants. We placed alcohol-based handenhance interactions between mathematicians andsanitizers at the 2 entrances of the conference

physicists. For example, each invited lecturer wasbuilding and asked everybody to use them. We

asked to prepare two hour lectures. The first hour ofbought 700 surgical masks for the conference,each lecture was to aim at a broad audience. Afterencouraged participants to get new masks every

a 10 minute break, latest results were presented inday and reminded them to wear them. In thethe second hour. This format turned out to be very

opening address, we made a special presentation topopular. Here is a typical quote from the comments

demonstrate how to wear and dispose of the masks

sent by one of the invited lecturers after the meeting:properly. In addition to asking all the participants toI really liked the idea of the format of the talk withmeasure their body temperatures in every morning

two hours, one geared to the general audience andand evening, we set up a thermographic camera at

one more specialized. Personally I've learned a lot

the exit of the conference hall and measured thefrom the first hours of the lectures.

body temperatures of all the participants before

In addition to invited lectures, we solicited talks bylunch every day. Participants cooperated with us

young researchers. Among them, 4 mathematicianswith a sense of humor; some called the meetingand 4 physicists were selected to give a 45 minuteas Conference in the Time of Influenza making

talk each. The total number of registered participantsa parody of the famous novel＊ by Gabriel García

was 66, among whom 32 were mathematicians

Márquez.

and 34 were physicists. It was genuinely an＊ Love in the Time of Cholera

Workshop17