Entrance hall of the ICMS

Lie algebras, vertex algebras and automorphic forms

Aug 31, 2009 - Sep 04, 2009

15 South College Street, Edinburgh EH8 9AA

Organisers

Name Institution
Höhn, Gerald Kansas State University
Nikulin, Viacheslav University of Liverpool
Scheithauer, Nils Technische Universität Darmstadt

Group photograph taken outside ICMS at 15 South College Street is available here.

In 1968, Kac and Moody introduced a new class of infinite dimensional Lie algebras, called Kac-Moody algebras, which generalize the finite dimensional simple Lie algebras in a natural way. These Lie algebras became very important in many areas of mathematics. In 1986, Borcherds defined vertex algebras as an axiomatic framework for conformal field theories. Certain subquotients of these algebras have a Lie algebra structure. The Lie algebras obtained in this way are very similar to Kac-Moody algebras but might have imaginary simple roots. These generalized Kac-Moody algebras describe symmetries in string theory. The most interesting generalized Kac-Moody algebras have denominator identities which are automorphic forms on orthogonal groups.

The relations between generalized Kac-Moody algebras, vertex algebras and automorphic forms have led to fascinating results e.g. Borcherds' proof of Conway and Norton's moonshine conjecture for the monster. However, many important problems are still open including:

  • Generalized Moonshine
  • Moonshine for other groups
  • Classification of vertex algebras and generalized Kac-Moody algebras
  • Construction of vertex algebras and generalized Kac-Moody algebras
  • Automorphic forms related to vertex algebras and generalized Kac-Moody algebras
  • Related areas as hyperbolic reflection groups and moduli spaces

In this workshop we present the recent achievements in these problems. We also want to discuss possible new approaches to them.


PDF file of full report.

Arrangements

Venue
The workshop will be held at 15 South College Street, Edinburgh. All lectures will be held in the Newhaven Lecture Theatre. To view this room and a list of the visual equipment available click here. In addition, an overhead projector will be available to use and two blackboards have recently been installed. Follow this link for a map showing the location of 15 South College Street.

Computer Access
The workshop venue, 15 South College Street, has wireless access throughout. On arrival at Registration you will be given instructions and a code for accessing the wireless network. If you wish to use a computer while in 15 South College Street please ask a member of the ICMS staff.

Travel
Information about travel to the UK and Edinburgh is available here. You may find this map useful.
Lothian buses charge 1.20 GBP for a single journey and 3.00 GBP for a day ticket. Please note that the exact fare is required and no change is given.

A taxi directly from the airport will cost approximately 15.00 to 20.00 GBP to the city centre for a one-way journey.

From the airport, you can take the airport bus Airlink to the city centre. It will cost you 3.50 GBP for a single ticket, or 6.00 GBP return. More details on the airport bus Airlink is available here. The bus will take you to Waverley Bridge (next to Waverley Railway Station).

If travelling by train, please note that Edinburgh has two railway stations - Waverley Railway Station being the main station and closest to the workshop venue at 15 South College Street. If you alight the train at Edinburgh Waverley, the workshop venue is an easy 10 minute walk over North and South Bridge map.

Accommodation

ICMS has arranged accommodation for participants who require it. En-suite rooms have been booked in two locations (unless otherwise requested): The Kenneth MacKenzie Suite in the city centre, and Chancellor's Court in the Pollock Halls area. A map of the location of the two accommodation venues and the location of the workshop venue can be found here.

Participants are also free to make their own arrangements. A list of Edinburgh accommodation of various sorts and prices can be found on the Accommodation Page on the ICMS web site. Section 4 is particularly relevant. September is busy in Edinburgh and early booking is recommended.

Meals and Refreshments
Morning and afternoon refreshments will be provided on each day of the workshop.

On Monday 31 August, the first day of the workshop, a light buffet lunch will be provided free of charge to participants in the in the Chapterhouse on the ground floor of 15 South College Street. For the remainder of the days, participants are free to go out for lunch and explore the many cafes, restaurants, sandwich shops and bars in the surrounding area. On arrival we will provide you with a ‘welcome’ pack which will contain information about eating places nearby.

Following the last talk on Tuesday 1 September, an informal wine reception, to which all all participants are invited, will take place in the Chapterhouse on the ground floor of 15 South College Street. The workshop dinner will take place at 19.30 at Blonde, 75 St. Leonards's Street, Edinburgh, EH8 9QR, on the evening of Thursday 3 September. The workshop grant will cover the cost of this catering.

Registration
Registration will take place on Monday 31 August from 9.00 to 10.00 in the Chapterhouse on the ground floor of 15 South College Street.

Financial Arrangements
Unless otherwise specified in your invitation letter, the workshop grant will cover the cost of your bed and breakfast accommodation, tea/coffee throughout the workshop, lunch on Monday only, the wine reception, and the Workshop Dinner on Thursday evening.

If we have agreed to pay some of your travel costs, you will be informed by email. Reimbursement will take place after the workshop and will involve payment directly into your bank account. At Registration you will be given an expenses claim form and this should be submitted to ICMS, with original receipts. It would be helpful if you could bring your bank details to the workshop. In addition to the bank account number, participants from the USA and Canada will require their bank’s routing number, those from the UK will be asked for the bank sort code, and those from Europe and the rest of the world, their IBAN and SWIFT/BIC code. We cannot reimburse any item without a signed claim form and original receipts.

Unless otherwise stated in your invitation or further correspondence, there will be a registration fee of 40.00 GBP for the workshop. We ask that this is paid in advance by using this credit or debit card payment form. The form should be printed out, completed and faxed back (as email is not a secure way of sending credit card information). The fax number is on the form. If it is not possible for you pay in advance, you may print out the credit/debit card form above and bring the completed form along to Registration. We prefer not to handle cash at Registration. Please note we do not accept American Express or Discover Card. Alternatively, we can accept sterling cheques. The cheque should be made payable to Heriot-Watt University and can be handed over at Registration.

Knowledge transfer event

Date 1 September
Time
16.45 to 20.00 (See the below programme)

As part of the workshop, Victor Kac (MIT) will deliver a lecture on Tuesday 1 September from 17.30 to 18.30 in the Newhaven lecture theatre of the e-Science building, 15 South College Street, Edinburgh. This talk will be opened up to mathematicians and mathematical physicists from all Scottish Universities.

Those interested in attending only the lecture (and not the rest of the workshop) should register their interest by emailing Johanna Kytöharju at ICMS, giving the following information:

  • your name and institution

A wine reception for all attendees will follow the lecture.

Please note the workshop participants have been automatically registered for this talk.


Programme

Monday 31 August

09.00 - 10.00Registration and coffee, Ground Floor, 15 South College Street
10.00 - 10.15Welcome
10.15 - 11.00Viacheslav Nikulin (University of Liverpool) PDF of Presentation
On the classification of arithmetic hyperbolic reflection groups
11.15 - 12.00Rudolf Scharlau (Technische Universität Dortmund) PDF of Presentation
On the explicit construction of arithmetic hyperbolic reflection groups
12.00 - 14.00Lunch provided in the Chapterhouse, Ground Floor, 15 South College Street
14.00 - 14.45Daniel Allcock (University of Texas at Austin)
On the classification of reflective Lorentzian lattices
15.00 - 15.45Tathagata Basak (University of Chicago) PDF of Presentation
A complex hyperbolic reflection group and the bimonster
15.45 - 16.15Tea/Coffee
16.15 - 17.00Robert Griess (University of Michigan)
Automorphism groups of VOAs

 

Tuesday 1 September

09.00 - 09.45Gerald Höhn (Kansas State University) PDF of Presentation
Vertex algebra constructions of generalized Kac-Moody algebras
09.45 - 10.15Tea/Coffee
10.15 - 11.00Scott Carnahan (Massachusetts Institute of Technology) PDF of Presentation
Singular commutative rings in a braided category
11.15 - 12.00Antun Milas (University at Albany, SUNY)
W-algebra extensions of minimal models
12.00 - 14.00Lunch break
14.00 - 14.45Masahiko Miyamoto (University of Tsukuba) PDF of presentation
Semirigidity and flatness of C_2-cofinite VOAs 
15.00 - 15.45Ching Hung Lam (Academia Sinica, Taipei) PDF of Presentation
Virasoro frames and structure codes of the moonshine vertex operator algebra
16.00 - 16.45John McKay (Concordia University)
Absorbing sporadics
 Knowledge Transfer event
16.45 - 17.30Tea/Coffee
17.30 - 18.30Victor Kac (MIT) PDF of Presentation
Quantization and chiralization
18.30 - 20.00Wine Reception in the Chapterhouse, 15 South College Street, Edinburgh

 

Wednesday 2 September

09.00 - 09.45Nils Scheithauer (Technische Universität Darmstadt) PDF of Presentation
On the classification of automorphic products and generalized Kac-Moody algebras
10.00 - 10.45Valery Gritsenko (Laboratoire Paul Painlevé)
The baby functions of the Borcherds form Phi_12
10.45 - 11.15Tea/Coffee
11.15 - 12.00Shigeyuki Kondo (Nagoya University) PDF of Presentation
The moduli spaces of Enriques, Coble surfaces and automorphic forms
12.15 - 13.00Nils-Peter Skoruppa (Universität Siegen)
Weil representations of finite quadratic modules and automorphic forms of critical weight
13.00 - Free afternoon

 

Thursday 3 September

09.00 - 09.45Urmie Ray (Université de Reims)
On Cartan subalgebras
09.45 - 10.15Tea/Coffee
10.15 - 11.00Chris Cummins (Concordia University)
Torsion free, genus one congruence subgroups, eta products and monstrous moonshine
11.15 - 12.00John Duncan (Harvard University)
Rademacher sums, monstrous moonshine and black holes
12.00 - 14.00Lunch break
14.00 - 14.45Miranda Cheng (Harvard University) PDF of Presentation
Wall-crossing, black holes and generalized Kac-Moody algebras
15.00 - 15.45Christoph Schweigert (Universität Hamburg) PDF of Presentation
Mapping class group representations and conformal boundary conditions
15.45 - 16.15Tea/Coffee
16.15 - 17.00Shrawan Kumar (University of North Carolina) PDF of Presentation
The Cachazo-Douglas-Seiberg-Witten conjecture for simple Lie algebras and its generalization to symmetric spaces
19.30 - Workshop Dinner at Blonde Restaurant, 75 St Leonard's Street, Edinburgh

 

Friday 4 September

09.00 - 09.45Peter Bantay (Eötvös Loránd University) PDF of Presentation
Vector-valued modular forms
09.45 - 10.15Tea/Coffee
10.15 - 11.00Terry Gannon (University of Alberta)
Mostly moonshine and vector-valued modular forms
11.15 - 12.00Michael Tuite (National University of Ireland, Galway) PDF of Presentation
Vertex operator superalgebras on a genus two Riemann surface
12.00 - 14.00Lunch break
14.00 - 14.45Hiroshi Yamauchi (Aichi University of Education) PDF of Presentation
n-transposition groups and vertex operator algebras
15.00 - 15.45Chongying Dong (University of California, Santa Cruz) PDF of Presentation
A characterization of the vertex operator algebra L(1/2,0) otimes L(1/2,0)
15.45 - 16.15Tea/Coffee
16.15 - 17.00Geoffrey Mason (University of California, Santa Cruz) PDF of Presentation
Vector-valued modular forms and logarithmic field theories

 

Presentations:

Presentation Details
Allcock, Daniel
On the classification of reflective Lorentzian lattices
View Abstract Down
Bantay, Peter
Vector-valued modular forms
View Abstract Down
Basak, Tathagata
A complex hyperbolic reflection group and the bimonster
View Abstract Down
Carnahan, Scott
Singular commutative rings in a braided category
View Abstract Down
Cheng, Miranda C.N.
Wall-crossing, black holes and generalized Kac-Moody algebras
View Abstract Down
Cummins, Chris
Torsion free, genus one congruence subgroups, eta products and monstrous moonshine
View Abstract Down
Dong, Chongying
A characterization of the vertex operator algebra L(1/2,0) otimes L(1/2,0)
View Abstract Down
Duncan, John
Rademacher sums, monstrous moonshine and black holes
View Abstract Down
Gannon, Terry
Mostly moonshine and vector-valued modular forms
View Abstract Down
Griess, Robert
Automorphism groups of VOAs
View Abstract Down
Gritsenko, Valery
The baby functions of the Borcherds form Phi_12
View Abstract Down
Höhn, Gerald
Vertex algebra constructions of generalized Kac-Moody algebras
View Abstract Down
Hung Lam, Ching
Virasoro frames and structure codes of the moonshine vertex operator algebra
View Abstract Down
Kac, Victor
Quantization and chiralization
View Abstract Down
Kondo, Shigeyuki
The moduli spaces of Enriques, Coble surfaces and automorphic forms
View Abstract Down
Kumar, Shrawan
The Cachazo-Douglas-Seiberg-Witten conjecture for simple Lie algebras and its generalization to symmetric spaces
View Abstract Down
Mason, Geoffrey
Vector-valued modular forms and logarithmic field theories
View Abstract Down
McKay, John
Absorbing sporadics
View Abstract Down
Milas, Antun
W-algebra extensions of minimal models
View Abstract Down
Miyamoto, Masahiko
Semirigidity and flatness of C_2-cofinite VOAs
View Abstract Down
Nikulin, Viacheslav
On the classification of arithmetic hyperbolic reflection groups
View Abstract Down
Ray, Urmie
On Cartan subalgebras
View Abstract Down
Scharlau, Rudolf
On the explicit construction of arithmetic hyperbolic reflection groups
View Abstract Down
Scheithauer, Nils
On the classification of automorphic products and generalized Kac-Moody algebras
View Abstract Down
Schweigert, Christoph
Mapping class group representations and conformal boundary conditions
View Abstract Down
Skoruppa, Nils-Peter
Weil representations of finite quadratic modules and automorphic forms of critical weight
View Abstract Down
Tuite, Michael
Vertex operator superalgebras on a genus two Riemann surface
View Abstract Down
Yamauchi, Hiroshi
n-transposition groups and vertex operator algebras
View Abstract Down

Participants

Name Institution
Allcock, Daniel University of Texas at Austin
Bantay, Peter Eötvös Loránd University
Barraclough, Richard Goldsmiths' College, University of London
Basak, Tathagata University of Chicago
Carnahan, Scott Massachusetts Institute of Technology
Cheng, Miranda C.N. Harvard University
Cooney, Nicholas University of Oxford
Cummins, Chris Concordia University
Docherty, Pamela University of Edinburgh
Dong, Chongying University of California, Santa Cruz
Duncan, John Harvard University
Gannon, Terry University of Alberta
Griess, Robert University of Michigan
Gritsenko, Valery Universite Lille 1
Höhn, Gerald Kansas State University
Hung Lam, Ching Academia Sinica, Taipei
Hurley, Donny National University of Ireland, Galway
Kac, Victor MIT
Kohnen, Winfried Universität Heidelberg
Kondo, Shigeyuki Nagoya University
Konechny, Anatoly Heriot-Watt University
Krauel, Matthew University of California, Santa Cruz
Kühn, Ulf Universität Hamburg
Kumar, Shrawan University of North Carolina
Lentner, Simon Ludwig-Maximillians-Universität München
Mason, Geoffrey University of California, Santa Cruz
McKay, John Concordia University
Milas, Antun University at Albany, SUNY
Miyamoto, Masahiko University of Tsukuba
Nikulin, Viacheslav University of Liverpool
Norton, Simon Cambridge University
Raulf, Nicole Université des Sciences et Technologies de Lille
Ray, Urmie Université de Reims
Scharlau, Rudolf Technische Universität Dortmund
Scheithauer, Nils Technische Universität Darmstadt
Schulze-Pillot, Rainer Universität des Saarlandes
Schweigert, Christoph Universität Hamburg
Skoruppa, Nils-Peter Universität Siegen
Spencer, Christopher University of Edinburgh
Tuite, Michael National University of Ireland, Galway
Yamauchi, Hiroshi Aichi University of Education
Zuevsky, Alexander National University of Ireland, Galway