Speakers and Schedule

Timetable and abstracts

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Monday, Sep 19

08:30 -  09.25      Registration

09.25 – 09.30      Welcome

09.30 – 10.30      Chongying Dong

Regularity of vertex operator algebras

Abstract: We will present our recent result on the regularity of vertex operator algebras. That is, if any Z-graded weak module for a vertex operator algebra V is completely reducible, then V is rational and C_{2}-cofinite. This gives a natural characterization of regular vertex operator algebras. The result can be regarded as the first step in proving a well known conjecture: rationality implies the C_2-cofiniteness.

10.30 – 11.00      Coffee

11.00 – 12.00      Jan Hendrik Bruinier

On the converse theorem for Borcherds products

Abstract: Richard Borcherds constructed a lift from vector valued weakly holomorphic elliptic modular forms of weight $1-n/2$ to meromorphic modular forms for the discriminant kernel subgroup $\Gamma(L)$ of the orthogonal group of an even lattice $L$ of signature $(2,n)$. The forms in the image of the lift have their zeros and poles on Heegner divisors with known multiplicities. Since they have particular infinite product expansions, they are often called "Borcherds products".

We consider a given meromorphic modular form $F$ for the group $\Gamma(L)$ whose zeros and poles lie on Heegner divisors. The converse theorem then states that, under certain assumptions on $L$, the form $F$ has to be the Borcherds lift of a weakly holomorphic modular form. We present some new results on this problem.

12.00 – 14.30      Lunch break and discussions

14.30 – 15.30      Michael Tuite

Vertex operator algebras on Riemann surfaces

Abstract: We describe recent progress in defining and computing the partition function and correlation functions for a Vertex Operator Algebra (VOA) on a Riemann surface formed by sewing together lower genus surfaces. We consider the Heisenberg VOA on a genus $g$ surface where such functions can be computed by application of a version of the MacMahon Master Theorem from classical combinatorics. We also discuss the modular properties of the partition function on a Riemann surface formed by multiple sewings of tori for which Siegel modular form like automorphic properties hold.

15.30 – 16.00      Coffee

16.00 – 17.00      Hiroshi Yamauchi

Trace identity and axial vectors for the Baby-monster

Abstract: We introduce a notion of the extended Griess algebra associated to a vertex operator superalgebra (SVOA) and derive a trace identy based on structure of conformal design.

Our trace identity is a generalization of Matsuo-Norton trace formula. Then we will apply our identity to the Baby-monster SVOA and exhibit that there exists a one-to-one correspondence between 2A-elements of the Baby-monster and N=1 c=7/10 Virasoro vectors of the Baby-monster SVOA.

Tuesday, Sep 20

09.00 – 10.00      Volker Schomerus

Abstract: The way strings perceive geometry is distinct from the usual geometric concepts that are associated with point particles. These differences have intrigued mathematicians and physicists for several decades. The celebrated AdS/CFT correspondence has added new relevance in particular to the string geometry of certain coset superspaces. After a brief overview over the concepts and motivations, I will review some recent progress. Links with the theory of affine Lie superalgebras, logarithmic conformal field theory and generalizations of theta functions are discussed.

Abstract:

10.00 – 10.30      Coffee

10.30 – 11.30      Terry Gannon

Vector-valued automorphic forms and the Riemann-Hilbert problem

Abstract: In this talk I'll sketch the basic theory of vector-valued automorphic forms for arbitrary finite-index subgroups of any genus-0 Fuchsian group of the first kind, and arbitrary representation and arbitrary weight. I'll describe the analogues here of Grothendieck's Thm, Riemann-Roch, Serre duality, etc and show they can be sharpened into effective tools (e.g. for finding dimensions and basis vectors). A crucial role is played by Fuchsian differential equations. I'll focus on the most familiar case of SL(2,Z), where there are plenty of direct applications to physics, geometry and algebra. This is joint work with Peter Bantay.

11.30 – 12.30      Gerald Höhn

Classification approaches for rational vertex operator algebras

Abstract: I will discuss classification approaches for rational vertex operator algebras and present some new results.

I will also describe an ongoing joint project of a database of vertex operator algebras and modular categories. A preliminary version of the database can be accessed at http://www.math.ksu.edu/~gerald/voas/

12.30 – 14.30      Lunch break and discussions

14.30 – 15.30      Scott Carnahan

Conformal blocks on nodal curves

Abstract: Frenkel and Ben-Zvi gave a method for attaching a space of conformal blocks to the data of a smooth complex algebraic curve, a quasi-conformal vertex algebra, and modules placed at points. Furthermore, when the vertex algebra has conformal structure, one obtains sheaves of conformal blocks with projectively flat connection on moduli spaces of smooth curves with marked points. I'll describe how logarithmic geometry can be employed to canonically extend these sheaves to the semistable locus, where the connection acquires at most logarithmic singularities. When one has a finite group G acting by automorphisms of the conformal vertex algebra, one may construct equivariant intertwining operators by varying ramified G-covers of the projective line.

15.30 – 16.00      Coffee

16.00 – 17.00      Viacheslav Nikulin

The transition constant for arithmetic hyperbolic reflection groups

Abstract: The transition constant was introduced in our 1981 paper and denoted as N(14). This constant is fundamental since if the degree of the ground field of an arithmetic hyperbolic reflection group is greater than N(14), then the field comes from special plane reflection groups. In recent paper, we gave its upper bound 56. Using similar but more difficult considerations, here we show that the upper bound is 25.

As applications, we show that the degree of ground fields of arithmetic hyperbolic reflection groups in dimensions at least 6 has the upper bound  25 (it was 56 before); in dimensions 5, 4, and 3 it has the upper bound 44 (in our papers, it was 138, and 909 before).

These results and developed methods will be important for further classification of these groups. See details in arXiv:0910.5217 .

Wednesday, Sep 21

09.00 – 10.00      Hiroki Shimakura

Towards the classification of framed holomorphic vertex operator algebras of central charge 24

Abstract: In this talk, we discuss our recent work on the classification of holomorphic framed vertex operator algebras of central charge 24 based on the classification of triply even codes of length 48.

The main topic is the classification of holomorophic simple current extensions of the tensor product of three copies of the vertex operator algebra $V_{\sqrt2E_8}^+$. In particular, we obtain new holomorphic framed vertex operator algebras of central charge 24.

10.00 – 10.30      Coffee

10.30 – 11.30      Antun Milas

Explicit construction of logarithmic modules in CFT

Abstract: I will present an explicit realization of logarithmic modules (including some projective covers) in the known rational logarithmic conformal field theories. This can be then used to construct one-point functions over elliptic curves. The talk is mostly based on recent joint work with D. Adamovic.

14.00                            Neckar River Cruise and Walk

Thursday, Sep 22

09.00 – 10.00      Matthias Gaberdiel

Mathieu Moonshine

Abstract: A brief introduction to the idea of Mathieu Moonshine will be given, i.e. to the observation of Eguchi, Ooguri and Tachikawa that the Fourier coefficients of the elliptic genus on K3 have an interpretation in terms of M24 representations.

I shall describe evidence in favour of this idea, and explain how some aspects of it can be understood by studying the symmetries of sigma model CFTs on K3. (This is based on joint work with S Hohenegger and R Volpato.)

10.00 – 10.30      Coffee

10.30 – 11.30      Nils Scheithauer

Modular forms for the Weil representation and Borcherds’ conjecture

Abstract: In 1995 Borcherds conjectured that the twisted denominator identities of the fake monster algebra under Conway's group are automorphic forms of singular weight on orthogonal groups. We describe the final steps in the proof.

11.30 – 12.30      Pierre Vanhove

Automorphic properties of string theory amplitudes in various dimensions

Abstract: Maximally supersymmetric string theory compactified on d-dimensional torus is invariant under the non-perturbative U-duality groups E_{d+1}(Z). These duality groups are the discrete version over the integers of the real split forms of the simply laced lie groups of rank d. The interactions of the theory are given by automorphic function under E_{d+1}(Z),  given in the simplest cases by the residue of maximal parabolic Eisenstein series. The dimensional reduction of the theory to three dimensions gives rise to the duality group E_8 from which all BPS couplings in higher dimensions can be extracted. We will present the construction of the automorphic  form relevant for the supersymmetric protected BPS  interactions in the low-energy expansion of maximal supersymmetric effective action. We will discuss the physical interpretation and the implication of these results, and as well as some (conjectured) relations between these automorphic forms and representation of nilpotent orbits.

This is based some work done in collaboration  with Michael B. Green, Stephen D. Miller, Jorge Russo

12.30 – 14.30      Lunch break and discussions

14.30 – 15.30      Alexander Zuevsky

Twisted correlation functions on self-sewn Riemann surfaces via generalized vertex algebra of intertwiners

Abstract: We introduce the intertwined partition and $n$-point correlation functions for a vertex operator superalgebra on a genus two Riemann surface formed by self-sewing of the torus.  In fundamental example of the free fermion vertex operator superalgebra we prove that the intertwined partition function is holomorphic in the sewing parameter. Using the explicit representation for vertex operators of a generalized vertex algebra for Heisenberg intetwiners [TZ2] we obtain a closed formula for the genus two intertwined partition function as an infinite dimensional determinant with entries arising from Szeg\H{o} kernel on original torus [TZ1]. We then compute the generating function for all genus two $n$-point twisted correlation functions in terms of the genus two Szeg\H{o} kernel determinant and discuss their modular properties. A genus two generalization for Jacobi triple product identity will be discussed [TZ1].

15.30 – 16.00      Coffee

Friday, Sep 23

09.00 – 10.00      Geoffrey Mason

Algebraic structure of strongly regular vertex operator algebras

Abstract: We present some results concerning the abstract structure of strongly regular vertex operator algebras (essentially, rational CFT). This suggests a reappraisal of the old idea of physicists (conformal bootstrap) of somehow classifying rational CFTs.

10.00 – 10.30      Coffee

10.30 – 11.30      Masahiko Miyamoto

Z_3-orbifold construction

Abstract: We prove that V_L^{\sigma} is C_2-cofinite for a lattice vertex operator algebra V_L associated with a positive definite even lattice L and an automorphism \sigma of V_L lifted from a triality automorphism of L. Using these results, we present two \Z_3-orbifold constructions as examples. One is the moonshine VOA V^{\natural} and the other is a new conformal field theory No.32 in Schellekens' list. We also talk about a few more generalized results.

11.30 – 12.30      Daniel Allcock

Finite presentation of Kac-Moody groups over Z

Abstract: Kac-Moody groups are analogues of the Chevalley groups, for infinite-dimensional Lie algebras.  Tits proved that they are characterized by certain natural properties (at least over fields), so one can speak of "the" Kac-Moody groups,  and he gave presentations for them.  These presentations are elaborations of Steinberg's presentation for Chevalley groups, and are "very infinite"---even over finite fields the relations are parameterized by pairs of not-necessarily simple roots with certain properties.  We have shown that in many cases, including the affine cases and some hyperbolic cases like E10, one may massage the presentation until it is a finite, if the base ring is finitely generated as a ring. The key ingredients are an understanding of centralizers in Coxeter groups and a geometric argument in hyperbolic space. (Joint work with Lisa Carbone)

Afternoon at free disposal

19.00                   Conference Dinner

Last Updated on Sunday, 02 October 2011 10:09