New moonshines, mock modular forms and string theory

London Mathematical Society -- EPSRC Durham Symposium

2015-08-03 to 2015-08-12

Schedule of Talks

(Click on title to view abstract.)

I will review the role of elliptic genus in study of topological invariants of singular algebraic varieties, McKay correspondence, modular properties of elliptic genus and recent study of elliptic genus of phases of N=2 theories. Several open problems also will be discussed.

In this talk I review some aspects of Monstrous Moonshine and Vertex Operator Algebras (VOAs). I also speculate on possible VOA approaches towards understanding Mathieu Moonshine.

The classification of the automorphism groups of K3 sigma models is reviewed. In many instances the symmetry group is a subgroup of M24, but there are examples where this is not the case. It is shown that most (if not all) of these exceptional examples are closely related to (cyclic) torus orbifolds. [This is based on joint work with Stefan Hohenegger and Roberto Volpato.]

The heterotic string compactified on T^2 has a large discrete symmetry group SO(2, 18; Z), which acts on the scalars in the theory in a natural way; there have been a number of attempts to construct models in which these scalars are allowed to vary by using SO(2, 18;Z)-invariant functions. In joint work with David Morrison, we give a more complete construction of these models in the special cases in which either there are no Wilson lines – and SO(2, 2;Z) symmetry – or there is a single Wilson line – and SO(2, 3; Z) symmetry. In those cases, the modular forms can be analyzed in detail and there turns out to be a precise theory of K3 surfaces with prescribed singularities which corresponds to the structure of the modular forms. This allows us to construct interesting examples of smooth Calabi-Yau threefolds as elliptic fibrations over Hirzebruch surfaces from pencils of irreducible genus-two curves.

We will discuss the important role played by weight one Jacobi forms in umbral moonshine.

In general, it is difficult to calculate the group of automorphisms of a K3 surface. In this talk, I will give a method to calculate it by using Conway's description of a fundamental domain of the reflection group of the even unimodular lattice $II_{1,25}$ of signature $(1,25)$. I will take a generic Jacobian Kummer surface as an example. The main idea of my talk is due to R. Borcherds.

In a joint work with Ben-Zvi and Sczcesny we showed the existence of an N=4 supercon- formal structures on the chiral de Rham complex of a Hyper- ̈ahler manifold in the smooth setting. This was later extended to show that in fact there exists two commuting N=4 structures on such a manifold. I will describe how from these seemingly old results it follows that the cohomology of the holomorphic chiral de Rham of such a manifold carries an N=4 superconformal structure.

In 1987, Norton proposed a strengthening of the Monstrous Moonshine conjecture. Among its assertions is the existence of a rule that produces special modular functions called Hauptmoduln from commuting pairs of elements of the monster. The Borcherds-Höhn program proposes a way to obtain such a rule by constructing infinite dimensional Lie algebras attached to elements of the monster. I will describe recent progress in this program.

We will begin this hour-long treat, with more than you ever wanted to hear on the orbifolds of holomorphic VOAs. Then the Truth about mock modular forms will be revealed. Finally, some steps towards a generalisation of umbral moonshine will be attempted.

Vector valued modular forms are typically studied as a module over the graded ring of level 1 modular forms. We consider the tensor product of vector valued modular forms and give two applications to congruence representations. First, we represent cusp forms as products of at most two Eisenstein series. Second, we obtain relations of Fourier coefficients of mock modular forms. Additionally, we discuss vector valued Hecke operators.

In this talk we give a simple cohomological identity between a weakly holomorphic form and a cusp form both of weight k obtained by applying certain differential operators to a given harmonic Maass form of weight 2-k. We derive several consequences. In particular, we give a cohomological interpretation for the equality of periods of the two weight k forms in question.

The Rogers-Ramanujan identities and Monstrous moonshine are important prototypes of results which occur at the interface of number theory, representation theory and physics. The speaker will discuss these identities, and describe recent work with Duncan, Griffin on Warnaar on their recent generalizations. This will include a comprehensive framework of Rogers-Ramanujan identities and singular moduli, and recent work umbral Moonshine.

Eventhough holography is relatively well understood in the case of AdS3/CFT2, only very few explicit examples of holographic 2d CFTs are known, most famously symmetric orbifolds. We generalize this construction to orbifolds by arbitrary permutation groups, and discuss which of those theories have holographic properties in the large N limit. We namely investigate their spectrum for Hawking-Page transitions and check if their correlation functions factorize.

I will consider N=4 CHL models obtained from orbifolds of heterotic string theory on T^6 by an order N symmetry. This class of theories exhibit a surprising “Fricke S-duality” acting as S —> -1/(NS) on the axio-dilaton modulus S. This is a novel symmetry that lies outside of the SL(2,Z)-symmetry of the parent theory. I will demonstrate that the counting of 1/2 BPS-states is invariant under Fricke S-duality and show how this connects to Mathieu moonshine. This is joint work with R. Volpato.

We propose a physical interpretation of the Monstrous Moonshine observations in terms of certain heterotic string compactifications based on the Frenkel-Lepowsky-Meurman Monstrous module.

I discuss a surprising relationship between the enumerative geometry of K3 surfaces, and the c=12 moonshine module recently investigated in connection with mock modular moonshine for the groups $M_{22}, M_{23}$ and $M_{24}$. The partition functions capturing curve counts on K3 -- the Yau-Zaslow, KKV, and KKP invariants -- can be recast as traces with appropriate insertions in the moonshine module. This allows us to make predictions for large classes of new, twined (or `equivariant') BPS invariants of K3.