Modular Invariance Research Articles

Genus two partition functions of extremal conformal field theories

Recently Witten conjectured the existence of a family of extremal conformal field theories (ECFTs) of central charge c=24k, which are supposed to be dual to three-dimensional pure quantum gravity in AdS3. Assuming their existence, we determine explicitly the genus two partition functions of k=2 and k=3 ECFTs, using modular invariance and the behavior of the partition function in degenerating limits of the Riemann surface. The result passes highly nontrivial tests and in particular provides a piece of evidence for the existence of the k=3 ECFT. We also argue that the genus two partition function of ECFTs with k<11 are uniquely fixed (if they exist).

Modular invariants for lattice polarized K3 surfaces

We study the class of complex algebraic K3 surfaces admitting an embedding of HE8 � E8 inside the Neron-Severi lattice. These special K3 surfaces are classified by a pair of modular invariants, in the same manner that ellipticcurves over C are classifiedby the J-invariant. Via the canonical Shioda-Inose structure we construct a geometric correspondence relating K3 surfaces of the above type with abelian surfaces realized as cartesian products of two elliptic curves. We then use this correspondence to determine explicit formulas for the modular invariants.

Twisted quantum modular data and braided subfactors

The modular data from twisted quantum double of finite groups G are studied, with cyclic groups. It is proved that the [k] and [ − k] modular data are the same up to relabelling of the primary fields and complex conjugation of the underlying representation of the modular group . Then we produce some lower bounds for the number of modular invariants of these models, and complete the study for the cases and at all twists, proving in particular that all their modular invariants are produced by braided subfactors.

From modular invariants to graphs: the modular splitting method

We start with a given modular invariant of a two-dimensional conformal field theory (CFT) and present a general method for solving the Ocneanu modular splitting equation and then determine, in a step-by-step explicit construction, (1) the generalized partition functions corresponding to the introduction of boundary conditions and defect lines; (2) the quantum symmetries of the higher ADE graph G associated with the initial modular invariant . Note that one does not suppose here that the graph G is already known, since it appears as a by-product of the calculations. We analyse several exceptional cases at levels 5 and 9.

Remarks on worldsheet theories dual to free largeNgauge theories

We continue to investigate properties of the worldsheet conformal field theories (CFTs) which are conjectured to be dual to free large $N$ gauge theories, using the mapping of Feynman diagrams to the worldsheet suggested in [R. Gopakumar, Phys. Rev. D 70, 025009 (2004); R. GopakumarPhys. Rev. D70, 025010 (2004); R. GopakumarC. R. Physique 5, 1111 (2004); R. GopakumarPhys. Rev. D 72, 066008 (2005)]. The modular invariance of these CFTs is shown to be built into the formalism. We show that correlation functions in these CFTs which are localized on subspaces of the moduli space may be interpreted as delta-function distributions, and that this can be consistent with a local worldsheet description given some constraints on the operator product expansion coefficients. We illustrate these features by a detailed analysis of a specific four-point function diagram. To reliably compute this correlator, we use a novel perturbation scheme which involves an expansion in the large dimension of some operators.

Higher genus superstring amplitudes and the measure on the moduli space

AbstractRecent derivations of completely gauge‐invariant two‐loop superstring amplitudes renewed the interest in the possible form of higher genus contributions. In this respect, two main points are discussed: a. At genus greater than 3, the Schottky problem actually represents a formidable obstruction for the explicit expression of the integration measure on the moduli space in terms of Riemann period matrices. As a first step toward such a construction, a modular invariant measure for all genera is derived, corresponding to the restriction to the Schottky locus of the Siegel metric on the upper half‐space. b. A class of natural generalizations of the 2 loop D'Hoker and Phong formula is discussed for the higher genus contributions to the 4‐gravitons amplitude in type II theories. Such expressions also satisfy the non‐trivial requirements of modular invariance and gauge slice independence.

KÄHLER STABILIZED, MODULAR INVARIANT HETEROTIC STRING MODELS

We review the theory and phenomenology of effective supergravity theories based on orbifold compactifications of the weakly-coupled heterotic string. In particular, we consider theories in which the four-dimensional theory displays target space modular invariance and where the dilatonic mode undergoes Kähler stabilization. A self-contained exposition of effective Lagrangian approaches to gaugino condensation and heterotic string theory is presented, leading to the development of the models of Binétruy, Gaillard and Wu. Various aspects of the phenomenology of this class of models are considered. These include issues of supersymmetry breaking and superpartner spectra, the role of anomalous U(1) factors, issues of flavor and R-parity conservation, collider signatures, axion physics, and early universe cosmology. For the vast majority of phenomenological considerations the theories reviewed here compare quite favorably to other string-derived models in the literature. Theoretical objections to the framework and directions for further research are identified and discussed.

New dimensions for wound strings: The modular transformation of geometry to topology

We show, using a theorem of Milnor and Margulis, that string theory on compact negatively curved spaces grows new effective dimensions as the space shrinks, generalizing and contextualizing the results in E. Silverstein, Phys. Rev. D 73, 086004 (2006).. Milnor's theorem relates negative sectional curvature on a compact Riemannian manifold to exponential growth of its fundamental group, which translates in string theory to a higher effective central charge arising from winding strings. This exponential density of winding modes is related by modular invariance to the infrared small perturbation spectrum. Using self-consistent approximations valid at large radius, we analyze this correspondence explicitly in a broad set of time-dependent solutions, finding precise agreement between the effective central charge and the corresponding infrared small perturbation spectrum. This indicates a basic relation between geometry, topology, and dimensionality in string theory.

Method of orbit sums in the theory of modular vector invariants

Let be a field, a finite-dimensional -vector space, a finite group, and the -fold direct sum with the diagonal action of . The group acts naturally on the symmetric graded algebra as a group of non-degenerate linear transformations of the variables. Let be the subalgebra of invariants of the polynomial algebra with respect to . A classical result of Noether [1] says that if , then is generated as an -algebra by homogeneous polynomials of degree at most , no matter how large can be. On the other hand, it was proved by Richman [2], [3] that this result does not hold when the characteristic of is positive and divides the order of . Let , , be a prime number, a finite field of elements, a linear -vector space of dimension , and a cyclic group of order generated by a matrix of a certain special form. In this paper we describe explicitly (Theorem 1) one complete set of generators of . After that, for an arbitrary complete set of generators of this algebra we find a lower bound for the highest degree of the generating elements of this algebra. This is a significant extension of the corresponding result of Campbell and Hughes [4] for the particular case of . As a consequence we show (Theorem 3) that if and is an arbitrary finite group, then each complete set of generators of contains an element of degree at least , where is a positive integer dependent on the structure of the generating matrix of the group . This result refines considerably the earlier lower bound obtained by Richman [3].

Simple current modular invariants from braided subfactors

Abstract We use the braided subfactor framework to further the study of modular partition functions, treating in detail the 22 modular invariants arising from the double of the cyclic group Z 4 .

Lower degree bounds for modular vector invariants

Let $G$ be a finite group of order divisible by a prime $p$ acting on an $\mathbb {F}$ vector space $V,$ where $\mathbb {F}$ is the field with $p$ elements and $\dim _{\mathbb {F}} V=n$. Consider the diagonal action of $G$ on $m$ copies of $V.$ This note sharpens a lower bound for $\beta (\mathbb {F}[\oplus _mV]^G)$ for groups which have an element of order $p$ whose Jordan blocks have sizes at most 2.

Statistics on the heterotic landscape: Gauge groups and cosmological constants of four-dimensional heterotic strings

Recent developments in string theory have reinforced the notion that the space of stable supersymmetric and nonsupersymmetric string vacua fills out a landscape whose features are largely unknown. It is then hoped that progress in extracting phenomenological predictions from string theory--such as correlations between gauge groups, matter representations, potential values of the cosmological constant, and so forth--can be achieved through statistical studies of these vacua. To date, most of the efforts in these directions have focused on type I vacua. In this note, we present the first results of a statistical study of the heterotic landscape, focusing on more than 10{sup 5} explicit nonsupersymmetric tachyon-free heterotic string vacua and their associated gauge groups and one-loop cosmological constants. Although this study has several important limitations, we find a number of intriguing features which may be relevant for the heterotic landscape as a whole. These features include different probabilities and correlations for different possible gauge groups as functions of the number of orbifold twists. We also find a vast degeneracy amongst nonsupersymmetric string models, leading to a severe reduction in the number of realizable values of the cosmological constant as compared with naieve expectations. Finally, we find strong correlations between cosmological constantsmore » and gauge groups which suggest that heterotic string models with extremely small cosmological constants are overwhelmingly more likely to exhibit the standard model gauge group at the string scale than any of its grand-unified extensions. In all cases, heterotic world sheet symmetries such as modular invariance provide important constraints that do not appear in corresponding studies of type I vacua.« less

Modular invariants for layered object structures

Classical specification and verification techniques support invariants for individual objects whose fields are primitive values, but do not allow sound modular reasoning about invariants involving more complex object structures. Such non-trivial object structures are common, and occur in lists, hash tables, and whenever systems are built in layers. A sound and modular verification technique for layered object structures has to deal with the well-known problem of representation exposure and the problem that invariants of higher layers are potentially violated by methods in lower layers; such methods cannot be modularly shown to preserve these invariants. We generalize classical techniques to cover layered object structures using a refined semantics for invariants based on an ownership model for alias control. This semantics enables sound and modular reasoning. We further extend this ownership technique to even more expressive invariants that gain their modularity by imposing certain visibility requirements.

Dimensional mutation and spacelike singularities

I argue that string theory compactified on a Riemann surface crosses over at small volume to a higher dimensional background of supercritical string theory. Several concrete measures of the count of degrees of freedom of the theory yield the consistent result that at finite volume, the effective dimensionality is increased by an amount of order $2h/V$ for a surface of genus $h$ and volume $V$ in string units. This arises in part from an exponentially growing density of states of winding modes supported by the fundamental group, and passes an interesting test of modular invariance. Further evidence for a plethora of examples with the spacelike singularity replaced by a higher dimensional phase arises from the fact that the sigma model on a Riemann surface can be naturally completed by many gauged linear sigma models, whose RG flows approximate time evolution in the full string backgrounds arising from this in the limit of large dimensionality. In recent examples of spacelike singularity resolution by tachyon condensation, the singularity is ultimately replaced by a phase with all modes becoming heavy and decoupling. In the present case, the opposite behavior ensues: more light degrees of freedom arise in the small radius regime. We comment on the emerging zoology of cosmological singularities that results.

Odd primary Steenrod algebra, additive formal group laws, and modular invariants

We give a conceptual clarification of Milnor's theorem, which tells us the Hopf algebra structure of the stable co-operations H * H in the odd primary ordinary cohomology. Directly connecting H * H with the quasi-strict automorphism group of some 1 -dimensional additive formal group law and modular invariants, we give a new proof of this theorem of Milnor.

CLOSED BOSONIC STRING PARTITION FUNCTION IN TIME INDEPENDENT EXACT pp-WAVE BACKGROUND

The modular invariance of the one-loop partition function of the closed bosonic string in four dimensions in the presence of certain homogeneous exact pp -wave backgrounds is studied. In the absence of an axion field, the partition function is found to be modular invariant and equal to the free field partition function. The partition function remains unchanged also in the presence of a fixed axion field. However, in this case, the covariant form of the action suggests summation over all possible twists generated by the axion field. This is shown to modify the partition function. In the light-cone gauge, the axion field generates twists only in the worldsheet σ-direction, so the resulting partition function is not modular invariant, hence wrong. To obtain the correct partition function one needs to sum over twists in the t-direction as well, as suggested by the covariant form of the action away from the light-cone gauge.

Twisted boundary states and representation of generalized fusion algebra

The mutual consistency of boundary conditions twisted by an automorphism group G of the chiral algebra is studied for general modular invariants of rational conformal field theories. We show that a consistent set of twisted boundary states associated with any modular invariant realizes a non-negative integer matrix representation (NIM-rep) of the generalized fusion algebra, an extension of the fusion algebra by representations of the twisted chiral algebra associated with the automorphism group G. We check this result for several concrete cases. In particular, we find that two NIM-reps of the fusion algebra for su ( 3 ) k ( k = 3 , 5 ) are organized into a NIM-rep of the generalized fusion algebra for the charge-conjugation automorphism of su ( 3 ) k . We point out that the generalized fusion algebra is non-commutative if G is non-Abelian and provide some examples for G ≅ S 3 . Finally, we give an argument that the graph fusion algebra associated with simple current extensions coincides with the generalized fusion algebra for the extended chiral algebra, and thereby explain that the graph fusion algebra contains the fusion algebra of the extended theory as a subalgebra.

DIFFERENTIAL EQUATIONS, DUALITY AND MODULAR INVARIANCE

We solve the problem of constructing all chiral genus-one correlation functions from chiral genus-zero correlation functions associated to a vertex operator algebra satisfying the following conditions: (i) V(n) = 0 for n &lt; 0 and V(0) = ℂ1, (ii) every ℕ-gradable weak V-module is completely reducible and (iii) V is C2-cofinite. We establish the fundamental properties of these functions, including suitably formulated commutativity, associativity and modular invariance. The method we develop and use here is completely different from the one previously used by Zhu and others. In particular, we show that the q-traces of products of certain geometrically-modified intertwining operators satisfy modular invariant systems of differential equations which, for any fixed modular parameter, reduce to doubly-periodic systems with only regular singular points. Together with the results obtained by the author in the genus-zero case, the results of the present paper solves essentially the problem of constructing chiral genus-one weakly conformal field theories from the representations of a vertex operator algebra satisfying the conditions above.

Higher Coxeter graphs associated with affine su(3) modular invariants

The affine $su(3)$ modular invariant partition functions in 2d RCFT are associated with a set of generalized Coxeter graphs. These partition functions fall into two classes, the block-diagonal (Type I) and the non block-diagonal (Type II) cases, associated, from spectral properties, to the subsets of subgroup and module graphs respectively. We introduce a modular operator $\hat{T}$ taking values on the set of vertices of the subgroup graphs. It allows us to obtain easily the associated Type I partition functions. We also show that all Type II partition functions are obtained by the action of suitable twists $\vartheta$ on the set of vertices of the subgroup graphs. These twists have to preserve the values of the modular operator $\hat{T}$.

Invariants of the diagonal [formula omitted]-action on [formula omitted

Let C p denote the cyclic group of order p where p ⩾ 3 is prime. We denote by V 3 the indecomposable three-dimensional representation of C p over a field F of characteristic p. We compute a set of generators, in fact a SAGBI basis, for the ring of invariants F [ V 3 ⊕ V 3 ] C p . Our main result confirms the conjecture of Shank [R.J. Shank, Classical covariants and modular invariants, in: H.E.A. Campbell, D.L. Wehlau (Eds.), Invariant Theory in All Characteristics, CRM Proc. Lecture Notes, vol. 35, Amer. Math. Soc., 2004, pp. 241–249], for this example, that all modular rings of invariants of C p are generated by rational invariants, norms and transfers.