Modular Invariance Research Articles

A relative tensor product of subfactors over a modular tensor category

We define and study a certain relative tensor product of subfactors over a modular tensor category. This gives a relative tensor product of two completely rational heterotic full local conformal nets with trivial superselection structures over a common chiral representation category. In particular, we have a new realization of fusion rules of modular invariants. This also gives a mathematical definition of a composition of two gapped domain walls between topological phases.

One loop amplitude from null string

We generalize the CHY formalism to one-loop level, based on the framework of the null string theory. The null string, a tensionless string theory, produces the same results as the ones from the chiral ambitwistor string theory, with the latter believed to give a string interpretation of the CHY formalism. A key feature of our formalism is the interpretation of the modular parameters. We find that the S modular transformation invariance of the ordinary string theory does not survive in the case of the null string theory. Treating the integration over the modular parameters this way enable us to derive the n-gons scattering amplitude in field theory, thus proving the n-gons conjecture.

Bootstrapping chiral CFTs at genus two

Genus two partition functions of 2d chiral conformal field theories are given by Siegel modular forms. We compute their conformal blocks and use them to perform the conformal bootstrap. The advantage of this approach is that it imposes crossing symmetry of an infinite family of four point functions and also modular invariance at the same time. Since for a fixed central charge the ring of Siegel modular forms is finite dimensional, we can perform this analytically. In this way we derive bounds on three point functions and on the spectrum of such theories.

F-rationality of the ring of modular invariants

Using the description of the Frobenius limit of modules over the ring of invariants under an action of a finite group on a polynomial ring over a field of characteristic p>0 developed by Symonds and the author, we give a characterization of the ring of invariants with a positive dual F-signature. Combining this result and Kemper's result on depths of the ring of invariants under an action of a permutation group, we give an example of an F-rational, but non-F-regular ring of invariants under the action of a finite group.

Conformal embeddings and higher-spin bulk duals

It is well-known that conformal embeddings can be used to construct non-diagonal modular invariants for affine lie algebras. This idea can be extended to construct infinite series of non-diagonal modular invariants for coset CFTs. In this paper, we systematically approach the problem of identifying higher-spin bulk duals for these kind of non-diagonal invariants. In particular, for a special value of the 't Hooft coupling, there exist a class of partition functions that have enhanced supersymmetry, which should be reflected in a bulk dual. As a illustration of this, we show that a partition function of a orthogonal group coset CFT has a $\mathcal N=1$ supersymmetric higher-spin bulk dual, in the 't Hooft limit. We also propose that two of the series of CFT partition functions, obtained from conformal embeddings, are equal, generalising the well-known dual interpretation of the 3-state Potts model as a $\frac{SU(2)_3 \otimes SU(2)_1}{SU(2)_4}$ and also as a $\frac{SU(3)_1 \otimes SU(3)_1}{SU(3)_2}$ coset model.

Hopf ring structure on the mod p cohomology of symmetric groups

We describe a Hopf ring structure on ⊕ n≥0H∗(Σn; ℤp), discovered by Strickland and Turner, where Σn is the symmetric group of n objects and p is an odd prime. We also describe an additive basis on which the cup product is explicitly determined, compute the restriction to modular invariants and determine the action of the Steenrod algebra on our Hopf ring generators. For p = 2 this was achieved in work of Giusti, Salvatore and Sinha, of which this work is an extension.

The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy

A central question in arrangement theory is to determine whether the characteristic polynomial Δ q of the algebraic monodromy acting on the homology group H q ( F ( A ) , C ) of the Milnor fiber of a complex hyperplane arrangement A is determined by the intersection lattice L ( A ) . Under simple combinatorial conditions, we show that the multiplicities of the factors of Δ 1 corresponding to certain eigenvalues of order a power of a prime p are equal to the Aomoto–Betti numbers β p ( A ) , which in turn are extracted from L ( A ) . When A defines an arrangement of projective lines with only double and triple points, this leads to a combinatorial formula for the algebraic monodromy. To obtain these results, we relate nets on the underlying matroid of A to resonance varieties in positive characteristic. Using modular invariants of nets, we find a new realizability obstruction (over C) for matroids, and estimate the number of essential components in the first complex resonance variety of A. Our approach also reveals a rather unexpected connection of modular resonance with the geometry of SL 2 ( C ) -representation varieties, which are governed by the Maurer–Cartan equation.

The fully packed loop model as a non-rational W3 conformal field theory

The fully packed loop (FPL) model is a statistical model related to the integrable vertex model. In this paper we study the continuum limit of the FPL. With the appropriate weight of non-contractible loops, we give evidence of an extended W3 symmetry in the continuum. The partition function on the torus is calculated exactly, yielding new modular invariants of W3 characters. The full conformal field theory spectrum is obtained, and is found to be in excellent agreement with exact diagonalisation.

Modular invariance and anomaly cancellation formulas in odd dimension II

By studying modular invariance properties of some characteristic forms, we get some generalized anomaly cancellation formulas on (4r − 1)-dimensional manifolds with no assumption that the 3rd de-Rham cohomology of manifolds vanishes. These anomaly cancellation formulas generalize our previous anomaly cancellation formulas on (4r − 1)-dimensional manifolds. We also generalize our previous anomaly cancellation formulas on (4r − 1)-dimensional manifolds and the Han–Yu rigidity theorem to the (a, b) case.

Multiple D3-Instantons and Mock Modular Forms I

We study D3-instanton corrections to the hypermultiplet moduli space in type IIB string theory compactified on a Calabi-Yau threefold. In a previous work, consistency of D3-instantons with S-duality was established at first order in the instanton expansion, using the modular properties of the M5-brane elliptic genus. We extend this analysis to the two-instanton level, where wall-crossing phenomena start playing a role. We focus on the contact potential, an analogue of the Kahler potential which must transform as a modular form under S-duality. We show that it can be expressed in terms of a suitable modification of the partition function of D4-D2-D0 BPS black holes, constructed out of the generating function of MSW invariants (the latter coincide with Donaldson-Thomas invariants in a particular chamber). Modular invariance of the contact potential then requires that, in case where the D3-brane wraps a reducible divisor, the generating function of MSW invariants must transform as a vector-valued mock modular form, with a specific modular completion built from the MSW invariants of the constituents. Physically, this gives a powerful constraint on the degeneracies of BPS black holes. Mathematically, our result gives a universal prediction for the modular properties of Donaldson-Thomas invariants of pure two-dimensional sheaves.

Modular invariants and singularity indices of hyperelliptic fibrations

The modular invariants of a family of curves are the degrees of the pullback of the corresponding divisors by the moduli map. The singularity indices were introduced by Xiao (1991) to classify singular fibers of hyperelliptic fibrations and to compute global invariants locally. In semistable case, the author shows that the modular invariants corresponding to the boundary divisor classes are just the singularity indices. As an application, the author shows that the formula of Xiao for relative Chern numbers is the same as that of Cornalba-Harris in semistable case.

Modular invariants of a vector and a covector: A proof of a conjecture of Bonnafé and Kemper

Consider a finite dimensional vector space V over a finite field Fq. We give a minimal generating set for the ring of invariants Fq[V⊕V⁎]GL(V), and show that this ring is a Gorenstein ring but is not a complete intersection. These results confirm a conjecture of Bonnafé and Kemper [3, Conjecture 3.1].

The Weak Gravity Conjecture in three dimensions

We study weakly coupled $U(1)$ theories in $AdS_3$, their associated charged BTZ solutions, and their charged spectra. We find that modular invariance of the holographic dual two-dimensional CFT and compactness of the gauge group together imply the existence of charged operators with conformal dimension significantly below the black hole threshold. We regard this as a form of the Weak Gravity Conjecture (WGC) in three dimensions. We also explore the constraints posed by modular invariance on a particular discrete $\mathbb{Z}_N$ symmetry which arises in our discussion. In this case, modular invariance does not guarantee the existence of light $\mathbb{Z}_N$-charged states. We also highlight the differences between our discussion and the usual heuristic arguments for the WGC based on black hole remnants.

Logarithmic intertwining operators and genus-one correlation functions

This is the first of two papers in which we study the modular invariance of pseudotraces of logarithmic intertwining operators. We construct and study genus-one correlation functions for logarithmic intertwining operators among generalized modules over a positive-energy and [Formula: see text]-cofinite vertex operator algebra [Formula: see text]. We consider grading-restricted generalized [Formula: see text]-modules which admit a right action of some associative algebra [Formula: see text], and intertwining operators among such modules which commute with the action of [Formula: see text] ([Formula: see text]-intertwining operators). We obtain duality properties, i.e. suitable associativity and commutativity properties, for [Formula: see text]-intertwining operators. Using pseudotraces introduced by Miyamoto and studied by Arike, we define formal [Formula: see text]-traces of products of [Formula: see text]-intertwining operators, and obtain certain identities for these formal series. This allows us to show that the formal [Formula: see text]-traces satisfy a system of differential equations with regular singular points, and therefore are absolutely convergent in a suitable region and can be extended to yield multivalued analytic functions, called genus-one correlation functions. Furthermore, we show that the space of solutions of these differential equations is invariant under the action of the modular group.

Modular invariant inflation

Modular invariance is a striking symmetry in string theory, which may keep stringy corrections under control. In this paper, we investigate a phenomenological consequence of the modular invariance, assuming that this symmetry is preserved as well as in a four dimensional (4D) low energy effective field theory. As a concrete setup, we consider a modulus field T whose contribution in the 4D effective field theory remains invariant under the modular transformation and study inflation drived by T. The modular invariance restricts a possible form of the scalar potenntial. As a result, large field models of inflation are hardly realized. Meanwhile, a small field model of inflation can be still accomodated in this restricted setup. The scalar potential traced during the slow-roll inflation mimics the hilltop potential Vht, but it also has a non-negligible deviation from Vht. Detecting the primordial gravitational waves predicted in this model is rather challenging. Yet, we argue that it may be still possible to falsify this model by combining the information in the reheating process which can be determined self-completely in this setup.

Small black holes and near-extremal CFTs

Pure theories of AdS$_3$ quantum gravity are conjectured to be dual to CFTs with sparse spectra of light primary operators. The sparsest possible spectrum consistent with modular invariance includes only black hole states above the vacuum. Witten conjectured the existence of a family of extremal CFTs, which realize this spectrum for all admissible values of the central charge. We consider the quantum corrections to the classical spectrum, and propose a specific modification of Witten's conjecture which takes into account the existence of "small" black hole states. These have zero classical horizon area, with a calculable entropy attributed solely to loop effects. Our conjecture passes various consistency checks, especially when generalized to include theories with supersymmetry. In theories with $\mathcal{N}=2$ supersymmetry, this "near-extremal CFT" proposal precisely evades the no-go results of Gaberdiel et al.

Representations of affine superalgebras and mock theta functions. III

We study modular invariance of normalized supercharacters of tame integrable modules over an affine Lie superalgebra, associated to an arbitrary basic Lie superalgebra . For this we develop a several step modification process of multivariable mock theta functions, where at each step a Zwegers' type `modifier' is used. We show that the span of the resulting modified normalized supercharacters is -invariant, with the transformation matrix equal, in the case the Killing form on is non- degenerate, to that for the basic defect 0 subalgebra of , orthogonal to a maximal isotropic set of roots of .

Revisiting noninteracting string partition functions in Rindler space

We revisit non-interacting string partition functions in Rindler space by summing over fields in the spectrum. In field theory, the total partition function splits in a natural way in a piece that does not contain surface terms and a piece consisting of solely the so-called edge states. For open strings, we illustrate that surface contributions to the higher spin fields correspond to open strings piercing the Rindler origin, unifying the higher spin surface contributions in string language. For closed strings, we demonstrate that the string partition function is not quite the same as the sum over the partition functions of the fields in the spectrum: an infinite overcounting is present for the latter. Next we study the partition functions obtained by excluding the surface terms. Using recent results of JHEP 1505 (2015) 106, this construction, first done by Emparan, can be put on much firmer ground. We generalize to type II and heterotic superstrings and demonstrate modular invariance. All of these exhibit an IR divergence that can be interpreted as a maximal acceleration close to the black hole horizon. Ultimately, since these partition functions are only part of the full story, divergences here should not be viewed as a failure of string theory: maximal acceleration is a feature of a faulty treatment of the higher spin fields in the string spectrum. We comment on the relevance of this to Solodukhin's recent proposal. A possible link with the firewall paradox is apparent.

Reflections on conformal spectra

We use modular invariance and crossing symmetry of conformal field theory to reveal approximate reflection symmetries in the spectral decompositions of the partition function in two dimensions in the limit of large central charge and of the four-point function in any dimension in the limit of large scaling dimensions $\Delta_0$ of external operators. We use these symmetries to motivate universal upper bounds on the spectrum and the operator product expansion coefficients, which we then derive by independent techniques. Some of the bounds for four-point functions are valid for finite $\Delta_0$ as well as for large $\Delta_0$. We discuss a similar symmetry in a large spacetime dimension limit. Finally, we comment on the analogue of the Cardy formula and sparse light spectrum condition for the four-point function.

KMS weights on higher rank buildings

We extend some of the results of Carey-Marcolli-Rennie on modular index invariants of Mumford curves to the case of higher rank buildings. We discuss notions of KMS weights on buildings, that generalize the construction of graph weights over graph C*-algebras.