arXiv subjects

Miranda C. N. Cheng

Publications and source records attributed to Miranda C. N. Cheng.

At least 55 records · Page 3Linked to original sources

arXiv2015

Elliptic Genera and 3d Gravity

We describe general constraints on the elliptic genus of a 2d supersymmetric conformal field theory which has a gravity dual with large radius in Planck units. We give examples of theories which do and do not satisfy the bounds we derive, by describing the elliptic genera of symmetric product orbifolds of $K3$, product manifolds, certain simple families of Calabi-Yau hypersurfaces, and symmetric products of the "Monster CFT." We discuss the distinction between theories with supergravity duals and those whose duals have strings at the scale set by the AdS curvature. Under natural assumptions we attempt to quantify the fraction of (2,2) supersymmetric conformal theories which admit a weakly curved gravity description, at large central charge.

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arXiv2015

Exceptional Algebra and Sporadic Groups at c=12

In earlier works, it was seen that a ${\mathbb Z}/2$ orbifold of the theory of 24 free two-dimensional chiral fermions admits various sporadic finite simple groups as global symmetry groups when viewed as an ${\cal N}=1$, ${\cal N}=2$, or ${\cal N}=4$ superconformal field theory. In this note, we show that viewing the same theory as an SCFT with extended ${\cal N}=1$ symmetry -- where the extension is the same one which arises in string compactification on manifolds of exceptional Spin$(7)$ holonomy -- yields theories which have global symmetry given by the sporadic groups $M_{24}, Co_2$ or $Co_3$. The partition functions twined by these symmetries, when decomposed into characters of the Spin(7) algebra, give rise to two-component vector-valued mock modular forms encoding an infinite-dimensional module for the corresponding sporadic groups.

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arXiv2014

Umbral Moonshine and the Niemeier Lattices

In this paper we relate umbral moonshine to the Niemeier lattices: the 23 even unimodular positive-definite lattices of rank 24 with non-trivial root systems. To each Niemeier lattice we attach a finite group by considering a naturally defined quotient of the lattice automorphism group, and for each conjugacy class of each of these groups we identify a vector-valued mock modular form whose components coincide with mock theta functions of Ramanujan in many cases. This leads to the umbral moonshine conjecture, stating that an infinite-dimensional module is assigned to each of the Niemeier lattices in such a way that the associated graded trace functions are mock modular forms of a distinguished nature. These constructions and conjectures extend those of our earlier paper, and in particular include the Mathieu moonshine observed by Eguchi-Ooguri-Tachikawa as a special case. Our analysis also highlights a correspondence between genus zero groups and Niemeier lattices. As a part of this relation we recognise the Coxeter numbers of Niemeier root systems with a type A component as exactly those levels for which the corresponding classical modular curve has genus zero.

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arXiv2013

Umbral Moonshine

We describe surprising relationships between automorphic forms of various kinds, imaginary quadratic number fields and a certain system of six finite groups that are parameterised naturally by the divisors of twelve. The Mathieu group correspondence recently discovered by Eguchi-Ooguri-Tachikawa is recovered as a special case. We introduce a notion of extremal Jacobi form and prove that it characterises the Jacobi forms arising by establishing a connection to critical values of Dirichlet series attached to modular forms of weight two. These extremal Jacobi forms are closely related to certain vector-valued mock modular forms studied recently by Dabholkar-Murthy-Zagier in connection with the physics of quantum black holes in string theory. In a manner similar to monstrous moonshine the automorphic forms we identify constitute evidence for the existence of infinite-dimensional graded modules for the six groups in our system. We formulate an umbral moonshine conjecture that is in direct analogy with the monstrous moonshine conjecture of Conway-Norton. Curiously, we find a number of Ramanujan's mock theta functions appearing as McKay-Thompson series. A new feature not apparent in the monstrous case is a property which allows us to predict the fields of definition of certain homogeneous submodules for the groups involved. For four of the groups in our system we find analogues of both the classical McKay correspondence and McKay's monstrous Dynkin diagram observation manifesting simultaneously and compatibly.

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arXiv2013

Mathieu Moonshine and N=2 String Compactifications

There is a `Mathieu moonshine' relating the elliptic genus of K3 to the sporadic group M_{24}. Here, we give evidence that this moonshine extends to part of the web of dualities connecting heterotic strings compactified on K3 \times T^2 to type IIA strings compactified on Calabi-Yau threefolds. We demonstrate that dimensions of M_{24} representations govern the new supersymmetric index of the heterotic compactifications, and appear in the Gromov--Witten invariants of the dual Calabi-Yau threefolds, which are elliptic fibrations over the Hirzebruch surfaces F_n.

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arXiv2013

On the Discrete Groups of Mathieu Moonshine

We prove that a certain space of cusp forms for the Hecke congruence group of a given level is one-dimensional if and only if that level is the order of an element of the second largest Mathieu group. As such, our result furnishes a direct analogue of Ogg's observation that the normaliser of a Hecke congruence group of prime level has genus zero if and only if that prime divides the order of the Fischer-Griess monster group. The significance of the cusp forms under consideration is explained by the Rademacher sum construction of the McKay-Thompson series of Mathieu moonshine. Our result supports a conjectural characterisation of the discrete groups and multiplier systems arising in Mathieu moonshine.

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arXiv2012

Rademacher Sums and Rademacher Series

We exposit the construction of Rademacher sums in arbitrary weights and describe their relationship to mock modular forms. We introduce the notion of Rademacher series and describe several applications, including the determination of coefficients of Rademacher sums and a very general form of Zagier duality. We then review the application of Rademacher sums and series to moonshine both monstrous and umbral and highlight several open problems. We conclude with a discussion of the interpretation of Rademacher sums in physics.

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arXiv2012

The Largest Mathieu Group and (Mock) Automorphic Forms

We review the relationship between the largest Mathieu group and various modular objects, including recent progress on the relation to mock modular forms. We also review the connections between these mathematical structures and string theory on K3 surfaces.

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arXiv2011

On Rademacher Sums, the Largest Mathieu Group, and the Holographic Modularity of Moonshine

Recently a conjecture has been proposed which attaches (mock) modular forms to the largest Mathieu group. This may be compared to monstrous moonshine, in which modular functions are attached to elements of the Monster group. One of the most remarkable aspects of monstrous moonshine is the following genus zero property: the modular functions turn out to be the generators for the function fields of their invariance groups. In particular, these invariance groups define genus zero quotients of the upper half plane. It is therefore natural to ask if there is an analogue of this property in the Mathieu case, and at first glance the answer appears to be negative since not all the discrete groups arising there have genus zero. On the other hand, in this article we prove that each (mock) modular form appearing in the Mathieu correspondence coincides with the Rademacher sum constructed from its polar part. This property, inspired by the AdS/CFT correspondence in physics, was shown previously to be equivalent to the genus zero property of monstrous moonshine. Hence we conclude that this "Rademacher summability" property serves as the natural analogue of the genus zero property in the Mathieu case. Our result constitutes further evidence that the Rademacher method provides a powerful framework for understanding the modularity of moonshine, and leads to interesting physical questions regarding the gravitational duals of the relevant conformal field theories.

math.RT↗

arXiv2011

Quantum Geometry of Refined Topological Strings

We consider branes in refined topological strings. We argue that their wave-functions satisfy a Schrödinger equation depending on multiple times and prove this in the case where the topological string has a dual matrix model description. Furthermore, in the limit where one of the equivariant rotations approaches zero, the brane partition function satisfies a time-independent Schroedinger equation. We use this observation, as well as the back reaction of the brane on the closed string geometry, to offer an explanation of the connection between integrable systems and N=2 gauge systems in four dimensions observed by Nekrasov and Shatashvili.

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arXiv2010

Non-Perturbative Topological Strings And Conformal Blocks

We give a non-perturbative completion of a class of closed topological string theories in terms of building blocks of dual open strings. In the specific case where the open string is given by a matrix model these blocks correspond to a choice of integration contour. We then apply this definition to the AGT setup where the dual matrix model has logarithmic potential and is conjecturally equivalent to Liouville conformal field theory. By studying the natural contours of these matrix integrals and their monodromy properties, we propose a precise map between topological string blocks and Liouville conformal blocks. Remarkably, this description makes use of the light-cone diagrams of closed string field theory, where the critical points of the matrix potential correspond to string interaction points.

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arXiv2010

Yukawa Couplings in F-theory and Non-Commutative Geometry

We consider Yukawa couplings generated by a configuration of intersecting seven-branes in F-theory. In configurations with a single interaction point and no fluxes turned on, the Yukawa matrices have rank one. This is no longer true when the three-form H-flux is turned on, which is generically the case for F-theory compactifications on Calabi-Yau fourfolds. In the presence of H-fluxes, the Yukawa coupling is computed using a non-commutative deformation of holomorphic Chern-Simons theory (and its reduction to seven-branes) and subsequently the rank of the Yukawa matrix changes. Such fluxes give rise to a hierarchical structure in the Yukawa matrix in F-theory GUTs of the type which has recently been proposed as a resolution of the flavor hierarchy problem.

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arXiv2010

K3 Surfaces, N=4 Dyons, and the Mathieu Group M24

A close relationship between K3 surfaces and the Mathieu groups has been established in the last century. Furthermore, it has been observed recently that the elliptic genus of K3 has a natural interpretation in terms of the dimensions of representations of the largest Mathieu group M24. In this paper we first give further evidence for this possibility by studying the elliptic genus of K3 surfaces twisted by some simple symplectic automorphisms. These partition functions with insertions of elements of M24 (the McKay-Thompson series) give further information about the relevant representation. We then point out that this new "moonshine" for the largest Mathieu group is connected to an earlier observation on a moonshine of M24 through the 1/4-BPS spectrum of K3xT^2-compactified type II string theory. This insight on the symmetry of the theory sheds new light on the generalised Kac-Moody algebra structure appearing in the spectrum, and leads to predictions for new elliptic genera of K3, perturbative spectrum of the toroidally compactified heterotic string, and the index for the 1/4-BPS dyons in the d=4, N=4 string theory, twisted by elements of the group of stringy K3 isometries.

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arXiv2009

Deformations of Lifshitz holography

The simplest gravity duals for quantum critical theories with z=2 `Lifshitz' scale invariance admit a marginally relevant deformation. Generic black holes in the bulk describe the field theory with a dynamically generated momentum scale Lambda as well as finite temperature T. We describe the thermodynamics of these black holes in the quantum critical regime where T >> Lambda^2. The deformation changes the asymptotics of the spacetime mildly and leads to intricate UV sensitivities of the theory which we control perturbatively in Lambda^2/T.

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arXiv2009

A Geometric Derivation of the Dyon Wall-Crossing Group

Recently, using supergravity analysis, a hyperbolic reflection group was found to underlie the structure of wall-crossing, or the discontinuous moduli dependence of the supersymmetric index due to the presence of walls of marginal stability, of the BPS dyons in the N=4, d=4 compactification. In this paper we work in the regime where four-dimensional gravity decouples and we show how the presence of such a group structure can be easily understood as a consequence of the supersymmetry of a system of (p,q) five-brane network, or equivalently the holomorphicity of the Riemann surface wrapped by the appropriate M5 branes in the Euclidean M-theory frame.

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arXiv2008

Dying Dyons Don't Count

The dyonic 1/4-BPS states in 4D string theory with N=4 spacetime supersymmetry are counted by a Siegel modular form. The pole structure of the modular form leads to a contour dependence in the counting formula obscuring its duality invariance. We exhibit the relation between this ambiguity and the (dis-)appearance of bound states of 1/2-BPS configurations. Using this insight we propose a precise moduli-dependent contour prescription for the counting formula. We then show that the degeneracies are duality-invariant and are correctly adjusted at the walls of marginal stability to account for the (dis-)appearance of the two-centered bound states. Especially, for large black holes none of these bound states exists at the attractor point and none of these ambiguous poles contributes to the counting formula. Using this fact we also propose a second, moduli-independent contour which counts the "immortal dyons" that are stable everywhere.

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arXiv2008

Wall Crossing, Discrete Attractor Flow and Borcherds Algebra

The appearance of a generalized (or Borcherds-) Kac-Moody algebra in the spectrum of BPS dyons in N=4, d=4 string theory is elucidated. From the low-energy supergravity analysis, we identify its root lattice as the lattice of the T-duality invariants of the dyonic charges, the symmetry group of the root system as the extended S-duality group PGL(2,Z) of the theory, and the walls of Weyl chambers as the walls of marginal stability for the relevant two-centered solutions. This leads to an interpretation for the Weyl group as the group of wall-crossing, or the group of discrete attractor flows. Furthermore we propose an equivalence between a "second-quantized multiplicity" of a charge- and moduli-dependent highest weight vector and the dyon degeneracy, and show that the wall-crossing formula following from our proposal agrees with the wall-crossing formula obtained from the supergravity analysis. This can be thought of as providing a microscopic derivation of the wall-crossing formula of this theory.

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arXiv2008

Borcherds-Kac-Moody Symmetry of N=4 Dyons

We consider compactifications of heterotic string theory to four dimensions on CHL orbifolds of the type T^6 /Z_N with 16 supersymmetries. The exact partition functions of the quarter-BPS dyons in these models are given in terms of genus-two Siegel modular forms. Only the N=1,2,3 models satisfy a certain finiteness condition, and in these cases one can identify a Borcherds-Kac-Moody superalgebra underlying the symmetry structure of the dyon spectrum. We identify the real roots, and find that the corresponding Cartan matrices exhaust a known classification. We show that the Siegel modular form satisfies the Weyl denominator identity of the algebra, which enables the determination of all root multiplicities. Furthermore, the Weyl group determines the structure of wall-crossings and the attractor flows of the theory. For N> 4, no such interpretation appears to be possible.

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