40 research outputs found
Comments on M representations and geometries
We show using string dualities that Mathieu moonshine controls Gromov-Witten
invariants and periods of the holomorphic 3-form for certain
manifolds. We also discuss how the period vectors appear in flux
compactifications on these manifolds and work out the connection between
the sporadic group M and the Yukawa couplings in four dimensional
theories that arise from heterotic string theory compactifications on these
manifolds.Comment: 27 pages, v2: minor additions, published versio
Twisted Supergravity and Koszul Duality: A Case Study in AdSâ
In this note, we study a simplified variant of the familiar holographic duality between supergravity on AdSâ Ă SÂł Ă Tâ´ and the SCFT (on the moduli space of) the symmetric orbifold theory Sym^N(Tâ´) as Nââ. This variant arises conjecturally from a twist proposed by the first author and Si Li. We recover a number of results concerning protected subsectors of the original duality working directly in the twisted bulk theory. Moreover, we identify the symmetry algebra arising in the Nââ limit of the twisted gravitational theory. We emphasize the role of Koszul dualityâa ubiquitous mathematical notion to which we provide a friendly introductionâin field theory and string theory. After illustrating the appearance of Koszul duality in the âtoyâ example of holomorphic Chern-Simons theory, we describe how (a deformation of) Koszul duality relates bulk and boundary operators in our twisted setup, and explain how one can compute algebra OPEs diagrammatically using this notion. Further details, results, and computations will appear in a companion paper
BPS Algebras, Genus Zero, and the Heterotic Monster
In this note, we expand on some technical issues raised in \cite{PPV} by the
authors, as well as providing a friendly introduction to and summary of our
previous work. We construct a set of heterotic string compactifications to 0+1
dimensions intimately related to the Monstrous moonshine module of Frenkel,
Lepowsky, and Meurman (and orbifolds thereof). Using this model, we review our
physical interpretation of the genus zero property of Monstrous moonshine.
Furthermore, we show that the space of (second-quantized) BPS-states forms a
module over the Monstrous Lie algebras ---some of the first and
most prominent examples of Generalized Kac-Moody algebras---constructed by
Borcherds and Carnahan. In particular, we clarify the structure of the module
present in the second-quantized string theory. We also sketch a proof of our
methods in the language of vertex operator algebras, for the interested
mathematician.Comment: 19 pages, 2 figure
No more walls! A tale of modularity, symmetry, and wall crossing for 1/4 BPS dyons
Abstract We determine the generating functions of 1/4 BPS dyons in a class of 4d N = 4 string vacua arising as CHL orbifolds of K3 Ă T 2, a classification of which has been recently completed. We show that all such generating functions obey some simple physical consistency conditions that are very often sufficient to fix them uniquely. The main constraint we impose is the absence of unphysical walls of marginal stability: discontinuities of 1/4 BPS degeneracies can only occur when 1/4 BPS dyons decay into pairs of 1/2 BPS states. Formally, these generating functions in spacetime can be described as multiplicative lifts of certain supersymmetric indices (twining genera) on the worldsheet of the corresponding nonlinear sigma model on K3. As a consequence, our procedure also leads to an explicit derivation of almost all of these twining genera. The worldsheet indices singled out in this way match precisely a set of functions of interest in moonshine, as predicted by a recent conjecture
Monstrous BPS-Algebras and the Superstring Origin of Moonshine
We provide a physics derivation of Monstrous moonshine. We show that the
McKay-Thompson series , , can be interpreted as
supersymmetric indices counting spacetime BPS-states in certain heterotic
string models. The invariance groups of these series arise naturally as
spacetime T-duality groups and their genus zero property descends from the
behaviour of these heterotic models in suitable decompactification limits. We
also show that the space of BPS-states forms a module for the Monstrous Lie
algebras , constructed by Borcherds and Carnahan. We argue that
arise in the heterotic models as algebras of spontaneously
broken gauge symmetries, whose generators are in exact correspondence with
BPS-states. This gives an interpretation as a kind of
BPS-algebra.Comment: 73 pages, with results summarized in introduction. v2: added a
discussion about coupling to gravity (section 3.3), additional references,
minor corrections and improvement
Twining Genera of (0,4) Supersymmetric Sigma Models on K3
Conformal field theories with (0,4) worldsheet supersymmetry and K3 target
can be used to compactify the E8xE8 heterotic string to six dimensions in a
supersymmetric manner. The data specifying such a model includes an appropriate
configuration of 24 gauge instantons in the E8xE8 gauge group to satisfy the
constraints of anomaly cancellation. In this note, we compute twining genera -
elliptic genera with appropriate insertions of discrete symmetry generators in
the trace - for (0,4) theories with various instanton embeddings. We do this by
constructing linear sigma models which flow to the desired conformal field
theories, and using the techniques of localization. We present several examples
of such twining genera which are consistent with a moonshine relating these
(0,4) models to the finite simple sporadic group M24.Comment: 22 pages, 3 tables. We thank T. Eguchi and K. Hikami for permission
to copy our Tables 2 and 3 (M24 character table and q-expansions of some
twining genera in the (4,4) sigma model with K3 target) from their article
arXiv:1008.492
(0,2) Dualities and the 4-Simplex
We propose that a simple, Lagrangian 2d duality
interface between the 3d XYZ model and 3d SQED
can be associated to the simplest triangulated 4-manifold: the 4-simplex. We
then begin to flesh out a dictionary between more general triangulated
4-manifolds with boundary and 2d interfaces. In
particular, we identify IR dualities of interfaces associated to local changes
of 4d triangulation, governed by the (3,3), (2,4), and (4,2) Pachner moves. We
check these dualities using supersymmetric half-indices. We also describe how
to produce stand-alone 2d theories (as opposed to interfaces) capturing the
geometry of 4-simplices and Pachner moves by making additional field-theoretic
choices, and find in this context that the Pachner moves recover abelian
trialities of Gadde-Gukov-Putrov. Our work provides new,
explicit tools to explore the interplay between 2d dualities and 4-manifold
geometry that has been developed in recent years.Comment: 50 pages, 15 figures. v2: minor edits, references adde
Twisted Supergravity and Koszul Duality: A case study in AdSâ
In this note, we study a simplified variant of the familiar holographic duality between supergravity on AdSâĂSÂłĂTâ´ and the SCFT (on the moduli space of) the symmetric orbifold theory Sym^N(Tâ´) as Nââ. This variant arises conjecturally from a twist proposed by the first author and Si Li. We recover a number of results concerning protected subsectors of the original duality working directly in the twisted bulk theory. Moreover, we identify the symmetry algebra arising in the Nââ limit of the twisted gravitational theory. We emphasize the role of Koszul duality---a ubiquitous mathematical notion to which we provide a friendly introduction---in field theory and string theory. After illustrating the appearance of Koszul duality in the "toy" example of holomorphic Chern-Simons theory, we describe how (a deformation of) Koszul duality relates bulk and boundary operators in our twisted setup, and explain how one can compute algebra OPEs diagrammatically using this notion. Further details, results, and computations will appear in a companion paper
Celestial holography meets twisted holography: 4d amplitudes from chiral correlators
We propose a new program for computing a certain integrand of scattering
amplitudes of four-dimensional gauge theories which we call the \textit{form
factor integrand}, starting from 6d holomorphic theories on twistor space. We
show that the form factor integrands can be expressed as sums of products of
1.) correlators of a 2d chiral algebra, related to the algebra of asymptotic
symmetries uncovered recently in the celestial holography program, and 2.) OPE
coefficients of a 4d non-unitary CFT. We prove that conformal blocks of the
chiral algebras are in one-to-one correspondence with local operators in 4d. We
use this bijection to recover the Parke-Taylor formula, the CSW formula, and
certain one-loop scattering amplitudes. Along the way, we explain and derive
various aspects of celestial holography, incorporating techniques from the
twisted holography program such as Koszul duality. This perspective allows us
to easily and efficiently recover the infinite-dimensional chiral algebras of
asymptotic symmetries recently extracted from scattering amplitudes of massless
gluons and gravitons in the celestial basis. We also compute some simple
one-loop corrections to the chiral algebras and derive the three-dimensional
bulk theories for which these 2d algebras furnish an algebra of boundary local
operators.Comment: 80 pages, 6 figures, 1 table. Version 2: several corrections, more
one-loop amplitudes compute
Heterotic sigma models on and the Borcherds automorphic form
We consider the spectrum of BPS states of the heterotic sigma model with
supersymmetry and target, as well as its second-quantized
counterpart. We show that the counting function for such states is intimately
related to Borcherds' automorphic form , a modular form which
exhibits automorphy for . We comment on possible
implications for Umbral moonshine and theories of AdS gravity.Comment: 12 pages; v2 error (involving fermion zero modes) correcte