Lara Anderson
Geometric Moduli Stabilization in Heterotic Calabi-Yau Vacua
Abstract
We present a new approach to moduli stabilization in Calabi-Yau compactifications of Heterotic M-theory. Our approach is based in a detailed mathematical understanding of the moduli dependence of supersymmetric heterotic vacua. We will discuss a scenario to stabilize all geometric moduli - that is, the complex structure, Kahler moduli and the dilaton - without Neveu-Schwarz three-form flux.
David Baraglia
Topological T-duality with monodromy
Abstract
In topological T-duality one usually considers principal torus bundles. Connections with mirror symmetry suggest that T-duality should be applicable to more general torus fibrations. In this direction we consider topological T-duality for torus bundles with non-trivial monodromy. In particular for general circle bundles we prove existence and uniqueness of the T-duals. We consider also the behavior of twisted cohomology and twisted K-theory under such T-dualities.
Nicolas Behr
Rational" matrix factorizations and defects via closed functor algebra
Abstract
The Landau-Ginzburg/CFT correspondence as first proposed by D. Gepner in 1990 relates certain 2d RCFTs (Kazama-Suzuki models) to Landau-Ginzburg theories (via a choice of superpotential). While originally a correspondence of bulk theories, a similar correspondence of solutions to boundary conditions has been constructed for a number of explicit models, including the A-type minimal model aka the SU(2)/U(1) KS model. Here, for B-type boundary conditions, maximally symmetric ("rational") solutions for the RCFT boundary problem (Cardy branes) are associated to certain elementary matrix factorizations in the LG theory. To find such a "dictionary" for more general setups is a very non-trivial problem, already for the SU(3)/U(2) KS models. While some partial "dictionary" can be obtained in a "pedestrian way" [NB,1005.2117], the full solution requires some more sophisticated methods. As I will present in this talk, one may construct a closed functor algebra which not only allows to obtain an explicit "dictionary" of Cardy branes to matrix factorizations, but which also induces a set of topological defects for the LG theory that precisely matches the structure of such defects for the RCFT. This provides in particular some to the best of our knowledge new, physically motivated structure on the triangulated category of matrix-(bi)-factorizations, and I will also comment on some possible generalizations.
Eric Bergshoeff
Dual Doubled Geometry
Abstract
It is well-known that a T-duality covariant formulation of the fundamental branes of toroidally compactified string theory with 32 supercharges requires a doubled geometry. We will probe this doubled geometry with dual fundamental branes, i.e. solitonic branes. Restricting ourselves first to solitonic branes with more than two transverse directions we find that the doubled geometry corresponds to an effective wrapping rule for the solitonic branes which is dual to the wrapping rule for fundamental branes. This dual wrapping rule can be understood by the presence of Kaluza-Klein monopoles. Extending our analysis to solitonic branes with less than or equal to two transverse directions we find that the solitonic wrapping rule suggests the existence of a class of generalized Kaluza-Klein monopoles in ten dimensions.
Chris Brav
Ping-pong and exceptional vector bundles
Abstract
We present a strategy for proving that full exceptional collections of vector bundles on projective n-space can be constructed from a standard collection of line bundles, reducing the question of constructibility to the problem of freeness of a certain finitely generated linear group. We use the ping-pong lemma of Fricke-Klein to solve this problem in low dimensions, thus providing a new proof of constructibility of exceptional collections in some cases. We expect a similar ping-pong argument to give constructibility on projective n-space and on some other Fano varieties of Picard rank one. Constructibility results are useful for understanding spaces of stability conditions associated to such varieties. This is joint work in progress with Hugh Thomas.
Nils Carqueville
Topological defects and Khovanov-Rozansky homology
Abstract
Topological defects in affine Landau-Ginzburg models are described by matrix factorisations. We motivate some of their properties such as defect fusion and action on bulk fields, which we then treat rigorously by proving that the defect category has a pivotal rigid monoidal structure that is compatible with the triangulated structure. Furthermore, having defect fusion under good control allows for a direct and explicit computation of Khovanov-Rozansky link invariants, and we shall present various examples. The two parts of my talk are based on joint work with Ingo Runkel and Daniel Murfet, respectively.
Shyamoli Chaudhuri
Matrix Theory and the Fundamental Symmetry $D_{11}\times \mathcal{E}_{8}\times \mathcal{E}_{8}$: A Planck Scale Generalized Electric-Magnetic Dual Initial State
Abstract
We sketch the line of reasoning that leads out from Peter West's well-established result in 2001-04, motivated by the exciting developments in generalized electric magnetic duality in the worldsheet formalism, identifying theories with 32 supercharges by the global symmetry group, $E_{11}$, demonstrating an explicit isomorphism to $E_{11}$ of each of the four theories with 32 supercharges: 11d supergravity, and the type IIA and type IIB, and massive type IIA theories. In subsequent work I have shown that this leads to the group $D_{11}$ for the chiral type IA, type IB, and heterotic anomalous supergravities, and anomaly cancelation with the introduction of Yang-Mills fields completes our identification of the fundamental symmetry group of String/M theory with sixteen conserved supercharges as: $D_{11}\times \mathcal{G}_{\text{Yang-Mills}}$, where $\mathcal{G}_{\text{Yang-Mills}}$ is the affine extension of $E_{8}\times E_{8}$ or $Spin(32)/\mathbb{Z}_{2}$; not the group $B_{11}$, suggested by Schnakenburg and West. Comparison of the half BPS states of this algebra, including both massive modes in the Kac Moody Algebra and the higher levels of the very extended group, with the BPS algebras of N=8 and N=4 string theories, using the partially complete 1993-96 pioneering analyses by Gregory Moore, and by Jeffrey Harvey and Moore, followed by many authors, of N=2 heterotic strings, which bring in many subtle new features, would give complementary insight, and enable significant progress leading from West's intriguing results for thesymmetry group of M theory. We sketch the implications for a straightforward matrix realization of a theory with this algebra demonstrating emergent spacetime, the early beginnings of an expanding universe cosmology, and chiral matter, incorporating also the principle of generalized 11d Hodge-Poincare electric-magnetic duality. We identify the Planck scale patch of initial state spacetime-matter with a massive monopole carrying all six species of generalized electric-magnetic charge, in a fundamental theory which incorporates generalized electric-magnetic duality in its initial conditions.
Rhys Davies
Hyperconifold singularities and transitions
Abstract
I will introduce hyperconifold singularities, which are point-like threefold singularities given by finite quotients of the conifold, and arise naturally in compact varieties which lie on the boundary of moduli spaces of smooth multiply-connected Calabi-Yau threefolds. These varieties admit projective crepant resolutions, giving rise to new topological transitions -- hyperconifold transitions -- between compact Calabi-Yau manifolds. They are related to ordinary conifold transitions by mirror symmetry, but can change the fundamental group as well as the Hodge numbers.
Jimmy Dillies
Generalized Borcea-Voisin Construction
Abstract
We generalize the construction of Borcea and Voisin of Calabi-Yau threefolds by allowing automorphisms of higher order and considering both three- and fourfolds. We classify all orbifolds / varieties obtained in this way and show that they come in mirror pairs only if constructed through involutions. We gave suggestions on the obstruction to the existence of 'classical' mirrors.
Mboyo Esole
The resolved geometry of SU(5) GUT
Abstract
I explore the resolution of singularities of elliptic fibrations describing SU(5) Grand Unified Theories in F-theory. I discuss some new properties like the presence of non-kodaira fibers above codimension three points in the base and the existence of transitions between different resolutions that are topologically not equivalent.
David Favero
Graded matrix factorizations and functor categories
Abstract
We provide a matrix factorization model for the derived internal Hom (continuous), in the homotopy category of k-linear dg-categories, between categories of graded matrix factorizations. This description is used to calculate the derived natural transformations between twists functors on categories of graded matrix factorizations. Furthermore, we combine our model with a theorem of Orlov to establish a geometric picture related to Kontsevich's Homological Mirror Symmetry Conjecture. As applications, we obtain new cases of a conjecture of Orlov concerning the Rouquier dimension of the bounded derived category of coherent sheaves on a smooth variety and a proof of the Hodge conjecture for n-fold products of a K3 surface closely related to the Fermat cubic fourfold.
James Fullwood
On generalized Sethi-Vafa-Witten formulas
Abstract
We present a formula for computing proper pushforwards of classes in the Chow ring of a projective bundle under the projection $\pi : \mathbb{P}(\mathcal{E})\rightarrow B$, for $B$ a non-singular compact complex algebraic variety of any dimension. Our formula readily produces generalizations of formulas derived by Sethi,Vafa, and Witten to compute the Euler characteristic of elliptically fibered Calabi-Yau fourfolds used for F-theory compactifications of string vacua. As an application, we show that their is a an orientifold limit of a $D_5$ model of F-theory admitting a unique configuration of smooth branes satisfying the tadpole matching condition between F-theory and type IIB.
Sergey Grigorian
Deformations of $G_{2}$ structures
Abstract
$G_2$ structures on 7-dimensional manifolds arise naturally in supersymmetric M-theory compactifications, in general with fluxes. An important special case is when fluxes are turned off - this gives torsion-free $G_2$ structures, which correspond to manifolds with holonomy $G_2$. Here we review recent results on deformations of $G_2$ structures with torsion, as well as results on the geometry of the moduli space of torsion-free $G_2$ structures.
Jim Halverson
Hilbert's 10th Problem and Computability in the String Landscape
Abstract
The systematic study of non-perturbative (and other) effects in string theory often leads one to the study of large systems of diophantine equations. I will give examples, relate these issues to Hilbert's 10th problem, and discuss implications for computability of the landscape.
Masashi Hamanaka
Non-commutative Solitons and Quasi-determinants
Abstract
We discuss extension of soliton theory and integrable systems to non-commutative (NC) spaces, focusing on integrable aspects of non-commutative anti-self-dual Yang-Mills (ASDYM) equations. We present Backlund transformations for the G=U(2) non-commutative anti-self-dual Yang-Mills equations and give wide class of exact solutions of them (not only instanton-type solutions with finite action). We find that one kind of non-commutative determinants,quasi-determinants, play crucial roles in the construction of non-commutative solutions. We also discuss twistor descriptions of the results, analysis of the exact solutions, and reduction of a non-commutative anti-self-dual Yang-Mills equation to non-commutative integrable equations such non-commutative KdV and Toda equations, if possible. This is partially based on collaboration with C. Gilson and J. Nimmo (Glasgow).
Tyler Jarvis
Moduli of Curves with W-structure and Mirror Symmetry
Abstract
The moduli of curves with W-structure and their corresponding cohomological field theories form an orbifolded Landau-Ginzburg A-model and are the subject of several beautiful mirror symmetry conjectures. Some of these conjectures have been proved, including the Witten ADE-integrable hierarchies conjecture, while others are still open. In this talk I will give an overview of the theory as well as discussing recent progress on some of the conjectures.
Benjamin Jurke
Cohomology of Toric Varieties and Applications
Abstract
Recent non-standard techniques for the computation of line bundle valued cohomology group dimensions on toric varieties are presented. Those methods can also be extended to compute the cohomology on quotient spaces, a highly relevant class of geometries encountered in string model building. Further applications in string phenomenology are briefly discussed.
Louis H. Kauffman
A Quantum Context for the Jones Polynomial and Khovanov Homology
Abstract
We give a quantum statistical interpretation for the bracket polynomial state sum $\langle K \rangle$, the Jones polynomial $V_K(t)$ and for virtual knot theory versions of the Jones polynomial, including the arrow polynomial. We use these quantum mechanical interpretations to give new quantum algorithms for these polynomials. In those cases where Khovanov homology is defined, the Hilbert space $C(K)$ of our model is isomorphic with the chain complex for Khovanov homology with coefficients in the complex numbers. There is a natural unitary transformation $U: C(K) \to C(K)$ such that $\langle K \rangle = \text{Tr}(U)$ where $\langle K\rangle$ denotes the evaluation of the state sum model for the corresponding polynonmial. We show that for the Khovanov boundary operator $d: C(K) \to C(K)$, we have the relationship $dU + Ud = 0.$ Consequently, the operator $U$ acts on the Khovanov homology, and we obtain a direct relationship between Khovanov homology and this quantum algorithm for the Jones polynomial. We raise the question of the relationship of this model with recent work of Witten.
Michael Kay
Bulk deformations of open topological string theory
Abstract
I will present a general method to construct bulk-deformed open topological string theories from Landau-Ginzburg models. The main ingredients in the construction are a "weak" version of deformation quantisation, and an application of the homological perturbation lemma. Together they allow to explicitly compute all bulk-deformed open topological string amplitudes at tree-level before tadpole-cancellation. The talk is based on work with N. Carqueville (arXiv:1104.5438).
Takashi Kimura
On stringy algebraic operations in equivariant K-theory
Abstract
We will introduce and explain some new algebraic structures associated to a smooth variety with a proper action of an algebraic group that are invariants of the associated quotient orbifold. Such algebraic structures arise from their orbifold K-theory ring, a K-theoretic variant of Chen-Ruan orbifold cohomology, and generalize more familiar operations in ordinary K-theory.
Peggy Kouroumalou
Supergravities in 5 and 6 dimensions and their possible realization in F-theory
Abstract
We review the basic features of 5D Magical and Minkowskian supergravities and of their parent theories in 6 dimensions. We comment on the possibility of embedding the 6D parent theories in a F-theory construction.
Thomas Kragh
Fibrancy of Symplectic Homology in Cotangent Bundles
Abstract
Let $M$ be any exact sub-Liouville domain of a cotangent bundle $T^{*}N$. In this talk I will describe how the symplectic homology of $M$ in a sense is "fibrant" over the base $N$, and how this implies that there is a Serre type spectral sequence with product converging to the symplectic homology ring of $M$. In particular this implies that for a closed Lagrangian brane in $T^{*}N$ this spectral sequence over $N$ converges to the loop space homology of $L$ equipt with the Chas-Sullivan string product. This puts new restrictions on $L$ and in fact proves that up to a finite covring spaces lift of $N$ the map $L \to T^{*}N \to N$ is a homology equivalence.
Andreas Malmendier
Heterotic/F-theory duality and lattice polarized K3 surfaces
Abstract
The heterotic string compactified on $T^2$ has a large discrete symmetry group $SO(2, 18; \mathbb{Z})$, 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 $SO(2, 18; \mathbb{Z})$-invariant functions. In our new work (which is 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 $SO(2, 2;\mathbb{Z})$ symmetry - or there is a single Wilson line - and $SO(2, 3; \mathbb{Z})$ 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. We work out precise relations between the function theory and geometry on these K3 surfaces. This allows us to compute explicitly the periods and period relations for their two-forms and many-valued modular forms, and describe the transcendental lattices and the Riemann matrices for associated Kuga-Satake varieties explicitly in terms of the periods.
Stefan Mendez-Diez
K-theoretic Aspects of String Theory Dualities
Abstract
This talk will discuss K-theoretic matching of D-brane charges in the string duality between the type I superstring theory on a 4-torus and the type IIA theory on a K3 surface. We will see that replacing K3 by its orbifold blow-down seems to resolve many of the apparent problems that arise in the theory from using a smooth K3, such as not having an isomorphism of the torsion charges. We will also discuss how using the equivariant K-theory classification of D-brane charges on the orbifold limit of K3 provides added information about the K-theory of the blow-down.
Samuel Monnier
The global gravitational anomaly of the self-dual field theory
Abstract
We will report on recent work on the global gravitational anomaly of the self-dual field theory (also called the abelian chiral p-form). The partition function of the self-dual field theory on a 4k+2 dimensional manifold M is a section of a certain line bundle over the space of metrics on M modulo diffeomorphisms. We will describe the isomorphism class of this bundle and propose a conjectural formula for the holonomy of a natural connection living on it. Mathematically, this story provides an interesting link between the theory of determinant line bundles of Dirac operators, Siegel theta functions and certain cobordism invariants defined by Hopkins and Singer.
El-kaïoum M. Moutuou
Twistings of $KR$ for Real groupoids
Abstract
Given a locally compact Hausdorff groupoid $\mathcal{G}$ with Haar system, the twistings of its $K$-groups are defined as Morita equivalence classes of graded Dixmier-Douady bundles, or equivalently, as equivalence classes of graded $\mathbb{S}^1$-central extensions of some groupoids Morita equivalent to $\mathcal{G}$. It is well known that these elements form an abelian group $\widehat{Br}(\mathcal{G})$, called the graded Brauer group of $\mathcal{G}$, which is isomorphic to $\check{H}^1(\mathcal{G}_\bullet,\mathbb{Z}_2) \oplus \check{H}^2(\mathcal{G}_\bullet,\mathbb{S}^1)$. In my talk I introduce the Real graded Brauer group $\widehat{BrR}(\mathcal{G})$ that constitutes the group of twistings for the twisted analog of Atiyah's $KR$-theory of locally compact Real groupoids (i.e. groupoids endowed with involutions). I will then give a cohomological formula of this group.
Daniel Pomerleano
Curved String Topology and Tangential Fukaya Categories
Abstract
In this talk, we will look at non-commutative versions of Landau-Ginzburg models. More precisely, given a simply connected manifold M such that it's cochain algebra, $C^{*}(M)$, is a pure Sullivan dga, we will consider curved deformations of the algebra $C_{*}(\Omega M)$ and consider when the category of curved modules over these algebras becomes fully dualizable. For simple manifolds, like products of spheres, we are able to give an explicit criterion, like the Jacobian criterion, for when the resulting category of curved modules is smooth,proper and CY and thus gives rise to a TQFT. We give Floer theoretic interpretations of these theories for projective spaces and their products, which involve defining a Fukaya category which counts holomorphic disks with prescribed tangencies to a divisor.
Anatoly Preygel
Matrix factorizations via group actions on categories, etc.
Abstract
We'll describe a method for getting a handle on the closed string sector of the Landau-Ginzburg B-model, i.e., the Hochschild invariants of the 2-periodic dg-categories of matrix factorizations, by relating it to the Hochschild invariants of the total space of the potential. The starting point will be a description of the 2-periodic dg-category of matrix factorizations as a 'Tate-construction' for a homotopy circle action (or better, $B\widehat{\mathbb{G}}_{a}$-action) on the dg-category of perfect complexes on the total space. Then, we'll explain some algebraic tools allowing one to leverage this description along with the well-known understanding of Hochschild invariants of perfect complexes (e.g., the various formalities).
Thorsten Rahn
Landscape Study of Target Space Duality of (0,2) Heterotic String Models
Abstract
Extending earlier ideas, in the framework of (0,2) gauged linear sigma models, I will present the proposal of a systematic procedure to generate dual models in non-geometric phases. I will explain how we calculated the massless chiral spectra and the dimensions of the moduli spaces for such pairs, moving to their respective geometric phases and how we found total agreement in thousands of cases. Our study includes Calabi-Yau geometries given by complete intersections of hyper surfaces in toric varieties equipped with SU(n) vector bundles defined via the monad construction. I will present the study of a whole landscape of about 50 thousand such (0,2) models which became feasible due to the efficient implementation cohomCalg for the computation of bundle valued cohomology classes over toric varieties.
Daniel Robbins
Higher derivative couplings on D-branes from T-duality
Abstract
D-brane actions are known to receive higher derivative corrections involving the bulk metric which can make key contributions to tadpole equations, and hence understanding these terms is important for determining the consistency of vacua. Consistency with T-duality implies that there should also be higher derivative couplings involving the B-field, and these terms, too, will be important for the consistency of certain classes of constructions. We use T-duality to predict some of these couplings, and then verify their form by computing disc amplitudes.
Fabio Ferrari Ruffino
Freed-Witten anomaly and D-brane gauge theories
Abstract
I will discuss the different nature of the gauge theories on a D-brane or a stack of D-branes, as follows from Freed-Witten anomaly. Usually on a D-brane world-volume there is a standard gauge theory, described by the A-field thought of as a connection on a complex vector bundle. Actually, this is a particular case, even if it is the most common one. In order to get a complete picture, within the framework provided by the geometry of gerbes with connection, it is necessary to give a joint geometrical description of the A-field and the B-field, via the language of Cech hypercohomology. A global world-sheet anomaly, called Freed-Witten anomaly, imposes some constraints on such fields: I will show case by case what is the nature of the corresponding gauge theory on the D-brane or stack of D-branes.
Jihye Seo
Singularity structure and massless dyons of pure N = 2 theories with SU(r+1) and Sp(2r) gauge groups
Abstract
We study hyper-elliptic Seiberg-Witten curves for N = 2 pure super Yang-Mills theories. We study the locus of vanishing discriminant of the curve and compute charges for all 2r + 1 and 2r + 2 massless dyons for Sp(2r) and SU(r + 1) with rank r > 1 in a region of moduli space. We apply a double-discriminant method to locate Argyres- Douglas singularity loci in the moduli space, by applying another discriminant operator, with respect to another variable to be one of the gauge-invariant moduli u. Two BPS dyons become massless where two different solutions of vanishing discriminant intersect. This complex codimension-2 locus in the moduli space is easily captured by demanding an extra condition of vanishing double discriminant, whose order of vanishing also determines whether dyons are Argyres-Douglas type or not. The intersection locus in the moduli space is cusp-like and node-like for mutually non-local and local pairs of BPS dyons respectively.
Artan Sheshmani
Higher rank stable pairs and virtual localization
Abstract
We introduce a higher rank analog of the Pandharipande-Thomas theory of stable pairs on a Calabi-Yau threefold $X$. More precisely, we develop a moduli theory for frozen triples given by the data $\mathcal{O}_{X}^{\oplus r}(-n) \to F$ where $F$ is a sheaf of pure dimension $1$. The moduli space of such objects does not naturally determine an enumerative theory: that is, it does not naturally possess a perfect symmetric obstruction theory. Instead, we build a zero-dimensional virtual fundamental class by hand, by truncating a deformation-obstruction theory coming from the moduli of objects in the derived category of $X$. This yields the first deformation-theoretic construction of a higher-rank enumerative theory for Calabi-Yau threefolds. We calculate this enumerative theory for local $\mathbb{P}^1$ using the Graber-Pandharipande virtual localization technique. In a sequel to this project (arXiv:1101.2251) we have shown how to compute similar invariants associated to frozen triples using Kontsevich-Soibelman, Joyce-Song wall-crossing techniques.
Nicolò Sibilla
Mirror Symmetry and Ribbon Graphs
Abstract
In this talk I will explain, through examples, how to construct a model for the Fukaya category of punctured Riemann surfaces in terms of a sheaf of dg-categories over a suitable category of ribbon graphs. The main ingredients will be Nadler and Zaslow's work on cotangent bundles, and Kontsevich' recent ideas on the locality of the Fukaya category of a Stein manifold. Further, I will explain applications to homological mirror symmetry for degenerate elliptic curves. This work is joint with David Treumann and Eric Zaslow.
Matt Szczesny
Feynman graphs, Hall algebras, and $F_1$-linear categories
Abstract
I will discuss several examples of categories, which while not additive, have enough structure to define their Hall algebras. These include categories of Feynman graphs, monoid representations, and others, which should be viewed as $F_1$ - analogues of abelian categories.
Justin Vazquez-Poritz
An Infinite Class of $G_{2}$ Holonomy Spaces
Abstract
Spaces of $G_{2}$ holonomy are of considerable interest since they enable one to construct a supersymmetric embedding of four-dimensional Minkowski spacetime in M-theory. I will discuss a countably infinite family of spaces with $G_{2}$ holonomy which are characterized by two winding numbers. These spaces are described by cohomogeneity one metrics with $S^{3}\times S^{3}$ principal orbits. The construction of these new $G_{2}$ holonomy spaces is analogous to that of Einstein-Kahler spaces which led to countably infinite Einstein-Sasaki spaces.
Junya Yagi
Vanishing Chiral Algebras and Höhn-Stolz Conjecture
Abstract
Given a two-dimensional quantum field theory with (0, 2) supersymmetry, one can construct a chiral (or vertex) algebra. The chiral algebra of a (0, 2) sigma model is, perturbatively, the cohomology of a sheaf of chiral differential operators on a string Kähler manifold. However, it vanishes in some cases when instantons are taken into account. I will discuss the implications of this phenomenon for the geometry of loop spaces and the Höhn-Stolz conjecture on the Witten genus.
Masahito Yamazaki
Hyperbolic Volume from Gauge Theories on Duality Walls
Abstract
We propose an equivalence of the two quantities defined from a punctured Riemann surface together with an element of the mapping class group. One is the partition function of the 3d N=2 theory on duality domain wall inside a 4d N=2 theory. Another is a hyperbolic volume of the mapping torus. We have proven that the classical limit of the former reproduces the latter in the case of the once-punctured torus. We will also explain this equality by a chain of connections involving Liouville/Toda theory and quantum Teichmuller theory. This suggests a ``categorification'' of the Alday-Gaiotto-Tachikawa relation.