Arithmetic of Calabi–Yau varieties and rational conformal field theory

It is proposed that certain techniques from arithmetic algebraic geometry provide a framework which is useful to formulate a direct and intrinsic link between the geometry of Calabi–Yau manifolds and the underlying conformal field theory. Specifically, it is pointed out how the algebraic number field determined by the fusion rules of the conformal field theory can be derived from the number theoretic structure of the cohomological Hasse–Weil L-function determined by Artin’s congruent zeta function of the algebraic variety. In this context, a natural number theoretic characterization arises for the quantum dimensions in this geometrically determined algebraic number field.     

Emergent Spacetime from Modular Motives

The program of constructing spacetime geometry from string theoretic modular forms is extended to Calabi-Yau varieties of dimensions three and four, as well as higher rank motives. Modular forms on the worldsheet can be constructed from the geometry of spacetime by computing the L?Cfunctions associated to omega motives of Calabi-Yau varieties, generated by their holomorphic n?forms via Galois representations. The modular forms that emerge in this way are related to characters of the underlying rational conformal field theory. The converse problem of constructing space from string theory proceeds in the class of diagonal theories by determining the motives associated to modular forms in the category of pure motives with complex multiplication. The emerging picture suggests that the L?Cfunction can be viewed as defining a map between the geometric category of motives and the category of conformal field theories on the worldsheet.     

Geometric Kac–Moody modularity

It is shown how the arithmetic structure of algebraic curves encoded in the Hasse–Weil L-function can be related to affine Kac–Moody algebras. This result is useful in relating the arithmetic geometry of Calabi–Yau varieties to the underlying exactly solvable theory. In the case of the genus three Fermat curve we identify the Hasse–Weil L-function with the Mellin transform of the twist of a number theoretic modular form derived from the string function of a non-twisted affine Lie algebra. The twist character is associated to the number field of quantum dimensions of the conformal field theory.     

Towards supersymmetric cosmology in M-theory

We present a new solution in the heterotic M-theory in which the metric depends on (cosmic) time. The solution preserves N=1 supersymmetry in 4 dimensions in the leading order of the κ^2/3 expansion. It is the first example of the time-dependent supersymmetric solution in M-theory on . It describes expanding 4-dimensional space–time with shrinking orientifold interval and static Calabi–Yau internal space.     

Scalar potential for the gauged Heisenberg algebra and a non-polynomial antisymmetric tensor theory

We study some issues related to the effective theory of Calabi–Yau compactifications with fluxes in type II theories. At first the scalar potential for a generic electric Abelian gauging of the Heisenberg algebra, underlying all possible gaugings of R–R isometries, is presented and shown to exhibit, in some circumstances, a “dual” no-scale structure under the interchange of hypermultiplets and vector multiplets. Subsequently a new setting of such theories, when all R–R scalars are dualized into antisymmetric tensors, is discussed. This formulation falls in the class of non-polynomial tensor theories considered long ago by Freedman and Townsend and it may be relevant for the introduction of both electric and magnetic charges.     

String modular phases in Calabi–Yau families

We investigate the structure of singular Calabi–Yau varieties in moduli spaces that contain a Brieskorn–Pham point. Our main tool is a construction of families of deformed motives over the parameter space. We analyze these motives for general fibers and explicitly compute the L$L$-series for singular fibers for several families. We find that the resulting motivic L$L$-functions agree with the L$L$-series of modular forms whose weight depends both on the rank of the motive and the degree of the degeneration of the variety. Surprisingly, these motivic L$L$-functions are identical in several cases to L$L$-series derived from weighted Fermat hypersurfaces. This shows that singular Calabi–Yau spaces of non-conifold type can admit a string worldsheet interpretation, much like rational theories, and that the corresponding irrational conformal field theories inherit information from the Gepner conformal field theory of the weighted Fermat fiber of the family. These results suggest that phase transitions via non-conifold configurations are physically plausible. In the case of severe degenerations we find a dimensional transmutation of the motives. This suggests further that singular configurations with non-conifold singularities may facilitate transitions between Calabi–Yau varieties of different dimensions.     

A-model and generalized Chern–Simons theory

The relation between open topological strings and Chern–Simons theory was discovered by Witten. He proved that A-model on T^*M where M is a three-dimensional manifold is equivalent to Chern–Simons theory on M and that A-model on arbitrary Calabi–Yau 3-fold is related to Chern–Simons theory with instanton corrections. In present Letter we discuss multidimensional generalization of these results.     

D-branes, exceptional sheaves and quivers on Calabi–Yau manifolds: from Mukai to McKay

We present a method based on mutations of helices which leads to the construction (in the large-volume limit) of exceptional coherent sheaves associated with the (∑_al_a=0) orbits in Gepner models. This is explicitly verified for a few examples including some cases where the ambient weighted projective space has singularities not inherited by the Calabi–Yau hypersurface. The method is based on two conjectures which lead to the analog, in the general case, of the Beilinson quiver for . We discuss how one recovers the McKay quiver using the gauged linear sigma model (GLSM) near the orbifold or Gepner point in Kähler moduli space.     

Eisenstein type series for Calabi–Yau varieties

In this article we introduce an ordinary differential equation associated to the one parameter family of Calabi–Yau varieties which is mirror dual to the universal family of smooth quintic three folds. It is satisfied by seven functions written in the q-expansion form and the Yukawa coupling turns out to be rational in these functions. We prove that these functions are algebraically independent over the field of complex numbers, and hence, the algebra generated by such functions can be interpreted as the theory of (quasi) modular forms attached to the one parameter family of Calabi–Yau varieties. Our result is a reformulation and realization of a problem of Griffiths around seventies on the existence of automorphic functions for the moduli of polarized Hodge structures. It is a generalization of the Ramanujan differential equation satisfied by three Eisenstein series.     

Conformal angular momentum and lie derivative of spinor field in Riemann space

It is shown that in Riemann space—time the conformal angular momentum and Lie derivative of spinors differ only by a part involving multiplication by a factor from the conformal Killing equations.Yakutsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 99–101, November, 1994.     

Generalized conformal invariance conditions on a sigma model of the open bosonic string including the first massive mode

Conformal invariance conditions on a sigma model of the open bosonic string including the tachyon, the abelian gauge field an the first excited massive mode are calculated up to order a′. Inner symmetries are used to compute conformal invariance conditions from renormalization group beta functions.     

Fusion rings and geometry

The algebraic structure of fusion rings in rational conformal field theories is analyzed in detail in this paper. A formalism which closely parallels classical tools in the study of the cohomology of homogeneous spaces is developed for fusion rings, in general, and for current algebra theories, in particular. It is shown that fusion rings lead to a natural orthogonal polynomial structure. The rings are expressed through generators and relations. The relations are then derived from some potentials leading to an identification of the fusion rings with deformations of affine varieties. In general, the fusion algebras are mapped to affine varieties which are the locus of the relations. The connection with modular transformations is investigated in this picture. It is explained how chiral algebras, arising inN=2 superconformal field theory, can be derived from fusion rings. In particular, it is argued that theories of the typeSU(N) _k /SU(n–1) are theN=2 counterparts of Grassmann manifolds and that there is a natural identification of the chiral fields with Schubert varieties, which is a graded algebra isomorphism.Supported in part by NSF grant PHY 89-04035 supplemented by funds from NASA     

A plasma analogy and Berry matrices for non-abelian quantum Hall states

We present an approach to the computation of the non-abelian statistics of quasiholes in quantum Hall states, such as the Pfaffian state, whose wavefunctions are related to the conformal blocks of minimal model conformal field theories. We use the Coulomb gas construction of these conformal field theories to formulate a plasma analogy for the quantum Hall states. A number of properties of the Pfaffian state follow immediately, including the Berry phases, which demonstrate the quasiholes' fractional charge, the abelian statistics of the two-quasihole state, and equal-time ground state correlation functions. The non-abelian statistics of multi-quasihole states follows from an additional assumption.     

Conformal Orbifold Theories and Braided Crossed G-Categories

The Berezin-Toeplitz deformation quantization of an abelian variety is explicitly computed by the use of Theta-functions. An SL(2n,)-equivariant complex structure dependent equivalence E between the constant Moyal-Weyl product and this family of deformations is given. This equivalence is seen to be convergent on the dense subspace spanned by the pure phase functions. The Toeplitz operators associated to the equivalence E applied to a pure phase function produces a covariant constant section of the endomorphism bundle of the vector bundle of Theta-functions (for each level) over the moduli space of abelian varieties.Applying this to any holonomy function on the symplectic torus one obtains as the moduli space of U(1)-connections on a surface, we provide an explicit geometric construction of the abelian TQFT-operator associated to a simple closed curve on the surface. Using these TQFT-operators we prove an analog of asymptotic faithfulness (see [A1]) in this abelian case. Namely that the intersection of the kernels for the quantum representations is the Toreilli subgroup in this abelian case.Furthermore, we relate this construction to the deformation quantization of the moduli spaces of flat connections constructed in [AMR1] and [AMR2]. In particular we prove that this topologically defined *-product in this abelian case is the Moyal-Weyl product. Finally we combine all of this to give a geometric construction of the abelian TQFT operator associated to any link in the cylinder over the surface and we show the glueing axiom for these operators.This research was conducted in part for the Clay Mathematics Institute at University of California, Berkeley.This work was supported by MaPhySto – A Network in Mathematical Physics and Stochastics, funded by The Danish National Research Foundation     

A Monoidal Category for Perturbed Defects in Conformal Field Theory

Starting from an abelian rigid braided monoidal category C{\mathcal{C}} we define an abelian rigid monoidal category C_F{\mathcal{C}_F} which captures some aspects of perturbed conformal defects in two-dimensional conformal field theory. Namely, for V a rational vertex operator algebra we consider the charge-conjugation CFT constructed from V (the Cardy case). Then C = Rep(V){\mathcal{C} = {\rm Rep}(V)} and an object in C_F{\mathcal{C}_F} corresponds to a conformal defect condition together with a direction of perturbation. We assign to each object in C_F{\mathcal{C}_F} an operator on the space of states of the CFT, the perturbed defect operator, and show that the assignment factors through the Grothendieck ring of C_F{\mathcal{C}_F}. This allows one to find functional relations between perturbed defect operators. Such relations are interesting because they contain information about the integrable structure of the CFT.     

Local conformal field algebras

The local conformal field algebras with the multiplication corresponding to the regularized pointwise product of fields in the operator algebras of the quantum conformal field theory are investigated.     

Relative periods and open-string integer invariants for a compact Calabi–Yau hypersurface

In this work we compute relative periods for B-branes, realized in terms of divisors in a compact Calabi–Yau hypersurface, by means of direct integration. Although we exemplify the method of direct integration with a particular Calabi–Yau geometry, the recipe automatically generalizes for divisors in other Calabi–Yau geometries as well. From the calculated relative periods we extract double-logarithmic periods. These periods qualify to describe disk instanton generated N=1 superpotentials of the corresponding compact mirror Calabi–Yau geometry in the large volume regime. Finally we extract the integer invariants encoded in these brane superpotentials.     

Rational Generalised Moonshine from abelian orbifoldings of the Moonshine Module

We consider orbifoldings of the Moonshine Module with respect to the abelian group generated by a pair of commuting Monster group elements with one of prime order p=2,3,5,7 and the other of order pk for k=1 or k prime. We show that constraints arising from meromorphic orbifold conformal field theory allow us to demonstrate that each orbifold partition function with rational coefficients is either constant or is a hauptmodul for an explicitly found modular fixing group of genus zero. We thus confirm in the cases considered the Generalised Moonshine conjectures for all rational modular functions for the Monster centralisers related to the Baby Monster, Fischer, Harada-Norton and Held sporadic simple groups. We also derive non-trivial constraints on the possible Monster conjugacy classes to which the elements of the orbifolding abelian group may belong.