Massive Orbifold

We study some aspects of 2d supersymmetric sigma models on orbifolds. It turns out that independently of whether the 2d QFT is conformal the operator products of twist operators are non-singular, suggesting that massive (non-conformal) orbifolds also `resolve singularities' just as in the conformal case. Moreover we recover the OPE of twist operators for conformal theories by considering the UV limit of the massive orbifold correlation functions. Alternatively, we can use the OPE of twist fields at the conformal point to derive conditions for the existence of non-singular solutions to special non-linear differential equations (such as Painleve III).Comment: 12 page

K\"ahler Potential of Moduli Space of Calabi-Yau dd-fold embedded in CPd+1CP^{d+1}

We study a kaehler potential K of a one parameter family of Calabi-Yau d-fold embedded in CP^{d+1}. By comparing results of the topological B-model and the data of the CFT calculation at Gepner point, the K is determined unambiguously. It has a moduli parameter psi that describes a deformation of the CFT by a marginal operator. Also the metric, curvature and hermitian two-point functions in the neighborhood of the Gepner point are analyzed. We use a recipe of tt^{*} fusion and develop a method to determine the K from the point of view of topological sigma model. It is not restricted to this specific model and can be applied to other Calabi-Yau cases.Comment: 10 pages, 2 figure

Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes

We develop techniques to compute higher loop string amplitudes for twisted $N=2$ theories with $\hat c=3$ (i.e. the critical case). An important ingredient is the discovery of an anomaly at every genus in decoupling of BRST trivial states, captured to all orders by a master anomaly equation. In a particular realization of the $N=2$ theories, the resulting string field theory is equivalent to a topological theory in six dimensions, the Kodaira--Spencer theory, which may be viewed as the closed string analog of the Chern--Simon theory. Using the mirror map this leads to computation of the `number' of holomorphic curves of higher genus curves in Calabi--Yau manifolds. It is shown that topological amplitudes can also be reinterpreted as computing corrections to superpotential terms appearing in the effective 4d theory resulting from compactification of standard 10d superstrings on the corresponding $N=2$ theory. Relations with $c=1$ strings are also pointed out.Comment: 178 pages, 20 figure

N=2 SUSY and Representation Theory: An Introduction

The determination of the exact BPS spectrum of a (large class of) four\u2013dimensional N=2 QFT\u2019s is equivalent to the Representation Theory of certain associative algebras, namely the path algebras of quivers bounded by the Jacobian ideal of a superpotential W. In this contest there is a detailed dictionary between QFT and Representation Theory. Properties which are elementary on one side are tipically subtle and hard to prove in the other. Hence both subjects have something to learn from the correspondence

Holomorphic Anomalies in Topological Field Theories

We study the stringy genus one partition function of $N=2$ SCFT's. It is shown how to compute this using an anomaly in decoupling of BRST trivial states from the partition function. A particular limit of this partition function yields the partition function of topological theory coupled to topological gravity. As an application we compute the number of holomorphic elliptic curves over certain Calabi-Yau manifolds including the quintic threefold. This may be viewed as the first application of mirror symmetry at the string quantum level.Comment: 32 pages. Appendix by S.Kat

Special arithmetic of flavor

We revisit the classification of rank-1 4d N= 2 QFTs in the spirit of Diophantine Geometry, viewing their special geometries as elliptic curves over the chiral ring (a Dedekind domain). The Kodaira-N\ue9ron model maps the space of non-trivial rank-1 special geometries to the well-known moduli of pairs (\u3b5, F 1e) where E is a relatively minimal, rational elliptic surface with section, and F 1e a fiber with additive reduction. Requiring enough Seiberg-Witten differentials yields a condition on (\u3b5, F 1e) equivalent to the \u201csafely irrelevant conjecture\u201d. The Mordell-Weil group of E (with the N\ue9ron-Tate pairing) contains a canonical root system arising from ( 121)-curves in special position in the N\ue9ron-Severi group. This canonical system is identified with the roots of the flavor group F: the allowed flavor groups are then read from the Oguiso-Shioda table of Mordell-Weil groups. Discrete gaugings correspond to base changes. Our results are consistent with previous work by Argyres et al

FQHE and tt * geometry

Cumrun Vafa [1] has proposed a microscopic description of the Fractional Quantum Hall Effect (FQHE) in terms of a many-body Hamiltonian H invariant under four supersymmetries. The non-Abelian statistics of the defects (quasi-holes and quasi-particles) is then determined by the monodromy representation of the associated tt* geometry. In this paper we study the monodromy representation of the Vafa 4-susy model. Modulo some plausible assumption, we find that the monodromy representation factors through a Temperley-Lieb/Hecke algebra with q = \ub1 exp (\u3c0i/\u3bd) as predicted in [1]. The emerging picture agrees with the other predictions of [1] as well. The bulk of the paper is dedicated to the development of new concepts, ideas, and techniques in tt* geometry which are of independent interest. We present several examples of these geometric structures in various contexts

Geometric classification of 4d N= 2 SCFTs

The classification of 4d N= 2 SCFTs boils down to the classification of conical special geometries with closed Reeb orbits (CSG). Under mild assumptions, one shows that the underlying complex space of a CSG is (birational to) an affine cone over a simply-connected \u211a-factorial log-Fano variety with Hodge numbers h p,q = \u3b4 p,q . With some plausible restrictions, this means that the Coulomb branch chiral ring [InlineMediaObject not available: see fulltext.] is a graded polynomial ring generated by global holomorphic functions u i of dimension \u394 i . The coarse-grained classification of the CSG consists in listing the (finitely many) dimension k-tuples \u394 1 , \u394 2 , ef , \u394 k which are realized as Coulomb branch dimensions of some rank-k CSG: this is the problem we address in this paper. Our sheaf-theoretical analysis leads to an Universal Dimension Formula for the possible \u394 1 , ef , \u394 k \u2019s. For Lagrangian SCFTs the Universal Formula reduces to the fundamental theorem of Springer Theory. The number N(k) of dimensions allowed in rank k is given by a certain sum of the Erd\uf6s-Bateman Number-Theoretic function (sequence A070243 in OEIS) so that for large kN(k)=2\u3b6(2)\u3b6(3)\u3b6(6)k2+o(k2). In the special case k = 2 our dimension formula reproduces a recent result by Argyres et al. Class Field Theory implies a subtlety: certain dimension k-tuples \u394 1 , ef , \u394 k are consistent only if supplemented by additional selection rules on the electro-magnetic charges, that is, for a SCFT with these Coulomb dimensions not all charges/fluxes consistent with Dirac quantization are permitted. Since the arguments tend to be abstract, we illustrate the various aspects with several concrete examples and perform a number of explicit checks. We include detailed tables of dimensions for the first few k\u2019s

Proofs of Two Conjectures Related to the Thermodynamic Bethe Ansatz

We prove that the solution to a pair of nonlinear integral equations arising in the thermodynamic Bethe Ansatz can be expressed in terms of the resolvent kernel of the linear integral operator with kernel exp(-u(theta)-u(theta'))/cosh[(1/2)(theta-theta')]Comment: 16 pages, LaTeX file, no figures. Revision has minor change

Surface Operators in N=2 4d Gauge Theories

N=2 four dimensional gauge theories admit interesting half BPS surface operators preserving a (2,2) two dimensional SUSY algebra. Typical examples are (2,2) 2d sigma models with a flavor symmetry which is coupled to the 4d gauge fields. Interesting features of such 2d sigma models, such as (twisted) chiral rings, and the tt* geometry, can be carried over to the surface operators, and are affected in surprising ways by the coupling to 4d degrees of freedom. We will describe in detail a relation between the parameter space of twisted couplings of the surface operator and the Seiberg-Witten geometry of the bulk theory. We will discuss a similar result about the tt* geometry of the surface operator. We will predict the existence and general features of a wall-crossing formula for BPS particles bound to the surface operator.Comment: 25 pages, 4 figure