Gauged Linear Sigma Models

These lecture notes cover my talks at the June 2009 "Geometry of Quantum Fields and Strings" summer school at the University of Pennsylvania. I begin by giving a very brief introduction to quantum field theory, quickly moving to nonlinear sigma models and the associated A and B model topological field theories, and the physical realization of derived categories. Later lectures will cover Landau-Ginzburg models, matrix factorization, gauged linear sigma models, and mirror symmetry.

This work introduces a novel modification of classical perturbation method (PM), denominated Optimized Distribution of Boundary Conditions Perturbation Method (ODBCPM) with the purpose to improve the performance of PM in the solution of ordinary differential equations (ODES). We will see that the main proposal of ODBCPM rests above all in the redistribution and optimization of the boundary conditions of the problem to be solved among the iterations of the proposed method. The solution of a couple of heat relevant problems indicates the potentiality of ODBCPM even for the case of large values of the perturbative parameter.

Continuing the investigation of CNM (chiral-nonminimal) hypermultiplet nonlinear σ-models, we propose extensions of the concept of the c-map which relate holomorphic functions to hyper-Kähler geometries. In particular, we show that a whole series of hyper-Kähler potentials can be derived by replacing the role of the 4D, N = 1 tensor multiplet in the original c-map by 4D, N = 1 non-minimal multiplets and auxiliary superfields. The resulting N = 2 models appear to have interesting connections to Calabi-Yau manifolds and algebraic varieties. These models also emphasize the fact that special hyper-Kähler manifolds (the analogs of special Kähler manifolds) without isometries exist.

We present a general scheme for generating (2, 2) symmetric fermionic stri~ models and classify the models in D = 8 and D = 6 space-time dimensions with one twist. It is pointed out that they allow the geometrical interpretation as generalised torus compactifications. Their relation to other compactifications is discussed and the overlap with orbifolds is determined.

Three-manifold invariantsẐ ("Z-hat") , also known as homological blocks, are q-series with integer coefficients. Explicit q-series form forẐ is known for SU (2) group, supergroup SU (2|1) and ortho-symplectic supergroup OSp(2|2). We focus onẐ for SO(3) group and orthosymplectic supergroup OSp(1|2) in this paper. Particularly, the change of variable relating SU (2) link invariants to the SO(3) & OSp(1|2) link invariants plays a crucial role in explicitly writing the q-series.

We consider superstring compactifications where both the classical description, in terms of a Calabi-Yau manifold, and also the quantum theory is known in terms of a Landau-Ginzburg orbifold model. In particular, we study (smooth) Calabi-Yau examples in which there are obstructions to parametrizing all of the complex structure cohomology by polynomial deformations thus requiring the analysis based on exact and spectral sequences. General arguments ensure that the Landau-Ginzburg chiral ring copes with such a situation by having a non-trivial contribution from twisted sectors. Beyond the expected final agreement between the mathematical and physical approaches, we find a direct correspondence between the analysis of each, thus giving a more complete mathematical understanding of twisted sectors. Furthermore, this approach shows that physical reasoning based upon spectral flow arguments for determining the spectrum of Landau-Ginzburg orbifold models finds direct mathematical justifica...

In this paper we discuss some of the effects of using "unidexterous" worldsheet superfields, which satisfy worldsheet differential constraints ∂ = Λ = 0 = ∂ = | Υ and so are partly on-shell, i.e., on half-shell. Most notably, this results in a stratification of the field space that reminds of "brane-world" geometries. Linear dependence on such superfields provides a worldsheet generalization of the super-Zeeman effect. In turn, non-linear dependence yields additional left-right asymmetric dynamical constraints on the propagating fields, again in a stratified fashion.

Continuing the investigation of CNM (chiral-nonminimal) hypermultiplet nonlinear σ-models, we propose extensions of the concept of the c-map which relate holomorphic functions to hyper-Kähler geometries. In particular, we show that a whole series of hyper-Kähler potentials can be derived by replacing the role of the 4D, N = 1 tensor multiplet in the original c-map by 4D, N = 1 non-minimal multiplets and auxiliary superfields. The resulting N = 2 models appear to have interesting connections to Calabi-Yau manifolds and algebraic varieties. These models also emphasize the fact that special hyper-Kähler manifolds (the analogs of special Kähler manifolds) without isometries exist.

We study fractional 2p-branes and their intersection numbers in noncompact orbifolds as well the continuation of these objects in Kähler moduli space to coherent sheaves in the corresponding smooth non-compact Calabi-Yau manifolds. We show that the restriction of these objects to compact Calabi-Yau hypersurfaces gives the new fractional branes in LG orbifolds constructed by Ashok et. al. in hep-th/0401135. We thus demonstrate the equivalence of the B-type branes corresponding to linear boundary conditions in LG orbifolds, originally constructed in hep-th/9907131, to a subset of those constructed in LG orbifolds using boundary fermions and matrix factorization of the world-sheet superpotential. The relationship between the coherent sheaves corresponding to the fractional twobranes leads to a generalization of the McKay correspondence that we call the quantum McKay correspondence due to a close parallel with the construction of branes on non-supersymmetric orbifolds. We also provide evidence that the boundary states associated to these branes in a conformal field theory description corresponds to a sub-class of the boundary states associated to the permutation branes in the Gepner model associated with the LG orbifold.

We study integrality of instanton numbers (genus zero Gopakumar-Vafa invariants) for quintic and other Calabi-Yau manifolds. We start with the analysis of the case when the moduli space of complex structures is one-dimensional; later we show that our methods can be used to prove integrality in general case. We give an expression of instanton numbers in terms of Frobenius map on p-adic cohomology; the proof of integrality is based on this expression.

This thesis is devoted to the study of gauged linear sigma models (GLSMs) and mirror symmetry. The first chapter of this thesis aims to introduce some basics of GLSMs and mirror symmetry. The second chapter contains the author's contributions to new exact results for GLSMs obtained by applying supersymmetric localization. The first part of that chapter concerns supermanifolds. We use supersymmetric localization to show that A-twisted GLSM correlation functions for certain supermanifolds are equivalent to corresponding Atwisted GLSM correlation functions for hypersurfaces. The second part of that chapter defines associated Cartan theories for non-abelian GLSMs by studying partition functions as well as elliptic genera. The third part of that chapter focuses on N =(0,2) GLSMs. For those deformed from N =(2,2) GLSMs, we consider A/2-twisted theories and formulate the genuszero correlation functions in terms of Jeffrey-Kirwan-Grothendieck residues on Coulomb branches, which generalize the Jeffrey-Kirwan residue prescription relevant for the N =(2,2) locus. We reproduce known results for abelian GLSMs, and can systematically calculate more examples with new formulas that render the quantum sheaf cohomology relations and other properties manifest. We also include unpublished results for counting deformation parameters. The third chapter is about mirror symmetry. In the first part of the third chapter, we propose an extension of the Hori-Vafa mirrror construction [25] from abelian (2,2) GLSMs they considered to non-abelian (2,2) GLSMs with connected gauge groups, a potential solution to an old problem. We formally show that topological correlation functions of B-twisted mirror LGs match those of A-twisted gauge theories. In this thesis, we study two examples, Grassmannians and two-step flag manifolds, verifying in each case that the mirror correctly reproduces details ranging from the number of vacua and correlations functions to quantum cohomology relations. In the last part of the third chapter, we propose an extension of the Hori-Vafa construction [25] of (2,2) GLSM mirrors to (0,2) theories obtained from (2,2) theories by special tangent bundle deformations. Our ansatz can systematically produce the (0,2) mirrors of toric varieties and the results are consistent with existing examples which were produced by laborious guesswork. The last chapter briefly discusses some directions that the author would like to pursue in the future.

I describe our understanding of physics near the planck length, in particular the great progress of the last four years in string theory. Superstring theory, and a recent extension called M theory, are leading candidates for a quantum theory that unifies gravity with the other forces. As such, they are certainly not ordinary quantum field theories. However, recent duality conjectures suggest that a more complete definition of these theories can be provided by the large N limits of suitably chosen U(N) gauge theories associated to the asymptotic boundary of spacetime.

We identify the exactly solvable theory of the conformal fixed point of (0,2) Calabi-Yau σ-models and their Landau-Ginzburg phases. To this end we consider a number of (0,2) models constructed from a particular (2,2) exactly solvable theory via the method of simple currents. In order to establish the relation between exactly solvable (0,2) vacua of the heterotic string, (0,2) Landau-Ginzburg orbifolds and (0,2) Calabi-Yau manifolds, we compute the Yukawa couplings of the exactly solvable model and compare the results with the product structure of the chiral ring which we extract from the structure of the massless spectrum of the exact theory. We find complete agreement between the two up to a finite number of renormalizations. For a particularly simple example we furthermore derive the generating ideal of the chiral ring from a (0,2) linear σ-model which has both a Landau-Ginzburg and a (0,2) Calabi-Yau phase.

We generalize the previously established (0,2) triality of exactly solvable models, Landau-Ginzburg theories and Calabi-Yau manifolds to a number of different classes of (0,2) compactifications derived from (2,2) vacua. For the resulting models we show that the known (2,2) mirror constructions induce mirror symmetry in the (0,2) context.

This thesis is devoted to the study of gauged linear sigma models (GLSMs) and mirror symmetry. The first chapter of this thesis aims to introduce some basics of GLSMs and mirror symmetry. The second chapter contains the author's contributions to new exact results for GLSMs obtained by applying supersymmetric localization. The first part of that chapter concerns supermanifolds. We use supersymmetric localization to show that A-twisted GLSM correlation functions for certain supermanifolds are equivalent to corresponding Atwisted GLSM correlation functions for hypersurfaces. The second part of that chapter defines associated Cartan theories for non-abelian GLSMs by studying partition functions as well as elliptic genera. The third part of that chapter focuses on N =(0,2) GLSMs. For those deformed from N =(2,2) GLSMs, we consider A/2-twisted theories and formulate the genuszero correlation functions in terms of Jeffrey-Kirwan-Grothendieck residues on Coulomb branches, which generalize the Jeffrey-Kirwan residue prescription relevant for the N =(2,2) locus. We reproduce known results for abelian GLSMs, and can systematically calculate more examples with new formulas that render the quantum sheaf cohomology relations and other properties manifest. We also include unpublished results for counting deformation parameters. The third chapter is about mirror symmetry. In the first part of the third chapter, we propose an extension of the Hori-Vafa mirrror construction [25] from abelian (2,2) GLSMs they considered to non-abelian (2,2) GLSMs with connected gauge groups, a potential solution to an old problem. We formally show that topological correlation functions of B-twisted mirror LGs match those of A-twisted gauge theories. In this thesis, we study two examples, Grassmannians and two-step flag manifolds, verifying in each case that the mirror correctly reproduces details ranging from the number of vacua and correlations functions to quantum cohomology relations. In the last part of the third chapter, we propose an extension of the Hori-Vafa construction [25] of (2,2) GLSM mirrors to (0,2) theories obtained from (2,2) theories by special tangent bundle deformations. Our ansatz can systematically produce the (0,2) mirrors of toric varieties and the results are consistent with existing examples which were produced by laborious guesswork. The last chapter briefly discusses some directions that the author would like to pursue in the future.

We study two-dimensional N =(0, 2) supersymmetric gauged linear sigma models (GLSMs) using supersymmetric localization. We consider N =(0, 2) theories with an R-symmetry, which can always be defined on curved space by a pseudo-topological twist while preserving one of the two supercharges of flat space. For GLSMs which are deformations of N =(2, 2) GLSMs and retain a Coulomb branch, we consider the A/2-twist and compute the genus-zero correlation functions of certain pseudo-chiral operators, which generalize the simplest twisted chiral ring operators away from the N =(2, 2) locus. These correlation functions can be written in terms of a certain residue operation on the Coulomb branch, generalizing the Jeffrey-Kirwan residue prescription relevant for the N =(2, 2) locus. For abelian GLSMs, we reproduce existing results with new formulas that render the quantum sheaf cohomology relations and other properties manifest. For non-abelian GLSMs, our methods lead to new results. As an example, we briefly discuss the quantum sheaf cohomology of the Grassmannian manifold.

We consider Landau-Ginzburg (LG) models with boundary conditions preserving A-type N=2 supersymmetry. We show the equivalence of a linear class of boundary conditions in the LG model to a particular class of boundary states in the corresponding CFT by an explicit computation of the open-string Witten index in the LG model. We extend the linear class of boundary conditions to general non-linear boundary conditions and determine their consistency with A-type N=2 supersymmetry. This enables us to provide a microscopic description of special Lagrangian submanifolds in C^n due to Harvey and Lawson. We generalise this construction to the case of hypersurfaces in P^n. We find that the boundary conditions must necessarily have vanishing Poisson bracket with the combination (W(ϕ)-W̅(ϕ̅)), where W(ϕ) is the appropriate superpotential for the hypersurface. An interesting application considered is the T^3 supersymmetric cycle of the quintic in the large complex structure limit.

Talk presented by P.N. at “Mathematics and Physics of Strings,” Berkeley and “Topology and Geometry in Theoretical Physics,” Turku (Finland), 1991. Disciplines Physical Sciences and Mathematics | Physics This conference paper is available at ScholarlyCommons: http://repository.upenn.edu/physics_papers/537

We study orbifolds of (0, 2) models and their relation to (0, 2) mirror symmetry. In the Landau-Ginzburg phase of a (0, 2) model the superpotential features a whole bunch of discrete symmetries, which by quotient action lead to a variety of consistent (0, 2) vacua. We study a few examples in very much detail. Furthermore, we comment on the application of (0, 2) mirror symmetry to the calculation of Yukawa couplings in the space-time superpotential.

Abstract: We consider Landau-Ginzburg (LG) models with boundary conditions preserving A-type N = 2 supersymmetry. We show the equivalence of a linear class of boundary conditions in the LG model to a particular class of boundary states in the corresponding CFT by an explicit computation of the open-string Witten index in the LG model. We extend the linear class of boundary conditions to general non-linear boundary conditions and determine their consistency with A-type N = 2 supersymmetry. This enables us to provide a microscopic description of special Lagrangian submanifolds in C n due to Harvey and Lawson. We generalise this construction to the case of hypersurfaces in P n. We find that the boundary conditions must necessarily have vanishing Poisson bracket with the combination (W(φ) − W(φ)), where W(φ) is the appropriate superpotential for the hypersurface. An interesting application considered is the T 3 supersymmetric cycle of the quintic in the large complex structure limit. Ke...

We review some recent results on D-branes on Calabi-Yau (CY) manifolds. We show the existence of structures (helices and quivers) which enable one to make statements about large families of D-branes in various phases of the Gauged Linear Sigma Model (GLSM) associated with the CY manifold. A comparison of the quivers of two phases leads to the prediction that certain D-brane configurations will decay as one moves across phases. We discuss how boundary fermions can be used to realise various D-brane configurations associated with coherent sheaves in the GLSM with boundary. This is based on the talk presented by S.G. at Strings 2001, Mumbai.

Abstract: We consider Landau-Ginzburg (LG) models with boundary conditions preserving A-type N = 2 supersymmetry. We show the equivalence of a linear class of boundary conditions in the LG model to a particular class of boundary states in the corresponding CFT by an explicit computation of the open-string Witten index in the LG model. We extend the linear class of boundary conditions to general non-linear boundary conditions and determine their consistency with A-type N = 2 supersymmetry. This enables us to provide a microscopic description of special Lagrangian submanifolds in C n due to Harvey and Lawson. We generalise this construction to the case of hypersurfaces in P n. We find that the boundary conditions must necessarily have vanishing Poisson bracket with the combination (W(φ) − W(φ)), where W(φ) is the appropriate superpotential for the hypersurface. An interesting application considered is the T 3 supersymmetric cycle of the quintic in the large complex structure limit. Ke...

We review some recent results on D-branes on Calabi-Yau (CY) manifolds. We show the existence of structures (helices and quivers) which enable one to make statements about large families of D-branes in various phases of the Gauged Linear Sigma Model (GLSM) associated with the CY manifold. A comparison of the quivers of two phases leads to the prediction that certain D-brane configurations will decay as one moves across phases. We discuss how boundary fermions can be used to realise various D-brane configurations associated with coherent sheaves in the GLSM with boundary. This is based on the talk presented by S.G. at Strings 2001, Mumbai.

It has been shown earlier [16, 17] that, in the mixed space, there is an unexpected simple relation between any finite temperature graph and its zero temperature counterpart through a multiplicative scalar operator (termed thermal operator) which carries the entire temperature dependence. This was shown to hold only in the imaginary time formalism and the closed time path (σ = 0) of the real time formalism (as well as for its conjugate σ = 1). We study the origin of this operator from the more fundamental Bogoliubov transformation which acts, in the momentum space, on the doubled space of fields in the real time formalisms [9, 10, 12]. We show how the (2 × 2) Bogoliubov transformation matrix naturally leads to the scalar thermal operator for σ = 0, 1 while it fails for any other value 0 < σ < 1. This analysis also suggests that a generalized scalar thermal operator description, in the mixed space, is possible even for 0 < σ < 1. We also show the existence of a scalar thermal operator relation in the momentum space.

We study instanton-corrected renormalization group flow in the two dimensional sigma models and four dimensional gauge theory. In two dimensions we do that by replacing the non-linear supersymmetric CP N −1 model by the gauged linear sigma model which is in the same universality class. We compare the moduli spaces of the instantons in the non-linear model and that of BPS field configurations in the linear model. We reduce the problem of matching of the parameters of the two systems to the intersection theory on the compact moduli space of the latter model. In four dimensions we find that the proper analogue of the linear sigma model moduli space is the space of torsion free sheaves. Both cases (2d and 4d) are unified by the notion of the freckled instantons. We also put an end to the discussion of the nature of the superpotentials W ∼ σlogσ in 2d and 4d and discover the surprising disconnectnessness of the effective target space.

We study instanton-corrected renormalization group flow in the two dimensional sigma models and four dimensional gauge theory. In two dimensions we do that by replacing the non-linear supersymmetric ${\IC\IP}^{N-1}$ model by the gauged linear sigma model which is in the same universality class. We compare the moduli spaces of the instantons in the non-linear model and that of BPS field

In this paper, a collocation method using sinc functions and Chebyshev wavelet method is implemented to solve linear systems of Volterra integro-differential equations. To test the validity of these methods, two numerical examples with known exact solution are presented. Numerical results indicate that the convergence and accuracy of these methods are in good a agreement with the analytical solution. However, according to comparison of these methods, we conclude that the Chebyshev wavelet method provides more accurate results.

We review some recent results on D-branes on Calabi-Yau (CY) manifolds. We show the existence of structures (helices and quivers) which enable one to make statements about large families of D-branes in various phases of the Gauged Linear Sigma Model (GLSM) associated with the CY manifold. A comparison of the quivers of two phases leads to the prediction that certain D-brane configurations will decay as one moves across phases. We discuss how boundary fermions can be used to realise various D-brane configurations associated with coherent sheaves in the GLSM with boundary. This is based on the talk presented by S.G. at Strings 2001, Mumbai.

AbsTRaCT: We consider Landau-Ginzburg (LG) models with boundary conditions preserving A-type N= 2 supersymmetry. We show the equivalence of a linear class of boundary conditions in the LG model to a particular class of boundary states in the corresponding CFT by an explicit computation of the open-string Witten index in the LG model. We extend the linear class of boundary conditions to general non-linear boundary conditions and determine their consistency with A-type N= 2 supersymmetry. This enables ...

In this paper we discuss some of the effects of using "unidexterous" worldsheet superfields, which satisfy worldsheet differential constraints ∂ = Λ = 0 = ∂ = | Υ and so are partly on-shell, i.e., on half-shell. Most notably, this results in a stratification of the field space that reminds of "brane-world" geometries. Linear dependence on such superfields provides a worldsheet generalization of the super-Zeeman effect. In turn, non-linear dependence yields additional left-right asymmetric dynamical constraints on the propagating fields, again in a stratified fashion.

In this paper we discuss some of the effects of using "unidexterous" worldsheet superfields, which satisfy worldsheet differential constraints ∂ = Λ = 0 = ∂ = | Υ and so are partly on-shell, i.e., on half-shell. Most notably, this results in a stratification of the field space that reminds of "brane-world" geometries. Linear dependence on such superfields provides a worldsheet generalization of the super-Zeeman effect. In turn, non-linear dependence yields additional left-right asymmetric dynamical constraints on the propagating fields, again in a stratified fashion.

It has been shown earlier [16, 17] that, in the mixed space, there is an unexpected simple relation between any finite temperature graph and its zero temperature counterpart through a multiplicative scalar operator (termed thermal operator) which carries the entire temperature dependence. This was shown to hold only in the imaginary time formalism and the closed time path (σ = 0) of the real time formalism (as well as for its conjugate σ = 1). We study the origin of this operator from the more fundamental Bogoliubov transformation which acts, in the momentum space, on the doubled space of fields in the real time formalisms [9, 10, 12]. We show how the (2 × 2) Bogoliubov transformation matrix naturally leads to the scalar thermal operator for σ = 0, 1 while it fails for any other value 0 < σ < 1. This analysis also suggests that a generalized scalar thermal operator description, in the mixed space, is possible even for 0 < σ < 1. We also show the existence of a scalar thermal operator relation in the momentum space.

The two-dimensional σ-model with the de Sitter target space is locally canonic in the north pole diamond of the Penrose diagram in the cosmological gauge. The left and right moving modes on the cylindrical base space are entangled among themselves and interact with the de Sitter metric. Firstly, we show that the untangled oscillators can be obtained from the entangled operators by applying a set of Bogoliubov transformations constrained by the requirement that the partial evolution generator be diagonal. Secondly, we determine the nonequilibrium dynamics of the untangled modes in the nonequilibrium thermofield dynamics formalism. The thermal modes are represented as thermal doublet oscillators that satisfy partial evolution equations of Heisenberg-type. From these we compute the local free one-body propagator of an arbitrary mode between two times. Thirdly, we discuss the field representation of the thermal modes. We show that there is a set of thermal doublet fields that satisfy th...

We consider the gauge neutral matter in the low{energy eective action for string theory compactication on a Calabi{Yau manifold with (2; 2) world{sheet supersymmetry. At the classical level these states (the 1's of E 6) correspond to the cohomology group H 1 (M; End T). We examine the rst order contribution of instantons to the mass matrix of these particles. In principle, these corrections depend on the K ahler parameters t i through factors of the form e 2it i and also depend on the complex structure parameters. For simplicity w e consider in greatest detail the quintic threefold IP 4 [5]. It follows on general grounds that the total mass is often, and perhaps always, zero. The contribution of individual instantons is however nonzero and the contribution of a given instanton may develop poles associated with instantons coalescing for certain values of the complex structure. This can happen when the underlying Calabi{Yau manifold is smooth. Hence these poles must cancel between the coalescing instantons in order that the superpotential be nite. We examine also the Yukawa couplings involving neutral matter 1 3 and neutral and charged elds 27.27.1, which h a v e been little investigated even though they are of phenomenological interest. We study the general conditions under which these couplings vanish classically. We also calculate the rst{order world{sheet instanton correction to these couplings and argue that these also vanish. \ Alexander von Humboldt Fellow [ On leave from the Institute \Rudjer Bo skovi c", Zagreb, Croatia. Contents 1. Introduction 2. Review of What is Known Classically 2.1 Deformations of the complex structure and of the tangent bundle 2.2 The number of singlets 2.3 Integrability and the vanishing of the 1 3 coupling 2.4 The vanishing of the 27.27.1 coupling 3. Preliminaries Concerning Instanton Contributions to Correlation Functions 3.1 The zero m o des 3.2 Vertex operators and powers of g 3.3 Decomposition of singlets along a line 4. Masses Generated by Instantons 4.1 The mass matrix 4.2 Other splitting types 5. Examples of the Eects of Lines in IP 4 [5] 5.1 General facts 5.2 An O(0) O (2) line as the limit of O(1) O (1) lines 5.3 The singlet mass matrix is probably always zero 6. The Yukawa Couplings 6.1 The couplings in terms of zero m o des 6.2 The form of the Yukawa couplings 7. Conclusion 7.1 Summary and discussion 7.2 Open questions A. Remarks on the Determinants and Resolution of an Apparent P aradox

We develop systematic string techniques to study brane world effective actions for models with magnetized (or equivalently intersecting) D-branes. In particular, we derive the dependence on all NS-NS moduli of the kinetic terms of the chiral matter in a generic non-supersymmetric brane configurations with non-commuting open string fluxes. Near a N = 1 supersymmetric point the effective action is consistent with a Fayet-Iliopoulos supersymmetry breaking and the normalization of the scalar kinetic terms is nothing else than the Kähler metric. We also discuss, from a stringy perspective, D and F term breaking mechanisms, and how, in this generic set up, the Kähler metric enters in the physical Yukawa couplings.

We discuss spontaneous supersymmetry breaking in N = 2 globally supersymmetric theories describing abelian vector multiplets. The most general form of the action admits, in addition to the usual Fayet-Iliopoulos term, a magnetic Fayet-Iliopoulos term for the auxiliary components of dual vector multiplets. In a generic case, this leads to a spontaneous breakdown of one of the two supersymmetries. In some cases however, dyon condensation restores N = 2 SUSY vacuum. This talk is based on the work done in collaboration with H. Partouche .

The most general homogeneous monodromy conditions in N=2 string theory are classified in terms of the conjugacy classes of the global symmetry group U(1,1)⊗openZ2. For classes which generate a discrete subgroup Γ, the corresponding target space backgrounds openC1,1/Γ include half spaces, complex orbifolds, and tori. We propose a generalization of the intercept formula to matrix-valued twists, but find massless physical states only for Γ=open1 (untwisted) and Γ=openZ2 (in the manner of Mathur and Mukhi), as well as for Γ being a parabolic element of U(1,1). In particular, the 16 openZ2-twisted sectors of the N=2 string are investigated, and the corresponding ground states are identified via bosonization and BRST cohomology. We find enough room for an extended multiplet of ``spacetime'' supersymmetry, with the number of supersymmetries being dependent on global ``spacetime'' topology. However, world-sheet locality for the chiral vertex operators does not permit interactions among all massless ``spacetime'' fermions.

We study instanton-corrected renormalization group flow in the two dimensional sigma models and four dimensional gauge theory. In two dimensions we do that by replacing the non-linear supersymmetric ${\IC\IP}^{N-1}$ model by the gauged linear sigma model which is in the same universality class. We compare the moduli spaces of the instantons in the non-linear model and that of BPS field

We study instanton-corrected renormalization group flow in the two dimensional sigma models and four dimensional gauge theory. In two dimensions we do that by replacing the non-linear supersymmetric ${\IC\IP}^{N-1}$ model by the gauged linear sigma model which is in the same universality class. We compare the moduli spaces of the instantons in the non-linear model and that of BPS field configurations in the linear model. We reduce the problem of matching of the parameters of the two systems to the intersection theory on the compact moduli space of the latter model. Both cases (2d and 4d) are unified by the notion of the {\it freckled instantons}. We also put an end to the discussion of the nature of the superpotentials $W \sim {\s} {\rm log} {\s}$ in 2d and 4d and discover the surprising disconnectnessness of the effective target space.

We show that one can express Frobenius transformation on middle-dimensional p-adic cohomology of Calabi-Yau threefold in terms of mirror map and instanton numbers. We express the mirror map in terms of Frobenius transformation on p-adic cohomology . We discuss a $p$-adic interpretation of the conjecture about integrality of Gopakumar-Vafa invariants.

We show that one can express Frobenius transformation on middle-dimensional p-adic cohomology of Calabi-Yau threefold in terms of mirror map and instanton numbers. We express the mirror map in terms of Frobenius transformation on p-adic cohomology . We discuss a p-adic interpretation of the conjecture about integrality of Gopakumar-Vafa invariants.

Constant curvature surfaces are constructed from the finite action solutions of the supersymmetric CP N−1 sigma model. It is shown that there is a unique holomorphic solution which leads to constant curvature surfaces: the generalized Veronese curve. We also present an explicit construction of non-holomorphic solutions for the supersymmetric CP 1 and CP 2 models and give a general discussion on the possible construction of these solutions for N > 3.

We show that one can express Frobenius transformation on middle-dimensional p-adic cohomology of Calabi-Yau threefold in terms of mirror map and instanton numbers. We express the mirror map in terms of Frobenius transformation on p-adic cohomology . We discuss a p-adic interpretation of the conjecture about integrality of Gopakumar-Vafa invariants.

Constant curvature surfaces are constructed from the finite action solutions of the supersymmetric CP N−1 sigma model. It is shown that there is a unique holomorphic solution which leads to constant curvature surfaces: the generalized Veronese curve. We also present an explicit construction of non-holomorphic solutions for the supersymmetric CP 1 and CP 2 models and give a general discussion on the possible construction of these solutions for N > 3.

We conjecture a relation between the sl(N) knot homology, recently introduced by Khovanov and Rozansky, and the spectrum of BPS states captured by open topological strings. This conjecture leads to new regularities among the sl(N) knot homology groups and suggests that they can be interpreted directly in topological string theory. We use this approach in various examples to predict the

We formulate four-dimensional N = 2 supersymmetric nonlinear sigma models in N = 1 superspace. We show how to add superpotentials consistent with N = 2 supersymmetry. We lift our construction to higher-dimensional spacetime and write five-dimensional nonlinear sigma models in N = 1 superspace.

It is shown that, classically, the W-algebras are directly related to the extrinsic geometry of the embedding of two-dimensional manifolds with chiral parametrisation (W-surfaces) into higher dimensional Kähler manifolds. We study the local and the global geometries of such embeddings, and connect them to Toda equations. The additional variables of the related KP hierarchy are shown to yield a specific coordinate system of the target-manifold, and this allows us to prove that W-transformations are simply particular diffeomorphisms of this target space. The W-surfaces are shown to be instantons of the corresponding non-linear σ-models.