Rolf Schimmrigk - Academia.edu (original) (raw)

Papers by Rolf Schimmrigk

Journal of High Energy Physics, Sep 1, 2020

Geometric modularity has recently been conjectured to be a characteristic feature of flux vacua w... more Geometric modularity has recently been conjectured to be a characteristic feature of flux vacua with W = 0. This paper provides support for the conjecture by computing motivic modular forms in a direct way for several string compactifications for which such vacua are known to exist. The analysis of some Calabi-Yau manifolds which do not admit supersymmetric flux vacua shows that the reverse of the conjecture does not hold.

International Journal of Modern Physics, Aug 30, 2012

arXiv (Cornell University), Dec 19, 2022

One of the fundamental questions in inflation is how to characterize the structure of different t... more One of the fundamental questions in inflation is how to characterize the structure of different types of models in the field theoretic landscape. Proposals in this direction include attempts to directly characterize the formal structure of the theory by considering complexity measures of the potentials. An alternative intrinsic approach is to focus on the behavior of the observables that result from different models and to ask whether their behavior differs among models. This type of analysis can be applied even to nontrivial multifield theories where a natural measure of the complexity of the model is not obvious and the analytical evaluation of the observables is often impossible. In such cases one may still compute these observables numerically and investigate their behavior. One interesting case is when observables show a scaling behavior, in which case theories can be characterized in terms of their scaling amplitudes and exponents. Generically, models have nontrivial parameter spaces, leading to exponents that are functions of these parameters. In such cases we consider an iterative procedure to determine whether the exponent functions in turn lead to a scaling behavior. We show that modular inflation models can be characterized by families of simple scaling laws and that the scaling exponents that arise in this way in turn show scaling in dependence of the varying energy scales.

Black holes in string theory compactified on Calabi-Yau varieties a priori might be expected to h... more Black holes in string theory compactified on Calabi-Yau varieties a priori might be expected to have moduli dependent features. For example the entropy of the black hole might be expected to depend on the complex structure of the manifold. This would be inconsistent with known properties of black holes. Supersymmetric black holes appear to evade this inconsistency by having moduli fields that flow to fixed points in the moduli space that depend only on the charges of the black hole. Moore observed in the case of compactifications with elliptic curve factors that these fixed points are arithmetic, corresponding to curves with complex multiplication. The main goal of this talk is to explore the possibility of generalizing such a characterization to Calabi-Yau varieties with finite fundamental groups.

Physics Letters B, Nov 1, 1996

In Calabi-Yau compactifications of the heterotic string there exist quantities which are universa... more In Calabi-Yau compactifications of the heterotic string there exist quantities which are universal in the sense that they are present in every Calabi-Yau string vacuum. It is shown that such universal characteristics provide numerical information, in the form of scaling exponents, about the space of ground states in string theory. The focus is on two physical quantities. The first is the Yukawa coupling of a particular antigeneration, induced in four dimensions by virtue of supersymmetry. The second is the partition function of the topological sector of the theory, evaluated on the genus one worldsheet, a quantity relevant for quantum mirror symmetry and threshold corrections. It is shown that both these quantities exhibit scaling behavior with respect to a new scaling variable and that a scaling relation exists between them as well.

Nuclear Physics B, Sep 1, 2003

We show how Moore's observation, in the context of toroidal compactifications in type IIB string ... more We show how Moore's observation, in the context of toroidal compactifications in type IIB string theory, concerning the complex multiplication structure of black hole attractor varieties, can be generalized to Calabi-Yau compactifications with finite fundamental groups. This generalization leads to an alternative general framework in terms of motives associated to a Calabi-Yau variety in which it is possible to address the arithmetic nature of the attractor varieties in a universal way via Deligne's period conjecture.

Communications in Mathematical Physics, Feb 25, 2011

The program of constructing spacetime geometry from string theoretic modular forms is extended to... more The program of constructing spacetime geometry from string theoretic modular forms is extended to Calabi-Yau varieties of dimensions two, 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-functions associated to omega motives of Calabi-Yau varieties, generated by their holomorphic n−forms via Galois representations. The modular forms that emerge from the Ω−motive and other motives of the intermediate cohomology 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 indicates that the L-function can be interpreted as a map from the geometric category of motives to the category of conformal field theories on the worldsheet.

WORLD SCIENTIFIC eBooks, Dec 1, 2012

Over the past few years the arithmetic Langlands program has found applications in two quite diff... more Over the past few years the arithmetic Langlands program has found applications in two quite different problems that arise in string physics. The first of these is concerned with the fundamental problem of deriving the geometry of spacetime from the worldsheet dynamics, leading to a realization of the notion of an emergent spacetime in string theory. The second problem is concerned with the idea of using automorphic black holes as probes of spacetime. In this article both of these applications of the Langlands program are described.

Physics Letters B, Oct 1, 1990

It is shown that subclasses of a recently constructed set of Landau-Ginzburg/Calabi-Yau string co... more It is shown that subclasses of a recently constructed set of Landau-Ginzburg/Calabi-Yau string compactifications can be described as orbifolds of simpler superpotentials. This construction allows for a systematic identification of dual pairs that appear naturally in this class of string vacua. The strange duality of Arnold appears as a special case of these constructions.

Nuclear Physics B, Sep 1, 1990

It has recently been recognized that the relation between exactly solvable conformal field theory... more It has recently been recognized that the relation between exactly solvable conformal field theory compactifications of the Heterotic String and Calabi-Yau manifolds necessarily involves the discussion of embeddings in weighted projective space. We therefore study this class of manifolds more closely. We have constructed a subclass of these spaces and find that this class features a surprising symmetry under chi

arXiv (Cornell University), Jan 11, 2013

In this paper automorphic motives are constructed and analyzed with a view toward the understandi... more In this paper automorphic motives are constructed and analyzed with a view toward the understanding of the geometry of compactification manifolds in string theory in terms of the modular structure of the worldsheet theory. The results described generalize a framework considered previously in two ways, first by relaxing the restriction to modular forms, and second by extending the construction of motives from diagonal varieties to nondiagonal spaces. The framework of automorphic forms and representations is described with a view toward applications, emphasizing the explicit structure of these objects.

Annals of the New York Academy of Sciences, Jun 1, 1993

The rôle in string theory of manifolds of complex dimension D crit +2(Q−1) and positive first Che... more The rôle in string theory of manifolds of complex dimension D crit +2(Q−1) and positive first Chern class is described. In order to be useful for string theory the first Chern class of these spaces has to satisfy a certain relation. Because of this condition the cohomology groups of such manifolds show a specific structure. A group that is particularly important is described by (D crit +Q−1, Q−1)-forms because it is this group which contains the higher dimensional counterpart of the holomorphic (D crit , 0)-form that figures so prominently in Calabi-Yau manifolds. It is shown that the higher dimensional manifolds do not, in general, have a unique counterpart of this holomorphic form of rank D crit. It is also shown that these manifolds lead, in general, to a number of additional modes beyond the standard Calabi-Yau spectrum. This suggests that not only the dilaton but also the other massless string modes, such as the antisymmetric torsion field, might be relevant for a possible stringy interpretation.

Springer eBooks, 1993

A recently introduced framework for the compactification of supersymmetric string theory involvin... more A recently introduced framework for the compactification of supersymmetric string theory involving noncritical manifolds of complex dimension 2k + D crit , k ≥ 1, is reviewed. These higher dimensional manifolds are spaces with quantized positive Ricci curvature and therefore do not, a priori, describe consistent string vacua. It is nevertheless possible to derive from these manifolds the massless spectra of critical string groundstates. For a subclass of these noncritical theories it is also possible to explicitly construct Calabi-Yau manifolds from the higher dimensional spaces. Thus the new class of theories makes contact with the standard framework of string compactification. This class of manifolds is more general than that of Calabi-Yau manifolds because it contains spaces which correspond to critical string vacua with no Kähler deformations, i.e. no antigenerations, hence providing mirrors of rigid Calabi-Yau manifolds. The constructions reviewed here lead to new insight into the relation between exactly solvable models and their mean field theories on the one hand and Calabi-Yau manifolds on the other, leading, for instance, to a modification of Gepner's conjecture. They also raise fundamental questions about the Kaluza-Klein concept of string compactification, in particular regarding the rôle played by the dimension of the internal theories.

International Journal of Modern Physics D, Oct 1, 2012

Over the past few years the arithmetic Langlands program has proven useful in addressing physical... more Over the past few years the arithmetic Langlands program has proven useful in addressing physical problems. In this paper it is shown how Langlands' reciprocity conjecture for automorphic forms, in combination with a representation theoretic notion of motives, suggests a framework in which the entropy of automorphic black holes can be viewed as a probe of spacetime that is sensitive to the geometry of the extra dimensions predicted by string theory. If it were possible to produce black holes with automorphic entropy in the laboratory their evaporation would provide us with information about the precise shape of the compact geometry.

Physics Letters B, 1991

We use the Landau-Ginzburg formulation of the Kazama-Suzuki series of exactly solvable models to ... more We use the Landau-Ginzburg formulation of the Kazama-Suzuki series of exactly solvable models to compute the spectrum of all N= 2 supersymmetric twisted G/H Landau-Ginzburg compactifications for hermitian symmetric spaces G/H. We show that the Landau-Ginzburg potentials associated to Kazama-Suzuki models all belong to the class of singularities we have constructed in previous work with P. Candelas.

Nuclear Physics B, Mar 1, 1988

arXiv (Cornell University), Oct 9, 2008

arXiv (Cornell University), Dec 9, 1992

A recently introduced method for constructing marginal singular flows between distinct Landau--Gi... more A recently introduced method for constructing marginal singular flows between distinct Landau--Ginzburg theories at fixed central charge is reviewed. The flows are constructed in an enlarged moduli space obtained by adding theories with zero central charge. This mechanism is used to construct flows between mirror pairs of string vacua described by N$=$2 superconformal Landau--Ginzburg fixed points. In contrast to previous methods this new construction of mirror theories does not depend on particular symmetries of the original theory. (Based in part on a talk presented at the Trieste Summer Workshop on Superstrings and Related Topics, July 1992)

arXiv (Cornell University), Nov 22, 1992

A new method for constructing flows between distinct Landau-Ginzburg theories at fixed central ch... more A new method for constructing flows between distinct Landau-Ginzburg theories at fixed central charge is presented. The essential ingredient of the construction is an enlarged moduli space obtained by adding theories with zero central charge. The flows involve only marginal directions hence they can be applied to transitions between string vacua, in particular to the construction of mirror pairs of string ground states described by RG fixed points of N=2 supersymmetric Landau--Ginzburg theories. In contrast to previous methods this new construction of mirror theories does not depend on particular symmetries of the original theory.

Journal of High Energy Physics, Sep 1, 2020

Geometric modularity has recently been conjectured to be a characteristic feature of flux vacua w... more Geometric modularity has recently been conjectured to be a characteristic feature of flux vacua with W = 0. This paper provides support for the conjecture by computing motivic modular forms in a direct way for several string compactifications for which such vacua are known to exist. The analysis of some Calabi-Yau manifolds which do not admit supersymmetric flux vacua shows that the reverse of the conjecture does not hold.

International Journal of Modern Physics, Aug 30, 2012

arXiv (Cornell University), Dec 19, 2022

One of the fundamental questions in inflation is how to characterize the structure of different t... more One of the fundamental questions in inflation is how to characterize the structure of different types of models in the field theoretic landscape. Proposals in this direction include attempts to directly characterize the formal structure of the theory by considering complexity measures of the potentials. An alternative intrinsic approach is to focus on the behavior of the observables that result from different models and to ask whether their behavior differs among models. This type of analysis can be applied even to nontrivial multifield theories where a natural measure of the complexity of the model is not obvious and the analytical evaluation of the observables is often impossible. In such cases one may still compute these observables numerically and investigate their behavior. One interesting case is when observables show a scaling behavior, in which case theories can be characterized in terms of their scaling amplitudes and exponents. Generically, models have nontrivial parameter spaces, leading to exponents that are functions of these parameters. In such cases we consider an iterative procedure to determine whether the exponent functions in turn lead to a scaling behavior. We show that modular inflation models can be characterized by families of simple scaling laws and that the scaling exponents that arise in this way in turn show scaling in dependence of the varying energy scales.

Black holes in string theory compactified on Calabi-Yau varieties a priori might be expected to h... more Black holes in string theory compactified on Calabi-Yau varieties a priori might be expected to have moduli dependent features. For example the entropy of the black hole might be expected to depend on the complex structure of the manifold. This would be inconsistent with known properties of black holes. Supersymmetric black holes appear to evade this inconsistency by having moduli fields that flow to fixed points in the moduli space that depend only on the charges of the black hole. Moore observed in the case of compactifications with elliptic curve factors that these fixed points are arithmetic, corresponding to curves with complex multiplication. The main goal of this talk is to explore the possibility of generalizing such a characterization to Calabi-Yau varieties with finite fundamental groups.

Physics Letters B, Nov 1, 1996

In Calabi-Yau compactifications of the heterotic string there exist quantities which are universa... more In Calabi-Yau compactifications of the heterotic string there exist quantities which are universal in the sense that they are present in every Calabi-Yau string vacuum. It is shown that such universal characteristics provide numerical information, in the form of scaling exponents, about the space of ground states in string theory. The focus is on two physical quantities. The first is the Yukawa coupling of a particular antigeneration, induced in four dimensions by virtue of supersymmetry. The second is the partition function of the topological sector of the theory, evaluated on the genus one worldsheet, a quantity relevant for quantum mirror symmetry and threshold corrections. It is shown that both these quantities exhibit scaling behavior with respect to a new scaling variable and that a scaling relation exists between them as well.

Nuclear Physics B, Sep 1, 2003

We show how Moore's observation, in the context of toroidal compactifications in type IIB string ... more We show how Moore's observation, in the context of toroidal compactifications in type IIB string theory, concerning the complex multiplication structure of black hole attractor varieties, can be generalized to Calabi-Yau compactifications with finite fundamental groups. This generalization leads to an alternative general framework in terms of motives associated to a Calabi-Yau variety in which it is possible to address the arithmetic nature of the attractor varieties in a universal way via Deligne's period conjecture.

Communications in Mathematical Physics, Feb 25, 2011

The program of constructing spacetime geometry from string theoretic modular forms is extended to... more The program of constructing spacetime geometry from string theoretic modular forms is extended to Calabi-Yau varieties of dimensions two, 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-functions associated to omega motives of Calabi-Yau varieties, generated by their holomorphic n−forms via Galois representations. The modular forms that emerge from the Ω−motive and other motives of the intermediate cohomology 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 indicates that the L-function can be interpreted as a map from the geometric category of motives to the category of conformal field theories on the worldsheet.

WORLD SCIENTIFIC eBooks, Dec 1, 2012

Over the past few years the arithmetic Langlands program has found applications in two quite diff... more Over the past few years the arithmetic Langlands program has found applications in two quite different problems that arise in string physics. The first of these is concerned with the fundamental problem of deriving the geometry of spacetime from the worldsheet dynamics, leading to a realization of the notion of an emergent spacetime in string theory. The second problem is concerned with the idea of using automorphic black holes as probes of spacetime. In this article both of these applications of the Langlands program are described.

Physics Letters B, Oct 1, 1990

It is shown that subclasses of a recently constructed set of Landau-Ginzburg/Calabi-Yau string co... more It is shown that subclasses of a recently constructed set of Landau-Ginzburg/Calabi-Yau string compactifications can be described as orbifolds of simpler superpotentials. This construction allows for a systematic identification of dual pairs that appear naturally in this class of string vacua. The strange duality of Arnold appears as a special case of these constructions.

Nuclear Physics B, Sep 1, 1990

It has recently been recognized that the relation between exactly solvable conformal field theory... more It has recently been recognized that the relation between exactly solvable conformal field theory compactifications of the Heterotic String and Calabi-Yau manifolds necessarily involves the discussion of embeddings in weighted projective space. We therefore study this class of manifolds more closely. We have constructed a subclass of these spaces and find that this class features a surprising symmetry under chi

arXiv (Cornell University), Jan 11, 2013

In this paper automorphic motives are constructed and analyzed with a view toward the understandi... more In this paper automorphic motives are constructed and analyzed with a view toward the understanding of the geometry of compactification manifolds in string theory in terms of the modular structure of the worldsheet theory. The results described generalize a framework considered previously in two ways, first by relaxing the restriction to modular forms, and second by extending the construction of motives from diagonal varieties to nondiagonal spaces. The framework of automorphic forms and representations is described with a view toward applications, emphasizing the explicit structure of these objects.

Annals of the New York Academy of Sciences, Jun 1, 1993

The rôle in string theory of manifolds of complex dimension D crit +2(Q−1) and positive first Che... more The rôle in string theory of manifolds of complex dimension D crit +2(Q−1) and positive first Chern class is described. In order to be useful for string theory the first Chern class of these spaces has to satisfy a certain relation. Because of this condition the cohomology groups of such manifolds show a specific structure. A group that is particularly important is described by (D crit +Q−1, Q−1)-forms because it is this group which contains the higher dimensional counterpart of the holomorphic (D crit , 0)-form that figures so prominently in Calabi-Yau manifolds. It is shown that the higher dimensional manifolds do not, in general, have a unique counterpart of this holomorphic form of rank D crit. It is also shown that these manifolds lead, in general, to a number of additional modes beyond the standard Calabi-Yau spectrum. This suggests that not only the dilaton but also the other massless string modes, such as the antisymmetric torsion field, might be relevant for a possible stringy interpretation.

Springer eBooks, 1993

A recently introduced framework for the compactification of supersymmetric string theory involvin... more A recently introduced framework for the compactification of supersymmetric string theory involving noncritical manifolds of complex dimension 2k + D crit , k ≥ 1, is reviewed. These higher dimensional manifolds are spaces with quantized positive Ricci curvature and therefore do not, a priori, describe consistent string vacua. It is nevertheless possible to derive from these manifolds the massless spectra of critical string groundstates. For a subclass of these noncritical theories it is also possible to explicitly construct Calabi-Yau manifolds from the higher dimensional spaces. Thus the new class of theories makes contact with the standard framework of string compactification. This class of manifolds is more general than that of Calabi-Yau manifolds because it contains spaces which correspond to critical string vacua with no Kähler deformations, i.e. no antigenerations, hence providing mirrors of rigid Calabi-Yau manifolds. The constructions reviewed here lead to new insight into the relation between exactly solvable models and their mean field theories on the one hand and Calabi-Yau manifolds on the other, leading, for instance, to a modification of Gepner's conjecture. They also raise fundamental questions about the Kaluza-Klein concept of string compactification, in particular regarding the rôle played by the dimension of the internal theories.

International Journal of Modern Physics D, Oct 1, 2012

Over the past few years the arithmetic Langlands program has proven useful in addressing physical... more Over the past few years the arithmetic Langlands program has proven useful in addressing physical problems. In this paper it is shown how Langlands' reciprocity conjecture for automorphic forms, in combination with a representation theoretic notion of motives, suggests a framework in which the entropy of automorphic black holes can be viewed as a probe of spacetime that is sensitive to the geometry of the extra dimensions predicted by string theory. If it were possible to produce black holes with automorphic entropy in the laboratory their evaporation would provide us with information about the precise shape of the compact geometry.

Physics Letters B, 1991

We use the Landau-Ginzburg formulation of the Kazama-Suzuki series of exactly solvable models to ... more We use the Landau-Ginzburg formulation of the Kazama-Suzuki series of exactly solvable models to compute the spectrum of all N= 2 supersymmetric twisted G/H Landau-Ginzburg compactifications for hermitian symmetric spaces G/H. We show that the Landau-Ginzburg potentials associated to Kazama-Suzuki models all belong to the class of singularities we have constructed in previous work with P. Candelas.

Nuclear Physics B, Mar 1, 1988

arXiv (Cornell University), Oct 9, 2008

arXiv (Cornell University), Dec 9, 1992

A recently introduced method for constructing marginal singular flows between distinct Landau--Gi... more A recently introduced method for constructing marginal singular flows between distinct Landau--Ginzburg theories at fixed central charge is reviewed. The flows are constructed in an enlarged moduli space obtained by adding theories with zero central charge. This mechanism is used to construct flows between mirror pairs of string vacua described by N$=$2 superconformal Landau--Ginzburg fixed points. In contrast to previous methods this new construction of mirror theories does not depend on particular symmetries of the original theory. (Based in part on a talk presented at the Trieste Summer Workshop on Superstrings and Related Topics, July 1992)

arXiv (Cornell University), Nov 22, 1992

A new method for constructing flows between distinct Landau-Ginzburg theories at fixed central ch... more A new method for constructing flows between distinct Landau-Ginzburg theories at fixed central charge is presented. The essential ingredient of the construction is an enlarged moduli space obtained by adding theories with zero central charge. The flows involve only marginal directions hence they can be applied to transitions between string vacua, in particular to the construction of mirror pairs of string ground states described by RG fixed points of N=2 supersymmetric Landau--Ginzburg theories. In contrast to previous methods this new construction of mirror theories does not depend on particular symmetries of the original theory.