We present the coset structure of the untwisted moduli space of heterotic (0, 2) Z N orbifold com... more We present the coset structure of the untwisted moduli space of heterotic (0, 2) Z N orbifold compactifications with continuous Wilson lines. For the cases where the internal 6-torus T 6 is given by the direct sum T 4 ⊕ T 2 , we explicitly construct the Kähler potentials associated with the underlying 2torus T 2 . We then discuss the transformation properties of these Kähler potentials under target space modular symmetries. For the case where the Z N twist possesses eigenvalues of −1, we find that holomorphic terms occur in the Kähler potential describing the mixing of complex Wilson moduli. As a consequence, the associated T and U moduli are also shown to mix under target space modular transformations.
The macroscopic entropy and the attractor equations for BPS black holes in fourdimensional N = 2 ... more The macroscopic entropy and the attractor equations for BPS black holes in fourdimensional N = 2 supergravity theories follow from a variational principle for a certain 'entropy function'. We present this function in the presence of R 2interactions and non-holomorphic corrections. The variational principle identifies the entropy as a Legendre transform and this motivates the definition of various partition functions corresponding to different ensembles and a hierarchy of corresponding duality invariant inverse Laplace integral representations for the microscopic degeneracies. Whenever the microscopic degeneracies are known the partition functions can be evaluated directly. This is the case for N = 4 heterotic CHL black holes, where we demonstrate that the partition functions are consistent with the results obtained on the macroscopic side for black holes that have a non-vanishing classical area. In this way we confirm the presence of a measure in the duality invariant inverse Laplace integrals. Most, but not all, of these results are obtained in the context of semiclassical approximations. For black holes whose area vanishes classically, there remain discrepancies at the semiclassical level and beyond, the nature of which is not fully understood at present.
We discuss the transformation properties of classical extremal N = 2 black hole solutions in S-T ... more We discuss the transformation properties of classical extremal N = 2 black hole solutions in S-T -U like models under S and T duality. Using invariants of (subgroups of) the triality group, which is the symmetry group of the classical BPS mass formula, the transformation properties of the moduli on the event horizon and of the entropy under these transformations become manifest.
We construct a family of five-dimensional gauged supergravity actions which describe flop transit... more We construct a family of five-dimensional gauged supergravity actions which describe flop transitions of M-theory compactified on Calabi-Yau threefolds.
We analyze a broad class of stationary solutions with residual N = 1 supersymmetry of four-dimens... more We analyze a broad class of stationary solutions with residual N = 1 supersymmetry of four-dimensional N = 2 supergravity theories with terms quadratic in the Weyl tensor. These terms are encoded in a holomorphic function, which determines the most relevant part of the action and which plays a central role in our analysis. The solutions include extremal black holes and rotating field configurations, and may have multiple centers. We prove that they are expressed in terms of harmonic functions associated with the electric and magnetic charges carried by the solutions by a proper generalization of the so-called stabilization equations. Electric/magnetic duality is manifest throughout the analysis. We also prove that spacetimes with unbroken supersymmetry are fully determined by electric and magnetic charges. This result establishes the so-called fixed-point behavior according to which the moduli fields must flow towards certain prescribed values on a fully supersymmetric horizon, but now in a more general context with higher-order curvature interactions. We briefly comment on the implications of our results for the metric on the moduli space of extremal black hole solutions.
Compactifications of M-theory on singular manifolds contain additional charged massless states de... more Compactifications of M-theory on singular manifolds contain additional charged massless states descending from M-branes wrapped on vanishing cycles. We construct the first explicit example of a complete supergravity Lagrangian that includes such extra states. This is done for a compactification on a Calabi-Yau threefold that develops a genus zero curve of A 1 singularities at the boundary of the Kähler cone with a resulting SU (2) gauge symmetry enhancement. The corresponding SU (2) gauged supergravity Lagrangian includes two charged and two neutral vector multiplets, and turns out to be uniquely fixed by the Calabi-Yau geometry and by the effective ungauged Lagrangian describing the Coulomb branch. One can see explicitly how resolving the singularity corresponds to a supersymmetric Higgs effect in the gauged supergravity description. The elementary transformation relating the two families of smooth Calabi-Yau resolutions of the singularity acts as the SU (2) Weyl twist. The resulting structure appears to be very rigid and is likely to apply to other types of singularities and manifolds as well.
We construct the general action for Abelian vector multiplets in rigid 4-dimensional Euclidean (i... more We construct the general action for Abelian vector multiplets in rigid 4-dimensional Euclidean (instead of Minkowskian) N = 2 supersymmetry, i.e., over space-times with a positive definite instead of a Lorentzian metric. The target manifolds for the scalar fields turn out to be para-complex manifolds endowed with a particular kind of special geometry, which we call affine special para-Kähler geometry.
It is shown that the asymptotic growth of the microscopic degeneracy of BPS dyons in four-dimensi... more It is shown that the asymptotic growth of the microscopic degeneracy of BPS dyons in four-dimensional N = 4 string theory captures the known corrections to the macroscopic entropy of four-dimensional extremal black holes. These corrections are subleading in the limit of large charges and originate both from the presence of interactions in the effective action quadratic in the Riemann tensor and from non-holomorphic terms. The presence of the non-holomorphic corrections and their contribution to the thermodynamic free energy is discussed. It is pointed out that the expression for the macroscopic entropy, written as a function of the dilaton field, is stationary at the horizon by virtue of the attractor equations.
We study the heterotic string compactified on K3 × T 2 near the line T = U, where the effective a... more We study the heterotic string compactified on K3 × T 2 near the line T = U, where the effective action becomes singular due to an SU(2) gauge symmetry enhancement. By 'integrating in' the light W ± vector multiplets we derive a quantum corrected effective action which is manifestly SU(2) invariant and non-singular. This effective action is found to be consistent with a residual SL(2, Z) quantum symmetry on the line T = U. In an appropriate decompactification limit, we recover the known SU(2) invariant action in five dimensions.
In this note we compare the moduli spaces of the heterotic string compactified on a two-torus and... more In this note we compare the moduli spaces of the heterotic string compactified on a two-torus and F-Theory compactified on an elliptic K3 surface for the case of an unbroken E 8 ×E 8 gauge group. The explicit map relating the deformation parameters α and β of the F-Theory K3 surface to the moduli T and U of the heterotic torus is found using the close relationship between the K3 discriminant and the discriminant of the Calabi-Yau-threefold X 1,1,2,8,12 (24) in the limit of a large base P 1 .
We consider four-dimensional charged black-holes occuring in toroidally compactified heterotic st... more We consider four-dimensional charged black-holes occuring in toroidally compactified heterotic string theory, whose ten-dimensional interpretation involves a Kaluza-Klein monopole and a five-brane. We show that these four-dimensional black-holes can be connected to two-dimensional charged heterotic black-holes upon removal of the constants appearing in the harmonic functions associated with the Kaluza-Klein monopole and the five-brane.
Based on special geometry, we consider corrections to N = 2 extremal blackhole solutions and thei... more Based on special geometry, we consider corrections to N = 2 extremal blackhole solutions and their entropies originating from higher-order derivative terms in N = 2 supergravity. These corrections are described by a holomorphic function, and the higher-order black-hole solutions can be expressed in terms of symplectic Sp(2n+2) vectors. We apply the formalism to N = 2 type-IIA Calabi-Yau string compactifications and compare our results to recent related results in the literature.
In this note, we comment on Calabi-Yau spaces with Hodge numbers h 1;1 = 3 and h 2;1 = 243. We fo... more In this note, we comment on Calabi-Yau spaces with Hodge numbers h 1;1 = 3 and h 2;1 = 243. We focus on the Calabi-Yau space W P 1 ; 1 ; 2 ; 8 ; 12 (24) and show how some of its instanton numbers are related to coecients of certain modular forms. We also comment on the relation of four dimensional exchange symmetries in certain N = 2 dual models to six dimensional heterotic/heterotic string duality.
We determine the corrections to the entropy of extremal black holes arising from terms quadratic ... more We determine the corrections to the entropy of extremal black holes arising from terms quadratic in the Riemann tensor in N = 2, D = 4 supergravity theories. We follow Wald's proposal to modify the Bekenstein-Hawking area law. The new entropy formula, whose value only depends on the electric/magnetic charges, is expressed in terms of a single holomorphic function and is consistent with electric-magnetic duality. For string effective field theories arising from Calabi-Yau compactifications, our result for the entropy of a certain class of extremal black-hole solutions fully agrees with the counting of microstates performed some time ago by Maldacena, Strominger, Witten and by Vafa.
In this paper we discuss the M -brane description for a N = 2 black hole. This solution is a resu... more In this paper we discuss the M -brane description for a N = 2 black hole. This solution is a result of the compactification of M -5-brane configurations over a Calabi-Yau threefold with arbitrary intersection numbers C ABC . In analogy to the D-brane description where one counts open string states we count here open M -2-branes which end on the M -5-brane.
In this paper we summarize our recent work about perturbative and non-perturbative effects in fou... more In this paper we summarize our recent work about perturbative and non-perturbative effects in four-dimensional heterotic strings with N = 2 space-time supersymmetry.
We address non-perturbative effects and duality symmetries in N = 2 heterotic string theories in ... more We address non-perturbative effects and duality symmetries in N = 2 heterotic string theories in four dimensions. Specifically, we consider how each of the four lines of enhanced gauge symmetries in the perturbative moduli space of N = 2 T 2 compactifications is split into 2 lines where monopoles and dyons become massless. This amounts to considering non-perturbative effects originating from enhanced gauge symmetries at the microscopic string level.
We study the BPS spectrum in D = 4, N = 4 heterotic string compactifications, with some emphasis ... more We study the BPS spectrum in D = 4, N = 4 heterotic string compactifications, with some emphasis on intermediate N = 4 BPS states. These intermediate states, which can become short in N = 2 compactifications, are crucial for establishing an S − T exchange symmetry in N = 2 compactifications. We discuss the implications of a possible S − T exchange symmetry for the N = 2 BPS spectrum. Then we present the exact result for the 1-loop corrections to gravitational couplings in one of the heterotic N = 2 models recently discussed by Harvey and Moore. We conjecture this model to have an S − T exchange symmetry. This exchange symmetry can then be used to evaluate non-perturbative corrections to gravitational couplings in some of the non-perturbative regions (chambers) in this particular model and also in other heterotic models. December 1995 1
We use heterotic/type-II prepotentials to study quantum/classical black holes with half the N = 2... more We use heterotic/type-II prepotentials to study quantum/classical black holes with half the N = 2, D = 4 supersymmetries unbroken. We show that, in the case of heterotic string compactifications, the perturbatively corrected entropy formula is given by the tree-level entropy formula with the treelevel coupling constant replaced by the perturbative coupling constant. In the case of type-II compactifications, we display a new entropy/area formula associated with axion-free black-hole solutions, which depends on the electric and magnetic charges as well as on certain topological data of Calabi-Yau three-folds, namely the intersection numbers, the second Chern class and the Euler number of the three-fold. We show that, for both heterotic and type-II theories, there is the possibility to relax the usual requirement of the nonvanishing of some of the charges and still have a finite entropy.
We present the coset structure of the untwisted moduli space of heterotic (0, 2) Z N orbifold com... more We present the coset structure of the untwisted moduli space of heterotic (0, 2) Z N orbifold compactifications with continuous Wilson lines. For the cases where the internal 6-torus T 6 is given by the direct sum T 4 ⊕ T 2 , we explicitly construct the Kähler potentials associated with the underlying 2torus T 2 . We then discuss the transformation properties of these Kähler potentials under target space modular symmetries. For the case where the Z N twist possesses eigenvalues of −1, we find that holomorphic terms occur in the Kähler potential describing the mixing of complex Wilson moduli. As a consequence, the associated T and U moduli are also shown to mix under target space modular transformations.
The macroscopic entropy and the attractor equations for BPS black holes in fourdimensional N = 2 ... more The macroscopic entropy and the attractor equations for BPS black holes in fourdimensional N = 2 supergravity theories follow from a variational principle for a certain 'entropy function'. We present this function in the presence of R 2interactions and non-holomorphic corrections. The variational principle identifies the entropy as a Legendre transform and this motivates the definition of various partition functions corresponding to different ensembles and a hierarchy of corresponding duality invariant inverse Laplace integral representations for the microscopic degeneracies. Whenever the microscopic degeneracies are known the partition functions can be evaluated directly. This is the case for N = 4 heterotic CHL black holes, where we demonstrate that the partition functions are consistent with the results obtained on the macroscopic side for black holes that have a non-vanishing classical area. In this way we confirm the presence of a measure in the duality invariant inverse Laplace integrals. Most, but not all, of these results are obtained in the context of semiclassical approximations. For black holes whose area vanishes classically, there remain discrepancies at the semiclassical level and beyond, the nature of which is not fully understood at present.
We discuss the transformation properties of classical extremal N = 2 black hole solutions in S-T ... more We discuss the transformation properties of classical extremal N = 2 black hole solutions in S-T -U like models under S and T duality. Using invariants of (subgroups of) the triality group, which is the symmetry group of the classical BPS mass formula, the transformation properties of the moduli on the event horizon and of the entropy under these transformations become manifest.
We construct a family of five-dimensional gauged supergravity actions which describe flop transit... more We construct a family of five-dimensional gauged supergravity actions which describe flop transitions of M-theory compactified on Calabi-Yau threefolds.
We analyze a broad class of stationary solutions with residual N = 1 supersymmetry of four-dimens... more We analyze a broad class of stationary solutions with residual N = 1 supersymmetry of four-dimensional N = 2 supergravity theories with terms quadratic in the Weyl tensor. These terms are encoded in a holomorphic function, which determines the most relevant part of the action and which plays a central role in our analysis. The solutions include extremal black holes and rotating field configurations, and may have multiple centers. We prove that they are expressed in terms of harmonic functions associated with the electric and magnetic charges carried by the solutions by a proper generalization of the so-called stabilization equations. Electric/magnetic duality is manifest throughout the analysis. We also prove that spacetimes with unbroken supersymmetry are fully determined by electric and magnetic charges. This result establishes the so-called fixed-point behavior according to which the moduli fields must flow towards certain prescribed values on a fully supersymmetric horizon, but now in a more general context with higher-order curvature interactions. We briefly comment on the implications of our results for the metric on the moduli space of extremal black hole solutions.
Compactifications of M-theory on singular manifolds contain additional charged massless states de... more Compactifications of M-theory on singular manifolds contain additional charged massless states descending from M-branes wrapped on vanishing cycles. We construct the first explicit example of a complete supergravity Lagrangian that includes such extra states. This is done for a compactification on a Calabi-Yau threefold that develops a genus zero curve of A 1 singularities at the boundary of the Kähler cone with a resulting SU (2) gauge symmetry enhancement. The corresponding SU (2) gauged supergravity Lagrangian includes two charged and two neutral vector multiplets, and turns out to be uniquely fixed by the Calabi-Yau geometry and by the effective ungauged Lagrangian describing the Coulomb branch. One can see explicitly how resolving the singularity corresponds to a supersymmetric Higgs effect in the gauged supergravity description. The elementary transformation relating the two families of smooth Calabi-Yau resolutions of the singularity acts as the SU (2) Weyl twist. The resulting structure appears to be very rigid and is likely to apply to other types of singularities and manifolds as well.
We construct the general action for Abelian vector multiplets in rigid 4-dimensional Euclidean (i... more We construct the general action for Abelian vector multiplets in rigid 4-dimensional Euclidean (instead of Minkowskian) N = 2 supersymmetry, i.e., over space-times with a positive definite instead of a Lorentzian metric. The target manifolds for the scalar fields turn out to be para-complex manifolds endowed with a particular kind of special geometry, which we call affine special para-Kähler geometry.
It is shown that the asymptotic growth of the microscopic degeneracy of BPS dyons in four-dimensi... more It is shown that the asymptotic growth of the microscopic degeneracy of BPS dyons in four-dimensional N = 4 string theory captures the known corrections to the macroscopic entropy of four-dimensional extremal black holes. These corrections are subleading in the limit of large charges and originate both from the presence of interactions in the effective action quadratic in the Riemann tensor and from non-holomorphic terms. The presence of the non-holomorphic corrections and their contribution to the thermodynamic free energy is discussed. It is pointed out that the expression for the macroscopic entropy, written as a function of the dilaton field, is stationary at the horizon by virtue of the attractor equations.
We study the heterotic string compactified on K3 × T 2 near the line T = U, where the effective a... more We study the heterotic string compactified on K3 × T 2 near the line T = U, where the effective action becomes singular due to an SU(2) gauge symmetry enhancement. By 'integrating in' the light W ± vector multiplets we derive a quantum corrected effective action which is manifestly SU(2) invariant and non-singular. This effective action is found to be consistent with a residual SL(2, Z) quantum symmetry on the line T = U. In an appropriate decompactification limit, we recover the known SU(2) invariant action in five dimensions.
In this note we compare the moduli spaces of the heterotic string compactified on a two-torus and... more In this note we compare the moduli spaces of the heterotic string compactified on a two-torus and F-Theory compactified on an elliptic K3 surface for the case of an unbroken E 8 ×E 8 gauge group. The explicit map relating the deformation parameters α and β of the F-Theory K3 surface to the moduli T and U of the heterotic torus is found using the close relationship between the K3 discriminant and the discriminant of the Calabi-Yau-threefold X 1,1,2,8,12 (24) in the limit of a large base P 1 .
We consider four-dimensional charged black-holes occuring in toroidally compactified heterotic st... more We consider four-dimensional charged black-holes occuring in toroidally compactified heterotic string theory, whose ten-dimensional interpretation involves a Kaluza-Klein monopole and a five-brane. We show that these four-dimensional black-holes can be connected to two-dimensional charged heterotic black-holes upon removal of the constants appearing in the harmonic functions associated with the Kaluza-Klein monopole and the five-brane.
Based on special geometry, we consider corrections to N = 2 extremal blackhole solutions and thei... more Based on special geometry, we consider corrections to N = 2 extremal blackhole solutions and their entropies originating from higher-order derivative terms in N = 2 supergravity. These corrections are described by a holomorphic function, and the higher-order black-hole solutions can be expressed in terms of symplectic Sp(2n+2) vectors. We apply the formalism to N = 2 type-IIA Calabi-Yau string compactifications and compare our results to recent related results in the literature.
In this note, we comment on Calabi-Yau spaces with Hodge numbers h 1;1 = 3 and h 2;1 = 243. We fo... more In this note, we comment on Calabi-Yau spaces with Hodge numbers h 1;1 = 3 and h 2;1 = 243. We focus on the Calabi-Yau space W P 1 ; 1 ; 2 ; 8 ; 12 (24) and show how some of its instanton numbers are related to coecients of certain modular forms. We also comment on the relation of four dimensional exchange symmetries in certain N = 2 dual models to six dimensional heterotic/heterotic string duality.
We determine the corrections to the entropy of extremal black holes arising from terms quadratic ... more We determine the corrections to the entropy of extremal black holes arising from terms quadratic in the Riemann tensor in N = 2, D = 4 supergravity theories. We follow Wald's proposal to modify the Bekenstein-Hawking area law. The new entropy formula, whose value only depends on the electric/magnetic charges, is expressed in terms of a single holomorphic function and is consistent with electric-magnetic duality. For string effective field theories arising from Calabi-Yau compactifications, our result for the entropy of a certain class of extremal black-hole solutions fully agrees with the counting of microstates performed some time ago by Maldacena, Strominger, Witten and by Vafa.
In this paper we discuss the M -brane description for a N = 2 black hole. This solution is a resu... more In this paper we discuss the M -brane description for a N = 2 black hole. This solution is a result of the compactification of M -5-brane configurations over a Calabi-Yau threefold with arbitrary intersection numbers C ABC . In analogy to the D-brane description where one counts open string states we count here open M -2-branes which end on the M -5-brane.
In this paper we summarize our recent work about perturbative and non-perturbative effects in fou... more In this paper we summarize our recent work about perturbative and non-perturbative effects in four-dimensional heterotic strings with N = 2 space-time supersymmetry.
We address non-perturbative effects and duality symmetries in N = 2 heterotic string theories in ... more We address non-perturbative effects and duality symmetries in N = 2 heterotic string theories in four dimensions. Specifically, we consider how each of the four lines of enhanced gauge symmetries in the perturbative moduli space of N = 2 T 2 compactifications is split into 2 lines where monopoles and dyons become massless. This amounts to considering non-perturbative effects originating from enhanced gauge symmetries at the microscopic string level.
We study the BPS spectrum in D = 4, N = 4 heterotic string compactifications, with some emphasis ... more We study the BPS spectrum in D = 4, N = 4 heterotic string compactifications, with some emphasis on intermediate N = 4 BPS states. These intermediate states, which can become short in N = 2 compactifications, are crucial for establishing an S − T exchange symmetry in N = 2 compactifications. We discuss the implications of a possible S − T exchange symmetry for the N = 2 BPS spectrum. Then we present the exact result for the 1-loop corrections to gravitational couplings in one of the heterotic N = 2 models recently discussed by Harvey and Moore. We conjecture this model to have an S − T exchange symmetry. This exchange symmetry can then be used to evaluate non-perturbative corrections to gravitational couplings in some of the non-perturbative regions (chambers) in this particular model and also in other heterotic models. December 1995 1
We use heterotic/type-II prepotentials to study quantum/classical black holes with half the N = 2... more We use heterotic/type-II prepotentials to study quantum/classical black holes with half the N = 2, D = 4 supersymmetries unbroken. We show that, in the case of heterotic string compactifications, the perturbatively corrected entropy formula is given by the tree-level entropy formula with the treelevel coupling constant replaced by the perturbative coupling constant. In the case of type-II compactifications, we display a new entropy/area formula associated with axion-free black-hole solutions, which depends on the electric and magnetic charges as well as on certain topological data of Calabi-Yau three-folds, namely the intersection numbers, the second Chern class and the Euler number of the three-fold. We show that, for both heterotic and type-II theories, there is the possibility to relax the usual requirement of the nonvanishing of some of the charges and still have a finite entropy.
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Papers by Thomas Mohaupt