Publications

Strings on Orbifolds

Lance J. Dixon, Jeffrey A. Harvey, C. Vafa, Edward Witten (Princeton U.)

Jul 1985 - 9 pages

Abstract: String propagation on the quotient of a flat torus by a discrete group is considered. We obtain an exactly soluble and more or less realistic method of string compactification.

Calabi-Yau Manifolds and Renormalization Group Flows

Brian R. Greene, C. Vafa (Harvard U.), N.P. Warner (MIT)

Nov 1988 - 20 pages

Abstract: Using ideas of renormalization group flows we show how to represent a large class of Calabi-Yau manifolds in terms of renormalization group fixed points of Landau-Ginzburg models. In particular we show why Gepner's construction yields Calabi-Yau compactifications. More precisely, we show that all of his models correspond to strings propagating on algebraic varieties in weighted projective spaces.

Chiral Rings in N=2 Superconformal Theories

Wolfgang Lerche (Caltech), Cumrun Vafa (Harvard U.), Nicholas P. Warner (MIT)

Feb 1989 - 48 pages

Abstract: String propagation on the quotient of a flat torus by a discrete group is considered. We obtain an exactly soluble and more or less realistic method of string compactification.

Stringy Cosmic Strings and Noncompact Calabi-Yau Manifolds

Brian R. Greene (Harvard U.), Alfred D. Shapere (Princeton, Inst. Advanced Study), Cumrun Vafa, Shing-Tung Yau (Harvard U.)

Oct 1989 - 36 pages

Abstract: We describe string vacuum configurations for which the radii (moduli) of the internal compact space vary in four-dimensional space-time, focusing on configurations which have the interpretation of cosmic strings. We will show that some of them admit Ricci-flat Kähler metrics (i.e. they correspond to noncompact Calabi-Yau manifolds), thus providing new vacuum solutions to the full string theory. One novel feature of some of these solutions is that the internal space decompactifies near the core of the cosmic string, without producing any physical singularities.

Exact Results for N=2 Compactifications of Heterotic Strings

Shamit Kachru, Cumrun Vafa (Harvard U.)

May 1995 - 21 pages

Abstract: We search for N=2, d=4 theories which can be realized both as heterotic string compactifications onK_{3}\times T~{2}and as type II string compactifications on Calabi-Yau threefolds. In such cases, the exact non-perturbative superpotential of one string theory is given in terms of tree level computations in the other string theory. In particular we find concrete examples which provide the stringy realization of the results of Seiberg and Witten on N=2 Yang-Mills theory, corrected by gravitational/stringy effects. We also discuss some examples which shed light on how the moduli spaces of different N=2 heterotic vacua are connected.

Evidence for F-Theory

Cumrun Vafa (Harvard U.)

Feb 1996 - 18 pages

https://arxiv.org/abs/hep-th/9602022

Abstract: We construct compact examples of D-manifolds for type IIB strings. The construction has a natural interpretation in terms of compactification of a 12 dimensional `F-theory'. We provide evidence for a more natural reformulation of type IIB theory in terms of F-theory. Compactification of M-theory on a manifoldKwhich admits elliptic fibration is equivalent to compactification of F-theory onK\times S~1. A large class ofN=1theories in 6 dimensions are obtained by compactification of F-theory on Calabi-Yau threefolds. A class of phenomenologically promising compactifications of F-theory is onSpin(7)holonomy manifolds down to 4 dimensions. This may provide a concrete realization of Witten's proposal for solving the cosmological constant problem in four dimensions.

Compactifications of F-Theory on Calabi-Yau threefolds I

David R. Morrison (Duke U.), Cumrun Vafa (Harvard U.)

Feb 1996 - 23 pages

http://arxiv.org/abs/hep-th/9602114

Abstract: We study compactifications of F-theory on certain Calabi--Yau threefolds. We find thatN=2dualities of type II/heterotic strings in 4 dimensions get promoted toN=1dualities between heterotic string and F-theory in 6 dimensions. The six dimensional heterotic/heterotic duality becomes a classical geometric symmetry of the Calabi--Yau in the F-theory setup. Moreover the F-theory compactification sheds light on the nature of the strong coupling transition and what lies beyond the transition at finite values of heterotic string coupling constant.

Compactifications of F theory on Calabi-Yau threefolds II

David R. Morrison (Duke U.), Cumrun Vafa (Harvard U.)

Mar 1996 - 33 pages

http://arxiv.org/abs/hep-th/9603161

Abstract: We continue our study of compactifications of F-theory on Calabi--Yau threefolds. We gain more insight into F-theory duals of heterotic strings and provide a recipe for building F-theory duals for arbitrary heterotic compactifications on elliptically fibered manifolds. As a byproduct we find that string/string duality in six dimensions gets mapped to fiber/base exchange in F-theory. We also construct a number of newN=1,d=6examples of F-theory vacua and study transitions among them. We find that some of these transition points correspond upon further compactification to 4 dimensions to transitions through analogues of Argyres--Douglas points ofN=2moduli. A key idea in these transitions is the notion of classifying(0,4)fivebranes of heterotic strings.

Geometric origin of Montonen-Olive duality

Cumrun Vafa (Harvard U.)

Jul 1997 - 8 pages

https://arxiv.org/abs/hep-th/9707131

Abstract: We show how N=4, D=4 duality of Montonen and Olive can be derived for all gauge groups using geometric engineering in the context of type II strings, where it reduces to T-duality. The derivation for the non-simply laced cases involves the use of some well known facts about orbifold conformal theories.

Lectures on Strings and Dualities

Cumrun Vafa (Harvard U.)

Feb 1997 - 53 pages

http://arxiv.org/abs/hep-th/9702201

Abstract: In this set of lectures I review recent developments in string theory emphasizing their non-perturbative aspects and their recently discovered duality symmetries. The goal of the lectures is to make the recent exciting developments in string theory accessible to those with no previous background in string theory who wish to join the research effort in this area. Topics covered include a brief review of string theory, its compactifications, solitons and D-branes, black hole entropy and web of string dualities.

CFT's from Calabi-Yau Four-Folds

Sergei Gukov (Princeton U.), Cumrun Vafa (Harvard U.), Edward Witten (Princeton, Inst. Advanced Study)

Jun 1999 - 49 pages

http://arxiv.org/abs/hep-th/9906070

Abstract: We consider F/M/Type IIA theory compactified to four, three, or two dimensions on a Calabi-Yau four-fold, and study the behavior near an isolated singularity in the presence of appropriate fluxes and branes. We analyze the vacuum and soliton structure of these models, and show that near an isolated singularity, one often generates massless chiral superfields and a superpotential, and in many instances in two or three dimensions one obtains nontrivial superconformal field theories. In the case of two dimensions, we identify some of these theories with certain Kazama-Suzuki coset models, such as the N=2 minimal models.

Superstrings and Topological Strings at Large N

Cumrun Vafa (Harvard U.)

Aug 2000 - 31 pages

http://arxiv.org/abs/hep-th/0008142

Abstract: We embed the large N Chern-Simons/topological string duality in ordinary superstrings. This corresponds to a largeNduality between generalized gauge systems with N=1 supersymmetry in 4 dimensions and superstrings propagating on non-compact Calabi-Yau manifolds with certain fluxes turned on. We also show that in a particular limit of the N=1 gauge theory system, certain superpotential terms in the N=1 system (including deformations if spacetime is non-commutative) are captured to all orders in 1/N by the amplitudes of non-critical bosonic strings propagating on a circle with self-dual radius. We also consider D-brane/anti-D-brane system wrapped over vanishing cycles of compact Calabi-Yau manifolds and argue that at largeNthey induce a shift in the background to a topologically distinct Calabi-Yau, which we identify as the ground state system of the Brane/anti-Brane system.

Holography and the KKLT scenario

Severin Lüst, Cumrun Vafa, Max Wiesner, Kai Xu

Nov 11, 2022- 30 pages

https://arxiv.org/abs/2204.07171

Abstract: The KKLT scenario, one of the few ideas to realize dS vacua in string theory, consists of two steps: the first involves the construction of a supersymmetric AdS vacuum with a small negative cosmological constant, and the second involves breaking supersymmetry and uplifting the energy to achieve dS. In this paper we use conventional holography to argue why it is not possible to complete the first step. We obtain this by putting a bound on the central charge of the dual theory which involves branes wrapping special Lagrangian cycles in CY 4-folds. We find that l2AdS≲χ(CY4). Since l2species≳χ(CY4) this leads to lAdS/lspecies≲1 leading at best to a highly curved AdS which is beyond the validity of the EFT.

The String Landscape and the Swampland

Cumrun Vafa (Harvard U., Phys. Dept.)

Sep 2005 - 10 pages

http://arxiv.org/abs/hep-th/0509212

Abstract: Recent developments in string theory suggest that string theory landscape of vacua is vast. It is natural to ask if this landscape is as vast as allowed by consistent-looking effective field theories. We use universality ideas from string theory to suggest that this is not the case, and that the landscape is surrounded by an even more vast swampland of consistent-looking semiclassical effective field theories, which are actually inconsistent. Identification of the boundary of the landscape is a central question which is at the heart of the meaning of universality properties of consistent quantum gravitational theories. We propose certain finiteness criteria as one relevant factor in identifying this boundary (based on talks given at the Einstein Symposium in Alexandria, at the 2005 Simons Workshop in Mathematics and Physics, and the talk to have been presented at Strings 2005).

The String Landscape, Black Holes and Gravity as the Weakest Force

Nima Arkani-Hamed, Lubos Motl, Alberto Nicolis, Cumrun Vafa (Harvard U., Phys. Dept.)

Jan 2006 - 20 pages

http://arxiv.org/abs/hep-th/0601001

Abstract: We conjecture a general upper bound on the strength of gravity relative to gauge forces in quantum gravity. This implies, in particular, that in a four-dimensional theory with gravity and a U(1) gauge field with gauge coupling g, there is a new ultraviolet scale Lambda=g M_{Pl}, invisible to the low-energy effective field theorist, which sets a cutoff on the validity of the effective theory. Moreover, there is some light charged particle with mass smaller than or equal to Lambda. The bound is motivated by arguments involving holography and absence of remnants, the (in) stability of black holes as well as the non-existence of global symmetries in string theory. A sharp form of the conjecture is that there are always light elementary electric and magnetic objects with a mass/charge ratio smaller than the corresponding ratio for macroscopic extremal black holes, allowing extremal black holes to decay. This conjecture is supported by a number of non-trivial examples in string theory. It implies the necessary presence of new physics beneath the Planck scale, not far from the GUT scale, and explains why some apparently natural models of inflation resist an embedding in string theory.

On the Geometry of the String Landscape and the Swampland

Hirosi Ooguri (Caltech), Cumrun Vafa (Harvard U., Phys. Dept.)

May 2006 - 18 pages

http://arxiv.org/abs/hep-th/0605264

Abstract: We make a number of conjectures about the geometry of continuous moduli parameterizing the string landscape. In particular we conjecture that such moduli are always given by expectation value of scalar fields and that moduli spaces with finite non-zero diameter belong to the swampland. We also conjecture that points at infinity in a moduli space correspond to points where an infinite tower of massless states appear, and that near these regions the moduli space is negatively curved. We also propose that there is no non-trivial 1-cycle of minimum length in the moduli space. This leads in particular to the prediction of the existence of a radially massive partner to the axion. These conjectures put strong constraints on inflaton potentials that can appear in a consistent quantum theory of gravity. Our conjectures are supported by a number of highly non-trivial examples from string theory. Moreover it is shown that these conditions can be violated if gravity is decoupled.

Non-supersymmetric AdS and the Swampland

Hirosi Ooguri (Caltech & Tokyo U., IPMU), Cumrun Vafa (Harvard U., Phys. Dept.)

Oct 5, 2016 - 15 pages

http://arxiv.org/abs/arXiv:1610.01533

Abstract: We propose to sharpen the weak gravity conjecture by the statement that, except for BPS states in a supersymmetric theory, the gravitational force is strictly weaker than any electric force and provide a number of evidences for this statement. Our conjecture implies that any non-supersymmetric anti-de Sitter vacuum supported by fluxes must be unstable, as is the case for all known attempts at such holographic constructions.

The String Landscape, the Swampland, and the Missing Corner

T. Daniel Brennan (Rutgers U., Piscataway), Federico Carta (Madrid, Autonoma U. & Madrid, IFT), Cumrun Vafa (Harvard U., Phys. Dept.)

Nov 2, 2017 - 63 pages

http://arxiv.org/abs/arXiv:1711.00864

Abstract: We give a brief overview of the string landscape and techniques used to construct string compactifications. We then explain how this motivates the notion of the swampland and review a number of conjectures that attempt to characterize theories in the swampland. We also compare holography in the context of superstrings with the similar, but much simpler case of topological string theory. For topological strings, there is a direct definition of topological gravity based on a sum over a "quantum gravitational foam." In this context, holography is the statement of an identification between a gravity and gauge theory, both of which are defined independently of one another. This points to a missing corner in string dualities which suggests the search for a direct definition of quantum theory of gravity rather than relying on its strongly coupled holographic dual as an adequate substitute (Based on TASI 2017 lectures given by C. Vafa).

De Sitter Space and the Swampland

Georges Obied (Harvard U., Phys. Dept.), Hirosi Ooguri (Caltech & Tokyo U., IPMU), Lev Spodyneiko (Caltech), Cumrun Vafa (Harvard U., Phys. Dept.)

Jun 21, 2018 - 21 pages

http://arxiv.org/abs/arXiv:1806.08362

Abstract: It has been notoriously difficult to construct a meta-stable de Sitter (dS) vacuum in string theory in a controlled approximation. This suggests the possibility that meta-stable dS belongs to the swampland. In this paper, we propose a swampland criterion in the form of |\nabla V|\geq\ c \cdot V for a scalar potential V of any consistent theory of quantum gravity, for a positive constant c. In particular, this bound forbids dS vacua. The existence of this bound is motivated by the abundance of string theory constructions and no-go theorems which exhibit this behavior. We also extend some of the well-known no-go theorems for the existence of dS vacua in string theory to more general accelerating universes and reinterpret the results in terms of restrictions on allowed scalar potentials.

On the Cosmological Implications of the String Swampland

Prateek Agrawal, Georges Obied (Harvard U., Phys. Dept.), Paul J. Steinhardt (Princeton U. & Princeton U., CTP & New York U., CCPP), Cumrun Vafa (Harvard U., Phys. Dept.)

Jun 25, 2018 - 6 pages

http://arxiv.org/abs/arXiv:1806.09718

Abstract: We study constraints imposed by two proposed string Swampland criteria on cosmology. These criteria involve an upper bound on the range traversed by scalar fields as well as a lower bound on |∇ϕV|/V when V>0 . We find that inflationary models are generically in tension with these two criteria. Applying these same criteria to dark energy in the present epoch, we find that specific quintessence models can satisfy these bounds and, at the same time, satisfy current observational constraints. Assuming the two Swampland criteria are valid, we argue that the universe will undergo a phase transition within a few Hubble times. These criteria sharpen the motivation for future measurements of the tensor-to-scalar ratio r and the dark energy equation of state w , and for tests of the equivalence principle for dark matter.

Trans-Planckian Censorship and the Swampland

Alek Bedroya, Cumrun Vafa

Sep 24, 2019 - 35 pages

http://arxiv.org/abs/arXiv:1909.11063

Abstract: In this paper, we propose a new Swampland condition, the Trans-Planckian Censorship Conjecture (TCC), based on the idea that in a consistent quantum theory of gravity sub-Planckian quantum fluctuations should remain quantum and never become larger than the Hubble horizon and freeze in an expanding universe. Applied to the case of scalar fields, it leads to conditions that are similar to the refined dS Swampland conjecture. For large field ranges, TCC is stronger than the dS Swampland conjecture but it is weaker for small field ranges. In particular for asymptotic regions of field space, TCC leads to a bound |V'|\geq {2\over \sqrt{(d-1)(d-2)}}V, which is consistent with all known cases in string theory. Like the dS Swampland conjecture, the TCC forbids long-lived meta-stable dS spaces, but it does allow sufficiently short-lived ones.

Compactness of Brane Moduli and the String Lamppost Principle in d > 6

Alek Bedroya, Yuta Hamada, Miguel Montero and Cumrun Vafa

Dec 28, 2021 - 35 pages

https://arxiv.org/abs/2110.10157

Abstract: We demonstrate the validity of the String Lamppost Principle – that all consistent theories of quantum gravity are in the String Landscape – for supersymmetric theories in d > 6 using compactness and connectedness of the moduli space of small instantons, as well as the classification of the associated Coulomb branch.

The Dark Dimension and the Swampland

Miguel Montero, Cumrun Vafa and Irene Valenzuela

Aug 30, 2022- 18 pages

https://arxiv.org/abs/2205.12293v2

Abstract: Motivated by principles from the Swampland program, which characterize requirements for a consistent UV completion of quantum gravity, combined with observational data, we are led to a unique corner of the quantum gravity landscape. In particular, using the Distance/Duality conjecture and the smallness of dark energy, we predict the existence of a light tower of states and a unique extra mesoscopic dimension of length l∼Λ−14∼10−6m, with extra massless fermions propagating on it. This automatically leads to a candidate for a tower of sterile neutrinos, and an associated active neutrino mass scale mν∼⟨H⟩2Λ−112M−23pl. Moreover, assuming the mechanism for stabilization of this dark dimension leads to similar masses for active and sterile neutrinos we are led to the prediction of a Higgs vev ⟨H⟩∼Λ16M13pl. Another prediction of the scenario is a species scale M̂ ∼Λ112M23pl∼109−1010GeV, corresponding to the higher-dimensional Planck scale. This energy scale may be related to the resolution of the instability of the Higgs effective potential present at a scale of ∼1011GeV. We also speculate about the interplay between this energy scale and the GZK limit on ultra-high energy cosmic rays.

Dark Dimension Gravitons as Dark Matter

Eduardo Gonzalo, Miguel Montero, Georges Obied, Cumrun Vafa

Sep 25, 2022 - 18 pages

https://arxiv.org/abs/2209.09249

Abstract: We consider cosmological aspects of the Dark Dimension (a mesoscopic dimension of micron scale), which has recently been proposed as the unique corner of the quantum gravity landscape consistent with both the Swampland criteria and observations. In particular we show how this leads, by the universal coupling of the Standard Model sector to bulk gravitons, to massive spin 2 KK excitations of the graviton in the dark dimension (the "dark gravitons") as an unavoidable dark matter candidate. Assuming a lifetime for the current de Sitter phase of our universe of order Hubble, which follows from both the dS Swampland Conjecture and TCC, we show that generic features of the dark dimension cosmology can naturally lead to the correct dark matter density and a resolution of the cosmological coincidence problem, where the matter/radiation equality temperature (T∼ 1 eV) coincides with the temperature where the dark energy begins to dominate. Thus one does not need to appeal to Weinberg's anthropic argument to explain this coincidence. The dark gravitons are produced at T∼ 4 GeV, and their composition changes as they mainly decay to lighter gravitons, without losing much total mass density. The mass of dark gravitons is mDM∼1−100 keV today.

Swamplandish Unification of the Dark Sector

Cumrun Vafa (Harvard U., Phys. Dept.)

Feb 1, 2024 - 13 pages

https://inspirehep.net/literature/2755023

Abstract: We provide a short overview of recent progress made in our understanding of the dark sector based on the Swampland program which in turn is rooted in lessons from string theory. We explain how the existence of one extra mesoscopic dimension (the "dark dimension") in the micron range emerges and how this can lead to a unification of the dark energy and dark matter. In particular the smallness of the dark energy leads to the prediction of the existence of a tower of weakly interacting light particles which can naturally play the role of dark matter. Moreover this unifies dark matter with gravity as dark matter ends up being excitations of graviton in the dark dimension. We also explain how in combination with other Swampland principles one finds an explanation of the "why now" and the "cosmological coincidence" problems. This model is consistent with the cosmological bounds as well as the Newton's inverse square law, but makes predictions which differ from ΛCDM. This review is intended for a broad audience of high energy theorists and cosmologists without prior knowledge of string theory and it explains the motivations and predictions of this program in a non-technical form.

Restrictions on Symmetry Breaking in Vector-Like Gauge Theories

C. Vafa, Edward Witten (Princeton U.)

Oct 1983 - 16 pages

Abstract: Modulo some highly plausible assumptions, we show that vector-like global symmetries (like isospin or baryon number) are not spontaneously broken in vector-like gauge theories with θ = 0 (like QCD). We also show that in these theories massless bound states do not form from massive constituents.

Eigenvalue Inequalities for Fermions in Gauge Theories

Cumrun Vafa (Princeton U.), Edward Witten (Princeton U. & Princeton, Inst. Advanced Study)

Apr 1984 - 39 pages

Abstract: We show that QCD with a sufficient number of fermions of zero bare mass has physical massless particles. That result also follows from triangle anomalies, so only our method is novel. Our method involves proving special cases of recently conjectured paramagnetic inequalities for fermions. The proofs are simple applications of the Atiyah-Patodi-Singer theorem on spectral flow.

Parity Conservation in QCD

Cumrun Vafa (Princeton U.), Edward Witten (Princeton U. & Princeton, Inst. Advanced Study)

May 1984 - 7 pages

Abstract: We show that in parity-conserving vectorlike theories such as QCD, parity conservation is not spontaneously broken.

Topological Antitopological Fusion

Sergio Cecotti (SISSA, Trieste & INFN, Trieste), Cumrun Vafa (Harvard U.)

Jun 1991 - 103 pages

Abstract: We study some non-perturbative aspects of N = 2 supersymmetric quantum field theories (both superconformal and massive deformations thereof). We show that the metric for the supersymmetric ground states, which in the conformal limit is essentially the same as Zamolodchikov's metric, is pseudo-topological and can be viewed as a result of fusion of the topological version of N = 2 theory with its conjugate. For special marginal/relevant deformations (corresponding to theories with factorizable S -matrix), the ground state metric satisfies classical Toda/Affine Toda equations as a function of perturbation parameters. The unique consistent boundary conditions for these differential equations seem to predict the normalized OPE of chiral fields at the conformal point. Also the subset of N = 2 theories whose chiral ring is isomorphic to SU( N ) κ Verlinde ring turns out to lead to affine Toda equations of SU( N ) type satisfied by the ground state metric.

A New Supersymmetric Index

Sergio Cecotti (SISSA, Trieste), Paul Fendley (Boston U.), Kenneth A. Intriligator, Cumrun Vafa (Harvard U.)

Apr 1992 - 48 pages

http://arxiv.org/abs/hep-th/9204102

Abstract: We show that {\rm Tr}(-1)~F F e~{-\beta H} is an index for N=2 supersymmetric theories in two dimensions, in the sense that it is independent of almost all deformations of the theory. This index is related to the geometry of the vacua (Berry's curvature) and satisfies an exact differential equation as a function of \beta. For integrable theories we can also compute the index thermodynamically, using the exact S-matrix. The equivalence of these two results implies a highly non-trivial equivalence of a set of coupled integral equations with these differential equations, among them Painleve III and the affine Toda equations.

Geometric Engineering of Quantum Field Theories

Sheldon H. Katz (Oklahoma State U.), Albrecht Klemm (Chicago U., EFI), Cumrun Vafa (Harvard U.)

Sep 1996 - 26 pages

http://arxiv.org/abs/hep-th/9609239

Abstract: Using the recent advances in our understanding of non-perturbative aspects of type II strings we show how non-trivial exact results for N=2 quantum field theories can be reduced to T-dualities of string theory. This is done by constructing a local geometric realization of quantum field theories together with a local application of mirror symmetry. This construction is not based on any duality conjecture and thus reduces non-trivial quantum field theory results to much better understood T-dualities of type II strings. Moreover it can be used in principle to construct in a systematic way the vacuum structure for arbitrary N=2 gauge theories with matter representations.

Geometric Origin of Montonen-Olive Duality

Cumrun Vafa (Harvard U.)

Jul 1997 - 8 pages

http://arxiv.org/abs/hep-th/9707131

Abstract: We show how N=4, D=4 duality of Montonen and Olive can be derived for all gauge groups using geometric engineering in the context of type II strings, where it reduces to T-duality. The derivation for the non-simply laced cases involves the use of some well known facts about orbifold conformal theories.

On Domain Walls of N=1 Supersymmetric Yang-Mills in Four-Dimensions

Bobby Samir Acharya (Rutgers U., Piscataway), Cumrun Vafa (Harvard U.)

Mar 2001 - 14 pages

https://arxiv.org/abs/hep-th/0103011

Abstract: We study the BPS domain walls of supersymmetric Yang-Mills for arbitrary gauge group. We describe the degeneracies of domain walls interpolating between arbitrary pairs of vacua. A recently proposed large N duality sheds light on various aspects of such domain walls. In particular, for the case of G = SU(N) the domain walls correspond to wrapped D-branes giving rise to a 2+1 dimensional U(k) gauge theory on the domain wall with a Chern-Simons term of level N. This leads to a counting of BPS degeneracies of domain walls in agreement with expected results.

A Large N Duality via a Geometric Transition

F. Cachazo (Harvard U.), Kenneth A. Intriligator (UC, San Diego), Cumrun Vafa (Harvard U.)

Mar 2001 - 46 pages

http://arxiv.org/abs/hep-th/0103067

Abstract: We propose a large N dual of 4d, N=1 supersymmetric, SU(N) Yang-Mills with adjoint field \Phi and arbitrary superpotential W(\Phi). The field theory is geometrically engineered via D-branes partially wrapped over certain cycles of a non-trivial Calabi-Yau geometry. The large N, or low-energy, dual arises from a geometric transition of the Calabi-Yau, where the branes have disappeared and have been replaced by suitable fluxes. This duality yields highly non-trivial exact results for the gauge theory. The predictions indeed agree with expected results in cases where it is possible to use standard techniques for analyzing the strongly coupled, supersymmetric gauge theories. Moreover, the proposed large N dual provides a simpler and more unified approach for obtaining exact results for this class of supersymmetric gauge theories.

A Perturbative Window into Nonperturbative Physics

Robbert Dijkgraaf (Amsterdam U. & Amsterdam U., Inst. Math.), Cumrun Vafa (Harvard U., Phys. Dept.)

Aug 2002 - 38 pages

http://arxiv.org/abs/hep-th/0208048

Abstract: We argue that for a large class of N=1 supersymmetric gauge theories the effective superpotential as a function of the glueball chiral superfield is exactly given by a summation of planar diagrams of the same gauge theory. This perturbative computation reduces to a matrix model whose action is the tree-level superpotential. For all models that can be embedded in string theory we give a proof of this result, and we sketch an argument how to derive this more generally directly in field theory. These results are obtained without assuming any conjectured dualities and can be used as a systematic method to compute instanton effects: the perturbative corrections up to n-th loop can be used to compute up to n-instanton corrections. These techniques allow us to see many non-perturbative effects, such as the Seiberg-Witten solutions of N=2 theories, the consequences of Montonen-Olive S-duality in N=1* and Seiberg-like dualities for N=1 theories from a completely perturbative planar point of view in the same gauge theory, without invoking a dual description.

tt^* Geometry in 3 and 4 Dimensions

Sergio Cecotti (SISSA, Trieste), Davide Gaiotto (Perimeter Inst. Theor. Phys.), Cumrun Vafa (Harvard U., Phys. Dept.)

Dec 3, 2013 - 112 pages

https://arxiv.org/abs/1312.1008

Abstract: We consider the vacuum geometry of supersymmetric theories with 4 supercharges, on a flat toroidal geometry. The 2 dimensional vacuum geometry is known to be captured by the tt^* geometry. In the case of 3 dimensions, the parameter space is (T^{2}\times {\mathbb R})^N and the vacuum geometry turns out to be a solution to a generalization of monopole equations in 3N dimensions where the relevant topological ring is that of line operators. We compute the generalization of the 2d cigar amplitudes, which lead to S^2\times S^1 or S^3 partition functions which are distinct from the supersymmetric partition functions on these spaces, but reduce to them in a certain limit. We show the sense in which these amplitudes generalize the structure of 3d Chern-Simons theories and 2d RCFT's. In the case of 4 dimensions the parameter space is of the form (T^3\times {\mathbb R})^M\times T^{3N}, and the vacuum geometry is a solution to a mixture of generalized monopole equations and generalized instanton equations (known as hyper-holomorphic connections). In this case the topological rings are associated to surface operators. We discuss the physical meaning of the generalized Nahm transforms which act on all of these geometries.

6d Conformal Matter

Michele Del Zotto (Harvard U., Phys. Dept.), Jonathan J. Heckman (Harvard U., Phys. Dept. & North Carolina U.), Alessandro Tomasiello (Milan Bicocca U.), Cumrun Vafa (Harvard U., Phys. Dept.)

Jul 23, 2014 - 61 pages

http://arxiv.org/abs/arXiv:1409.1603

Abstract: A single M5-brane probing G, an ADE-type singularity, leads to a system which has G × G global symmetry and can be viewed as “bifundamental” (G, G) matter. For the A_{N} series, this leads to the usual notion of bifundamental matter. For the other cases it corresponds to a strongly interacting (1, 0) superconformal system in six dimensions. Similarly, an ADE singularity intersecting the Hořava-Witten wall leads to a superconformal matter system with E_{8} × G global symmetry. Using the F-theory realization of these theories, we elucidate the Coulomb/tensor branch of (G, G′) conformal matter. This leads to the notion of fractionalization of an M5-brane on an ADE singularity as well as fractionalization of the intersection point of the ADE singularity with the Hořava-Witten wall. Partial Higgsing of these theories leads to new 6d SCFTs in the infrared, which we also characterize. This generalizes the class of (1, 0) theories which can be perturbatively realized by suspended branes in IIA string theory. By reducing on a circle, we arrive at novel duals for 5d affine quiver theories. Introducing many M5-branes leads to large N gravity duals.

Atomic Classification of 6D SCFTs

Jonathan J. Heckman (UNC, Chapel Hill), David R. Morrison (UC, Santa Barbara & UC, Santa Barbara, Dept. Math.), Tom Rudelius, Cumrun Vafa (Harvard U., Phys. Dept.)

Feb 18, 2015 - 63 pages

http://arxiv.org/abs/arXiv:1502.05405

Abstract: We use F-theory to classify possibly all six-dimensional superconformal field theories (SCFTs). This involves a two step process: We first classify all possible tensor branches allowed in F-theory (which correspond to allowed collections of contractible spheres) and then classify all possible configurations of seven-branes wrapped over them. We describe the first step in terms of 'atoms' joined into 'radicals' and 'molecules,' using an analogy from chemistry. The second step has an interpretation via quiver-type gauge theories constrained by anomaly cancellation. A very surprising outcome of our analysis is that all of these tensor branches have the structure of a linear chain of intersecting spheres with a small amount of possible decoration at the two ends. The resulting structure of these SCFTs takes the form of a generalized quiver consisting of ADE-type nodes joined by conformal matter. A collection of highly non-trivial examples involving E8 small instantons probing an ADE singularity is shown to have an F-theory realization. This yields a classification of homomorphisms from ADE subgroups of into E8 in purely geometric terms, matching results obtained in the mathematics literature from an intricate group theory analysis.

4d N=1 from 6d (1,0)

Shlomo S. Razamat (Technion), Cumrun Vafa (Harvard U., Phys. Dept.), Gabi Zafrir (Tokyo U., IPMU & Technion)

Oct 28, 2016 - 101 pages

http://arxiv.org/abs/arXiv:1610.09178

Abstract: We study the geometry of 4d N=1 SCFT's arising from compactification of 6d (1,0) SCFT's on a Riemann surface. We show that the conformal manifold of the resulting theory is characterized, in addition to moduli of complex structure of the Riemann surface, by the choice of a connection for a vector bundle on the surface arising from flavor symmetries in 6d. We exemplify this by considering the case of 4d N=1 SCFT's arising from M5 branes probing Z_k singularity compactified on a Riemann surface. In particular, we study in detail the four dimensional theories arising in the case of two M5 branes on Z_2 singularity. We compute the conformal anomalies and indices of such theories in 4d and find that they are consistent with expectations based on anomaly and the moduli structure derived from the 6 dimensional perspective.

E‐String Theory on Riemann Surfaces

Hee-Cheol Kim (Harvard U., Phys. Dept.), Shlomo S. Razamat (Technion), Cumrun Vafa (Harvard U.), Gabi Zafrir (Tokyo U., IPMU)

Sep 7, 2017 - 85 pages

http://arxiv.org/abs/arXiv:1709.02496

Abstract: We study compactifications of the 6d E-string theory, the theory of a small E8 instanton, to four dimensions. In particular we identify N=1 field theories in four dimensions corresponding to compactifications on arbitrary Riemann surfaces with punctures and with arbitrary non-abelian flat connections as well as fluxes for the abelian sub-groups of the E8 flavor symmetry. This sheds light on emergent symmetries in a number of 4d N=1 SCFTs (including the ‘E7 surprise’ theory) as well as leads to new predictions for a large number of 4-dimensional exceptional dualities and symmetries.

On Geometric Classification of 5d SCFTs

Patrick Jefferson (Harvard U., Phys. Dept.), Sheldon Katz (Illinois U., Urbana, Math. Dept.), Hee-Cheol Kim (Harvard U., Phys. Dept. & POSTECH, Pohang), Cumrun Vafa (Harvard U., Phys. Dept.)

Jan 11, 2018 - 87 pages

http://arxiv.org/abs/arXiv:1801.04036

Abstract: We formulate geometric conditions necessary for engineering 5d superconformal field theories (SCFTs) via M-theory compactification on a local Calabi-Yau 3-fold. Extending the classification of the rank 1 cases, which are realized geometrically as shrinking del Pezzo surfaces embedded in a 3-fold, we propose an exhaustive classification of local 3-folds engineering rank 2 SCFTs in 5d. This systematic classification confirms that all rank 2 SCFTs predicted using gauge theoretic arguments can be realized as consistent theories, with the exception of one family which is shown to be non-perturbatively inconsistent and thereby ruled out by geometric considerations. We find that all rank 2 SCFTs descend from 6d (1,0) SCFTs compactified on a circle possibly twisted with an automorphism together with holonomies for global symmetries around the Kaluza-Klein circle. These results support our conjecture that every 5d SCFT can be obtained from the circle compactification of some parent 6d (1,0) SCFT.

Microscopic Origin of the Bekenstein-Hawking Entropy

Andrew Strominger (UC, Santa Barbara), Cumrun Vafa (Harvard U.)

Jan 1996 - 10 pages

https://arxiv.org/abs/hep-th/9601029

Abstract: The Bekenstein-Hawking area-entropy relation S_{BH}=A/4 is derived for a class of five-dimensional extremal black holes in string theory by counting the degeneracy of BPS soliton bound states.

D-branes and Spinning Black Holes

J.C. Breckenridge, Robert C. Myers (McGill U.), A.W. Peet (Princeton U.), C. Vafa (Harvard U.)

Feb 1996 - 10 pages

http://arxiv.org/abs/hep-th/9602065

Abstract: We obtain a new class of spinning charged extremal black holes in five dimensions, considered both as classical configurations and in the Dirichlet(D)--brane representation. The degeneracy of states is computed from the D--brane side and the entropy agrees perfectly with that obtained from the black hole side.

Macroscopic and Microscopic Entropy of Near Extremal Spinning Black Holes

J.C. Breckenridge (McGill U.), D.A. Lowe (UC, Santa Barbara), Robert C. Myers (McGill U.), A.W. Peet (Princeton U.), A. Strominger (UC, Santa Barbara), C. Vafa (Harvard U.)

Mar 1996 - 7 pages

http://arxiv.org/abs/hep-th/9603078

Abstract: A seven parameter family of five-dimensional black hole solutions depending on mass, two angular momenta, three charges and the asymptotic value of a scalar field is constructed. The entropy is computed as a function of these parameters both from the Bekenstein-Hawking formula and from the degeneracies of the corresponding D-brane states in string theory. The expressions agree at and to leading order away from extremality.

Minimal Cycles, Black Holes and QFT's

Cumrun Vafa (Harvard U.)

May 1998 - 6 pages

http://arxiv.org/abs/hep-th/9805213

Abstract: We review some aspects of minimal cycles in string compactifications and their role in constructing new critical theories in six and lower dimensions as well as in accounting for black hole entropy. (Based on a talk presented at the Salam Memorial Meeting, the Abdus Salam International Center for Theoretical Physics, Fall 1997)

Black Holes and Calabi-Yau Threefolds

David R. Morrison (Duke U.), Cumrun Vafa (Harvard U.)

Nov 1997 - 11 pages

http://arxiv.org/abs/hep-th/9711067

Abstract: We compute the microscopic entropy of certain 4 and 5 dimensional extermal black holes which arise for compactification of M-theory and type IIA on Calabi-Yau 3-folds. The results agree with macroscopic predictions, including some subleading terms. The macroscopic entropy in the 5 dimensional case predicts a surprising growth in the cohomology of moduli space of holomorphic curves in Calabi-Yau threefolds which we verify in the case of elliptic threefolds.

M-Theory, Topological Strings and Spinning Black Holes

Sheldon H. Katz (Oklahoma State U.), Albrecht Klemm (Princeton, Inst. Advanced Study), Cumrun Vafa (Harvard U.)

Oct 1999 - 76 pages

http://arxiv.org/abs/hep-th/9910181

Abstract: We consider M-theory compactification on Calabi-Yau threefolds. The recently discovered connection between the BPS states of wrapped M2 branes and the topological string amplitudes on the threefold is used both as a tool to compute topological string amplitudes at higher genera as well as to unravel the degeneracies and quantum numbers of BPS states. Moduli spaces of k-fold symmetric products of the wrapped M2 brane play a crucial role. We also show that the topological string partition function is the Calabi-Yau version of the elliptic genus of the symmetric product of K3's and use the macroscopic entropy of spinning black holes in 5 dimensions to obtain new predictions for the asymptotic growth of the topological string amplitudes at high genera.

Black Hole Attractors and the Topological String

Hirosi Ooguri (Caltech), Andrew Strominger, Cumrun Vafa (Harvard U., Phys. Dept.)

May 2004 - 32 pages

http://arxiv.org/abs/hep-th/0405146

Abstract: A simple relationship of the form Z_BH = |Z_top|^2 is conjectured, where Z_BH is a supersymmetric partition function for a four-dimensional BPS black hole in a Calabi-Yau compactification of Type II superstring theory and Z_top is a second-quantized topological string partition function evaluated at the attractor point in moduli space associated to the black hole charges. Evidence for the conjecture in a perturbation expansion about large graviphoton charge is given. The microcanonical ensemble of BPS black holes can be viewed as the Wigner function associated to the wavefunction defined by the topological string partition function.

F-Theory, Spinning Black Holes and Multi-string Branches

Babak Haghighat (Harvard U., Phys. Dept. & Harvard U., Dept. Math.), Sameer Murthy (King's Coll. London, Dept. Math), Cumrun Vafa (Harvard U., Phys. Dept.), Stefan Vandoren (Utrecht U.)

Sep 1, 2015 - 58 pages

http://arxiv.org/abs/arXiv:1509.00455

Abstract: We study 5d supersymmetric black holes which descend from strings of generic \mathcal{N}=\left(1,\kern0.5em 0\right) supergravity in 6d. These strings have an F-theory realization in 6d as D3 branes wrapping smooth genus g curves in the base of elliptic 3-folds. They enjoy (0, 4) worldsheet supersymmetry with an extra SU(2)_{L} current algebra at level g realized on the left-movers. When the smooth curves degenerate they lead to multi-string branches and we find that the microscopic worldsheet theory flows in the IR to disconnected 2d CFTs having different central charges. The single string sector is the one with maximal central charge, which when wrapped on a circle, leads to a 5d spinning BPS black hole whose horizon volume agrees with the leading entropy prediction from the Cardy formula. However, we find new phenomena where this branch meets other branches of the CFT. These include multi-string configurations which have no bound states in 6 dimensions but are bound through KK momenta when wrapping a circle, as well as loci where the curves degenerate to spheres. These loci lead to black hole configurations which can have total angular momentum relative to a Taub-Nut center satisfying J^{2} > M^{3} and whose number of states, though exponentially large, grows much slower than those of the large spinning black hole.

Superstrings in the Early Universe

Robert H. Brandenberger (Brown U.), C. Vafa (Harvard U.)

Aug 1988 - 20 pages

Abstract: We investigate some aspects of thermodynamics and cosmology for superstrings. By a rather delicate computation using the microcanonical ensemble we show that the thermodynamic description of strings is sound (specific heat is positive at large energies) only for strings propagating in spaces where all the spatial directions are compact. Using this result and by considering a simple model, we show how strings resolve the initial singularity of the Big Bang. We also discuss a cosmological scenario which has the potential of explaining the space-time dimensionality.

Elements of String Cosmology

Arkady A. Tseytlin (Johns Hopkins U.), C. Vafa (Harvard U.)

Sep 1991 - 24 pages

http://arxiv.org/abs/hep-th/9109048

Abstract: Aspects of string cosmology for critical and non-critical strings are discussed emphasizing the necessity to account for the dilaton dynamics for a proper incorporation of ``large - small" duality. This drastically modifies the intuition one has with Einstein's gravity. For example winding modes, even though contribute to energy density, oppose expansion and if not annihilated will stop the expansion. Moreover we find that the radiation dominated era of the standard cosmology emerges quite naturally in string cosmology. Our analysis of non-critical string cosmology provides a reinterpretation of the (universal cover of the) recently studied two dimensional black hole solution as a conformal realization of cosmological solutions found previously by Mueller.

String Gas Cosmology and Structure Formation

Robert H. Brandenberger (McGill U.), Ali Nayeri (Harvard U., Phys. Dept.), Subodh P. Patil (McGill U.), Cumrun Vafa (Harvard U., Phys. Dept.)

Aug 2006 - 12 pages

http://arxiv.org/abs/hep-th/0608121

Abstract: It has recently been shown that a Hagedorn phase of string gas cosmology may provide a causal mechanism for generating a nearly scale-invariant spectrum of scalar metric fluctuations, without the need for an intervening period of de Sitter expansion. A distinctive signature of this structure formation scenario would be a slight blue tilt of the spectrum of gravitational waves. In this paper we give more details of the computations leading to these results.

Cosmology of F-theory GUTs

Jonathan J. Heckman, Alireza Tavanfar, Cumrun Vafa (Harvard U.)

Dec 2008 - 92 pages

http://arxiv.org/abs/arXiv:0812.3155

Abstract: In this paper we study the interplay between the recently proposed F-theory GUTs and cosmology. Despite the fact that the parameter range for F-theory GUT models is very narrow, we find that F-theory GUTs beautifully satisfy most cosmological constraints without any further restrictions. The viability of the scenario hinges on the interplay between various components of the axion supermultiplet, which in F-theory GUTs is also responsible for breaking supersymmetry. In these models, the gravitino is the LSP and develops a mass by eating the axino mode. The radial component of the axion supermultiplet known as the saxion typically begins to oscillate in the early Universe, eventually coming to dominate the energy density. Its decay reheats the Universe to a temperature of ~ 1 GeV, igniting BBN and diluting all thermal relics such as the gravitino by a factor of ~ 10^(-4) - 10^(-5) such that gravitinos contribute a sizable component of the dark matter. In certain cases, non-thermally produced relics such as the axion, or gravitinos generated from the decay of the saxion can also contribute to the abundance of dark matter. Remarkably enough, this cosmological scenario turns out to be independent of the initial reheating temperature of the Universe. This is due to the fact that the initial oscillation temperature of the saxion coincides with the freeze out temperature for gravitinos in F-theory GUTs. We also find that saxion dilution is compatible with generating the desired baryon asymmetry from standard leptogenesis. Finally, the gravitino mass range in F-theory GUTs is 10-100 MeV, which interestingly coincides with the window of values required for the decay of the NLSP to solve the problem of Li(7) over-production.

On the Cosmological Implications of the String Swampland

Prateek Agrawal, Georges Obied (Harvard U., Phys. Dept.), Paul J. Steinhardt (Princeton U. & Princeton U., CTP & New York U., CCPP), Cumrun Vafa (Harvard U., Phys. Dept.)

Jun 25, 2018 - 6 pages

http://arxiv.org/abs/arXiv:1806.09718

Abstract: We study constraints imposed by two proposed string Swampland criteria on cosmology. These criteria involve an upper bound on the range traversed by scalar fields as well as a lower bound on |∇ϕV|/V when V>0 . We find that inflationary models are generically in tension with these two criteria. Applying these same criteria to dark energy in the present epoch, we find that specific quintessence models can satisfy these bounds and, at the same time, satisfy current observational constraints. Assuming the two Swampland criteria are valid, we argue that the universe will undergo a phase transition within a few Hubble times. These criteria sharpen the motivation for future measurements of the tensor-to-scalar ratio r and the dark energy equation of state w , and for tests of the equivalence principle for dark matter.

Topological Gravity as the Early Phase of Our Universe

Prateek Agrawal, Sergei Gukov, Georges Obied, Cumrun Vafa

Oct 20, 2020 - 27 pages

https://arxiv.org/abs/2009.10077

Abstract: Motivated by string dualities we propose topological gravity as the early phase of our universe. The topological nature of this phase naturally leads to the explanation of many of the puzzles of early universe cosmology. A concrete realization of this scenario using Witten's four dimensional topological gravity is considered. This model leads to the power spectrum of CMB fluctuations which is controlled by the conformal anomaly coefficients a,c. In particular the strength of the fluctuation is controlled by 1/a and its tilt by cg2 where g is the coupling constant of topological gravity. The positivity of c, a consequence of unitarity, leads automatically to an IR tilt for the power spectrum. In contrast with standard inflationary models, this scenario predicts (1) non-Gaussianities for four- and higher-point correlators and the absence of tensor modes in the CMB fluctuations.

Dark Dimension Gravitons as Dark Matter

Eduardo Gonzalo, Miguel Montero, Georges Obied, Cumrun Vafa

Sep 25, 2022- 18 pages

https://arxiv.org/abs/2209.09249

Abstract: We consider cosmological aspects of the Dark Dimension (a mesoscopic dimension of micron scale), which has recently been proposed as the unique corner of the quantum gravity landscape consistent with both the Swampland criteria and observations. In particular we show how this leads, by the universal coupling of the Standard Model sector to bulk gravitons, to massive spin 2 KK excitations of the graviton in the dark dimension (the "dark gravitons") as an unavoidable dark matter candidate. Assuming a lifetime for the current de Sitter phase of our universe of order Hubble, which follows from both the dS Swampland Conjecture and TCC, we show that generic features of the dark dimension cosmology can naturally lead to the correct dark matter density and a resolution of the cosmological coincidence problem, where the matter/radiation equality temperature (T∼ 1 eV) coincides with the temperature where the dark energy begins to dominate. Thus one does not need to appeal to Weinberg's anthropic argument to explain this coincidence. The dark gravitons are produced at T∼ 4 GeV, and their composition changes as they mainly decay to lighter gravitons, without losing much total mass density. The mass of dark gravitons is mDM∼1−100 keV today.

Swamplandish Unification of the Dark Sector

Cumrun Vafa (Harvard U., Phys. Dept.)

Feb 1, 2024 - 13 pages

https://inspirehep.net/literature/2755023

Abstract: We provide a short overview of recent progress made in our understanding of the dark sector based on the Swampland program which in turn is rooted in lessons from string theory. We explain how the existence of one extra mesoscopic dimension (the "dark dimension") in the micron range emerges and how this can lead to a unification of the dark energy and dark matter. In particular the smallness of the dark energy leads to the prediction of the existence of a tower of weakly interacting light particles which can naturally play the role of dark matter. Moreover this unifies dark matter with gravity as dark matter ends up being excitations of graviton in the dark dimension. We also explain how in combination with other Swampland principles one finds an explanation of the "why now" and the "cosmological coincidence" problems. This model is consistent with the cosmological bounds as well as the Newton's inverse square law, but makes predictions which differ from ΛCDM. This review is intended for a broad audience of high energy theorists and cosmologists without prior knowledge of string theory and it explains the motivations and predictions of this program in a non-technical form.

Fractional Quantum Hall Effect and M-Theory

Cumrun Vafa (Harvard U., Phys. Dept.)

Nov 10, 2015 - 29 pages

http://arxiv.org/abs/arXiv:1511.03372

Abstract: We propose a unifying model for FQHE which on the one hand connects it to recent developments in string theory and on the other hand leads to new predictions for the principal series of experimentally observed FQH systems with filling fraction \nu ={n\over 2n\pm1} as well as those with \nu ={m\over m+2}. Our model relates these series to minimal unitary models of the Virasoro and superVirasoro algebra and is based on SL(2, {\bf C}) Chern-Simons theory in Euclidean space or SL(2,{\bf R})\times SL(2,{\bf R}) Chern-Simons theory in Minkowski space. This theory, which has also been proposed as a soluble model for 2+1 dimensional quantum gravity, and its N=1 supersymmetric cousin, provide effective descriptions of FQHE. The principal series corresponds to quantized levels for the two SL(2,{\bf R})'s such that the diagonal SL(2,{\bf R}) has level 1. The model predicts, contrary to standard lore, that for principal series of FQH systems the quasiholes possess non-abelian statistics. For the multi-layer case we propose that complex ADE Chern-Simons theories provide effective descriptions, where the rank of the ADE is mapped to the number of layers. Six dimensional (2,0) ADE theories on the Riemann surface \Sigma provides a realization of FQH systems in M-theory. Moreover we propose that the q-deformed version of Chern-Simons theories are related to the anisotropic limit of FQH systems which splits the zeroes of the Laughlin wave function. Extensions of the model to 3+1 dimensions, which realize topological insulators with non-abelian topologically twisted Yang-Mills theory is pointed out.

Chern-Simons Theory and Wilson Loops in the Brillouin Zone

Biao Lian (Stanford U., Phys. Dept.), Cumrun Vafa (Harvard U., Phys. Dept.), Farzan Vafa (UC, Santa Barbara), Shou-Cheng Zhang (Stanford U., Phys. Dept.)

Oct 18, 2016 - 13 pages

http://arxiv.org/abs/arXiv:1610.05810

Abstract: Berry connection is conventionally defined as a static gauge field in the Brillouin zone. Here we show that for three-dimensional (3D) time-reversal invariant superconductors, a generalized Berry gauge field behaves as a fluctuating field of a Chern-Simons gauge theory. The gapless nodal lines in the momentum space play the role of Wilson loop observables, while their linking and knot invariants modify the gravitational theta angle. This angle induces a topological gravitomagnetoelectric effect where a temperature gradient induces a rotational energy flow. We also show how topological strings may be realized in the six-dimensional phase space, where the physical space defects play the role of topological D-branes.

Topological Chern-Simons/Matter Theories

Mina Aganagic (UC, Berkeley), Kevin Costello (Perimeter Inst. Theor. Phys.), Jacob McNamara, Cumrun Vafa (Harvard U., Phys. Dept.)

Jun 29, 2017 - 51 pages

http://arxiv.org/abs/arXiv:1706.09977

Abstract: We propose a new partially topological theory in three dimensions which couples Chern-Simons theory to matter. The 3-manifolds needed for this construction admit transverse holomorphic foliation (THF). The theory depends only on the choice of such a structure, but not on a choice of metric and in this sense, it is topological. We argue that this theory arises in topological A-model string theory on Lagrangian 3-branes in the presence of additional parallel coisotropic 5-branes. The theory obtained in this way is equivalent to an N=2 supersymmetric Chern-Simons matter theory on the same 3-manifold, which also only depends on the THF structure. The theory is a realization of a topological theory of class H, which allows splitting of a temporal direction from spatial directions. We briefly discuss potential condensed matter applications.

Topological Landau-Ginzburg Models

Cumrun Vafa (Harvard U.)

Nov 1990 - 14 pages

Abstract: We derive a general expression for correlation functions of topological Landau-Ginzburg models on an arbitrary genus Riemann surface. The expressions we find for the correlation functions suggest that for ĉ>1 the perturbation of the theory by chiral primary fields of dimensions bigger than one is rather singular, though perturbation by relevant chiral primary fields seems sensible regardless of the value of ĉ.

Topological Reduction of 4d SYM to 2d Sigma Models

M. Bershadsky, A. Johansen, V. Sadov, C. Vafa (Harvard U.)

Jan 1995 - 26 pages

http://arxiv.org/abs/hep-th/9501096

Abstract: By considering a (partial) topological twisting of supersymmetric Yang-Mills compactified on a 2d space with `t Hooft magnetic flux turned on we obtain a supersymmetric \sigma-model in 2 dimensions. For N=2 SYM this maps Donaldson observables on products of two Riemann surfaces to quantum cohomology ring of moduli space of flat connections on a Riemann surface. For N=4 SYM it maps S-duality to T-duality for \sigma-models on moduli space of solutions to Hitchin equations.

Topological Mirrors and Quantum Rings

Cumrun Vafa (Harvard U.)

Nov 1991 - 21 pages

http://arxiv.org/abs/hep-th/9111017

Abstract: Aspects of duality and mirror symmetry in string theory are discussed. We emphasize, through examples, the importance of loop spaces for a deeper understanding of the geometrical origin of dualities in string theory. Moreover we show that mirror symmetry can be reformulated in very simple terms as the statement of equivalence of two classes of topological theories: Topological sigma models and topological Landau-Ginzburg models. Some suggestions are made for generalization of the notion of mirror symmetry.

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

M. Bershadsky, S. Cecotti, H. Ooguri, C. Vafa (Harvard U.)

Sep 1993 - 155 pages

http://arxiv.org/abs/hep-th/9309140

Abstract: 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.

Geometric Physics

Cumrun Vafa (Harvard U.)

Aug 1998 - 25 pages

http://arxiv.org/abs/hep-th/9810149

Abstract: Talk given at ICM '98, Berlin, reviewing some of the recent developments in understanding of string theory for a mathematical audience (to appear in Documenta Mathematica).

M-Theory and Topological Strings--I

Rajesh Gopakumar, Cumrun Vafa (Harvard U.)

Sep 1998 - 14 pages

http://arxiv.org/abs/hep-th/9809187

Abstract: The R^2 F^{2g-2} terms of Type IIA strings on Calabi-Yau 3-folds, which are given by the corresponding topological string amplitudes (a worldsheet instanton sum for all genera), are shown to have a simple M-theory interpretation. In particular, a Schwinger one-loop computation in M-theory with wrapped M2 branes and Kaluza-Klein modes going around the loop reproduces the all genus string contributions from constant maps and worldsheet instanton corrections. In the simplest case of an isolated M2 brane with the topology of the sphere, we obtain the contributions of small worldsheet instantons (sphere ``bubblings'') which extends the results known or conjectured for low genera. Surprisingly, the 't Hooft expansion of large N Chern-Simons theory on S^3 can also be used in a novel way to compute these gravitational terms at least in special cases.

On the Gauge Theory / Geometry Correspondence

Rajesh Gopakumar, Cumrun Vafa (Harvard U.)

Nov 1998 - 24 pages

http://arxiv.org/abs/hep-th/9811131

Abstract: The 't Hooft expansion of SU(N) Chern-Simons theory on S^3 is proposed to be exactly dual to the topological closed string theory on the S^2 blow up of the conifold geometry. The B-field on the S^2 has magnitude Ng_s=\lambda, the 't Hooft coupling. We are able to make a number of checks, such as finding exact agreement at the level of the partition function computed on {\it both} sides for arbitrary \lambda and to all orders in 1/N. Moreover, it seems possible to derive this correspondence from a linear sigma model description of the conifold. We propose a picture whereby a perturbative D-brane description, in terms of holes in the closed string worldsheet, arises automatically from the coexistence of two phases in the underlying U(1) gauge theory. This approach holds promise for a derivation of the AdS/CFT correspondence.

M-Theory and Topological Strings--II

Rajesh Gopakumar, Cumrun Vafa (Harvard U.)

Dec 1998 - 20 pages

http://arxiv.org/abs/hep-th/9812127

Abstract: It is shown how the topological string amplitudes encode the BPS structure of wrapped M2 branes in M-theory compactification on Calabi-Yau threefolds. This in turn is related to a twisted supersymmetric index in 5 dimensions which receives contribution only from BPS states. The spin dependence of BPS states in 5 dimensions is captured by the string coupling constant dependence of topological string amplitudes.

M-Theory, Topological Strings and Spinning Black Holes

Sheldon H. Katz (Oklahoma State U.), Albrecht Klemm (Princeton, Inst. Advanced Study), Cumrun Vafa (Harvard U.)

Oct 1999 - 76 pages

http://arxiv.org/abs/hep-th/9910181

Abstract: We consider M-theory compactification on Calabi-Yau threefolds. The recently discovered connection between the BPS states of wrapped M2 branes and the topological string amplitudes on the threefold is used both as a tool to compute topological string amplitudes at higher genera as well as to unravel the degeneracies and quantum numbers of BPS states. Moduli spaces of k-fold symmetric products of the wrapped M2 brane play a crucial role. We also show that the topological string partition function is the Calabi-Yau version of the elliptic genus of the symmetric product of K3's and use the macroscopic entropy of spinning black holes in 5 dimensions to obtain new predictions for the asymptotic growth of the topological string amplitudes at high genera.

Knot Invariants and Topological Strings

Hirosi Ooguri (Caltech), Cumrun Vafa (Harvard U.)

Dec 1999 - 24 pages

http://arxiv.org/abs/hep-th/9912123

Abstract: We find further evidence for the conjecture relating large N Chern-Simons theory on S^3 with topological string on the resolved conifold geometry by showing that the Wilson loop observable of a simple knot on S^3 (for any representation) agrees to all orders in N with the corresponding quantity on the topological string side. For a general knot, we find a reformulation of the knot invariant in terms of new integral invariants, which capture the spectrum (and spin) of M2 branes ending on M5 branes embedded in the resolved conifold geometry. We also find an intriguing link between knot invariants and superpotential terms generated by worldsheet instantons in N=1 supersymmetric theories in 4 dimensions.

Mirror Symmetry

Kentaro Hori, Cumrun Vafa (Harvard U.)

Feb 2000 - 90 pages

http://arxiv.org/abs/hep-th/0002222

Abstract: We prove mirror symmetry for supersymmetric sigma models on Kahler manifolds in 1+1 dimensions. The proof involves establishing the equivalence of the gauged linear sigma model, embedded in a theory with an enlarged gauge symmetry, with a Landau-Ginzburg theory of Toda type. Standard R -> 1/R duality and dynamical generation of superpotential by vortices are crucial in the derivation. This provides not only a proof of mirror symmetry in the case of (local and global) Calabi-Yau manifolds, but also for sigma models on manifolds with positive first Chern class, including deformations of the action by holomorphic isometries.

Mirror Symmetry

K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, E. Zaslow

Jun 19, 2006 - 929 pages

http://www.claymath.org/library/monographs/cmim01.pdf

Abstract: This book is a product of a month-long school on mirror symmetry that CMI held at Pine Manor College in Brookline, Massachusetts in the Spring of 2000. The aim of the book is to provide a pedagogical introduction to the field of mirror symmetry from both a mathematical and physical perspective. After covering the relevant background material, the main part of the monograph is devoted to the proof of mirror symmetry from various viewpoints. More advanced topics are also discussed. In particular, topological strings at higher genera and the notion of holomorphic anomaly.

The Topological Vertex

Mina Aganagic (Harvard U.), Albrecht Klemm (Humboldt U., Berlin), Marcos Marino (CERN), Cumrun Vafa (Harvard U. & Caltech)

May 2003 - 70 pages

http://arxiv.org/abs/hep-th/0305132

Abstract: We construct a cubic field theory which provides all genus amplitudes of the topological A-model for all non-compact Calabi-Yau toric threefolds. The topology of a given Feynman diagram encodes the topology of a fixed Calabi-Yau, with Schwinger parameters playing the role of Kahler classes of Calabi-Yau. We interpret this result as an operatorial computation of the amplitudes in the B-model mirror which is the Kodaira-Spencer quantum theory. The only degree of freedom of this theory is an unconventional chiral scalar on a Riemann surface. In this setup we identify the B-branes on the mirror Riemann surface as fermions related to the chiral boson by bosonization.

Quantum Calabi-Yau and Classical Crystals

K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, E. Zaslow

Sep 2003 - 26 pages

http://arxiv.org/abs/hep-th/0309208

Abstract: We propose a new duality involving topological strings in the limit of large string coupling constant. The dual is described in terms of a classical statistical mechanical model of crystal melting, where the temperature is inverse of the string coupling constant. The crystal is a discretization of the toric base of the Calabi-Yau with lattice length g_s. As a strong evidence for this duality we recover the topological vertex in terms of the statistical mechanical probability distribution for crystal melting. We also propose a more general duality involving the dimer problem on periodic lattices and topological A-model string on arbitrary local toric threefolds. The (p,q) 5-brane web, dual to Calabi-Yau, gets identified with the transition regions of rigid dimer configurations.

Topological Strings and their Physical Applications

Andrew Neitzke, Cumrun Vafa (Harvard U., Phys. Dept.)

Oct 2004 - 77 pages

http://arxiv.org/abs/hep-th/0410178

Abstract: We give an introductory review of topological strings and their application to various aspects of superstrings and supersymmetric gauge theories. This review includes developing the necessary mathematical background for topological strings, such as the notions of Calabi-Yau manifold and toric geometry, as well as physical methods developed for solving them, such as mirror symmetry, large N dualities, the topological vertex and quantum foam. In addition, we discuss applications of topological strings to N=1,2 supersymmetric gauge theories in 4 dimensions as well as to BPS black hole entropy in 4 and 5 dimensions. (These are notes from lectures given by the second author at the 2004 Simons Workshop in Mathematics and Physics.)

Khovanov-Rozansky Homology and Topological Strings

Sergei Gukov (Harvard U., Phys. Dept.), Albert S. Schwarz (UC, Davis, Dept. Math.), Cumrun Vafa (Harvard U., Phys. Dept.)

Dec 2004 - 25 pages

http://arxiv.org/abs/hep-th/0412243

Abstract: 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 sl(N) knot homology groups for all values of N. We verify that our predictions pass some non-trivial checks.

The Refined Topological Vertex

Amer Iqbal, Can Kozcaz (Washington U., Seattle), Cumrun Vafa (Harvard U., Phys. Dept. & MIT, LNS)

Jan 2007 - 70 pages

http://arxiv.org/abs/hep-th/0701156

Abstract: We define a refined topological vertex which depends in addition on a parameter, which physically corresponds to extending the self-dual graviphoton field strength to a more general configuration. Using this refined topological vertex we compute, using geometric engineering, a two-parameter (equivariant) instanton expansion of gauge theories which reproduce the results of Nekrasov. The refined vertex is also expected to be related to Khovanov knot invariants.

Link Homologies and the Refined Topological Vertex

Sergei Gukov (UC, Santa Barbara), Amer Iqbal, Can Kozcaz (Washington U., Seattle), Cumrun Vafa (MIT, LNS & Jefferson Lab)

2010 - 38 pages

http://arxiv.org/abs/arXiv:0705.1368

Abstract: We establish a direct map between refined topological vertex and sl(N) homological invariants of the of Hopf link, which include Khovanov-Rozansky homology as a special case. This relation provides an exact answer for homological invariants of the of Hopf link, whose components are colored by arbitrary representations of sl(N). At present, the mathematical formulation of such homological invariants is available only for the fundamental representation (the Khovanov-Rozansky theory) and the relation with the refined topological vertex should be useful for categorifying quantum group invariants associated with other representations (R_1, R_2). Our result is a first direct verification of a series of conjectures which identifies link homologies with the Hilbert space of BPS states in the presence of branes, where the physical interpretation of gradings is in terms of charges of the branes ending on Lagrangian branes.

BPS Degeneracies and Superconformal Index in Diverse Dimensions

Amer Iqbal (Lahore U. Mgt. Sci.), Cumrun Vafa (Harvard U., Phys. Dept.)

Oct 2012 - 27 pages

http://arxiv.org/abs/arXiv:1210.3605

Abstract: We present a unifying theme relating BPS partition functions and superconformal indices. In the case with complex supersymmetry central charges (as in N=2 in d=4 and N=(2,2) in d=2), the known results can be reinterpreted as the statement that the BPS partition functions can be used to compute a specialization of the superconformal indices. We argue that in the case with real central charge in the supersymmetry algebra, as in N=1 in d=5 (or the N=2 in d=3), the BPS degeneracy captures the full superconformal index. Furthermore, we argue that refined topological strings, which capture five-dimensional (5d) BPS degeneracies of M theory on Calabi-Yau 3-folds, can be used to compute a 5d supersymmetric index including in the sectors with three-dimensional defects for a large class of 5d superconformal theories. Moreover, we provide evidence that distinct Calabi–Yau singularities which are expected to lead to the same Superconformal field theory yield the same index.

Tangles, Generalized Reidemeister Moves, and Three-Dimensional Mirror Symmetry

Clay Cordova (Harvard U.), Sam Espahbodi, Babak Haghighat, Ashwin Rastogi, Cumrun Vafa (Harvard U., Phys. Dept.)

Nov 2012 - 70 pages

http://arxiv.org/abs/arXiv:1211.3730

Abstract: Three-dimensional N=2 superconformal field theories are constructed by compactifying M5-branes on three-manifolds. In the infrared the branes recombine, and the physics is captured by a single M5-brane on a branched cover of the original ultraviolet geometry. The branch locus is a tangle, a one-dimensional knotted submanifold of the ultraviolet geometry. A choice of branch sheet for this cover yields a Lagrangian for the theory, and varying the branch sheet provides dual descriptions. Massless matter arises from vanishing size M2-branes and appears as singularities of the tangle where branch lines collide. Massive deformations of the field theory correspond to resolutions of singularities resulting in distinct smooth manifolds connected by geometric transitions. A generalization of Reidemeister moves for singular tangles captures mirror symmetries of the underlying theory yielding a geometric framework where dualities are manifest.

Topological Strings, D-Model, and Knot Contact Homology

Mina Aganagic (UC, Berkeley & UC, Berkeley, Math. Dept.), Tobias Ekholm (Uppsala U. & Mittag-Leffler Inst.), Lenhard Ng (Duke U.), Cumrun Vafa (Harvard U. & Jefferson Lab)

Apr 21, 2013 - 130 pages

http://arxiv.org/abs/arXiv:1304.5778

Abstract: We study the connection between topological strings and contact homology recently proposed in the context of knot invariants. In particular, we establish the proposed relation between the Gromov-Witten disk amplitudes of a Lagrangian associated to a knot and augmentations of its contact homology algebra. This also implies the equality between the Q-deformed A-polynomial and the augmentation polynomial of knot contact homology (in the irreducible case). We also generalize this relation to the case of links and to higher rank representations for knots. The generalization involves a study of the quantum moduli space of special Lagrangian branes with higher Betti numbers probing the Calabi-Yau. This leads to an extension of SYZ, and a new notion of mirror symmetry, involving higher dimensional mirrors. The mirror theory is a topological string, related to D-modules, which we call the 'D-model'. In the present setting, the mirror manifold is the augmentation variety of the link. Connecting further to contact geometry, we study intersection properties of branches of the augmentation variety guided by the relation to D-modules. This study leads us to propose concrete geometric constructions of Lagrangian fillings for links. We also relate the augmentation variety with the large N limit of the colored HOMFLY, which we conjecture to be related to a Q-deformation of the extension of A-polynomials associated with the link complement.

Fivebranes and 3-Manifold Homology

Sergei Gukov (Caltech), Pavel Putrov (Princeton, Inst. Advanced Study), Cumrun Vafa (Harvard U., Phys. Dept.)

Feb 17, 2016 - 86 pages

http://arxiv.org/abs/arXiv:1602.05302

Abstract: Motivated by physical constructions of homological knot invariants, we study their analogs for closed 3-manifolds. We show that fivebrane compactifications provide a universal description of various old and new homological invariants of 3-manifolds. In terms of 3d/3d correspondence, such invariants are given by the Q-cohomology of the Hilbert space of partially topologically twisted 3d \mathcal{N}=2 theory T[M_{3}] on a Riemann surface with defects. We demonstrate this by concrete and explicit calculations in the case of monopole/Heegaard Floer homology and a 3-manifold analog of Khovanov-Rozansky link homology. The latter gives a categorification of Chern-Simons partition function. Some of the new key elements include the explicit form of the S-transform and a novel connection between categorification and a previously mysterious role of Eichler integrals in Chern-Simons theory.

Quadrality for Supersymmetric Matrix Models

Sebastian Franco (City Coll., N.Y. & CUNY, Graduate School - U. Ctr.), Sangmin Lee (Seoul Natl. U.), Rak-Kyeong Seong (Uppsala U.), Cumrun Vafa (Harvard U., Phys. Dept.)

Dec 20, 2016 - 47 pages

http://arxiv.org/abs/arXiv:1612.06859

Abstract: We introduce a new duality for \mathcal{N} = 1 supersymmetric gauged matrix models. This 0d duality is an order 4 symmetry, namely an equivalence between four different theories, hence we call it Quadrality. Our proposal is motivated by mirror symmetry, but is not restricted to theories with a D-brane realization and holds for general \mathcal{N} = 1 matrix models. We present various checks of the proposal, including the matching of: global symmetries, anomalies, deformations and the chiral ring. We also consider quivers and the corresponding quadrality networks. Finally, we initiate the study of matrix models that arise on the worldvolume of D(-1)-branes probing toric Calabi-Yau 5-folds.

BPS Spectra and 3-Manifold Invariants

Sergei Gukov (Caltech), Du Pei (Caltech & Aarhus U., Math. Sci.), Pavel Putrov (Princeton, Inst. Advanced Study), Cumrun Vafa (Harvard U., Phys. Dept.)

Jan 23, 2017 - 72 pages

http://arxiv.org/abs/arXiv:1701.06567

Abstract: We provide a physical definition of new homological invariants \mathcal{H}_a (M_3) of 3-manifolds (possibly, with knots) labeled by abelian flat connections. The physical system in question involves a 6d fivebrane theory on M_3 times a 2-disk, D^2, whose Hilbert space of BPS states plays the role of a basic building block in categorification of various partition functions of 3d \mathcal{N}=2 theory T[M_3]: D^2\times S^1 half-index, S^2\times S^1 superconformal index, and S^2\times S^1 topologically twisted index. The first partition function is labeled by a choice of boundary condition and provides a refinement of Chern-Simons (WRT) invariant. A linear combination of them in the unrefined limit gives the analytically continued WRT invariant of M_3. The last two can be factorized into the product of half-indices. We show how this works explicitly for many examples, including Lens spaces, circle fibrations over Riemann surfaces, and plumbed 3-manifolds.

Topological Chern-Simons/Matter Theories

Mina Aganagic (UC, Berkeley), Kevin Costello (Perimeter Inst. Theor. Phys.), Jacob McNamara, Cumrun Vafa (Harvard U., Phys. Dept.)

Jun 29, 2017 - 51 pages

http://arxiv.org/abs/arXiv:1706.09977

Abstract: We propose a new partially topological theory in three dimensions which couples Chern-Simons theory to matter. The 3-manifolds needed for this construction admit transverse holomorphic foliation (THF). The theory depends only on the choice of such a structure, but not on a choice of metric and in this sense, it is topological. We argue that this theory arises in topological A-model string theory on Lagrangian 3-branes in the presence of additional parallel coisotropic 5-branes. The theory obtained in this way is equivalent to an N=2 supersymmetric Chern-Simons matter theory on the same 3-manifold, which also only depends on the THF structure. The theory is a realization of a topological theory of class H, which allows splitting of a temporal direction from spatial directions. We briefly discuss potential condensed matter applications.

4-Manifolds and Topological Modular Forms

Sergei Gukov, Du Pei, Pavel Putrov, Cumrun Vafa

Nov 19, 2018 - 74 pages

http://arxiv.org/abs/arXiv:1811.07884

Abstract: We build a connection between topology of smooth 4-manifolds and the theory of topological modular forms by considering topologically twisted compactification of 6d (1,0) theories on 4-manifolds with flavor symmetry backgrounds. The effective 2d theory has (0,1) supersymmetry and, possibly, a residual flavor symmetry. The equivariant topological Witten genus of this 2d theory then produces a new invariant of the 4-manifold equipped with a principle bundle, valued in the ring of equivariant weakly holomorphic (topological) modular forms. We describe basic properties of this map and present a few simple examples. As a byproduct, we obtain some new results on 't Hooft anomalies of 6d (1,0) theories and a better understanding of the relation between 2d (0,1) theories and TMF spectra.

On Upper Bounds in Dimension Gaps of CFT's

Tristan C. Collins, Daniel Jafferis, Cumrun Vafa, Kai Xu, Shing-Tung Yau

Jan 10, 2022- 14 pages

https://arxiv.org/abs/2201.03660

Abstract: We consider CFT's arising from branes probing singularities of internal manifolds. We focus on holographic models with internal space including arbtirary Sasaki-Einstein manifolds coming from CY as well as arbitrary sphere quotients. In all these cases we show that there is a universal upper bound (depending only on the spacetime dimension) for the conformal dimension of the first non-trivial spin 2 operator in the dual CFT and a minimal diameter (in AdS units) for the internal space of the holographic dual and conjecture it holds for all CFT's.