C. Vafa and E. Witten, Nucl. Phys. B431 (1994) 3; hep-th/9408074.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....states. If however, the path integral is performed on a non trivial compact or noncompact manifold with supersymmetry preserving boundary conditions, then the 16 Witten index depends non trivially both on the manifold and the coupling constant . The Witten index for N=4 Yang Mills was computed [43] on K3 and on ALE manifolds and gave a result that was covariant under SL(2,ZZ) duality. ffl In string theory, the M O duality of N=4 super Yang Mills is equivalent to T duality (a perturbative duality of string theory that is well understood) via a string string duality that has had its own ....

C. Vafa and E. Witten, Nucl. Phys. B431 (1994) 3; hep-th/9408074.

.... string frame metric is ds 2 10 = fF Gamma1 ) 1=2 [F f Gamma1 (dudv Kdu 2 ) f Gamma1 dy n dy n dx m dx m ] 8) The number of associated excited BPS states and thus the statistical entropy can then be evaluated by representing this R R background in terms of D branes [11 13]. There are various other U dual representations in terms of configurations of intersecting NS NS (solitonic) or R R p branes in D = 10. These can be also described as anisotropic 6 branes (in D = 4 black hole case) or 5 branes (in D = 5 black hole case) with all internal coordinates being ....

A. Strominger and C. Vafa, HUTP-96-A002, hep-th/9601029.

....in [30, 18] it was shown in detail how it works when the gauge group is a product of U(1) s. In practice, however, it is difficult to apply this idea directly in non abelian cases. This is because, as we will see below, we would need to find non abelian generalization of the conifold transition [40], 41] It turns out that the task is significantly simplified if we use mirror symmetry for Calabi Yau threefolds. Let us describe our strategy to analyze the non abelian case by first reviewing the abelian case. 2.1 Duality in the Abelian Case As is well known, D branes wrapped around cycles of ....

....2.1 Duality in the Abelian Case As is well known, D branes wrapped around cycles of Calabi Yau give rise to solitons in this geometry. In particular if one considers type IIB with an S 3 inside a Calabi Yau threefold W , by wrapping a D3 brane around S 3 we obtain a charged hypermultiplet [40] (charged under the U(1) obtained by decomposition of the 4 form RR gauge potential as the volume form on 1 In four dimensions the vector multiplets do not receive any quantum string corrections whereas the hypermultiplet moduli do. However if we go down to three dimensions on a further circle ....

A. Strominger, Nucl. Phys. B451 (1995) 96, hep-th/9504090.

....Grav. Talk presented by B. de Wit at Strings 99, Potsdam, July 19 24 1999 Area law corrections from state counting and supergravity 2 1. Introduction A microscopic derivation of the entropy of certain extremal black holes has recently become available in the context of string theory [1] [6]. For four dimensional extremal black holes in the limit of large electric magnetic charges Q, the microscopic entropy is generically of the form S micro q Q 4 : 1) This result agrees with the one obtained from macroscopic calculations based on the corresponding effective field theories. ....

....by a (0; 4) two dimensional conformal field theory. Hence one obtains entropy formulae similar to (3) We will return to them later. Inspection of (3) shows that, for large charges p A , there are subleading corrections (proportional to c 2A ) to the microscopic entropy. It was argued in [5, 6] that these deviations in the entropy formula should, at the macroscopic level, arise from terms Area law corrections from state counting and supergravity 3 in the effective action proportional to the square of the Weyl tensor, with coefficients linearly related to the second Chern class of the ....

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Vafa C 1998 Adv. Theor. Math. Phys. 2 207, hep-th/9711067

....Quantum Grav. Talk presented by B. de Wit at Strings 99, Potsdam, July 19 24 1999 Area law corrections from state counting and supergravity 2 1. Introduction A microscopic derivation of the entropy of certain extremal black holes has recently become available in the context of string theory [1] [6] For four dimensional extremal black holes in the limit of large electric magnetic charges Q, the microscopic entropy is generically of the form S micro q Q 4 : 1) This result agrees with the one obtained from macroscopic calculations based on the corresponding effective field ....

Strominger A and Vafa A 1996 Phys. Lett. B379 99, hep-th/9601029

.... the past year [1, 2] Many of their remarkable properties have been elucidated [3, 4] and the coupling of their bosonic degrees of freedom to bosonic background fields has been worked out [5, 6] In particular, they have provided a powerful tool for the study of black holes in string theory [7]. Very recently an interesting proposal for understanding non perturbative 11d (M theory) physics in terms of ensembles of D 0 branes has been put forward [8] For these reasons and more it is desirable to achieve as thorough an understanding of D branes as possible. It has been known for some ....

A. Strominger and C. Vafa, Phys. Lett. B379 (1996) 99, hep-th/9601029.

....II: Extremal Black Hole Entropy Another important application has been the demonstration that the Bekenstein Hawking entropy area law[19] of Black Hole Thermodynamics can be derived as a truly statistical result. This was first done for five dimensional Riessner Nordstrom black holes[20], and later shown for the four dimensional case[21,22] The basic idea is simple[20] The black holes of interest were embedded into (say) K3 ThetaT 2 compactified type IIB string theory as a BPS state. The abelian (Maxwell) fields they carry were embedded into the R R sector of the ....

....demonstration that the Bekenstein Hawking entropy area law[19] of Black Hole Thermodynamics can be derived as a truly statistical result. This was first done for five dimensional Riessner Nordstrom black holes[20] and later shown for the four dimensional case[21,22] The basic idea is simple[20]. The black holes of interest were embedded into (say) K3 ThetaT 2 compactified type IIB string theory as a BPS state. The abelian (Maxwell) fields they carry were embedded into the R R sector of the compactified theory. The final embedding resembles a macroscopic string in six dimensions with a ....

A. Strominger and C. Vafa, hep-th/9601029.

....PACS: 11.15. q, 11.15.Kc, 11.30.Pb, 14.80. Martin Cederwall and Magnus Holm: Monopole and Dyon Spectra: 2 1. Introduction The last years have seen a tremendous progress in the understanding of nonperturbative aspects of four dimensional field theory. New techniques [1,2,3,4,5] enable calculation of exact results valid beyond the perturbative level. It was long ago conjectured [6,7] that the N=4 supersymmetric Yang Mills (SYM) theories should possess some kind of strong weak coupling duality. These theories are perturbatively finite [8,9,10,11,12,13] and actually ....

C. Vafa and E. Witten, Nucl.Phys. B431 (1994) 3 (hep-th/9408074).

....This talk contains a short summary of recent results [1] The last years have seen big progress in the understanding of non perturbative aspects of quantum field theory and string theory. In supersymmetric enough theories, some non perturbative properties may actually be calculated exactly [2,3,4,5,6]. Intimately connected with the new techniques developed is the somewhat older concept of duality [7,8] Finite quantum field theories may, and seem to, exhibit a duality symmetry, which is the realization and extension of the old puzzling symmetry between electricity and magnetism in Maxwell ....

C. Vafa and E. Witten, Nucl.Phys. B431 (1994) 3 (hep-th/9408074).

.... theories with terms quadratic in the Riemann tensor, they were determined in [1] where it was shown that, for a particular class of black holes arising in compactifications of M theory and type IIA string theory, the modified macroscopic entropy exactly matches the microscopic entropy computed in [2, 3]. The four dimensional supersymmetric black hole solutions considered in [1] are static, rotationally symmetric solitonic interpolations between two N = 2 supersymmetric groundstates: flat Minkowski spacetime at spatial infinity and Bertotti Robinson spacetime at the horizon [4] The crucial ....

....macroscopic entropy formula can be generalized to the case of N = 4 supersymmetric black holes. In order to appreciate the consequences of our results we will return to the counting of micro states in the dual type II or M theory version of these black holes, following the same approach as in [2, 3]. In this analysis certain special features of K3 Theta T 2 emerge which are not present for generic Calabi Yau threefolds, implying that the micro state counting in the former case is more subtle than in the latter case. Based on this analysis we propose a modification of the state counting ....

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C. Vafa, Adv. Theor. Math. Phys. 2 (1998) 207, hep-th/9711067.

....area law and more general definitions of macroscopic black hole entropy are immaterial. Only recently has it become apparent that microscopic entropy formulae contain subleading corrections which on the macroscopic level are due to higher derivative interactions in the field theory [10, 11, 12]. In this case the correct macroscopic definition of entropy is crucial for finding agreement, as we will review below. The structure of this paper is as follows. First we will review the derivation of the Noether potential for Yang Mills theories and for gravity in the context of field theories ....

....from (38) reads [12] S = 2 q 1 6 jq 0 j(C ABC p A p B p C c 2A p A ) 41) where q 0 = q 0 1 12 D AB q A q B ; DAB = DABC p C ; DABD BC = ffi C A . The expression (41) for the macroscopic entropy is in exact agreement with the microscopic entropy formula computed in [10, 11] via state counting. Next, let us consider black hole solutions arising in heterotic string compactifications on K 3 Theta T 2 . The associated tree level function is given by F (Y; Upsilon) Gamma Y 1 Y a j ab Y b Y 0 c Y 1 Y 0 Upsilon ; a = 2; n ; 42) where the real ....

C. Vafa, Adv. Theor. Math. Phys. 2 (1998) 207, hep-th/9711067.

....infinity, to the horizon area which is defined at the inner boundary of the black hole solution. Also, the connection with thermodynamics suggests a possible interpretation of the entropy in terms of microstates. Such an interpretation has recently been provided in the context of string theory [3]. From the point of view of the field theory, be it fundamental or effective, it is rather surprising that variations near the outer boundary at spatial infinity are related to variations near the inner boundary at the horizon. Moreover, an effective field theory action will contain more than just ....

A. Strominger and C. Vafa, Phys. Lett. B379 (1996) 99, hep-th/9601029.

.... came through our improved understanding of the Ramond Ramond (RR) charged string solitons through the Dirichlet brane (D brane) description [14 20] This rapid progress has led to a microscopic state counting of the black hole entropy in an explicit string computation using the D brane technology [21]. The results of the D brane counting of the black hole entropy were curiously all limited to five dimensions at the beginning [21 24] The reason for this was that suitable fourdimensional black holes with non zero horizon which are bound states of D branes alone were not known 1 . One needs ....

.... [14 20] This rapid progress has led to a microscopic state counting of the black hole entropy in an explicit string computation using the D brane technology [21] The results of the D brane counting of the black hole entropy were curiously all limited to five dimensions at the beginning [21 24]. The reason for this was that suitable fourdimensional black holes with non zero horizon which are bound states of D branes alone were not known 1 . One needs to add to the weak coupling bound state equivalent (of the semiclassical four dimensional black holes) a Kaluza Klein monopole like in ....

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. A. Strominger and C. Vafa, hep-th/9601029.

....1 m 2 U 0 T 0 ( m 2 ) Gamma m 1 U 0 j 2 4 S 0 2 Gamma W 0 2 2 2U 0 2 T 0 2 U 0 2 (8.14) which gives the correct tree level type II mass formula in the large T 0 2 limit, taking into account (8.7) and the duality map. Owing to the adiabatic argument of ref. [33], we can obtain new dual heterotic type II pairs by orbifolding both the N = 4 heterotic and N = 4 type II strings, by the same freely acting symmetry. Thus we would like to identify the duals of the heterotic models constructed in the previous sections with spontaneously broken supersymmetry. ....

....acts on the type II side on the magnetically charged states of the momentum gauge fields of the two torus; it is thus, not visible in type II perturbation theory. The type II duals have 20 vector multiplets and 4 hypermultiplets; thus they are mirrors of the type II models discussed in ref. [33] with 4 vector multiplets and 20 hypermultiplets. Therefore, the perturbative partition function of the type II models dual to the heterotic ones is Z 4 2 II = 1 Im jjj 4 1 2 1 X h;g; h g=0 Gamma R 2;2 [ h g ] jjj 4 Z twist (4;4) h g ] Theta 1 2 1 X ff;fi=0 ....

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C. Vafa and E. Witten, hep-th/9507050.

.... [2] entropy of black holes has been under intensive scrutiny for the last couple of years, following the derivation of the entropy of certain extremal charged black hole solutions of toroidally compactified heterotic string and also type IIB superstring from the underlying string theories [3] [4]. In the former case of the heterotic string, the entropy was shown to be proportional to the area of the stretched horizon of the corresponding extremal black hole, while in the latter case it turned out to be precisely the B H result. The latter result was soon generalized to a large number of ....

A. Strominger and C. Vafa, Phys. Lett. B379, 99 (1996), hep-th/9601029.

....Fock space describing the large N limit of gauge theory should contains states describing black holes which obey the quantum Boltzmann statistics, i.e. black hole can be represented as a Boltzmann gas of branes. This suggestion was based on [8] on the computation of the black hole entropy [12] and on the idea [13] about condensate of D0 branes in the large N limit for the matrix regularisation of membrane. In [14, 15, 16, 17] it was shown how to compute entropy of black hole by using the Boltzmann gas model in Matrix theory [18] Recently an exciting new development in the study of the ....

A. Strominger and C. Vafa, hep-th/9601029

....are broken by generic values of the hyper multiplets. Now we consider the decompactification limit to six dimensions by making the base P 1 large, which is common to all three models (regarding K3 H locally as P 1 Theta T K3 2 ) In this way, using the (reverse) adiabatic argument of [16], we are dealing with a heterotic string on T T;U 2 Theta T K3 2 where the non Abelian gauge symmetries E 8 Theta E 8 are now broken by the Wilson line vector multiplets. The six dimensional heterotic string is in turn dual to the type IIA string on K3 CY or respectively dual to F theory ....

C. Vafa and E. Witten, hep-th/9507050

.... of evidence for this symmetry [42] and some of it has been rigorized [43, 44] Moreover there are hints that this symmetry is related to the more familiar T duality (R 1=R) symmetry of toroidal compactification where one views the threefold as a T 3 fibered over S 3 [45, 46, 47] see also [48]. We will consider type IIA on a local model of Calabi Yau 3 fold, MK whose Kahler deformations give the Coulomb branch of an N = 2 theory in d = 4 (the subscript K is there to remind us that we are considering varying the Kahler structure) We also consider the completely Higgsed branch which ....

C. Vafa and E. Witten, J. Geom. Phys. 15 (1995) 189, hep-th/9409188.

....the T1 duality of the circle to exchange type IIA and type IIB strings. The main ingredient needed in this description is a precise understanding of how the Coulomb Higgs phases of the gauge system are realized geometrically. Realization of Coulomb branches have been understood in the type IIA [32, 33, 34, 35, 36], and the type IIB setup [37, 38, 3, 4] However much less is known about the Higgs branch. In this paper we will develop techniques to describe the Higgs branch in a geometrical way. This construction not only allows us to rederive the N = 4 dualities in d = 3 from perturbative symmetries of ....

....gauge system. If we wish to get matter we will obtain it by colliding singularities which means that we consider intersecting P 1 s over which we have A D E singularities. Depending on what singularity is on top of intersecting P 1 s we will get matter in various representations [34, 35] 2 . For example if we wish to get U(n) Theta U(m) with matter in bi fundamental (n; m) we consider a type IIA geometry with two intersecting P 1 s over one having an A n Gamma1 singularity and over the other an Am Gamma1 singularity and at the intersection point an A n m Gamma1 singularity. ....

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S. Katz and C. Vafa, hep-th/9606086.

....N = 1 theories in five dimensions [10, 11, 12] and N = 1 theories in six dimensions [13, 14, 15, 16] have been engineered. In certain cases constructions can also be done using D branes in the presence of NS 5 branes [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27] and often there is a T duality [28] which connects the two pictures (see in particular [7] An interesting duality was proposed for three dimensional theories with N = 4 in [29] This was further extended to a large number of non abelian gauge theories in [30, 31] So far, the only approach from string theory involving a ....

....in the 4 fold S Theta C 2 . This 3 fold can be considered as an elliptic (or C ) fibration over S where the fibre acquires A Gamma1 type singularity at the zero locus of F . Now, we use the correspondence of Type IIB on A Gamma1 type singularity with Type IIA with NS fivebranes [28]. Then we can identify the Type IIB on W Theta S 1 as the theory on the NS fivebrane with worldvolume fF = 0g Theta S 1 Theta R 3 where we note that fF = 0g is a Riemann surface embedded in the surface S. This is the compactification on S 1 of the d = 4 N = 2 supersymmetric gauge theory ....

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H. Ooguri and C. Vafa, Nucl. Phys. B463 (1996) 55, hep-th/9511164.

....group [1, 2] Similarly exact results for N = 2 gauge systems can be obtained by geometric engineering of type II strings on Calabi Yau threefolds [3, 4] and by using mirror symmetry which is a classical symmetry of type II strings. This approach has been extended to N = 1 theories in d = 4 in [5, 6, 7, 8, 9] in which the dualities are realized as classical symmetries of strings. Similarly higher dimensional critical theories (with tensionless strings) have also been constructed from this viewpoint and in particular N = 1 theories in five dimensions [10, 11, 12] and N = 1 theories in six dimensions ....

S. Katz and C. Vafa, hep-th/9611090.

....studying QFT s is called geometric engineering. On the other hand there has been another direction of construction of field theories involving branes (see for example [7] 8] Some of these cases are already known to be equivalent, by T dualities to the geometrical cases (see for example [9] 10][11][12] Here we try to extend this dictionary to a more general class of theories and in particular to 5 dimensional critical theories constructed by Hanany and Aharony [13] and studied further in [14] 15] 16] The approach we follow will also lead to a simple geometric realization of the ....

H. Ooguri and C. Vafa, hep-th/9702180

....internal compactification manifold encode a great deal of information about quantum field theories. Turning things around we can engineer quantum field theories by suitably choosing singularities under consideration and use them to gain insight into quantum field theories (see for example [1] 2][3][4] 5] 6] This program of studying QFT s is called geometric engineering. On the other hand there has been another direction of construction of field theories involving branes (see for example [7] 8] Some of these cases are already known to be equivalent, by T dualities to the geometrical ....

S. Katz and C. Vafa, hep-th/9611090.

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C. Vafa and E. Witten, Nucl.Phys. B431 (1994) 3 (hep-th/9408074).

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. G. Horwitz and A. Strominger, hep-th/9602051.

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