University of Pennsylvania ScholarlyCommons
Department of Physics Papers Department of Physics
12-27-2007
Flux, Gaugino Condensation, and Antibranes in Heteroic M Theory
James Gray Université de Paris
André Lukas University of Oxford
Burt A. Ovrut University of Pennsylvania, [email protected]
Follow this and additional works at: https://repository.upenn.edu/physics_papers
Part of the Physics Commons
Recommended Citation Gray, J., Lukas, A., & Ovrut, B. A. (2007). Flux, Gaugino Condensation, and Antibranes in Heteroic M Theory. Retrieved from https://repository.upenn.edu/physics_papers/118
Suggested Citation: J. Gray, A. Lukas and B. Ovrut. (2007). "Flux, gaugino condensation, and antibranes in heterotic M theory." Physical Review D. 76, 126012.
© The American Physical Society http://dx.doi.org/10.1003/PhysRevD.76.126012
This paper is posted at ScholarlyCommons. https://repository.upenn.edu/physics_papers/118 For more information, please contact [email protected]. Flux, Gaugino Condensation, and Antibranes in Heteroic M Theory
Abstract We present the potential energy due to flux and gaugino condensation in heterotic M theory compactifications with antibranes in the vacuum. For reasons which we explain in detail, the contributions to the potential due to flux are not modified from those in supersymmetric contexts. The discussion of gaugino condensation is, however, changed by the presence of antibranes. We show how a careful microscopic analysis of the system allows us to use standard results in supersymmetric gauge theory in describing such effects—despite the explicit supersymmetry breaking which is present. Not surprisingly, the significant effect of antibranes on the threshold corrections to the gauge kinetic functions greatly alters the potential energy terms arising from gaugino condensation.
Disciplines Physical Sciences and Mathematics | Physics
Comments Suggested Citation: J. Gray, A. Lukas and B. Ovrut. (2007). "Flux, gaugino condensation, and antibranes in heterotic M theory." Physical Review D. 76, 126012.
© The American Physical Society http://dx.doi.org/10.1003/PhysRevD.76.126012
This journal article is available at ScholarlyCommons: https://repository.upenn.edu/physics_papers/118 PHYSICAL REVIEW D 76, 126012 (2007) Flux, gaugino condensation, and antibranes in heterotic M theory
James Gray,1 Andre´ Lukas,2 and Burt Ovrut3 1Institut d’Astrophysique de Paris and APC, Universite´ de Paris 7, 98 bis, Bd. Arago 75014, Paris, France 2Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, United Kingdom 3Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania 19104a˜ 6395, USA (Received 9 October 2007; published 27 December 2007) We present the potential energy due to flux and gaugino condensation in heterotic M theory compactifications with antibranes in the vacuum. For reasons which we explain in detail, the contributions to the potential due to flux are not modified from those in supersymmetric contexts. The discussion of gaugino condensation is, however, changed by the presence of antibranes. We show how a careful microscopic analysis of the system allows us to use standard results in supersymmetric gauge theory in describing such effects—despite the explicit supersymmetry breaking which is present. Not surprisingly, the significant effect of antibranes on the threshold corrections to the gauge kinetic functions greatly alters the potential energy terms arising from gaugino condensation.
DOI: 10.1103/PhysRevD.76.126012 PACS numbers: 11.25.Yb
up new avenues for heterotic phenomenology. For ex- I. INTRODUCTION ample, one can proceed to look at more detailed properties In [1], the perturbative four-dimensional effective action of these models such as terms [27], Yukawa couplings of heterotic M theory compactified on vacua containing [28], the number of moduli [29], and so forth. Other groups both branes and antibranes in the bulk space was derived. are also making strides in heterotic model building, see for That paper concentrated specifically on those aspects of the example [30–35]. perturbative low energy theory which are induced by the The addition of G-flux to the formalism described in [1] inclusion of antibranes. Hence, for clarity of presentation, is, as we will show in this paper, relatively straightforward. the effect of background G-flux on the effective theory was In contrast, it is not at first obvious how to incorporate not discussed. Furthermore, nonperturbative physics, gaugino condensation into this explicitly nonsupersym- namely, gaugino condensation and membrane instantons, metric compactification of M theory. The reason is that was not included. However, all three of these contributions almost everything we know about this nonperturbative are required for a complete discussion of moduli stabiliza- phenomenon is based on the dynamics of unbroken N tion, N 1 supersymmetry breaking, and the cosmologi- 1 supersymmetric gauge theories. However, by carefully cal constant. Therefore, in this paper, we extend the results analyzing the limit where the usual discussions of gaugino of [1] to include the effects of flux and gaugino condensa- condensation are applied, we will show that our nonsuper- tion. Another piece of the effective theory, that is, the symmetric system reverts to a globally supersymmetric contribution of membrane instantons, will be presented gauge theory. This fact allows us to construct the conden- elsewhere [2]. There is of course vast literature on the sation induced potential energy terms in the presence of subject of moduli stabilization, flux, and gaugino conden- antibranes using a component action approach similar to sation in heterotic theories. Some recent discussions of that of [36]. various aspects of these topics appear in [3–12]. The plan of this paper is as follows. In the next section, A strong motivation for attempting to find stable vacua we briefly review those aspects of [1] which are required in Calabi-Yau compactifications of heterotic theories for the current work. In Sec. III, we review the subject of comes from the advantages that such constructions enjoy flux in supersymmetric heterotic M theory. In Sec. IV, the in particle physics model building. For example, models effects of G-flux in heterotic M theory vacua which include with an underlying SO 10 grand unified theory (GUT) antibranes are explicitly computed. Gaugino condensation symmetry can be constructed where one right-handed neu- in the presence of antibranes is then introduced in Sec. V. trino per family occurs naturally in the 16 multiplet and We begin by describing how the potential energy terms gauge unification is generic due to the universal gauge induced by this effect can be calculated in terms of the kinetic functions in heterotic theories. Recent progress in condensate itself. It is then shown how one can explicitly the understanding of nonstandard embedding models [13– evaluate the condensate as a function of the moduli fields. 16] and the associated mathematics of vector bundles on Finally, we conclude in Sec. VI by writing out, in full, the Calabi-Yau spaces [17–20] has led to the construction of effective potential energy we have obtained by compacti- effective theories close to the minimal supersymmetric fying heterotic M theory in the presence of antibranes, standard model (MSSM), see [21–26]. This has opened G-flux, and gaugino condensation.
1550-7998=2007=76(12)=126012(22) 126012-1 © 2007 The American Physical Society JAMES GRAY, ANDRE´ LUKAS, AND BURT OVRUT PHYSICAL REVIEW D 76, 126012 (2007) 1;1 k have the graviton, g , h Abelian vector fields A with II. ANTIBRANES IN HETEROTIC M THEORY k 1;1 field strengths F , a real scalar field V, h real scalar The low energy effective Lagrangian of heterotic M k i j k fields b which obey the condition dijkb b b 6 (the dijk theory compactified on vacua containing bulk space branes being the intersection numbers on the Calabi-Yau three- and antibranes, but ignoring G-flux and nonperturbative fold) and, therefore, constitute h1;1 1 degrees of free- effects, was constructed in [1]. In this section, we provide dom, h2;1 complex scalar fields za, 2 h2;1 1 real scalar a brief summary of the aspects of [1] which are important A ~ A fields , B with their field strengths X , XB and the in the current paper. The basic vacuum configuration which three-form C with its field strength G . Of these we consider, as viewed from five dimensions, is depicted a k k in Fig. 1. We include an arbitrary number of three-branes bulk fields, V, z , g , gyy, b , C y, and Ay are even A ~ k in the vacuum, but, for clarity of notation, restrict the under the Z2 orbifold projection, while g y, , B, A , discussion to a single anti three-brane. Our results extend C are odd. almost trivially to the case where multiple antibranes are In addition, there are extra degrees of freedom living on present. The (anti) three-branes arise as the low energy the extended sources in the vacuum. On each of the two limit of (anti) M5-branes wrapped on holomorphic curves four-dimensional fixed planes, one finds N 1 gauge in the internal Calabi-Yau threefold. We will often refer to supermultiplets. These contain gauge fields A p indexed the (anti) three-branes simply as (anti)branes. A more de- over the adjoint representation of the unbroken gauge tailed discussion of this setup is provided in [1]. As shown group H p E8 for p 0, N 1. The associated fields in Fig. 1, we label the extended objects with a bracketed strengths are denote by F p . Furthermore, on the fixed index (p) ranging from (0) to (N 1). The values (0) and planes there are N 1 chiral matter supermultiplets with (N 1) correspond to the orbifold fixed planes, (p ) labels Rx scalar components C p , p 0, N 1 transforming in the antibrane, and the remaining (p) values are associated various representations R, with components x, of this with the branes. The bulk regions are labeled by the same gauge group. Details of the origin and structure of the index as the brane which borders them on the left. Fields matter sector can be found in the appendix of [1]. The associated with the world volume of a given extended world volume fields associated with the three-branes are as object or with a certain region of the bulk are often labeled follows. First, each brane (p) has an embedding coordi- with the associated index. nate, that is, the brane position, y and a world volume In [1], our starting point was the five-dimensional action p scalar s . Second, each brane supports an N 1 gauge describing the compactification of Horˇava-Witten theory p supermultiplet with structure group U 1 g p where g is on a Calabi-Yau threefold X in the presence of both M5 p branes and an anti-M5 brane. The bosonic field content of the genus of the curve wrapped by the brane (p). These contain the Abelian gauge fields Au , where u this theory is as follows. Let ; ; ... 0; ...; 4 be the p five-dimensional bulk space indices, ; ; ... 0; ...; 3 1; ...;g p . The associated field strengths are denoted u label the four-dimensional Minkowski space indices, and by E p . In general, there will be additional chiral multip- 1 y be the coordinate of the S =Z2 orbifold interval. lets describing the moduli space of the curves around Additionally, we let i; j; k; ... 1; ...;h1;1, a; b; ... which the M5 and anti-M5 branes are wrapped. 1; ...;h2;1, and A; B; ... 1; ...;h2;1 1, where h1;1 and Furthermore, there may be non-Abelian generalizations h2;1 are the dimensions of the H1;1 X and H2;1 X coho- of the gauge field degrees of freedom when branes are mology groups, respectively. Then, in the bulk space we stacked. However, these multiplets are not vital to our
FIG. 1 (color online). The brane configuration in five-dimensional heterotic M theory.
126012-2 FLUX, GAUGINO CONDENSATION, AND ANTIBRANES IN ... PHYSICAL REVIEW D 76, 126012 (2007) discussion and, therefore, we shall not explicitly take them into account. Given this field content, the five-dimensional action describing Horˇava-Witten theory compactified on an arbitrary Calabi-Yau threefold in the presence of M5 branes and an anti-M5 brane is given by
Z 1 5 p 1 1 k l 1 k l 1 2 2 i j k S 2 d x g R Gkl b @b @b Gkl b F F V @V dijkb b b 6 2 5 2 4 2 4 1 a b 1 ~ B 1 A C ~ D 1 2 K z @z @z V X M z X Im M z X M z X V G G 4 a b A A B C C D 4! Z 1 2 2 2 kl ^ ^ k l m A ~ ~ A ^ k m V G b k l 2 dklmA ^ F ^ F 2G ^ XA AX 2m kA 2 5 3 Z q m 1 @W @W d5x y h V 1bk 0 V tr F2 D CRxD C S V 1 RS 0 0 tr D2 0 2 k 0 G 0 RS 0 0 x G 0 Rx S 0 5 8 GUT @C 0 @C 0 x Z q m 1 5 1 k N 1 2 Rx S d x y h N 1 2 V b k V tr F N 1 G 0 RSD C N 1 D C N 1 x 5 8 GUT Z q @W @W 1 XN V 1 RS N 1 N 1 tr D2 d5x y y y y h mV 1 p bk G 0 Rx S N 1 2 p p p k @C N 1 @C N 1 x 2 5 p 1 2 nk p 2 p k u w p k u w j j Im E E 4C^ ^ d n s 2 Re E ^ E : (1) p l p p p uw p p p k p p p uw p p V l b
Here 5 and GUT are given by their precise form is not required for the analysis in this paper. The quantities 2 2 2=3 2 11 4 11 5 ; GUT ; (2) p v v k k n 1;1 (4) p h p 2 where 5 and 11 are 5- and 11-dimensional Planck con- l 1 l stants, respectively, v is the reference Calabi-Yau volume, are the normalized version of the three-brane charges and and constant m is the reference mass scale we have used the definition 2=3 2 11 p
m : (3) 2=3 k k ^ k v 4 j p d n s p A p : (5) nl p p The above expressions describe, in the order presented, the p l bulk space, the fixed planes, and the world volumes of the Hats on field quantities denote pullbacks of the correspond- M5 and anti-M5 branes. The metrics Gkl b , Ka b z as ing bulk variable onto the extended object under consid- well as the matrix MA B z appearing in the bulk space term eration. The matrix p uw appearing in the world volume are defined in Appendix A of [1]. Their precise form is not brane terms is the period matrix of the curve on which the required in this paper. The dklm coefficients are the inter- pth brane or antibrane is wrapped. This is defined in section numbers of the Calabi-Yau threefold. The integers Appendix C of [1]. Finally, the quantities h p are the p k are the tensions of the branes, antibranes, and fixed induced metrics on the boundary planes, branes, and the planes. Charges on individual branes are simply labeled by antibrane. It is important to note that despite the appear- p k and are equal to the associated tensions in the case of ance of an anti-M5-brane in the vacuum, action (1)is branes and fixed planes and equal in magnitude, but oppo- N 1 supersymmetric, both in the five-dimensional site in sign, in the case of the antibrane. The antibrane is bulk space and on all four-dimensional fixed planes and taken to be associated with a purely antieffective curve for branes. The reason for this is that the dimensional reduc- ^ simplicity. The k coefficients are the sum of the charges tion was carried out with respect to the supersymmetry on all extended sources to the left of where the bulk theory preserving Calabi-Yau threefold, and did not explicitly is being considered. We will frequently denote quantities involve the anti-M5-brane. However, as we now briefly associated with the antibrane by a bar. Hence, the tension discuss, the anti-M5-brane does enter the dimensional p of the antibrane is denoted by k k . The metrics reduction of the theory to four dimensions, explicitly G p RS, the matter field superpotentials W p , and the breaking its N 1 supersymmetry. D-terms D p with p 0, N 1 appearing in the bound- In [1], we described how the presence of an antibrane in ary plane terms are defined in Appendix B of [1]. Again, the bulk space of a heterotic M theory vacuum changes the
126012-3 JAMES GRAY, ANDRE´ LUKAS, AND BURT OVRUT PHYSICAL REVIEW D 76, 126012 (2007) k warping in the extra dimensions. We showed that this on a0, b0, V0, and b0 denotes quantities which are functions backreaction is manifested by a quadratic contribution to of the four uncompactified coordinates only. It is these the warping. It is interesting to note that this contribution functions that become the moduli when we dimensionally derives simply from the failure of the tensions of the reduce on this ansatz to obtain the effective four- extended objects to sum to zero in the case where anti- dimensional action. Note that we have not given an ex- branes are present. This property of the source terms is pression for the warping of the metric coefficient b. This is responsible for all of the major changes to the perturbative because this y dependence can be removed by a coordinate theory. The only bulk fields involved in the generalized redefinition and, hence, does not enter the calculation of domain wall solution including an antibrane are the metric the four-dimensional effective action. k g , the volume modulus V, and the Ka¨hler moduli b .We We emphasize several points about this result which will take as our metric ansatz be important in this paper. First, the z independent factors k 2 2 2 2 have been defined such that a0, b0, V0, and b0 are simply ds a x ;y g x dx dx b x ;y dy : (6) 5 4 the orbifold averages of a, b, V, and bk, respectively. This It turns out to be helpful to introduce a function which proves to be useful in performing the dimensional reduc- encodes the standard linear warping functions of heterotic tion around this configuration. Second, upon taking k ! M theory in a usefully normalized manner. It is given by 0 one recovers the conventional results for heterotic M theory, given, for example, in [37]. In particular, the qua- Xp NX 1 q 1 q dratic pieces of the warping correctly disappear as we turn
h z z z z z 2 p k k q k q q the antibrane into an M5 brane in this manner. Note that in q 0 2 q 0 each of the above expressions the warping occurs with the b k: (7) same multiplicative factor, specifically 0 .In[1,38], it 0 V0 The z which appears here is defined by z y= , where was shown that the four-dimensional effective theory is an is the reference length of the orbifold interval and y is expansion in the parameters S and R given by the coordinate of the fifth dimension. Each z q is the 2=3 1=6 1=2 11 2 b0 v V0 normalized position modulus for the qth brane or anti- ; : (13) S 2=3 R 4 v V0 b0 brane. Note that the fixed planes are located at z 0 0 and z N 1 1, respectively. In expression (7), The first of these is the ‘‘strong coupling’’ parameter which 1 specifies the size of the warping in the orbifold dimension. : (8) k 2 k k k The second term is the expansion parameter which controls the size of the Calabi-Yau Kaluza-Klein modes. Using (3) The quantity h p k has been defined so that its orbifold average is zero. This property enables us to extract defini- and (12), we see that tions of the zero modes of the compactification which lead b0
; (14) to a four-dimensional action in a sensible form, without the S 0 V need to use field redefinitions. 0 Using these definitions and the equations of motion for as it should be. V and bk, we find the following nonsupersymmetric do- Further results from [1] will be reproduced as and when main wall ansatz around which the theory will be reduced required in the following sections. to four dimensions. It is given by a p b0 k 2 1 III. FLUX IN SUPERSYMMETRIC HETEROTIC M 1 0 b0 h p k k z ; (9) THEORY a0 3V0 3 In this section, we review the subject of flux in super- V p b0 k 2 1 symmetric Calabi-Yau compactifications of heterotic M 1 2 0 b0 h p k k z ; (10) V0 V0 3 theory. Initially, we will consider heterotic theory from an 11-dimensional perspective so that a complete under-
k k b0 k 1 k l standing is obtained of all of the different fluxes present, b p b0 2 0 h p h p lb0b0 V0 3 including those whose nature is somewhat obscured by the five-dimensional description. Later on in this section, how- k 1 k l 2 1 lb b z ; (11) ever, having gained this understanding, we revert to the 3 0 0 3 five-dimensional picture which is technically more conve- where nient for the discussion of the addition of antibranes. The full 11-dimensional M theory indices will be denoted by
m (12) 0 I; J; K; ... 0; ...; 9; 11, whereas ten-dimensional indices with m defined in (3). The remaining notation is as ex- orthogonal to the orbifold direction y are written as plained earlier, with the one addition that the ‘‘0’’ subscript I; J; K; ... 0; ...; 9. Furthermore, the six-dimensional
126012-4 FLUX, GAUGINO CONDENSATION, AND ANTIBRANES IN ... PHYSICAL REVIEW D 76, 126012 (2007) real coordinates on the Calabi-Yau manifold (CY3) will be components are odd under the Z2 orbifolding. specified by A; B; C; ... 4; ...; 9. The associated holo- (3) One index on the orbifold and the rest on the CY3:In morphic and antiholomorphic complex coordinates on the this case there are four possible complex index threefold are then written as a; b; c; ... 1; 2; 3 and structures, Gabcy, Ga b cy and Gabcy , Ga bcy , for the a; b; c; ... 1; 2; 3, respectively. Finally, the indices GABCy components. All of these are, however, even ^ ^ ^ i; j; k; ... 4; ...; 9; 11 run over both the CY3 and orbifold under the Z2 due to the y index which they carry. directions. The material on the nonzero mode of heterotic In summary, if we are interested in stable vacua of M theory reviewed here was first presented in [38–41]. heterotic M theory with maximal space-time symmetry, we need only consider four-form fluxes with all indices A. The components of G relevant to stable vacua internal to the compactification seven-orbifold. Of these, the G are odd and the G even under the Z action. Let us find the components of the 11-dimensional back- ABCD ABCy 2 ground four-form flux GIJKL that can be nonvanishing in a realistic, stable vacuum. To begin, note that there are three B. The Bianchi identity and its sources possible types of index structure for the four-form G. The components of G are constrained by the require- (1) One or more 4D indices: The possibilities are ment of anomaly freedom to satisfy the Bianchi identity of G , G i^, G i^j^, and G i^j^ k^. 11-dimensional heterotic M theory with N bulk M5 branes. (a) G i^, G i^j^, and G i^j^ k^ all obviously break It follows from the above discussion that the nonvanishing maximal space-time symmetry in four dimen- components of G are contained in the seven-dimensional sions (that is, Poincare´, de Sitter, or anti-de restriction of this identity given by
Sitter symmetry) if present in the vacuum. As 2=3 11 0 N 1 such, they are not relevant for a discussion of dG yABCD 4 y J y J realistic stabilized vacua and will be set to 4 zero. 1 XN J p y y (b) A priori, G could have an expectation p 2 value proportional to the four-dimensional p 1 volume element without breaking maximal y y p : (15) space-time symmetry. However, G is intrinsi- ABCD cally odd under the Horˇava-Witten Z2 and so, The four-form charges localized on the orbifold fixed since G has no y index, this component is also odd under the orbifolding. Given this, if planes and bulk M5 branes in this expression are G nonzero, must jump at the orbifold 1 1 0 1 1 fixed planes. Were there to be such a discon- J 2 trF ^ F trR ^ R ; 16 2 y 0 tinuity, the exterior derivative dG y 1 1 would be a delta function at each fixed plane N 1 2 2 J 2 trF ^ F trR ^ R ; and, hence, require charges of the appropriate 16 2 y index for global consistency. On the fixed p J p C ; (16) planes we have two options, either an F ^ 2 F or R ^ R source. However, giv- p respectively, where C2 is a delta-function four-form ing an expectation value to a four- p dimensional vector gauge field clearly breaks localized on the curve C2 wrappedR by the pth M5R brane. p our requirement of maximal space-time sym- For any two-form ,wehave CY ^ C2 p . 3 C2 metry. Furthermore, it turns out that R ^ Note that since d dG 0, it follows that the sources J p , R for a space of maximal space-time p 0; ...;N 1 are closed four-forms on the CY3. symmetry vanishes. Therefore, a purely ex- Integrability conditions arise from the Bianchi identity ternal component of the four-form in a real- by integrating both sides of expression (15) over closed istic, stable vacuum must also be chosen to be five-cycles. Since the components of the Bianchi identity zero. We conclude that, if they are to be have a y index, one of the dimensions of the cycle must be nonvanishing, all indices of G must lie on the orbifold direction. The remaining dimensions form the internal seven-orbifold. some closed four-cycle in the Calabi-Yau threefold. The (2) All indices on the CY3: Written in terms of the left-hand side of (15), being exact, gives zero upon inte- complex coordinates on the CY3, there are three gration, whereas each term on the right-hand side becomes possible index structures for the GABCD compo- a topological invariant. We thus obtain the well-known nents; Gabcd , Ga b cd , and Gabc d . Since G is intrinsi- cohomology condition of heterotic M theory for each cally odd and GABCD has no y index, all of these choice of four-cycle C4,
126012-5 JAMES GRAY, ANDRE´ LUKAS, AND BURT OVRUT PHYSICAL REVIEW D 76, 126012 (2007)
Z Z 1 1 nonharmonic modes on the CY . There is no similar con- tr F 1 ^ F 1 tr F 2 ^ F 2 3 2 2 dition on the sum of the coefficients of the nonharmonic 16 C4 16 C4 Z XN Z components of the charges. 1 p The closed four-form charges in heterotic M theory have 2 tr R ^ R J 0: (17) 16 C4 p 1 C4 a definite index structure in terms of the complex structure of CY3. The gauge field in a supersymmetric compactifi- The physical interpretation of this condition is simply that cation is obtained by solving the killing spinor equations the sum of the charges on the compact space must vanish for the gauginos—the so-called Hermitian Yang-Mills since there is nowhere for the ‘‘field lines’’ to go. equations [42]. This results in a field strength which is a One can expand the charges (16) in terms of the eigen- (1, 1) form. Thus F ^ F is a (2, 2) form. For the Ricci flat modes of the Laplacian on the CY [38]. Consider the two- 3 metric on a CY3, R ^ R is similarly a (2, 2) form. Finally, forms satisfying the eigenmode equation solving the killing spinor equations for the world volume 2 fermions on the bulk five-branes tells us that these object 4 !hmiAB !hmiAB; (18) hmi wrap holomorphic curves. This implies that the J p four- where 4 is the Laplacian and the index hmi labels the form charges of the bulk branes are also (2, 2) forms, via 2 eigenmode on the threefold with eigenvalue hmi. Note the definition given in, and underneath, Eq. (16). It follows that eigenmodes with different values of hmi may have that, written in terms of complex indices, Bianchi identity identical eigenvalues. For example, consider the number of (15) decomposes into two conditions given by 2 two-form eigenmodes with hmi 0, that is, the zero 2=3 11 0 N 1 dG 4 y J y J modes of 4. Since by definition these two-forms are yabcd 4 harmonic, there are dimH2 X dimH 1;1 of them. We 1 XN will denote these harmonic modes by !k, where k p J y y p y y p 1;1 1; ...;h . Note that eigenmodes corresponding to a non- 2 p 1 abcd vanishing eigenvalue 2 Þ 0 are neither harmonic nor, hmi (24) in general, closed forms. The metric on the space of eigenmodes, and Z 1 dG abcdy 0 (25) Ghmihni !hmi ^ C Y3!hni; (19) 2v CY 3 and its Hermitian conjugate, respectively. where C Y3 is the Poincare´ duality operator on the CY3, In summary, in complex coordinates we have two com- and is used to raise and lower hmi-type indices. ponents of the Bianchi identity (15). The first of these, (24), Each of the four-form charges J p , p 0; ...;N 1 in contains closed (2, 2) form charges which are, in general, (16) can then be expanded as composed of a series of harmonic and nonharmonic modes X on the CY3. The coefficients of the harmonic modes in the p 1 p hmi
C Y J ! : expansion of these charges sum to zero, as shown in 3 2=3 hmi (20) 2v hmi Eq. (23). The remaining component of the Bianchi identity, given in (25), has vanishing sources. We now turn to the Using metric (19), it follows that solutions to these two constraints. Z p 1 p
! ^ J : hmi 1=3 hmi (21) v CY3 C. Solving for the background flux p 1. The nonzero mode In particular, for the coefficients k corresponding to the zero modes one finds Let us begin by considering the first constraint, Eq. (24). Z Expanding the left-hand side in components yields p p
@ G 2@ G 2@ G . Recall from k J ; (22) y abcd a jbjc dy b jacjd y C4k subsection III B that the source terms in the Bianchi iden- 1;1 tity all involve delta functions with y in the argument. It where C4k, k 1; ...;h are the four-cycles dual to the follows that the sources on the right-hand side of (24) can harmonic (1, 1) forms !k. It then follows from the coho- mology condition (17) that only be obtained from a nonsingular, if discontinuous, G as a y-derivative of Gabc d . We now solve for the (2, 2) NX 1 p component of G that saturates these charges.
0 (23) k The standard approach to finding this component is to p 0 restrict attention to the harmonic modes in the expansion of for each integer k. In general, as indicated in (20), the four- the charges (20). This is justified by the fact [38] that the form charges in (15) are built out of both harmonic and effects of the nonharmonic modes are suppressed relative
126012-6 FLUX, GAUGINO CONDENSATION, AND ANTIBRANES IN ... PHYSICAL REVIEW D 76, 126012 (2007) to those of the harmonic ones by powers of the parameter ponents of a closed three-form on the CY3. Again, these R. A solution to the Bianchi identity (24) in this approxi- closed forms are important, and will be discussed in detail mation is given by having a nonzero expectation value for in the next subsection. Here, however, we simply note that Gabc d which jumps at each extended object by an amount setting them to zero is sufficient to satisfy the Bianchi proportional to that objects charge and is constant in y identity (24) and (25). everywhere else. These requirements are satisfied by nzm Gabc d in Eq. (26) is the famous nonzero mode of heter- Xh1;1 otic M theory—hence our choice of labeling superscript. nzm 1
G y CY ! ; abc d 1=3 k 3 k bc d (26) The fact that, in the harmonic approximation, there is no V k 1 GABCy component to the expectation value, together with where the lack of bulk y dependence of the nonzero mode, means
2 2=3 that if we compactify the 11-dimensional theory down to y 11 0 y N 1 y k v2=3 4 k k five dimensions we will obtain an action in the bulk, XN between any two extended objects, which does not depend p y k y y p y y p : (27) explicitly on . This structure is the basis for the very p 1 existence of 5D heterotic M theory [41]. In the low energy Taking the y-derivative of this expression exactly reprodu- limit, heterotic M theory further reduces to an effective, ces the charges on the right-hand side of (24) in the four-dimensional, N 1 supersymmetric theory. It is harmonic mode approximation. Note that G nzm in (26) well known that the harmonic part of the nonzero mode abc d does not give rise to a potential in four dimensions. This is a closed (2, 2) form on the CY . 3 has been shown many times by explicit dimensional re- Now consider the (1, 2) and (2, 1) components of G, duction. The 4D superpotential contribution due to flux in G , and G , respectively, within the harmonic ap- bc dy acdy heterotic M theory takes the Gukov-Vafa-Witten form proximation. For values of y in the bulk space, that is, Z away from the extended sources, identity (24) becomes WGVW / ^ G: (33) M nzm 7 @yG 2@ aG 2@ G 0: (28) abc d jbjc dy b jacjd y Here is the holomorphic three-form on the CY3.Itis nzm nzm obvious that if we substitute G into this expression we Since G in (26) is constant between the sources, it abcd abcd get zero, since neither the flux nor carry a y index. Note nzm follows that @yGabc d 0 in (28) and, hence, nzm that this will continue to be the case for Gabc d even when CY3 2@ aGjb jc dy 2@ b Gjacjd y d G abc dy 0; (29) nonharmonic modes are included. Thus far, we have solved for the nonzero mode of CY3 where d is the exterior derivative operator on the CY3. heterotic M theory in the approximation that the charges Nontrivial solutions to Eq. (29) are both possible and C Y J p , p 0; ...;N 1 in (24) are harmonic (2, 2) important, and we will discuss them in detail in the follow- 3 forms. In the remainder of this subsection, we generalize ing subsection. Here, however, we simply note that, in the these results to include the effects of the nonharmonic harmonic approximation to the sources, the nonvanishing contributions to the charges. To balance the delta function component G nzm is sufficient to satisfy the identity (24) abc d on the right-hand side of (24), one still requires the field G everywhere in the orbifold interval, both at each extended to jump as it crosses each extended object in the y direc- source and in the bulk space. tion. Even including the nonharmonic components of the We now examine the second constraint equation, given charges, all of the four-forms on the right-hand side of (24) in (25). In components, this becomes have a (2, 2) index structure. Hence, it is the (2, 2) compo- nent of the four-form field strength which must jump at the @yGabcd 3@ aGbc dy @d Gabcy 0: (30) extended sources, as it did in the harmonic approximation As discussed previously, the (3, 1) component Gabcd is odd (26). However, recall that while the sum of the harmonic under the Z2 orbifolding. It follows that, if nonvanishing, contributions to the charges is zero, Eq. (23), the same is this component would require source charges on the right- not true for the nonharmonic modes. Because of this, when hand side of (30). Since none exist, we must set the nonharmonic modes are included, it no longer suffices to simply have the (2, 2) component of G jump at the
G 0: (31) abcd extended objects and be constant everywhere else. It was Equation (30) then becomes shown in [38] that to globally obey all of the boundary nzm CY3 conditions, the G component of the four-form has to 3@ aGbc dy @d Gabcy d G abcdy 0: (32) abcd evolve in y in the bulk space so that it may undergo the Combining (32) with (29) and their complex conjugates, correct jump at each charged object. we see that the (2, 1) and (3, 0) components of G, Gbcdy , This observation has implications for the (1, 2) and (2, 1) and Gabcy, respectively, and their conjugates are the com- forms, Gbc dy and Gacdy . In between the extended sources,
126012-7 JAMES GRAY, ANDRE´ LUKAS, AND BURT OVRUT PHYSICAL REVIEW D 76, 126012 (2007) nzm sentatives of the 2 h2;1 1 -dimensional cohomology identity (24) is given by Eq. (28). Clearly, if G is a abcd 3 nonconstant function of y in the bulk, we require non- space H X . It is these closed forms that we will refer to as the flux contributions to heterotic vacua. vanishing G and G components which are not bcdy acdy This GH contribution to G does give rise to a super- closed on the CY3. These components of G, which we nzm nzm potential in the four-dimensional theory; specifically, for denote by G and G , respectively, correspond in bcdy acdy the complex structure moduli of the CY3. We will repro- the ten-dimensional theory to the H flux resulting from a duce the derivation of this, and present, for the first time, its nonstandard embedding. The details of the solution of the extension to the case where antibranes are present, in the Bianchi identity (24) and the equations of motion for the next section. For now, we simply note that the harmonic case where nonharmonic modes are included can be found flux includes a (0, 3) form component Ga b cy , which can in [38]. The explicit functional form of G nzm , G nzm , and abc d bc dy Rgive a nonzero contribution to a superpotential WGVW / nzm ^ G when combined with the holomorphic (3, 0) Gacdy , including their exact y-dependence, is given in that M7 paper. The expressions are somewhat large and, since no form . The reader should not think that the above index 2;1 additional information beyond the comments made above structure considerations imply that only one of the 2 h 0;3 are required in this paper, we will not reproduce them here. 1 flux parameters, that corresponding to the H compo- Can the new G nzm and G nzm components of the non- nent, contributes to the superpotential. The superpotential bc dy acdy depends, in general, on the complex structure moduli. As zero mode now give rise to a contributionR to the heterotic these change their values, the component of the flux which M theory 4D superpotential: W ^ G? Since GVW M7 corresponds to the ‘‘(0, 3)’’ piece also changes. For ex- these forms have a y index, they can at least saturate the ample, as is well known and will be shown again below, in interval part of the integral. However, recall that these the large complex structure limit components are (1, 2) and (2, 1) forms on the CY3, re- p spectively. Since is a holomorphic (3, 0) form, such flux 2 v1=6 1 1 W d~ zazbzcn0 d~ zazbnc cannot give a nonvanishing contribution to the Gukov- flux 2 0 2 a b c a b c 4 6 2 Vafa-Witten expression for the superpotential. In addition a to the argument presented here, based upon previous mi- z na n0 : (35) croscopic derivations of the superpotential, valid to some order in the expansions of heterotic M theory, there are a Here n0, n , na, and n are the flux parameters and the d~are variety of macroscopic arguments based upon nonrenorm- 0 a the intersection numbers on the mirror CY . For any given alization theorems of [42–44]. These say that the nonzero 3 value of the complex structure fields za, we obtain a super- mode, including its nonharmonic pieces, should never potential depending upon a single combination of the contribute to the superpotential of supersymmetric heter- parameters. However, as the za change, the combination otic M theory at any order. of parameters which appears in the above expression also changes. The superpotential then, as a function on field 2. Harmonic flux space, depends on all 2 h2;1 1 parameters. Henceforth, nzm nzm nzm we will label the GH contribution to the Gukov-Vafa- The G and G , G forms comprising the abcd bcdy acdy Witten 4D superpotential as W . nonzero mode are not the most general solution to the flux Bianchi identity (24) and (25), and the equations of motion. Recall that the components of G nzm with a y index, when 3. Flux quantization and the derivation of the 4D evaluated using both harmonic and nonharmonic contribu- potential tions to the charges, are not closed on the CY3. As dis- To summarize: within the context of supersymmetric cussed above, one may add to G any closed forms with heterotic M theory we have considered two contributions index structure to the vacuum expectation value of G. The first is the nzm nzm nzm nonzero mode Gabc d and Gbc dy , Gacdy ; that is, the G ;Gabcy ;Gabcy;G ; (34) ab cy a b cy form-flux sourced by the extended objects in the theory. that is, any element of dimH3 X dimH1;2 dimH2;1 The nonzero mode is, in general, composed of both har- dimH3;0 dimH0;3, and still satisfy the Bianchi identity. monic and nonharmonic modes on the CY3. The second H By choosing these forms to be not just closed but also contribution to G we refer to as the harmonic flux G . This harmonic in seven dimensions, they continue to satisfy the is a ‘‘free field’’ background flux which is composed ex- 2;1 equations of motion. For this to be the case, one must clusively of the 2 h 1 harmonic forms Gab cy , Gabcy , choose the harmonic representative in each CY3 cohomol- Gabcy, Ga b cy on the internal manifold. Of these two con- ogy class and restrict these to be constant in y. We denote tributions to G, only the harmonic flux gives rise to a this harmonic flux contribution to G by GH. That is, the superpotential, and so a potential, in four dimensions. In components of GH are the y-independent harmonic repre- this section, we show that the 2 h2;1 1 forms in GH are
126012-8 FLUX, GAUGINO CONDENSATION, AND ANTIBRANES IN ... PHYSICAL REVIEW D 76, 126012 (2007) quantized and derive the four-dimensional potential which A more explicit expression for the quantization of the the harmonic flux gives rise to. harmonic flux GH can be found by expanding its compo- As shown in [45,46] and generalized here, the four-form nents (34) in terms of the a basis of the associated coho- B vacuum expectation value G obeys a quantization condi- mology groups. Specifically, let A and be a basis of the tion derived by demanding that the supermembrane path cohomology H2;1 H3;0 and H1;2 H0;3 of the Calabi-Yau integral be well defined in the background under consid- threefold. Written, for simplicity, in terms of the real eration. This condition is found to be indices on the CY3, one has 2=3 Z Z X2 Z 4 G 1 YM i A ~ B ! GABCy Xy AABC XyB ABC: (40) 2 2 2 i 11 C4 C4 16 i 1 @C4 XN Z By definition, as was discussed explicitly in [37], these p p A ~ !3 y y n; (36) coefficients are related to the fields and B in the five- p 1 C4 dimensional theory through the y-component of their field where C4 is any four-cycle in M7 and n is an arbitrary strengths integer. The four-form is half of the first Pontryagin class A A A ~ ~ ~ of the compactification manifold. It is associated with the X Xy @y ; XB XBy @y B: (41) 1 ^ L form 16 2 R R and locally can be written as d! , where !L is the Lorentz Chern-Simon three-form. In our application, we must take YM i Similarly, ! is the Yang-Mills Chern-Simon term on A ~ p X constant; XB constant (42) the ith orbifold fixed plane and the three-form !3 is p p defined locally by d!3 J on the pth M5 brane. in order for the associated GABCy to be harmonic in the We begin by choosing the cycle C4 to lie entirely in the seven-dimensional sense. CY3. Since such a cycle has no boundary in the Calabi-Yau Inserting expression (40) into condition (39) gives rise to threefold and no component in the orbifold y direction, it the quantization of the 2 h2;1 1 constants in Eq. (42) follows that condition (36) simplifies to when one integrates over the appropriate cycles. Let us Z Z define the three-cycles aA and b in the Calabi-Yau three- 4 2=3 G B
n: (37) fold by 2 2 11 C4 C4 Z Z nzm A 1=2 A ^ v v ; Recall that the G component of the nonzero mode is B B B abcd X aA the only background component of G with all indices in the Z Z A 1=2 A A CY3. Fixing the Pontryagin class of the CY3, we conclude ^ B v v B; (43) that G nzm is quantized as in (37). X bB abc d Z Z Now choose the cycle C4 in (36) to be composed of a B 0; A 0: A three-cycle in the CY3 and a component in the y direction. a bB nzm nzm Clearly this singles out the Gbc dy and Gacdy components They form a basis for the homology H1;2 H0;3 and H2;1 of the nonzero mode, as well as the harmonic forms Gab cy , B H ; which is dual to A and . First consider a four-cycle G , G , and G . First consider the nonzero mode 3 0 abcy abcy a b cy composed of three-cycle aA and the orbifold direction. nzm nzm C4 Gbc dy , Gacdy . As was shown in [46,47], Bianchi identity When the harmonic component of the flux given in (15) guarantees that these two components satisfy Eq. (40) is integrated over this four-cycle, condition (39) becomes, using (43), 2=3 Z Z X2 Z 4 G 1 YM i ! Z B 2 i 4 2=3 X B 11 C4 2 C4 2 16 @C A i 1 4 n : (44) A 2 XN Z 11 a p p !3 y y : (38) Referring to (43) again to perform the final integral, as well C p 1 4 as to the definitions in Eqs. (3) and (12), we find that That is, for fixed background CY3 and Yang-Mills gauge nzm 1 2 G A A connection the nonzero mode components and n : (45) bcdy 1=6 X nzm 0 v Gacdy are determined and cancel out of quantization con- A dition (36). This condition now simplifies to Here the n ’s are arbitrary integers for each A. A similar calculation, where the aA part of the four-cycle is replaced 2=3 Z H 4 G by b , demonstrates that the ~ ’s are quantized in a similar n: (39) B X 11 C4 2 manner. That is,
126012-9 JAMES GRAY, ANDRE´ LUKAS, AND BURT OVRUT PHYSICAL REVIEW D 76, 126012 (2007) 2 type superpotential of the form 1 ~ n ; p 1=6 XB B (46) Z 0 v 2 1
Wflux ^ G: (49) 2 1=2 1 where the nB are arbitrary integers for each B. 4 v X S =Z2 Let us now consider the dimensional reduction from the Substituting (40) into this expression gives five- to the four-dimensional supersymmetric theory in- p 2 A ~ B
cluding this quantized flux, using the methods introduced Wflux 2 X GA XBZ ; (50) in [1]. The starting point is the five-dimensional action 4 given in (1) with N M5-branes but no anti-M5-branes.We where the complex structure moduli space is parametrized B 1 see from (1) that there are only two terms in the five- by the periods Z ; GA Z defined as dimensional action which contain the scalar fields A and Z Z B ~ Z ; GA Z : (51) B. These are B a bA Z 1 p C It is now necessary to show that the term (48) in the four- d5x g V 1 ~ M z 2 X B B C X dimensional action is actually of this form. Applying the 2 5 usual supergravity formalism, using the Ka¨hler potential 1 B A ~ D Im M z X A MA D z X found in [1] and reproduced in (A3) of Appendix A, we Z 1 find that this is indeed the case. However, we postpone a 2G ^ AX~ ~ XA : (47) 2 2 A A proof of this until the next section and Appendix B, where 5 we also include anti-M5-branes. Here, instead, we will To find the terms in the four-dimensional theory which assume that (49) is the correct four-dimensional flux super- depend on the fluxes, therefore, it suffices to study the potential and use (50) and (51) to calculate its explicit form dimensional reduction of these two terms. The quantiza- in terms of the affine complex coordinates za in the large tion conditions (45) and (46) show that the fluxes them- complex structure limit. 2=3 selves are already first order in the 11 expansion. This We proceed as follows. It is a well-known fact (follow- makes the first term in (47) second order in this expansion. ing from an examination of the large Ka¨hler modulus limit In counting these orders, one should ignore, as usual [48], of the mirror compactification) that the prepotential, G, 2 the overall prefactor of the action proportional to 11 . takes the following form in the large complex structure Explicit calculation reveals that the terms involving flux limit: in the second component of (47) will also be at least second ~ a b c 2=3 1 da b cZ Z Z order in 11 and, even at this order, contain at least one G 0 : (52) four-dimensional derivative. Since we are interested here 6 Z in obtaining potential energy terms only, we discard these Note here we are splitting up the index A into an index a contributions henceforth. and the remaining possible value 0. Substituting this ex- A straightforward dimensional reduction of the first term pression into (50) we find the following: p of Eq. (47), using the reduction ansatz described in [1,49], ~ a b c ~ a b 2 1 da b cZ Z Z 1 da b cZ Z gives the following as the flux-dependent contribution to W X0 Xc flux 2 6 Z0 2 2 Z0 the potential energy in four dimensions. 4 Z p ~ a ~ 0 1 4 1 ~ B XaZ X0Z : (53)
2 d x g4 3 XA MA B z X 2 4 b0V0 We now use the definition of the affine coordinates, za 1 A C ~ D Za Im M z XC MC D z X : (48) . We also use the scale invariance of the physical theory Z0 under rescalings of the homogeneous coordinates Z to set Here 4 is the four-dimensional Planck constant defined by 0 to 1. 2 2 ~ Z
4 5= , and X, X are the quantized quantities de- p 2 1 1 scribed in (45) and (46). An examination of the parameters W d~ zazbzcX0 d~ zazbXc appearing in this action, and the quantized quantities flux 2 6 a b c 2 a b c 4 therein, reveals that these terms are of order 2 2 . The S R ~ a ~ fact that they are already second order in the strong cou- Xaz X0 : (54) pling parameter S, means that we will not consider higher order corrections to this flux potential arising from the warping. Such contributions are small and beyond the 1Note that the ZB denote a set of projective coordinates on the order to which we calculate the four-dimensional effective complex structure moduli space. One can obtain a set of affine coordinates by the usual procedure of picking one nonvanishing theory in this paper. homogeneous coordinate and dividing the others with respect to It is well known that flux potentials in supersymmetric it: that is, za Za=Z0, a 1; ...;h2;1 when Z0 is not zero. It is theories should be derivable from a Gukov-Vafa-Witten this set of affine coordinates that appears in action (1).
126012-10 FLUX, GAUGINO CONDENSATION, AND ANTIBRANES IN ... PHYSICAL REVIEW D 76, 126012 (2007) Using Eqs. (45) and (46) we obtain, finally, (35) which we IV. FLUX IN HETEROTIC M THEORY WITH repeat here: ANTIBRANES
p In this section, the explicit contributions of flux to the 2 v1=6 1 1 W d~ zazbzcn0 d~ zazbnc four-dimensional effective action of heterotic M theory flux 2 0 2 6 a b c 2 a b c 4 with anti-M5-branes will be derived. As in the previous a section, we find it most transparent to begin the discussion z na n0 : (55) in the 11-dimensional context. The role of flux in five and four dimensions is then derived by dimensional reduction. The results in this section were derived implicitly as- The exposition is similar to that given in the supersym- suming two important constraints on the magnitude of the metric case. Hence, we use Sec. III as a template, explicitly G-form expectation values. The first of these concerns the showing how anti-M5-branes alter the conclusions therein. nzm value of the nonzero mode, Gabc d . The smallness of the S As discussed in Sec. II, we will assume there are N 1 and R parameters ensures that the backreaction of this flux M5-branes and a single anti-M5-brane indexed by (p )in is adequately described by the warping it induces along the the bulk space. The extension of these results to an arbi- orbifold direction. Hence, one can continue to perform the trary number of antibranes is straightforward. analysis on a Calabi-Yau threefold, despite the presence of To begin, note that the discussion of the components of this component of the background flux, although the ge- background four-form flux G that are relevant to stable ometry of this space does change along the orbifold direc- vacua with maximal space-time symmetry, tion. This assumption is standard in the discussion of all Subsection III A, is unchanged by the addition of anti- strong coupling heterotic vacua. The second assumption branes. However, an anti-M5-brane located in the bulk concerns the magnitude of the G-flux corresponding to the space does alter the Bianchi identity and its sources dis- various nonvanishing GABCy harmonic components. For cussed in Subsection III B. Consider Bianchi identity (15). these components, we are making the standard ‘‘weak The right-hand side is sourced by the four-form charges flux’’ approximation. That is, we assume that, despite the localized on the orbifold fixed planes and N M5-branes presence of these G-fluxes, one can still compactify on a given in (16). In the case of N 1 M5-branes and a single Calabi-Yau threefold and does not require a more general anti-M5-brane, the form of expression (15) remains un- manifold of SU 3 structure. This approximation is par- changed. However, the sum on the right-hand side now ticularly easy to control from the point of view of the five- includes an anti-M5-brane charge J p . The charges on the dimensional theory. The fluxes are expectation values of two orbifold planes and the N 1 M5-branes given in (16) the y-derivative of certain five-dimensional moduli. Thus, remain unchanged. However, the anti-M5-brane four-form the Calabi-Yau approximation is valid whenever the five- charge is given by dimensional y-derivatives of the associated moduli are p p
small compared to the Calabi-Yau compactification scale. J C2 : (56) This will be the case whenever the number of units of flux p Note that C2 remains a holomorphic curve on which the is chosen to be sufficiently small and the moduli take anti-M5-brane is wrapped, the reverse orientation of the appropriate values. brane being expressed by the minus sign. As with all the Finally, one may ask about the effect of the diffuse p other four-forms in (16), J is closed on the CY3. With the source of curvature, which the flux represents, on the p bulk warping. The above discussion makes it clear that, proviso that J be given by (56) rather than the last line in for the case where we have a small number of units of flux (16), every expression and conclusion of Subsection III B quanta, this warping modification can only come in at remains unchanged. second order in S. This, as was explained in [1], is at a higher order than is needed to calculate the action of the A. The nonzero mode theory to the orders we consider here. Therefore, we can Now reconsider Subsection III C, where we solve for the consistently neglect this contribution to the warping.2 background flux, in the presence of an anti-M5-brane. Let Having completed our review of flux in supersymmetric us begin with the nonzero mode discussed in heterotic M theory let us now proceed to examine how the Subsection III C 1. As above, all expressions and equations above discussion changes in the presence of antibranes. in this subsection remain unchanged with the proviso that J p be given by (56) everywhere. Be that as it may, the 2It is interesting to note that this correction to the warping is of physical conclusions for the four-dimensional theory the same size as the potential term we are keeping. This is change dramatically. To see this, first consider the har- consistent because of the normalization we choose for our monic approximation to the nonzero mode given in (26) moduli fields. This normalization ensures that the correction to p the action, which comes from terms linear in this warping, and (27), where, now, k k is defined by expressions vanishes and only the quadratic and higher contributions are (22) and (56). It was stated in [1] that, when the 11- present. dimensional theory is dimensionally reduced on the CY3
126012-11 JAMES GRAY, ANDRE´ LUKAS, AND BURT OVRUT PHYSICAL REVIEW D 76, 126012 (2007) in the background of this nonzero mode, the effective five- the coupling of their field strengths to moduli, is modified dimensional theory is given by the action in Eq. (1). As to include nonholomorphic terms. Both of these effects discussed in Sec. II, the N 1 M5-branes and the anti-M5- explicitly break supersymmetry and both vanish if the brane contributions to the nonzero mode enter action (1) anti-M5-brane is removed from the theory. In this section, p ^ k we will discuss the first of these, that is, the supersymmetry through the coefficients k which appear in k, n p , and j . It is important to note that despite the appearance of breaking potential energy. We defer the discussion of the p modified gauge couplings to the following section on an anti-M5-brane in the vacuum, action (1)isN 1 gaugino condensation, where it becomes relevant. supersymmetric, both in the five-dimensional bulk space The explicit form of the supersymmetry breaking mod- and on all four-dimensional fixed planes and branes. The uli potential energy for an arbitrary number of Ka¨hler and reason for this is that the dimensional reduction was carried complex structure moduli, N 1 M5-branes and one anti out with respect to the supersymmetry preserving Calabi- M5-brane was given in Sec. 5 of [1]. The potential, written Yau threefold, and did not explicitly involve the anti-M5- in terms of the complex scalar components of the moduli brane. However, as discussed in [1] and Sec. II of this superfields, was found to be paper, the dimensional reduction to four dimensions does involve the anti-M5-brane. Hence, one expects the four- V V 1 V 2; (57) dimensional effective action to explicitly break N 1 supersymmetry. This is indeed the case, as was shown in where [1]. Remarkably, the effect of the antibrane was found to 2 appear in only two specific places in the action, to the order 2 0 4 k k KT KD V T T e 4 (58) at which we calculate. First, the four-dimensional theory 1 2 k exhibits a potential energy for the moduli. Second, the kinetic energy functions for the gauge fields, specifically, and
2 pX 1 NX 1 pX 1 2 Z p Z p Z p Z p Z p Z p 2 0 KT 2KD kl p p p V e 4 K 2 4 2 T l k m m k m m k p m m p 0 m T T p p 1 m T T p 0 m T T NX 1 Z Z NX 1 Z Z Z Z 2 p p p p 1 p p p p (59) k p m m k p m m p n n k p p 1 m T T p 0 m T T n T T 3 are the 2=3 and 4=3 contributions, respectively. The com- nzm 11 11 anti-M5-brane, the nonzero mode component G can- k 1;1 abcd plex scalar fields T , k 1; ...;h and Z p , p not contribute to the Gukov-Vafa-Witten superpotential 0; ...; p; ...;N 1, corresponding to the Ka¨hler moduli (33) and, hence, leads to vanishing potential energy for and the location moduli of the M5-branes and anti-M5- the moduli fields in the four-dimensional theory. The van- brane, respectively, are defined, along with the complex ishing of (57) in the case of no anti-M5-brane and, hence, dilaton scalar S, in Eq. (A1) of Appendix A. Similarly, the k ! 0, constitutes an explicit proof by dimensional re- Ka¨hler potentials KT and KD are given in (A4) of that duction of this conclusion. Note, however, that when an appendix. The key point is that both contributions (58) and anti-M5-brane is present, the potential energy is nonsuper- (59)toV are proportional to the anti-M5-brane tension symmetric and need not be derived from a superpotential. 1 Let us now include the nonharmonic contributions to the k 2 k k k; (60) four-form charges of the orbifold planes, the N 1 M5- where we have used the notation, introduced earlier, that branes and the anti-M5-brane. The effect of this is to add an p p k k and k k . Hence, the nonvanishing of this additional contribution to the G nzm component of the potential is due entirely to the existence of the anti-M5- abc d nonzero mode. Furthermore, as discussed in brane. We conclude that when sourced by an anti-M5- Subsection III C 1, the nonharmonic modes will induce brane, the harmonic contribution to the nonzero mode nzm nonvanishing values for the (1, 2) and (2, 1) components, Gabc d , together with a series of other contributions such nzm nzm as the sum of the tensions of the extended objects, does Gbc dy and Gacdy , respectively. One expects these addi- induce a nonvanishing supersymmetry breaking potential tional nonharmonic contributions to the nonzero mode to energy in the four-dimensional theory. induce corrections to the four-dimensional supersymmetry Were the anti-M5-brane to be removed and replaced by breaking potential (57). This is indeed the case. However, an M5-brane, all k ! 0 and, hence, V would vanish. This as discussed in [38] such additional contributions are sup- result is completely consistent with the statement given in pressed relative to the harmonic contributions by powers of Subsection III C 1 that in the supersymmetric case with no the parameter R. For that reason, we do not display their
126012-12 FLUX, GAUGINO CONDENSATION, AND ANTIBRANES IN ... PHYSICAL REVIEW D 76, 126012 (2007) explicit form here. Suffice it to say that these terms remain Sec. II of this paper. Remarkably, despite the fact that proportional to the anti-M5-brane tension k k. Once this background contains an anti-M5-brane, we find that again, if the anti-M5-brane is removed and replaced with the flux-dependent contribution to the potential energy in an M5-brane, then k ! 0 and these terms vanish. Again, four dimensions is given by this is completely consistent with the statement in Z nzm nzm 1 p 1 Subsection III C 1 that neither Gbc dy nor Gacdy can con- 4 ~ B
2 d x g4 3 XA MA B z X tribute to the Gukov-Vafa-Witten superpotential (33). 2 b V0 4 0 1 A C ~ D B. Harmonic flux and flux quantization Im M z XC MC D z X ; (61) Having discussed the nonzero mode, let us now recon- sider Subsection III C 2 and Subsection III C 3, the har- that is, the same expression (48) as in the supersymmetric monic flux and its flux quantization and contribution to case, up to the order in our expansions which we work to. the 4D effective action, respectively, in the presence of an Hence, although the anti-M5-brane does give rise to ex- anti-M5-brane. This is easy to do since, remarkably, the plicit supersymmetry breaking terms in the four- antibrane does not alter any of the conclusions of those dimensional theory, the flux sector of the effective theory subsections. First, consider Subsection III C 2. Since they remains N 1 supersymmetric. 2;1 are closed on the CY3, the 2 h 1 harmonic forms Since this term appears in a four-dimensional N 1 H Gab cy , Gabcy , Gabcy, Ga b cy in G are not sourced by the supersymmetric action (in the situation without anti- fixed planes, N 1 M5-branes, or the anti-M5-brane. branes), it must be possible to express the Lagrangian Hence, the addition of the anti-M5-brane does not affect density of (61) in the form the harmonic flux in any way. Nor does it affect the con- 2 clusions about the Gukov-Vafa-Witten superpotential and Kmod ij 2 2 V e 4 K D W D W 3 jW j ; expression (35) in that subsection. Since supersymmetry is flux mod i flux j flux 4 flux broken in the four-dimensional effective theory by the anti- (62) M5-brane, this last statement requires further discussion, which we will return to shortly. where Kmod is the Ka¨hler potential of the moduli and Wflux Now consider Subsection III C 3. First note that the flux is the holomorphic superpotential generated by the har- quantization condition (36), as well as the constraint monic flux. This is indeed the case. The proof, as originally nzm given in [37], is somewhat intricate, so we relegate it to Eq. (38) for the nonzero mode components Gbc dy and nzm Appendix B. The result is that the Lagrangian density of Gacdy , remain unchanged in the presence of an anti-M5- (61) can be written in the form (62) where the Ka¨hler p potential K is given in expression (A3)of brane, with the proviso that the three-form !3 associated mod with the antibrane is defined locally by d! p J p where Appendix A and Wflux is the Gukov-Vafa-Witten super- 3 potential J p is given by (56). It follows that the quantization nzm p condition (37) for the nonzero mode Gabc d and the quan- Z H 2 1 tization condition (39) for the harmonic modes G are also Wflux ^ G 2 1=2 1 unchanged. Furthermore, the presence of the anti-M5- 4 v X S =Z2 p brane does not alter the definitions of the four-dimensional 2 A ~ XAG X~ ZB ; (63) flux constants X , XB or their quantization conditions 2 A B given in (45) and (46), respectively. 4 One must now consider the dimensional reduction of the where the complex structure moduli space is parametrized five- to the four-dimensional theory including this quan- by the periods (ZB, G Z ) defined by tized flux. Here, however, this must be carried out in the A presence of an anti-M5-brane. Again, the starting point is Z Z the five-dimensional action given in (1), now, however, B Z ; GA Z : (64) B with N 1 M5-branes and an anti-M5-brane. As discussed a bA earlier, the form of this action does not change when an antibrane is included in the vacuum. Hence, the two terms This conclusion is valid for both the supersymmetric case in the five-dimensional action containing the scalar fields and when there is an anti-M5-brane in the vacuum. It is A ~ and B are still given by expression (47). As discussed consistent with, and proves, the form of the flux super- previously, only the first term in this expression can con- potential presented in (49) and (50) at the end of tribute to the potential energy. We now perform a dimen- Subsection III C 3. Similarly, the expression given in sional reduction of the first term in (47) with respect to the Subsection III C 3 for the superpotential in the large com- supersymmetry breaking vacuum described in [1] and plex structure limit,
126012-13 JAMES GRAY, ANDRE´ LUKAS, AND BURT OVRUT PHYSICAL REVIEW D 76, 126012 (2007)
p 2 v1=6 1 1 term is familiar from both the 11-dimensional theory and W d~ zazbzcn0 d~ zazbnc flux 2 0 2 6 a b c 2 a b c the ten-dimensional weakly coupled heterotic string 4 [36,51]. The first step in including gaugino condensates a in the reduction of the five-dimensional theory is to define z na n0 ; (65) a new set of fields. This avoids the appearance of squares of remains unchanged in the presence of an anti-M5-brane. delta functions in the discussion. In five dimensions, the relevant field redefinitions are
V. GAUGINO CONDENSATION IN HETEROTIC M X X ; (67) THEORY WITH ANTIBRANES 2 In this section, we discuss gaugino condensation in the 1 5 0 N 1 Xy Xy y y ; presence of antibranes. Our exposition will proceed in 32 GUT several steps. First, we show that the form of the potential (68) energy terms arising from gaugino condensation, when written in terms of the condensate itself, are unchanged where the quantities 0 and N 1 are the condensates, from the result one obtains without antibranes. Second, we that is, the fermion bilinears, themselves. For complete- argue that a gaugino condensate will indeed occur in this ness, we have introduced a condensate on each orbifold nonsupersymmetric setting, despite the fact that most of plane. One of these can always be set to zero, if so our knowledge of this effect is based on the structure of required.3 unbroken N 1 supersymmetric gauge theory. Finally, we In terms of these new quantities, the complete square in compute the condensate as an explicit function of the the five-dimensional action simply becomes a kinetic term moduli fields. We conclude that the significant changes for X , as can be seen using Eq. (1). Therefore, the only that antibranes induce in the gauge kinetic functions of the place where the condensate explicitly appears is in the orbifold gauge fields lead, via the gaugino condensate, to Bianchi identity of X . This is given by [50] important modifications of the potential energy. 2 5 0 0 dX 4J @ y y 32 A. The potential as a function of the condensate GUT N 1 N 1 We begin by reviewing how gaugino condensates are 4J @ y ; (69) included in the dimensional reduction of five-dimensional i heterotic M theory, as was first presented in [50]. We then where J is proportional to @ W i , the derivative of the show that, when written in terms of the condensate itself, matter field superpotential on the ith orbifold plane. Note the presence of antibranes does not alter the form of the that, in deriving (69), it is essential to realize that the potential. Unlike the flux contribution to the potential condensate can depend on the four-dimensional moduli. i energy, which is most naturally described in terms of the Otherwise, one would have @ 0 and the source 2;1 A ~ terms in the Bianchi identity would be condensate inde- 2 h 1 real scalar fields and B, the gaugino con- densate contribution is most simply expressed in terms of pendent. This would lead to vital terms in the dimensional the flux of a single complex scalar field . These fields are reduction being missed. It is also important to note that the related by antibranes do not appear in Bianchi identity (69). As was ! ! discussed in [1], antibranes simply do not source the bulk A A A Z fa fields involved in the present discussion. This is one of the ~ a H:c:; (66) B GB ha B essential features of antibranes. It ensures that the potential due to flux and gaugino condensation is of the same form A where the periods Z ; GB were defined in (51) and the when antibranes are present as it is in the supersymmetric A expressions for fa and ha B are found in the Appendix of case. [37]. The field supplies two of the four bosonic compo- To take into account the presence of a gaugino conden- nents of the ‘‘universal’’ hypermultiplet in five dimensions, sate, we simply have to solve the above system and then whereas the complex scalar fields a are half of the bo- perform a dimensional reduction about the result. A solu- sonic components of the remaining h2;1 hypermultiplets. tion for X has, in fact, several contributions. There is one contribution induced by matter field fluctuations on the The relevant quantity for gaugino condensation is X @ , the five-dimensional field strength associated with . As obtained by dimensional reduction from 11 dimen- 3It should be noted that the gauge groups which appear on sions, the five-dimensional action of heterotic M theory, various stacks of branes and antibranes in the bulk could also where the fields A, ~ are written in terms of , a using give rise to condensation in situations where they are strongly B coupled and non-Abelian. We will ignore this possibility here, (66), contains a term, consisting of -flux and gaugino although it should be kept in mind in a detailed discussion of bilinears, which is a ‘‘complete square.’’ This specific moduli stabilization.
126012-14 FLUX, GAUGINO CONDENSATION, AND ANTIBRANES IN ... PHYSICAL REVIEW D 76, 126012 (2007) boundaries, as indicated in the above by the source terms This result can be processed into a more user friendly i J in the Bianchi identity. Next there is a contribution due form by employing the results on special geometry given in to whatever harmonic flux we may choose to turn on, as Appendix B. Using (72), identities (B16) and (B17), the described in the previous section. Finally, there is the fact that M z is a symmetric matrix and the definitions of contribution obtained from the magnetic charges for this the flux superpotential and Ka¨hler potentials given in (63), field proportional to the derivative of the condensate. Since (A3), and (A4), respectively, we find that the potential all of these contributions are small in our approximations, energy is we can treat each of them separately, simply adding the 1 2 2 KD KT K z 2
4 4 results to obtain the full expression for X . In this section, Vc c=f f 2 e 2e jXy j 2 4 we discuss the last of these, that is, the contribution to X p p 2 2 due to the condensate. This contribution will be denoted by iXy 2 4Wflux i 2 4WfluxXy X . We find that 2 ij 4Kmod e KmodDiWfluxDj Wflux 2 5 0 N 1 2 2 Xy 2 ; (70) 3 4jWfluxj : (74) 64 GUT Note that, as in the pure flux potential discussed in the 2 previous section, the pure gaugino condensation and flux- 5 0 y 0 N 1 X @ : gaugino condensation cross term contributions to the po- 64 GUT tential for the moduli are independent of both the M5- (71) branes and anti-M5-branes in the vacuum, up to second Having found the complex -flux induced by gaugino order in S and the small condensate scale. Hence, the condensation, we would like to reexpress it in the same functional form of expression (74) is the same whether or field basis as in the previous section, that is, in terms of the not antibranes are present. How, then, does the action for theories with and without flux XA and ~ of the real scalar fields A and ~ . This is XB B antibranes differ. The answer lies in the explicit form of the easily done using relation (66). Differentiating this rela- condensates 0 and N 1 when expressed in terms of tion, we find that the gaugino condensate makes the fol- the moduli fields. It follows from (70) that these conden- lowing contributions to these fluxes: sates determine Xy , which we must now specify. Before A A A ~ doing this, first notice from (70) that Xy is proportional to X Z Xy Z X y ; XB GBXy GBX y : the sum of the two condensates. It turns out that one can (72) calculate Xy for each condensate separately, simply adding This result uses the fact that the warping in the complex the results at the end of the computation. We can, therefore, structure moduli is first order in our expansions. Therefore, restrict the discussion to a single boundary wall. since the condensate contribution to the -flux is already small, this warping can be neglected here. B. Condensate scales I: Supersymmetric case Let us insert this contribution to the X flux, along with In this subsection, we review the computation of the the contributions discussed in the previous section, into condensate Xy for the supersymmetric case without anti- (61). This gives us the four-dimensional potential energy branes. Let the gauge bundle in this sector have structure due to both gaugino condensation and harmonic flux. The group G. Then the low energy theory contains a super result is Yang-Mills connection with structure group H, where H
Z 1 p 1 is the commutant of G in E8. For simplicity, we will d4x g X~ X~ 2 2 4 b3V A A assume that H is a simple Lie group. This Yang-Mills 4 0 0 theory is coupled in the usual way to supergravity. B B 1 A C MA B z X X Im M z Therefore, the gauge coupling is determined by the values of certain moduli. There can also be matter multiplets in ~ ~ D D XC XC MC D z X X : (73) this sector. However, they are irrelevant to the discussion in this paper and, with the exception of their contribution to Note that, as in the previous section, the corrections to the the beta function, we will ignore them. Such a theory is warping of the bulk fields due to the presence of antibranes known to undergo gaugino condensation under certain do not change the form of the above expression from the conditions [52]. supersymmetric result. This is due to (1) the order at which Gaugino condensation induces a superpotential for the the above terms appear in the expansion parameters of moduli fields entering the gauge coupling. This superpo- heterotic M theory and (2) the low scale of the condensate. tential has been calculated, in [53] for example, using an These considerations make such corrections outside of the effective low energy field theory where the condensate approximations to which we work. itself is the lowest component of a superfield. For the
126012-15 JAMES GRAY, ANDRE´ LUKAS, AND BURT OVRUT PHYSICAL REVIEW D 76, 126012 (2007) simple case where the gauge coupling depends on the global supersymmetry result. One should, therefore, also dilaton modulus S only, this superpotential is found to be introduce gravitational corrections to the value of the con- S densate once we include these terms. Gravitationally, of W Ae ; gaugino (75) course, the condensate is coupled to everything in the where, at tree level, A is a constant of order v 1=2 , theory and so it will take a very complicated form at this level. Second, component analyses of the kind performed 6 in the previous subsection and [36] are somewhat naive. (76) b0 GUT We should also include the effects of integrating out the strongly coupled non-Abelian gauge fields and so forth. and b is the beta-function coefficient for the effective 0 This could also introduce new terms which would contrib- gauge theory. The dilaton field S is defined in our context ute to the problematic polynomial prefactor. in (A1) of Appendix A. For example, for H E there are 8 From a physical point of view, the inability to reproduce no matter fields and the polynomial prefactor for arbitrary moduli expectation
b 0 90: (77) values is not very important. In any regime of moduli space which is well described by effective supergravity, the real Adding Wgaugino to Wflux, and inserting them into the parts of the moduli fields, in particular, the dilaton and usual supergravity formalism, yields the complete moduli Ka¨hler moduli, should be much greater than unity (taking potential energy induced by gaugino condensation and the relevant reference volumes to be string scale in size). In flux. To evaluate the condensate Xy , one need only com- this regime, one need only keep the leading terms in an pare this potential energy to expression (74). This is most expansion in the inverse of the real parts of these moduli. easily done at low energy and for large values of the This is precisely the gaugino condensation limit. As dis- moduli. We will refer to this as the ‘‘gaugino condensation cussed above, the component action approach can then limit.’’ In this limit, we find that the leading terms for the successfully reproduce the supersymmetric potential en- pure gaugino and gaugino/flux contributions to the super- ergy using the simple form for the condensate given in gravity potential simplify to the following result: (79).
2K 2 2 2 2 S S The above result was derived assuming the gauge cou- V e 4 mod S S jAj e c c=f 4 pling depends on the dilaton S only. In this case, the gauge 2 S 4 S S Ae Wflux H:c : (78) kinetic function is given by
This expression is precisely reproduced by Eq. (74) if one f S: (80) chooses the leading order contribution to the condensate to It will be useful to rewrite superpotential (75)as be f W Ae : (81) 2 A gaugino K z 2 S
4 X y ie p 4 S S e : (79) 2 We now want to extend this result to the cases where there Note that it is perfectly consistent for the chiral condensate, are threshold corrections, including those due to super- which is the lowest component of a chiral superfield in the symmetric five-branes. In these circumstances, it is well effective theory, to take a nonholomorphic form. This is known [37,49,54] that the gauge kinetic functions on the because this relation simply describes the value of the field (0) and (N 1) boundary walls generalize to in vacuum and is not an identity on field space. XN N 1 k Expression (79) was computed in the gaugino conden- f 0 S 0 k T 2 Z p^ ; (82) sation limit. Can one find an expression for Xy at any p^ 1 energy and for arbitrary moduli expectation values? To do this, the complete supergravity moduli potential in- N 1 k f N 1 S 0 k T : (83) duced by Wgaugino and Wflux must be compared to expres- sion (74) and the functional form of the condensate Here Tk are the h1;1 Ka¨hler moduli, the p^ index runs over inferred. One finds that the condensate continues to be all N 1 supersymmetric five-branes but excludes the proportional to the same exponential factor of the gauge antibrane, and Z p^ are the location moduli of these five- k kinetic function as in the gaugino condensation limit. branes. The complex fields T and Z p^ are defined in (A1) However, in general, the prefactor is now a very compli- of Appendix A. The generalization of the gaugino conden- cated function of the moduli. sate superpotential is now straightforward. It is simply That it is difficult to reproduce the lower order terms in given by expression (81), where f takes the form (82) the large moduli expectation values, as was first noted and (83) on the (0) and (N 1) boundary walls, within the context of the weakly coupled heterotic string respectively. in [36], should not come as a surprise. First, parts of this One can now find the gaugino condensate in the pres- potential correspond to gravitational corrections to the ence of threshold effects by comparing expression (74)
126012-16 FLUX, GAUGINO CONDENSATION, AND ANTIBRANES IN ... PHYSICAL REVIEW D 76, 126012 (2007) with the complete potential energy derived from this modi- In the gaugino condensation limit, two types of terms fied superpotential. As above, this turns out to be difficult survive in the Yang-Mills sector of the low energy effective in the general case of arbitrary moduli expectation values. action. These are the kinetic terms for the gauge fields and Once again, the problem greatly simplifies in the gaugino those for the gauginos. We presented the gauge field kinetic condensate limit of low energy and large moduli values. terms, including the contribution of the antibrane, in [1]. The threshold corrections do, however, further complicate These are of the same form as in the supersymmetric case. the polynomial prefactor. Happily, in the physically inter- However, the gauge kinetic functions for the (0) and (N esting region of moduli space further simplification is 1) boundary walls are now given by possible. Note, using the definition of the superfields Z p^ XN 2 N 1 k p k k in Eq. (A1) of Appendix A, that each of the two types of f 0 S 0 k T 2 Z k T T threshold corrections in Eqs. (82) and (83) is suppressed p 1 3 relative to the leading dilaton term by factors of S,as Z Z 2 k k p p defined in (3), (12), and (14). Therefore, since we require k T T k k k T T S 1 for the validity of the four-dimensional effective theory, we find that Z Z 2 p p ; (89) k k N 1 k k T T S> 0 k T ; (84)
N 1 XN f S 1 Tk Tk T k N 1 0 k 3 k S> 0 2 Z p^ : (85) p^ 1 Z Z 2 k k p p k T T k k ; (90) It follows that, in the polynomial prefactor, it is a very good k T T approximation to simply ignore the threshold corrections, respectively. Similarly, we find that the gaugino kinetic yielding the same prefactor as discussed previously. Note, terms are of the same form as in the supersymmetric case, however, that even though conditions (84) and (85) but with the gauge kinetic functions replaced by expres- strongly hold, the values of the right-hand sides of these sions (89) and (90). expressions are generally much larger than unity. Hence, The reason the gaugino kinetic terms retain their super- one should never drop the threshold and five-brane correc- symmetric form is the following. In the five-dimensional tions to f in the exponential. theory, there is only one set of kinetic terms for the Putting this all together, we conclude that in the gaugino gauginos. These are localized on the appropriate orbifold condensate, S 1 limit, the gaugino condensate is given fixed plane. Upon dimensional reduction, there are three by possible sources of four-dimensional kinetic terms for 2 A K z 2 f these fields. The first is the direct dimensional reduction X y ie 4 p S S e ; (86) 2 4 of the five-dimensional kinetic terms. The second arises where the gauge kinetic function f is when the contribution to the warping of the bulk fields, which is proportional to the gaugino kinetic term, is sub- XN N 1 k p^ stituted into the tension terms of the various extended f 0 S 0 k T 2 Z ; (87) p^ 1 objects. Finally, a contribution arises when the warping terms due to the tension of the extended objects and the N 1 k f N 1 S 0 k T (88) gaugino fluctuations on the fixed planes are substituted into the bulk action. A simple argument reveals that two of for a condensate on the (0) and (N 1) boundary walls, these contributions always cancel. respectively. Consider the simple system of just the bulk action and the tension terms on the extended objects. This action has a C. Condensate scales II: Antibrane case reduction ansatz given by the warped antibrane back- Let us now turn to the case of heterotic M theory in the ground presented in [1] and in expressions (9)–(11) above. presence of antibranes. At first glance, it is not obvious Now add, as a perturbation, the gaugino terms in the action how to proceed. As we have just seen, the arguments used and the correction they give rise to in the reduction ansatz. in a discussion of gaugino condensation are firmly rooted Substituting the reduction ansatz into the action and inte- in the assumption that the theory is supersymmetric. How, grating, we obtain the same four-dimensional action as then, does one proceed when supersymmetry is broken by before plus the new four-dimensional gaugino kinetic antibranes? The key observation we will make is that when terms. The term which arises from substituting the new one takes the low energy, large modulus limit of a theory warping piece into the zeroth order action clearly vanishes. with antibranes, the system returns to a supersymmetric This is because, by definition, the zeroth order background form. Hence, one can continue to apply the usual argu- extremizes the action in the absence of gauginos. One is ments for gaugino condensation. left with the direct reduction of the five-dimensional gau-
126012-17 JAMES GRAY, ANDRE´ LUKAS, AND BURT OVRUT PHYSICAL REVIEW D 76, 126012 (2007) gino kinetic term as the four-dimensional kinetic term for tion for heterotic M theory in the presence of antibranes. these fields. This reproduces exactly the supersymmetric The result is
gaugino kinetic terms written, however, in terms of the 2 1 4Kmod 2 f 2 2 gauge kinetic function (89) and (90). Vc c=f f 2e 4jA S S e j A 4 S S We conclude that the action for the gauge theories on the e f W 2W A S S e f boundaries is, in the gaugino condensation limit, of exactly flux 4 flux 2 ij 4Kmod 2 2 the same form as in globally supersymmetric Yang-Mills e KmodDiWfluxDj Wflux 3 4jWfluxj : theory. The gauge kinetic functions which appear in all of (94) the kinetic terms are, however, given by the nonholomor- phic combination of moduli derived in [1] and presented in In this expression Wflux is given in (63), f is presented in (89) and (90). Because of this nonholomorphicity, the (92) and (93), and K is defined in (A3) of Appendix A. Yang-Mills sector of the theory is not, in general, super- mod symmetric. However, in any situation where the moduli are treated as constant, that is, independent of space-time, the VI. CONCLUSIONS Yang-Mills sector is supersymmetric. The nonholomorphy In this paper, we have included the effects of flux and of the gauge kinetic function is not manifest if the moduli, gaugino condensation in the four-dimensional effective and, hence, the gauge coupling, are simply regarded as description of heterotic M theory including antibranes numbers. Therefore, in this region of moduli space one can [1]. While the parts of the resulting potential which are apply exactly the same analysis of the gauge condensate as due purely to flux are unchanged from the supersymmetric in the supersymmetric case discussed in the previous sub- result, those which are caused by gaugino condensation are section. That is, one simply needs to replace f by the modified in important ways. correct gauge kinetic function for the case at hand. We It is not even obvious, a priori, that gaugino condensa- conclude, therefore, that in the gaugino condensate, S 1 tion would occur in such a nonsupersymmetric setting. limit, the condensate in the presence of antibranes is given However, because in a certain limit the system still looks by like globally supersymmetric gauge theory, we have ar- 2 A K z 2 f gued that indeed it does. We have also argued that we can X y ie 4 p S S e ; (91) 2 4 calculate an approximation to the potential for the moduli where the gauge kinetic function f is which it gives rise to. It should be noted that, in addition to the points explicitly mentioned in the proceeding sections, XN 2 N 1 k p k k the threshold corrections which antibranes give rise to in f 0 S 0 k T 2 Z k T T p 1 3 the gauge kinetic functions can completely change which extended objects in the higher dimensional theory are Z Z 2 Tk T k p p strongly coupled—and so, which exhibit gaugino k k k k T T condensation. Z Z Let us summarize the results derived in Secs. IV and V. 2 p p ; (92) Tk T k We have shown that the moduli potential energy that arises k from (1) perturbative effects, (2) gaugino condensation,
N 1 and (3) flux in heterotic M theory vacua with both M5- f S 1 Tk Tk T k N 1 0 k 3 k branes and anti-M5-branes is, to our order of approxima- Z Z 2 tion, given by Tk T k p p (93) k Tk T k k V V 1 V 2 Vc c=f f; (95) for a condensate on the (0) and (N 1) boundary walls, respectively. 2 2 0 4 k k KT KD This condensate can now be substituted into (74)togive V T T e 4 ; (96) 1 2 k the combined potential due to flux and gaugino condensa-
2 4 pX 1 NX 1 pX 1 2 Z p Z p Z p Z p Z p Z p 0 4 KT 2KD kl p p p V e 4 K 2 2 T l k m m k m m k p m m p 0 m T T p p 1 m T T p 0 m T T NX 1 Z Z NX 1 Z Z Z Z 2 p p p p 1 p p p p ; (97) k p m m k p m m p n n k p p 1 m T T p 0 m T T n T T 3
126012-18 FLUX, GAUGINO CONDENSATION, AND ANTIBRANES IN ... PHYSICAL REVIEW D 76, 126012 (2007)
2 Kmod 2 f 2 antibranes is still given, in terms of these fields, by the Vc c=f f e 4 4jA S S e j A S S usual N 1 supersymmetric formula with the appropri- f 2 2 f e 4Wflux 4WfluxA S S e 2=3 ate Ka¨hler potentials. Up to order 11 , these Ka¨hler po- 2 ij 4Kmod 2 2 tentials are found to be e KmodDiWfluxDj Wflux 3 4jWfluxj : 2 2 2 (98) 4Kscalar 4Kmod 4Kmatter; (A2) Here the superpotential for the flux is given by the ex- where pression K K K K (A3) mod D T p Z p 2 2 A ~ B and W ^ G X G X Z ; (99) flux 2 2 A B 4 X 4 XN Z Z 2 2 p p KD ln S S ; (A4) the appropriate gauge kinetic function, f, should be chosen 4 0 p k k p 1 k T T from (89)or(90), and the Ka¨hler potential Kmod is defined in (A3) of Appendix A. 2 1 k k l l m m
The contribution of nonperturbative membrane instan- 4KT ln dklm T T T T T T ; ton effects will be added to this potential in future work, as 48 will an analysis of the vacua of the resulting system [2]. (A5) ACKNOWLEDGMENTS 2 a a @G @G K z ln 2i G G i z z ; 4 a a A. L. is supported by the EC 6th Framework Programme @z @z MRTN-CT-2004-503369. B. A. O. is supported in part by (A6) the DOE under Contract No. DE-AC02-76-ER-03071 and X 2 KT =3 Mx N
4 by the NSF Focused Research Grant No. DMS0139799. Kmatter e G p MNC p C p x: (A7) J. G. is supported by CNRS. J. G. and A. L. would like to p 0;N 1 thank the Department of Physics, University of The symbol G in K z is the N 2 prepotential of the Pennsylvania where part of this work was being carried h2;1 sector. It is defined in terms of the periods G in (64) out for generous hospitality. A by @ . The significance of the 2=3 expansion GA @ZA G 11 relative to that in is described in [1]. APPENDIX A: FIELD DEFINITIONS AND KA¨ HLER S POTENTIALS FOR HETEROTIC VACUA WITH ANTIBRANES APPENDIX B: REPRODUCING THE FLUX POTENTIAL FROM THE GUKOV-VAFA-WITTEN In this appendix, we briefly state some results from [1] SUPERPOTENTIAL which are required in the main text. Despite the explicit supersymmetry breaking introduced by the anti-M5-brane Here we present a proof that, in heterotic M theory vacua in our vacuum, the kinetic terms in the four-dimensional with N 1 M5-branes and an anti-M5-brane, the Gukov- theory can still be expressed, as in the supersymmetric Vafa-Witten superpotential case, in terms of a Ka¨hler potential and complex structure. p 2 A ~ A
Let us define the complex scalar fields Wflux 2 X GA XAZ ; (B1)
4 XN p 2 k S e i 0 k z p T ; along with Ka¨hler potential Kmod given in (A3), reproduces p 1 the four-dimensional scalar potential energy (61) when Tk e bk 2i k; they are inserted into the supersymmetric expression for 0 the potential. The manipulations presented below were first p k p k Z p z p k T 2i k n p p ; (A1) presented in [37] in the context of vacua without anti- branes. The scalar potential energy generated by Wflux where V e , b e , bk, and z are specified in 0 0 0 p and Kmod is k Sec. II and , , and p are their real scalar super- 2 Kmod ij 2 2 partners given in [1]. Note that in the case with no anti- Vflux e 4 KmodDiWfluxDjWflux 3 4jWfluxj ; branes, each of these complex scalars would be the lowest (B2) component of an N 1 chiral superfield. This is no longer the case when the vacuum contains an anti-M5- where all of the complex fields (A1) are collectively de- i brane. Be that as it may, we find that the kinetic terms of noted by Y and the Ka¨hler covariant derivative is DiW 2 @Kscalar the bosonic sector of heterotic M theory in the presence of @iW 4 @Yi W. Using the form of the Ka¨hler potential
126012-19 JAMES GRAY, ANDRE´ LUKAS, AND BURT OVRUT PHYSICAL REVIEW D 76, 126012 (2007)
2 3 and the fact that the superpotential depends only on the D F D E 2 4 GC DU GF E GC DU 5 complex structure moduli za, it can be shown that XC; X~ 2 C C F C E 4 U G U 2 2 F E KT KD K a b ! V e 4 e 4 K D W D W flux a flux b flux XE : 4K~uv @ K~ @ K~ 2 jW j2 ~ (B7) 4 u v 3 4 flux (B3) XE
1 2K a b 2 2
4 D F 3 e K DaWfluxDbWflux 4jWfluxj : Here, U is given by the expression 2b0V0 (B4)
D F a b A B D 2 D U K @aZ @b Z A 4Z KA Here, we have defined K~ KD KT, the subscripts u, v run over all moduli indices except for a, a , and we have F 2 F B 4Z KB : (B8) ~uv ~ ~ 2 used the fact that K @uK @vK 4 4 .Asin[1], K is the Ka¨hler potential of the complex structure moduli. We now @ @ZA @ A We now define use @ a a @ @ and @ to a @z @z @ZA aZ A KA AK write a b a b A B MA B GA B TA B; K DaWfluxDbWflux K @aZ @b Z @AWflux Im G ZCIm G ZD (B9) 2 W A C B D 4KA flux TA B 2i E F : Z Im GE FZ 2 @BWflux 4KBWflux (B5) One can then use the relations
2 a b A B C ~ D D 2 2 K @aZ @b Z X GC D XD A Z 4KA A B A B B C B C U M U ; U M U ; (B10) 4 B C GB C GA B A B E ~ F 2 F X GE F XF B 4Z KB 2 3 D E D F D E together with the explicit form for U , GC DU GF E GC DU 2 C ~ 4 5 X ; XC 2 C F C E 4 U GF E U A B 1 2K 1A B A B
4 ! U 2e Im M Z Z ; (B11) XE ~ (B6) XE to write
2 3 2 3 D F D E D F D E GC DU GF E GC DU M C DU MF E M C DU 4 5 4 5 C F C E C F C E U G U U MF E U F E 2 3 2K 1 D F 1 D E e 4 MC D Im M MF E MC D Im M 4 5 1 C F 1 C E 2 Im M MF E Im M 2 3 E GCGE GCZ 4 5: (B12) C C E Z GE Z Z
In the last matrix, the relation M G ZB 1 A B A B V C; ~ 2iIm G ZB has also been employed. Finally, using the flux 2 3 X XC A B 2 4b0V0 identity 2 3 1 DF 1 DE "# ! 4MCD ImM MFE MCD ImM 5 E E 2 2 C ~ GCGE GCZ X 1 CF 1 CE jW j ; ; ImM M ImM flux 4 X XC C C E ~ FE 4 Z GE Z Z XE ! XE (B13) : (B14) X~ the scalar potential becomes E
126012-20 FLUX, GAUGINO CONDENSATION, AND ANTIBRANES IN ... PHYSICAL REVIEW D 76, 126012 (2007) This is easily rewritten as in the vacuum. It is, of course, valid in the purely super- 1 symmetric case as well. ~ B Vflux 2 3 XA MA BX Finally, we take the opportunity to state some identities 2 4b0V0 that we use in the body of the paper. These are Im M 1A C X~ M XD ; (B15) C C D B G A MA BZ (B16) which is exactly the four-dimensional flux potential (61) obtained by dimensional reduction. This completes the A B 1 2K
4 proof. We emphasize again that this proof was carried Im MA BZ Z 2e : (B17) out in the presence of both M5-branes and anti-M5-branes
[1] J. Gray, A. Lukas, and B. Ovrut, Phys. Rev. D 76, 066007 [25] V. Braun, Y.H. He, B. A. Ovrut, and T. Pantev, J. High (2007). Energy Phys. 05 (2006) 019. [2] J. Gray, A. Lukas, and B. Ovrut (unpublished). [26] V. Braun, Y.H. He, and B. A. Ovrut, J. High Energy Phys. [3] E. I. Buchbinder and B. A. Ovrut, Phys. Rev. D 69, 086010 06 (2006) 032. (2004). [27] V. Braun, Y.H. He, B. A. Ovrut, and T. Pantev, J. High [4] S. Gukov, S. Kachru, X. Liu, and L. McAllister, Phys. Rev. Energy Phys. 03 (2006) 006. D 69, 086008 (2004). [28] V. Braun, Y.H. He, B. A. Ovrut, and T. Pantev, J. High [5] M. Becker, G. Curio, and A. Krause, Nucl. Phys. B693, Energy Phys. 04 (2006) 019. 223 (2004). [29] V. Braun, Y.H. He, B. A. Ovrut, and T. Pantev, J. High [6] S. Gurrieri, A. Lukas, and A. Micu, Phys. Rev. D 70, Energy Phys. 01 (2006) 025. 126009 (2004). [30] V. Bouchard and R. Donagi, Phys. Lett. B 633, 783 (2006). [7] G. Curio, A. Krause, and D. Lust, Fortschr. Phys. 54, 225 [31] V. Bouchard, M. Cvetic, and R. Donagi, Nucl. Phys. B745, (2006). 62 (2006). [8] B. de Carlos, S. Gurrieri, A. Lukas, and A. Micu, J. High [32] B. Andreas and G. Curio, arXiv:hep-th/0602247. Energy Phys. 03 (2006) 005. [33] A. E. Faraggi, E. Manno, and C. Timirgaziu, Eur. Phys. J. [9] L. Anguelova and K. Zoubos, Phys. Rev. D 74, 026005 C 50, 701 (2007). (2006). [34] O. Lebedev, H. P. Nilles, S. Raby, S. Ramos-Sanchez, M. [10] V. Braun and B. A. Ovrut, J. High Energy Phys. 07 (2006) Ratz, P.K. S. Vaudrevange, and A. Wingerter, Phys. Lett. 035. B 645, 88 (2007). [11] M. Serone and A. Westphal, J. High Energy Phys. 08 [35] R. Blumenhagen, S. Moster, R. Reinbacher, and T. (2007) 080. Weigand, J. High Energy Phys. 05 (2007) 041. [12] F. P. Correia and M. G. Schmidt, arXiv:0708.3805. [36] M. Dine, R. Rohm, N. Seiberg, and E. Witten, Phys. Lett. [13] A. Lukas, B. A. Ovrut, and D. Waldram, Phys. Rev. D 59, B 156, 55 (1985). 106005 (1999). [37] M. Bra¨ndle, Ph.D. thesis, Humbolt University, Berlin, [14] R. Donagi, A. Lukas, B. A. Ovrut, and D. Waldram, J. 2003. High Energy Phys. 05 (1999) 018. [38] A. Lukas, B. A. Ovrut, and D. Waldram, Phys. Rev. D 59, [15] R. Donagi, A. Lukas, B. A. Ovrut, and D. Waldram, J. 106005 (1999). High Energy Phys. 06 (1999) 034. [39] E. Witten, Nucl. Phys. B471, 135 (1996). [16] R. Donagi, B. A. Ovrut, T. Pantev, and D. Waldram, J. [40] A. Lukas, B. A. Ovrut, and D. Waldram, Nucl. Phys. High Energy Phys. 08 (2001) 053. B532, 43 (1998). [17] R. Friedman, J. W. Morgan, and E. Witten, Commun. [41] A. Lukas, B. A. Ovrut, K. S. Stelle, and D. Waldram, Nucl. Math. Phys. 187, 679 (1997). Phys. B552, 246 (1999). [18] R. Donagi, Asian J. Math. 1, 214 (1997). [42] M. B. Green, J. H. Schwarz, and E. Witten, Cambridge [19] R. Donagi, B. A. Ovrut, T. Pantev, and D. Waldram, Adv. Monographs On Mathematical Physics (Cambridge Theor. Math. Phys. 5, 563 (2002). University Press, Cambridge, England, 1987), p. 596. [20] R. Donagi, B. A. Ovrut, T. Pantev, and D. Waldram, Adv. [43] E. Witten, Nucl. Phys. B268, 79 (1986). Theor. Math. Phys. 5, 499 (2002). [44] M. Dine and N. Seiberg, Phys. Rev. Lett. 57, 2625 (1986). [21] R. Donagi, Y.H. He, B. A. Ovrut, and R. Reinbacher, J. [45] R. Rohm and E. Witten, Ann. Phys. (N.Y.) 170, 454 High Energy Phys. 06 (2005) 070. (1986). [22] V. Braun, Y.H. He, B. A. Ovrut, and T. Pantev, Phys. Lett. [46] E. Witten, J. Geom. Phys. 22, 1 (1997). B 618, 252 (2005). [47] A. Lukas, B. A. Ovrut, and D. Waldram, Phys. Rev. D 57, [23] V. Braun, Y.H. He, B. A. Ovrut, and T. Pantev, J. High 7529 (1998). Energy Phys. 06 (2005) 039. [48] P. Horava and E. Witten, Nucl. Phys. B475, 94 (1996). [24] V. Braun, Y.H. He, B. A. Ovrut, and T. Pantev, Adv. Theor. [49] M. Brandle and A. Lukas, Phys. Rev. D 65, 064024 Math. Phys. 10, 4 (2006). (2002).
126012-21 JAMES GRAY, ANDRE´ LUKAS, AND BURT OVRUT PHYSICAL REVIEW D 76, 126012 (2007) [50] A. Lukas, B. A. Ovrut, and D. Waldram, J. High Energy (1982). Phys. 04 (1999) 009. [53] T. R. Taylor, Phys. Lett. B 252, 59 (1990). [51] P. Horava, Phys. Rev. D 54, 7561 (1996). [54] J. P. Derendinger and R. Sauser, Nucl. Phys. B598,87 [52] G. Veneziano and S. Yankielowicz, Phys. Lett. B 113, 231 (2001).
126012-22