arXiv:hep-th/9608086v2 19 Aug 1996 esuytehprutpe ouisaeo h yeI strin II type the of space space Yau moduli hypermultiplet the study We on uut1996 August n tigCmatfiaint he Dimensions Three to Compactification and X × S 1 X [email protected],[email protected] [email protected], n yuigrcn eut ntredmninlfil theory field dimensional three on results recent using by and ed hsb sn I/I ult nacmatfiaino t of compactification a in duality IIA/IIB using by this do We . yemliltMdl Space Moduli Hypermultiplet eateto hsc n Astronomy and Physics of Department ahnSiegadSehnShenker Stephen and Seiberg Nathan ictwy J08855-0849 NJ Piscataway, ugr University Rutgers opcie naCalabi- a on compactified g U9-8 hep-th/9608086 RU-96-68, . .edu esm theory same he The moduli of the type II theory compactified on a Calabi-Yau space X are in hy- permultiplets and vector multiplets of N = 2 . The has a product structure1 H×V. H is a quaternionic manifold and V is a special Kahler mani- fold. Since the is in a hypermultiplet, it appears only in H and not in V. Therefore, V is determined exactly in the classical theory. H on the other hand can receive quantum corrections. Strominger’s resolution of the singularity [1] left open the question of the behavior of H near a conifold point and in particular the question of whether the space is singular or not. It was suggested in [2] that non-perturbative effects – string – smooth out the singularity. It became immediately clear [2] that this question is intimately related to the behavior of the theory after compactification on another circle to three dimensions [2,3]. Recently the compactification of such N = 2 field theories from four to three dimen- sions was analyzed [4]. The Coulomb branch of the moduli space of the four dimensional theory, V, becomes in the three dimensional theory a hyper-Kahler manifold V˜. Con- sider for simplicity the case where V is one complex dimensional. Then it is a base of an SL(2, Z) vector bundle [5]. V˜ is the four real dimensional vector bundle with the same complex structure. The Kahler form of the fiber scales as the inverse of the radius R of the compactification and that of the base scales as R so that the volume form is independent of the radius. Equivalently, we can rescale the entire space and have a fiber whose volume 1 is R2 . Applying this to , consider compactifying the type IIA theory on X with a moduli space M(4) H ×V A = A A. (1)

1 Upon further compactification to three dimensions on S of radius RA the moduli space becomes2 M(3) H × V˜ A = A A. (2)

Locally, and in particular near conifold singularities, the field theoretic analysis of [4] 1 applies. The volume of the fiber torus is 2 2 where Mp is the four dimensional Planck Mp RA

1 To get such a product structure one might need to consider a multiple cover of V or H. 2 We are not sure that the moduli space is globally such a product (even if we consider multiple covers of HA and V˜A). Below we will need this fact only locally where field theory is valid. Then, the product structure follows from the fact that two different SU(2)’s rotate the complex structures of the two factors.

1 mass. In terms of the IIA string coupling λA and the string scale Ms it is

λ 2 A . (3)  MsRA 

Now perform a T-duality transformation on the circle. This leads to the IIB theory compactified on X × S1 with radius and coupling 1 R = B M 2R s A (4) λA λB = . MsRA The moduli space of the IIB theory is

M(3) H × V˜ B = B B. (5)

M(3) M(3) By T-duality A = B but HB =V˜A (6) HA =V˜B.

For the three dimensional theory a convenient coordinate system on the space of λA λA and RA is λA and λB = . HA = V˜B depends only on λA and HB = V˜A depends only MsRA on λB.

As RA goes to zero, RB goes to infinity and the IIB theory becomes four dimensional with moduli space M(4) H ×V B = B B (7)

where V˜B is related to VB as explained above. Therefore, all these moduli spaces are

uniquely determined by VA and VB. In particular, HB is easily constructed out of VA 2 where the volume of the fiber is λB. Using this information we can easily conclude a few facts about the singularities

in HB. Near a simple conifold, VA has a singularity associated with a single massive hypermultiplet becoming massless [1]. The corresponding three dimensional theory is U(1) with one electron whose moduli space V˜A is smooth [6,4]. Therefore, HB at that point is smooth. If N massive hypermultiplets become massless at a point in VA, V˜A has an AN−1 2 singularity [6,4] – it is locally C /ZN . Therefore, HB should also have such a singularity at that point. More complicated like those discussed in [7] can be analyzed in a similar way. Recently [8] some of these results have been derived from a different point of view. The corrections they compute have a simple interpretation in the four to three

2 dimensional language. They are simply the world line windings of the light hypermultiplet around the S1 [3]. To see this, we note that the metric for the U(1) field theories with N hypermultiplets is one loop exact. This fact can be derived as follows. If the gauge theory is Abelian, there are no magnetic monopoles in four dimensions and there are no instantons in the three dimensional theory. Therefore, there is no need to perform a duality transformation in the three dimensional Lagrangian. Writing it in terms of the U(1) gauge field, an argument similar to the one in four dimensions shows that only one loop corrections are possible. 2 2 Therefore, 1/e (z)=1/e0 + NI(z) where I(z) is the one loop charge renormalization of |z| 3 × 1 one hypermultiplet of mass λ in the space R S

∼ 1 3 1 ∼ 1 I(z) d p z + const. + const. (8) R Z 2 n 2 | | 2 2 |z|R Xn (p +( R ) +( λ ) ) Xn n2 +( )2 q λ For simplicity we set other scalar vevs to zero. The (infinite) additive constant in (8) 2 corresponds to renormalization of 1/e0. The function appearing in the metric for the 2 2 e0 dualized photon is e (z) = 2 and appears to have all order corrections. This is the 1+e0NI(z) origin of the positive mass Taub-NUT factors found in [4]. So all instanton effects come from the R dependence of I(z). For large R this dependence comes from the world lines |z| of particles of mass λ winding around the circle l times, producing an effect of weight −2πRl|z| exp( λ ). These effects can be isolated by Poisson resummation of (8). If (8) is cut off at mass M, e.g., by a heavy Pauli-Villars hypermultiplet, there will be additional R dependence of weight exp(−2πRlM) that signals the cutoff scale [3].

Acknowledgements This work was supported in part by DOE grant DE-FG02-96ER40559.

3 References

[1] A. Strominger, “Massless Black Holes and Conifolds in String Theory,” Nucl.Phys B451 (1995) 97, hep-th/9504090. [2] K. Becker, M. Becker and A. Strominger, “Fivebranes, Membranes and Non- Perturbative String Theory,” Nucl.Phys. B456 (1995) 130, hep-th/9507158. [3] S. Shenker, “Another length Scale in String Theory?” hep-th/9509132. [4] N. Seiberg and N. Witten, “Gauge Dynamics and Compactification to Three Dimen- sions,” hep-th/9607163. [5] N. Seiberg and E. Witten, “Electric-Magnetic Duality, Monopole Condensation, And Confinement In N = 2 Supersymmetric Yang-Mills Theory,” Nucl. Phys. B426 (1994) 19, hep-th/9407087; “Monopoles, Duality, And Chiral Symmetry Breaking In N =2 Supersymmetric QCD,” Nucl. Phys. B431 (1994) 484, hep-th/9408099. [6] N. Seiberg, “IR Dynamics on and Space-Time Geometry,” hep-th/9606017. [7] A. Klemm and P. Mayr, “Strong Coupling Symmetries and Nonabelian Gauge Sym- metries in N=2 String Theory,” hep-th/9601014; S. Katz, D. Morrison, and R. Plesser, “Enhanced Gauge Symmetry in Type II String Theory,” hep-th/9601108. [8] B.R. Greene, D.R. Morrison and C. Vafa, “A Geometric Realization of Confine- ment,” hep-th/9608039; H. Ooguri and C. Vafa, “Summing up D-Instantons,” hep- th/9608079.

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