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A Mysterious Duality

We establish a correspondence between toroidal compactifications of M-theory and del Pezzo surfaces. M-theory on T^k corresponds to P^2 blown up at k generic points; Type IIB corresponds to P^1\times P^1. The moduli of compactifications of M-theory on rectangular tori are mapped to Kahler moduli of del Pezzo surfaces.The U-duality group of M-theory corresponds to a group of classical symmetries of the del Pezzo represented by global diffeomorphisms. The half-BPS brane charges of M-theory correspond to spheres in the del Pezzo, and their tension to the exponentiated volume of the corresponding spheres. The electric/magnetic pairing of branes is determined by the condition that the union of the corresponding spheres represent the anticanonical class of the del Pezzo. The condition that a pair of half-BPS states form a bound state is mapped to a condition on the intersection of the corresponding spheres. We present some speculations about the meaning of this duality.


Introduction
The discovery of duality symmetries in string theory has led to spectacular progress in our understanding of non-perturbative aspects of the theory. However, we still do not have a deep understanding of the meaning of these symmetries. Indeed, we find ourselves in the strange situation that the quantum corrected physics has more symmetries than the classical strings had any right to expect! A clear appreciation of symmetry principles is a sacred principle of physics. Given any physical system, we should formulate the theory in a way that makes all of the symmetries manifest. But despite many years of work on a non-perturbative definition of string theory, we are no closer to making duality symmetries manifest than when the dualities were first discovered!
We would like to have a formulation of string theory in which all of the duality symmetries are classically visible. The aim of this paper is to develop a mysterious duality which points to the existence of such a formulation.
It was noted in [1] that there is a classical geometric system which shares all the Uduality symmetries of M-theory compactified on rectangular tori. The relevant geometric objects are del Pezzo surfaces, which are complex 2-dimensional Kähler manifolds with c 1 > 0. In this correspondence, M-theory in 11 dimensions is mapped to P 2 , and subsequent compactifications of M-theory on rectangular tori T k are mapped to P 2 blown up at k generic points (i.e. a manifold in which k generic points on P 2 are replaced with P 1 's), which is a del Pezzo surface denoted B k . Type IIB is mapped to P 1 × P 1 , which can be viewed as a blow down of B 2 . The U-duality group of M-theory on T k , which for rectangular compactifications with no C-field vevs is given by the Weyl group of E k (with a suitable definition for small k), is mapped to a subgroup of the global diffeomorphisms of the del Pezzo. For example, the S-duality of type IIB in 10 dimensions is realized as the exchange of the two P 1 's in P 1 × P 1 .
We will show in this paper that one can take this idea quite far, constructing a precise dictionary relating the two sides. In particular, the 1 2 -BPS p-branes of M-theory will be mapped to rational curves on the del Pezzo. Furthermore, we find a map which relates the moduli of the (extended) Kähler metric on B k , which is determined by k + 1 real parameters (one for the overall size of the original P 2 and k for the volumes of the blown-up P 1 's) with the k + 1 moduli of M-theory on T k (k moduli for the compactification radii and one for the Planck scale). The moduli space of Kähler classes also carries an action of a group of global diffeomorphisms of the del Pezzo surface, which exchange the various embedded 2-spheres; as noted above, this group is isomorphic to the Weyl group of E k . Moreover, this symmetry action is compatible with the identification of the moduli on the two sides. The map relates the tension of a 1 2 -BPS state with the exponential of the volume of the corresponding P 1 . In addition, electric/magnetic duality and the condition for a pair of branes to form a bound state admit a nice geometric interpretation on the del Pezzo side.
Some aspects of the dictionary we construct are listed in the table below: del Pezzo B k M-theory on T k Element of H 2 (B k , R) Point in moduli space of M-theory on T k Global diffeomorphisms U-duality group preserving the canonical class K 2-sphere C with volume V C and 1 2 -BPS p-brane state with tension 2π exp V C degree p + 1 Volume V K of canonical class Compactified Planck length, l p 9−k = exp V K Volume V H of hyperplane class 11-dimensional Planck length, l −3 p = exp V H Volume V E of exceptional curve Radius 2πR = exp −V E H, line in P 2 M2-brane 2H, conic in P 2 M5-brane 2-spheres C 1 , C 2 with C 1 + C 2 = −K Electric-magnetic dual objects Here H denotes the class of a hyperplane (a complex line) in P 2 , and K denotes the "canonical class," a 2-cycle class which is dual to the negative of the first Chern class of the del Pezzo.
The organization of this paper is as follows: In Section 2 we review some relevant geometric aspects of del Pezzo surfaces. In Section 3 we present the map between the two sides. In Section 4 we discuss some speculations as to the meaning of this duality and raise some natural questions.

Del Pezzo surfaces 2.1 Basic properties
The term "del Pezzo surface" refers to any manifold of complex dimension 2 such that the first Chern class is positive [2]. These surfaces admit metrics of positive scalar curvature. They may be classified as follows: either take P 2 and blow up k ≤ 8 generic points, or take P 1 × P 1 and blow up k ≤ 7 generic points. We call the resulting del Pezzos B k and F k respectively. In fact B k+1 ≃ F k for k ≥ 1, so it is enough to consider only B k and P 1 × P 1 . Now let us describe the homology of del Pezzo surfaces, beginning with P 2 . Since P 2 is simply connected the only interesting homology will be in dimension 2; in fact H 2 (P 2 , Z) is simply Z, generated by the class of a line, which we write H. When we blow up a point we replace it with a P 1 (a 2-sphere), which gives a new generator in H 2 (B k , Z) for each of the k points we blow up. Thus dim H 2 (B k , R) = k + 1; a natural basis to choose is {H, E 1 , . . . , E k }, where the E a are the "exceptional curves" obtained by blow-ups and H represents the pullback of the generator of H 2 (P 2 , Z) under the projection which simply collapses each exceptional curve E a to the corresponding point p a ∈ P 2 which was blown up. The intersection numbers are given by [3] H · H = 1 , H · E a = 0 , E a · E b = −δ ab , a, b = 1, · · · , k .
For F k−1 , dim H 2 (F k−1 , R) = k + 1 and the natural basis is given by {l 1 , l 2 , e 1 , . . . , e k−1 } (the l i come from P 1 × P 1 and the e j are the blown-up P 1 's) satisfying Since B k ≃ F k+1 we can write the two bases just described in terms of one another; the map is given by The inverse map is then The canonical class, defined to be minus the first Chern class of the tangent bundle, will play an important role in our correspondence to M-theory; it is given by where c 1 (X) is the first Chern class of X. Incidentally, from (6) we see directly that the condition of positive first Chern class is not satisfied for B k or F k−1 with k > 8, since We denote by d C the degree of a class C ∈ H 2 (B k ), which by definition is the intersection of C with the anticanonical class, i.e.
The terminology "degree" is explained by the fact that one can use the anticanonical class to give a map of B k into projective space, and if C is represented by a holomorphic curve then its image will have degree precisely d C .
If the class C ∈ H 2 (B k ) is realized as a genus g curve then we have an important relation between its self-intersection and its degree, known as the adjunction formula [4]:

E k root lattice
We now review an important relation between del Pezzo surfaces and exceptional Lie algebras. Namely, the lattice H 2 (B k , Z) endowed with the intersection product has a natural sublattice obtained by restricting to the orthogonal complement of K B k , which turns out to be isomorphic to the root lattice of E k . To see this (in case k ≥ 3 -for k = 1, 2 the story is slightly irregular) a convenient basis for the sublattice is the following, which is also a set of simple roots: It is easy to see that these classes are orthogonal to the canonical class, and that their intersection numbers are given by where A ab represents the Cartan matrix of the Lie algebra E k . The Dynkin diagrams of the Lie algebras E k are shown in Fig. 1. To each C ∈ H 2 (B k , Z) we can associate an E k weight vector C ⊥ , obtained by projecting onto the orthogonal complement of K B k . The Dynkin labels of C ⊥ are given by In terms of the weight vector, the self-intersection of C ∈ H 2 (B k , Z) is given by Because of the minus sign in (12) we will have C 2 ⊥ ≤ 0. Thus the lattice H 2 (B k , Z) has signature (1, k), Here Γ E k is the root lattice of E k , with the sign of the inner product reversed.

Weyl group
Next we show that the Weyl group of E k acts naturally on H 2 (B k , R), preserving the intersection form and K B k . Given any α ∈ H 2 (B k , Z) with α 2 = −2 and K B k · α = 0 (a root) we can define a transformation w α , which is the reflection in α and hence acts orthogonally: The elements w α corresponding to simple roots have a particularly interesting action on H 2 (B k , R). The simple root α i = E i − E i+1 exchanges E i and E i+1 (we will see in a later section that this corresponds to exchanging two of the circles on which we compactify Mtheory.) The action of Note that this transformation can only be defined for k ≥ 3. We will see later that it corresponds to a double T-duality, which from the 11-dimensional point of view requires three compact directions. The w α i together generate the Weyl group of E k ; in fact this is the group of all automorphisms of H 2 (B k , Z) which preserve the intersection form and K [2].
This action of the Weyl group on H 2 (B k , Z) can actually be realized by global diffeomorphisms which act on the del Pezzo surface exchanging the exceptional curves and preserving K. For the roots α i with i < k this is easy to see: we only have to take a diffeomorphism of P 2 which exchanges two of the blown-up points while fixing the rest. Such diffeomorphisms certainly exist and can be extended to the full del Pezzo. What is more interesting is the reflection in α k : this action is realized by a mechanism which is less obvious from the perspective of P 2 , which we discuss in Section 2.7.
Since the Weyl group action preserves K it preserves the degree of any curve. Thus curves of a given degree form a representation of the Weyl group.

Rational curves
In our correspondence it will be especially important to understand the genus zero curves, also called rational curves, in H 2 (B k , Z). Such a curve satisfies the adjunction formula (10) with g = 0. Hence if C = nH − k a=1 m a E a is represented by a rational curve we have or equivalently In terms of the E k weight vector C ⊥ of a rational curve C we have From (21) we see that the degree 1 rational curves (exceptional curves) transform under the Weyl group in the fundamental representation of E k , since for d C = 1 we would have which is just what we would expect for weight vectors of the fundamental.

Toric geometry
Toric geometry provides an interesting way to visualize some of the del Pezzo surfaces as well as a neat diagrammatic method of obtaining the intersection numbers of curves on toric del Pezzo surfaces. We therefore give a review of the relevant aspects of toric geometry applied to del Pezzo surfaces. P 1 : We start with a simple example: the representation of P 1 in toric geometry, following the discussion in [5]. Since P 1 = S 3 /U(1), we can describe it as [6] The complex variables Φ i are related to the projective coordinates [z 1 , z 2 ] of P 1 by The geometry defined by (23) can be understood in a slightly different way as well, which will be important for us when discussing toric del Pezzo surfaces. Namely, rewrite (23) and (24) as follows: Thus we see that 0 ≤ |Φ 1 | 2 ≤ r and therefore the range of |Φ 1 | 2 is an interval with endpoints given by |Φ 1 | 2 = 0, r. Fibered over every point of the interval except the endpoints we have two circles, given by the phases of Φ 1 and Φ 2 , the magnitudes being fixed by (26). However, because of the U(1) identification only the relative phase survives; at every point of the interval except the endpoints we have a finite size circle parametrized by this phase. At one of the endpoints, say |Φ 1 | = 0, we have |Φ 2 | 2 = r with the phase arbitrary; however, this phase is completely fixed by the U(1) quotient. Thus at the endpoints of the interval the circle fiber shrinks to zero radius. As shown in Fig. 2 the total geometry is topologically a P 1 . P 2 : Now that we understand the representation of P 1 as a circle fibration over an interval, we can try to understand the analogous picture for toric del Pezzo surfaces, starting with P 2 . In terms of complex variables Φ 1 , Φ 2 and Φ 3 , P 2 can be represented as [6] The complex variables are related to the projective coordinates [z 1 , z 2 , z 3 ] of P 2 , The equtaion (27) defines an S 5 of radius √ r, as can be seen easily by writing the equation in terms of the real and imaginary parts of the Φ i . The U(1) identification then gives us P 2 as S 5 /U(1) with Kähler parameter r. In analogy with the P 1 case we then see that the toric representation of P 2 will be an S 1 × S 1 fibration over some base, with the S 1 × S 1 given by the relative phases among Φ 1 , To obtain the base of the fibration we rewrite the equation defining S 5 as Since |Φ 3 | ≥ 0 the base is a triangular region in the plane, with coordinate axes parametrized by |Φ 1 | 2 and |Φ 2 | 2 and boundary given by three intervals as shown in Fig. 3(a), At every point inside the triangle the magnitudes of Φ 1 , Φ 2 and Φ 3 are fixed but the phases are not, giving a T 2 for the relative phases. At the boundary the situation is different since some of the Φ i are zero and hence the corresponding circles have collapsed. From (31) we see that at the boundary component I i we have Φ i = 0. When one of the Φ j vanishes (30) reduces to (26); thus the interval represents a P 1 inside P 2 , as shown in Fig. 3(b). At points where two of the I i intersect (the vertices of the triangle) only one Φ i is non-zero; its magnitude is determined by (30), and the phase is completely fixed by the U(1) quotient.
Thus we see that inside the triangle we have a T 2 fibration, at the edges the T 2 collapses to an S 1 , and at the vertices the S 1 collapses to a point. The three intervals I i represent three The toric diagram of P 2 , b) P 2 as a T 2 fibration with collapsing fibers at the boundary, c) The three boundary components as P 1 's.
B 1 : Next we consider the case of P 2 blown up at one point. In this case we have four complex variables Φ 1 , Φ 2 , Φ 3 and Φ 4 , satisfying The first equation in (32) defines a P 2 and the second defines a P 1 ; r 1 and r 2 are the two Kähler parameters of the surface. We can draw the toric picture in the plane, parametrized as before by |Φ 1 | 2 and |Φ 2 | 2 . Eliminating |Φ 3 | 2 and |Φ 4 | 2 we get linear constraints on |Φ 1 | 2 and |Φ 2 | 2 which define the base of the torus fibration: So now we have four boundary components, given by The region bounded by the I i is the base of the fibration, shown in Fig. 4. If we had used |Φ 2 | 2 in the second line of (32) then a different vertex of the triangle would have been replaced by a line segment; the choice among |Φ 1 | 2 , |Φ 2 | 2 and |Φ 3 | 2 corresponds in the toric picture to the choice of which vertex to replace by a line segment representing the exceptional curve.
B 2 : Now let us consider the case of P 2 blown up at two points. The equations representing B 2 are In the above equations r 1 , r 2 and r 3 are the three Kähler parameters of the surface B 2 . As before, we solve these equations in the plane parametrized by |Φ 1 | 2 and |Φ 2 | 2 . Eliminating |Φ 3 | 2 , |Φ 4 | 2 and |Φ 5 | 2 leads to inequalities in terms of |Φ 1 | 2 and |Φ 2 | 2 whose solutions bound a region in the plane, with boundary components given by The base of the torus fibration is shown in Fig. 5.
B 3 : Next we consider P 2 blown up at three points. We omit the equations but they are similar to those in the previous examples; there are three exceptional curves now, described torically by three intervals which replace the vertices of the triangle. The picture is shown in Fig. 6.
From the diagrams we have seen so far it is clear that the toric representation of the operation of blowing up a point which is a vertex of the toric diagram is the replacement of that point by a line segment. In case k ≤ 3 any triplet of points on P 2 is equivalent in the sense that there is an automorphism of P 2 which moves any given triplet to the three vertices of the triangle; hence it is always sufficient to consider blowing up the vertices. For k > 3 we encounter a difficulty: after blowing up three points all the curves appearing on the boundary of the toric diagram are exceptional curves, hence rigid. A generic point does not lie on any of these exceptional curves, so we cannot give a toric description of the process of blowing up a fourth point; hence B k does not admit a toric description for k > 3. From the toric diagrams we can easily see why P 2 blown up at two points is the same as P 1 × P 1 blown up at one point, in other words B 2 ≃ F 1 . To see this consider first the case of P 1 × P 1 , which is given by the equations: The toric diagram is shown in Fig. 7. Then we can consider P 1 × P 1 blown up at one point, which is given by After a change of coordinates the equations (41), (42) are equivalent to (36), (37), showing that B 2 ≃ F 1 ; but it is much easier to see this directly from the toric diagram, as shown in Fig. 8.

Toric description of curves and intersection numbers
As remarked earlier, each boundary line segment of the toric diagram corresponds to an exceptional curve. Let us denote the curve corresponding to I i by D i ; then we can see from Fig. 9 that H-E -E 1 3 H-E -E An equivalent way of representing curves in the del Pezzo is motivated from the string web picture in the context of (p, q) 5-branes [7], which is related to del Pezzos as discussed in [5]. Namely, the del Pezzos have a symplectic form represented by the Kähler class. The base of the toric fibration can be viewed as the 'x' space and the T 2 fibers can be viewed as 'p' directions, where we represent the symplectic form as 2 i=1 dx i ∧ dp i = 2 i=1 d|Φ i | 2 ∧ dθ i , where θ i represents the phase of Φ i . This implies that each direction in the base is naturally paired with a circle in the fiber. Consider for example a line in the base ending on a boundary. If the line ends orthogonally on the boundary, considering the total space of the line and the corresponding circle in the fiber one obtains a piece of a holomorphic curve. We can connect these pieces together to obtain closed curves by drawing trivalent graphs with external legs ending orthogonally on the boundary components I i . Consider first the case of P 2 ; let us try to represent in this way the class H of a line. If we can find a curve of genus zero with selfintersection one then it is clear that this curve represents H. As shown in Fig. 10(a), such a curve corresponds to the simplest trivalent graph in the triangle; this can be understood from the fact that the boundary components are in the same class as H and therefore which implies that the graph corresponding to H should have one external leg ending on each I i . In fact note that there is a moduli space for this curve (which is again a P 2 ) which can be viewed in the base as the choice for the position of the trivalent vertex (this description itself gives the base of a toric realization of the moduli space). One can see that if we go to a point on the moduli space of this curve corresponding to putting the trivalent point of the graph at one of the three vertices of the triangle we obtain the curve z 1 = 0 which represents H. Fig. 10(b) shows the case of a conic in P 2 , which has genus zero and in homology is just 2H. It is easy to see by deforming the graph and making it trivalent that it indeed represents a curve of genus zero. On the other hand, a cubic curve will have genus one, as one sees from the "hole" appearing in the rightmost picture in Fig. 10(c).

Quadratic transformation
Now we discuss the geometric realization of the nontrivial part of the Weyl group, the reflection in α k = H − E 1 − E 2 − E 3 . First consider the case k = 3 and observe that starting from B 3 we can obtain P 2 in two distinct ways: either blow down the set as shown in Fig. 11. The two sets of exceptional curves are related to each other precisely by the reflection in the root α 3 . Correspondingly, the P 2 's obtained this way are related by a birational map φ 0 called the "quadratic transformation," [4] given by The expression above makes sense only when at least two of the z i are nonzero (otherwise we get [0, 0, 0] which is not a point of P 2 ); so φ 0 is defined on P 2 minus the three points . By blowing up those three points to E 1 , E 2 , E 3 we can make φ 0 defined everywhere. Similarly, we can make φ 0 1-1 by blowing up the same three points in the image P 2 . We therefore obtain an automorphism of B 3 . Now how does φ 0 act on H 2 (B 3 , Z)? It is easy to see that and similarly for the other exceptional curves. This is precisely the action of the reflection in the root α 3 , as desired.
For k > 3 the situation is similar: the presence of additional blown-up points at generic positions on P 2 does not change anything, except that the quadratic transformation will move these points, so that we have to compose with a diffeomorphism of P 2 to move them back to their original positions. So for all k ≥ 3 the reflection in α k is realized by a global diffeomorphism.
The description of the quadratic transformation above was given in terms of the realization of B k as P 2 blown up at k points. However, as discussed above, B k ≃ F k+1 so we can equally well think of P 1 × P 1 blown up at k − 1 points. From this perspective the action of the quadratic transformation is simple to describe: namely, it corresponds to exchanging two of the k − 1 blown-up points. This description makes it obvious that the quadratic transformation is realized by a global diffeomorphism.

The correspondence
We have discussed in the previous section how the moduli spaces of del Pezzos are interrelated via the operations of blowing up and down. Moreover, as we have discussed, the group of global diffeomorphisms of B k which fix K is the Weyl group of the corresponding exceptional group E k .
On the other hand, if we consider compactifications of M-theory on rectangular tori T k with no vev for the C-field, the U-duality group acting on this class of compactifications is also realized as the Weyl group of E k [8] (see also [9].) It is thus natural to ask if there is a map between such M-theory compactifications and del Pezzo geometries. In fact, the story is already rather interesting for small k's, as shown in Fig. 12.
Note that in Fig. 12 each time we blow up a point to a P 1 in the del Pezzo, we compactify a circle on the M-theory side. It is remarkable that the exotic role type IIB plays in the chain of dualities of M-theory exactly matches the role P 1 ×P 1 plays among del Pezzos. In particular, if we wish to get from M-theory in 11-dimensions to type IIB in 10 dimensions, we have to compactify two circles and then let the resulting torus shrink to zero size; this is precisely mirrored by the fact that to get to P 1 × P 1 from P 2 we first have to blow up two points and then blow down another P 1 . Note also that the symmetries of the corresponding theories are manifest as classical symmetries of the del Pezzos. For example, the S-duality of type IIB corresponds to the exchange of the two sides of the rectangle above (i.e. the exchange of the two P 1 's). Moreover, the non-trivial part of S-duality in M-theory compactified to 9 dimensions is still Z 2 , which is reflected in the Z 2 symmetry of the rectangle with one corner cut off. The non-trivial part of S-duality in d = 8 is Z 2 × S 3 , which is manifest as the symmetries of the del Pezzo hexagon depicted in the above figure. Indeed this identification of Weyl groups continues to make sense for all del Pezzos, even those which are not toric (namely the B k with k > 3). It is thus natural to expect that we can get a more detailed map between the two sides and promote the above correspondence to the level of moduli and objects on both sides. We now consider this question.

Matching the moduli spaces
We consider the moduli space for M-theory on a geometry T k × R 10−k,1 for k ≤ 8, where the torus has a rectangular metric of the form with the x i periodic, x i ∼ x i + 2π, and all background C fields set to zero. At least when all R i are small compared to the 11 dimensional Planck scale l p , the low energy dynamics are described by an 11 − k dimensional supergravity theory with 32 supercharges. This theory is determined by the k + 1 parameters l p , R 1 , . . . , R k which are positive real numbers, so naively one would expect its moduli space to be R k+1 + . However, if we want the space of physically inequivalent theories then we have to take the quotient of R k+1 + by the duality group consisting of transformations which leave the physics invariant. For M-theory on a T k of arbitrary shape with arbitrary C-field the duality group is well known to be E k(k) (Z) [10]. In our case we want to consider only dualities which respect the "rectangular, no C-field" condition. These include the symmetric group S k permuting the k radii; this group is generated by the k − 1 exchanges In addition one also has the action of T-duality (M2/M5 exchange, after T 3 compactification) which when expressed in M-theory language involves three radii, so we need to include one extra generator in case k ≥ 3, namely As we will see below, this corresponds to the quadratic transformation discussed before in the context of del Pezzos. It is known [9,11] that the w α i generate the full duality group acting on "rectangular, no C-field" compactifications; just as in Section 2.3 this group is isomorphic to the Weyl group of E k . Moreover, the other elements of the U-duality group, such as periodicity of the C field or shifting complex moduli of tori by τ → τ +1, are manifest symmetries of string theory. So the non-obvious part of the full U-duality group is already captured by the Weyl group W (E k ). Now we are ready to map the moduli. Using the notation M k for the naive moduli space R k+1 + , we may write It is convenient to switch to a linear representation by taking logarithms: namely, we think of M k as a (k + 1)-dimensional real vector space, with a typical element (log l p , log(2πR 1 ), . . . , log(2πR k )).
Then U k acts linearly on M k . To establish our correspondence we need one more piece of structure on M k . Namely, by considering the log-tension formulas for 1 2 -BPS states we obtain a lattice Λ in the dual vector space M k * , spanned by k + 1 basis vectors (3 log l p ) and (log 2πR a ) (a = 1, . . . , k).
In sum, we have a (k + 1)-dimensional vector space M k , carrying an action of the Weyl group U k which furthermore preserves a lattice Λ in the dual space. Precisely this structure is also present on the del Pezzo surface: namely, we have the (k + 1)-dimensional cohomology H 2 (B k , R), carrying an action of the Weyl group by global diffeomorphisms preserving the canonical class, which furthermore preserves the homology lattice H 2 (B k , Z). The natural thing to do, then, is to identify the two vector spaces H 2 (B k , R) and M k * in a way which identifies the lattices H 2 (B k , Z) and Λ while preserving the action of the Weyl group. This can be done in an essentially unique way: in the notation of Section 2.1, we map Dually, we have an identification between M k and H 2 (B k , R), described by We think of ω ∈ H 2 (B k , R) as a kind of generalized Kähler class. If ω came from an ordinary (positive) Kähler metric, then ω(C) would be simply the volume of a holomorphic curve in the class C. Our ω need not come from a Kähler metric, since ω(C) may be negative (e.g. if 2πR a > 1 then ω(E a ) < 0.) The choice (52) is not quite unique -apart from the unimportant freedom to make a Weyl group transformation on one side, we could also have taken plus signs instead of minus. Our choice was dictated by the original intuition that blowing up E a corresponds to compactification from R a = ∞ to R a = finite, so that the volume of E a should be inversely related to R a .
Before going further, let us remark on the appearance of the group of diffeomorphisms preserving the canonical class K in the above correspondence. Recall from (6) that Applying (52) we find the correspondence where l p is the Planck length in the compactified theory, which is given by Since l p 9−k can be observed in the dimensionally reduced theory as the inverse of the gravitational coupling constant, it must be invariant under every element in the U-duality group; this is why we have to consider only diffeomorphisms preserving K.

Brane charges and rational curves
Next we consider the tensions of 1 2 -BPS states in M-theory. We will show that a holomorphic rational curve C in the del Pezzo corresponds to a 1 2 -BPS p-brane charge. Moreover, the BPS tension can be identified as where ω ∈ H 2 (B k , R) is the generalized Kähler class introduced in the last section. Given this formula for the tension, one easily checks that p is determined by the intersection product with K (recall that K corresponds to (9 − k) log l p , so we can think of it as in some sense "carrying the units of mass"): Furthermore, electric-magnetic duality has a simple interpretation in this framework: dual pairs are related simply by In particular this will imply, using (57) and (55), that for all electric-magnetic pairs of 1 2 -BPS branes. Before considering the general setup, let us consider the simple case of uncompactified M-theory.
M-theory in d = 11. Recall that this theory corresponds to P 2 . The lattice H 2 (P 2 , Z) is spanned by the single element H, so consider a curve in P 2 given by the class A priori we could choose any integral value for n. However, not every class in H 2 (P 2 , Z) is actually realized by a holomorphic curve: one has to choose n > 0. Choosing n = 1, H is the class of a line in P 2 , and the tension formula gives T = 2π exp ω(H) = 2π/l 3 p . Similarly, 2H is the class of a conic and gives T = 2π/l 6 p , the tension of the M5 brane. These are the only genus zero curves in P 2 , since by (10) the genus of C = nH is given by The toric representations of H and 2H are shown in Fig. 13. Note that the fact that the What about 3H? This is the class of a cubic curve in P 2 . If it plays a role in our correspondence it should correspond to a tension T = 2π/l 9 p , giving some hypothetical M8 brane. On the other hand, as remarked earlier, a cubic curve in P 2 has genus one. So if we restrict to rational curves we obtain only the M2 and M5.
It is intriguing that the fact that the worldvolume dimensions of the 1 2 -BPS states in 11 dimensions are a multiple of 3 directly follows from the fact that the canonical class of P 2 is a multiple of 3 given by −3H. This is indeed a remarkable map! Before going on to a more complicated example, we discuss some general aspects of the relevant physics. States which are 1 2 -BPS satisfy a formula relating their tensions to the SUSY central charges. These charges in turn are proportional to gauge charges in 11 − k dimensions, with the constant of proportionality depending on the parameters R a and l p . For example, a state with one unit of momentum in the compact 1 direction would satisfy while an M2-brane wrapped on the 1 − 2 directions would have By "gauge charge" we include charges under p-form symmetries as well as 1-forms, so we can also consider e.g. an M5 wrapped on the 1 − 2 − 3 directions. In this case we get a tension formula rather than a mass formula, which reads All three tension formulas (63), (64), (65) are of the form with n and all m a integral. It is not quite true that this is the most general situation for a 1 2 -BPS state -for example, a state with one unit of momentum in each of the 1 and 2 directions would have M = 1/R 2 1 + 1/R 2 2 . Put differently, the two states given by one unit of momentum in each of the 1 and 2 directions form a bound state which has mass smaller than the sum of the two masses. We will discuss below the geometric condition under which a pair of 1 2 -BPS states form a bound state; at this point we concentrate on a particular basis for the lattice of charges, such that every state in the basis has a tension formula of the form (66).
So let us relate the formula (66) to the del Pezzo surface B k . Setting we can rewrite (66) as On the other hand, through the map (53) we can rewrite (67) as where for some n and m satisfying (73). The tension is given by T n,m = 2π exp(ω(C n,m )) , From (74) Thus we get correct tensions for all the Dp-brane as well as the fundamental string and the NS5-brane, as shown in the above table.
The electric/magnetic pairing works as described by (59), with It is also interesting that the del Pezzo knows about ALF space, i.e. about the D6 brane, which does not come from simple dimensional reduction from 11 dimensions. Similarly the appearance of the D8 brane here, which does not correspond to any object in P 2 , is remarkable.
Type II in d = 8. For brevity we now skip d = 9 and go directly to d = 8. We search for curves which could represent p-branes with 0 ≤ p ≤ 6; using (10) as above and (58), this amounts to finding integer solutions of The results are summarized in the table below, where we include the interpretation of the states in terms of M-theory as well as Type IIA. (One could also interpret these states from the perspective of compactified Type IIB, using the relations (5) between blow-ups of P 2 and of P 1 × P 1 .) p homology class type IIA meaning M-theory meaning In the above table we have singled out one of the exceptional curves as the "M-theory direction" and called it E M , while the indices i, j run over the values 1, 2 for the other two circles. The entries labeled "exotic state" correspond to states satisfying exotic tension formulas which are not straightforward to interpret; nevertheless, these states are required by the U-duality symmetry, as reviewed e.g. in [11].
From looking at the table we notice a simple pattern, namely, if C is a 1 2 -BPS object not involving E i , then C − E i will be the same object wrapped on the i direction. One can also see this directly from the tension formulas since −E i corresponds to log 2πR i .
Type IIB in d = 10. In our discussion so far we have mostly stuck to the surfaces B k , but the discussion goes through essentially unchanged for P 1 × P 1 . Recall that a basis for H 2 (P 1 × P 1 , Z) is given by the classes of the two factors, which we write l 1 and l 2 , with l 1 · l 1 = l 2 · l 2 = 0, l 1 · l 2 = 1.
The canonical class is K = −2l 1 − 2l 2 , and the analog of (52) in this setting is where l s and g s now denote the Type IIB quantities. From (9) and (10), a genus zero curve C = nl 1 + ml 2 of degree p + 1 satisfies There are two integer solutions for each p ∈ 2Z + 1: The tension is then given by Then restricting to 0 ≤ p ≤ 8, we obtain homology class tension / 2π type IIB meaning

NS7-brane
Note that the fact that the worldvolume dimension of the Type IIB 1 2 -BPS branes are all even follows from the fact that K for P 1 × P 1 is even, i.e. K = −2l 1 − 2l 2 .
Type II in general d ≥ 3. In parallel to the discussion above, one can check that all the 1 2 -BPS p-branes with codimension at least 1 in uncompactified spacetime in compactifications of M-theory on T k for k ≤ 8 are in 1-1 correspondence with rational curves C with d C = p+1. One finds all the branes which appeared in the table for d = 8, plus their compactifications, plus various extra exotic states (again, all required to exist by U-duality.) It is not actually difficult to determine genus zero curves of a given degree satisfying the adjunction formula.
We can write the self-intersection of C in terms of the weight vector C ⊥ as discussed in section 2.2, The results are summarized in the following table in which we give the representation of E k to which the curve belongs, the size of the Weyl orbit and a representative curve. The uncompactified dimension of spacetime is d = 11 − k.
rep of E k Weyl orbit size curve threshold which is still 1 2 -BPS, ii) they form a bound state, not at threshold, which is 1 2 -BPS, iii) they form a bound state at threshold which is 1 4 -BPS. The first possibility is exemplified by a bound state of a pair of D0-branes. The second is exemplified by a bound state of a D0-brane with a 2-cycle wrapped D2-brane, and the last occurs e.g. for the bound state of a D0-brane with a 4-cycle wrapped D4-brane. It is not too difficult to show that on B 6 , C 1 · C 2 is the only Weyl invariant of a pair of exceptional curves and can only be −1, 0 or 1. These correspond to the above three possibilities: The mass of the bound state can be written in all three cases as Representatives of these three different cases can be chosen from the list of possible 0-brane classes (20):

Changes of scale
In our discussion to this point we have implicitly assumed that some energy scale µ has been fixed, with all quantities measured in units of µ. Indeed, such a choice is necessary in order to make sense of our main formula (57), T = 2π exp ω(C), which requires that T be dimensionless. To understand the significance of this choice let us write the µ dependence explicitly: then for a p-brane state corresponding to a curve C, Now consider changing scale from µ to µ ′ ; then we will have log(T /2πµ ′ p+1 ) = log(T /2πµ p+1 ) + (p + 1) log(µ/µ ′ ) = ω(C) + (p + 1) log(µ/µ ′ ).
Hence a change in µ is equivalent to a shift in the generalized Kähler class ω along the direction of K. In particular, shifts along the K direction do not affect any dimensionless quantities one might compute on the M-theory side (e.g. ratios of masses of 1 2 -BPS states.) So it is natural to think of ω as decomposed into where ω ⊥ is orthogonal to K; then ω ⊥ controls all the dimensionless quantities and λ sets the units of measurement. Setting µ = l p would correspond to fixing λ = 0 in (89). The situation just described is somewhat counterintuitive: one might have expected that the choice of units in M-theory would correspond to the overall volume of the del Pezzo, but the exponential relation between volumes and masses spoils that idea. In particular, given any ω ∈ H 2 (X, R), there is some λ for which ω + λK is in the Kähler cone; so if we interpret ω as a generalized metric, then whether volumes of curves on the del Pezzo are positive or not is dependent on the units we choose.

Concluding discussion
The duality we have described is mathematically rather striking. Of course, it is important to uncover its physical interpretation. It is hard to believe that the correspondence is purely accidental (it is reminiscent of the purely "accidental" appearance of Dynkin diagrams in the intersection matrix of vanishing 2-cycles in K3). One possibility is that the del Pezzo is the moduli space of some probe in M-theory. If so this probe must be a U-duality invariant probe, and should be unlike any brane with which we are familiar. However, we can obtain a hint about what it should be from the del Pezzo side. Recall that the U-duality group was mapped to the group of global diffeomorphisms of del Pezzo which preserve K. Thus a curve in the class given by −K will be U-duality invariant. Such a curve would have genus one (it is the elliptic curve in P 2 , with the blow-up classes added.) Up to now we have only considered rational curves, but if we assume there is a p-brane associated to this duality invariant elliptic curve, then we get p + 1 = (−K) · (−K) = 9 − k i.e., it would be an 8 − k-brane, which would have codimension 2 in the uncompactified spacetime. It would be interesting to see if such a U-duality invariant brane exists in some sense, and if so whether its moduli space is given by the del Pezzo.
A rather interesting suggestion has been made by Motl: perhaps the del Pezzos should be viewed as target spaces of (2, 1) strings, thus providing a concrete realization of the proposal [13]. There are a number of obstacles to overcome to make this precise, but there are encouraging signs that this idea may be on the right track [15].
Up to now, we have considered only rectangular compactifications of M-theory, with the C fields vanishing. It is natural to ask how we can relax these conditions. If we fix l p then by a naive dimension count we find that the tangent space to the moduli space of M-theory compactifications could correspond to p=1,2,3,6 where the symbol ⊥ means we restrict to the subspace orthogonal to K. The rectangular compactifications correspond to p = 1 above; this is the map we have already described. The p = 2 can be viewed as the choice of making parallelogram compactification of tori, p = 3 is turning on the C-field and p = 6 corresponds to turning on the 6-form field coupling to the M5-brane (for compactifications to 3-dimensions some other exotic moduli appear). It would be interesting to interpret the p = 1 deformations in some geometric way. Similarly, it would be useful to have a clearer geometric understanding of the role of 1 2 -BPS states whose masses are not of the simple form (66).
It would also be interesting to extend this kind of duality to other compactifications of M-theory. A natural candidate to study in this case is the duality web with 16 supercharges.
group. This is essentially the structure which was uncovered in Section 3.1 and identified with H 2 (B k , Z) ⊂ H 2 (B k , R) (except that in that section we had a lattice of dimension k + 1 rather than k; the one-dimensional extension arises because we allow l p to vary in order to sweep out all of H 2 (B k , R).) The fact that the tension formulas involve the exponential is also natural from this point of view since the BPS mass formulas can be expressed in terms of the eigenvalues of the E k action on states [14].