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The SYZ conjecture is an attempt to understand the
mirror symmetry In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D ther ...
conjecture, an issue in theoretical physics and mathematics. The original conjecture was proposed in a paper by Strominger, Yau, and Zaslow, entitled "Mirror Symmetry is ''T''-duality".. Along with the homological mirror symmetry conjecture, it is one of the most explored tools applied to understand mirror symmetry in mathematical terms. While the homological mirror symmetry is based on
homological algebra Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
, the SYZ conjecture is a geometrical realization of mirror symmetry.


Formulation

In
string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interac ...
, mirror symmetry relates type IIA and
type IIB In theoretical physics, type II string theory is a unified term that includes both type IIA strings and type IIB strings theories. Type II string theory accounts for two of the five consistent superstring theory, superstring theories in ten dimens ...
theories. It predicts that the effective field theory of type IIA and type IIB should be the same if the two theories are compactified on mirror pair manifolds. The SYZ conjecture uses this fact to realize mirror symmetry. It starts from considering
BPS state BPS, Bps or bps may refer to: Science and mathematics *Plural of bp, base pair, a measure of length of DNA *Plural of bp, basis point, one one-hundredth of a percentage point - ‱ *Battered person syndrome, a physical and psychological condition ...
s of type IIA theories compactified on ''X'', especially 0-branes that have
moduli space In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spac ...
''X''. It is known that all of the BPS states of type IIB theories compactified on ''Y'' are 3-branes. Therefore, mirror symmetry will map 0-branes of type IIA theories into a subset of 3-branes of type IIB theories. By considering
supersymmetric In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories ...
conditions, it has been shown that these 3-branes should be
special Lagrangian submanifold In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sympl ...
s... On the other hand,
T-duality In theoretical physics, T-duality (short for target-space duality) is an equivalence of two physical theories, which may be either quantum field theories or string theories. In the simplest example of this relationship, one of the theories descr ...
does the same transformation in this case, thus "mirror symmetry is T-duality".


Mathematical statement

The initial proposal of the SYZ conjecture by Strominger, Yau, and Zaslow, was not given as a precise mathematical statement. One part of the mathematical resolution of the SYZ conjecture is to, in some sense, correctly formulate the statement of the conjecture itself. There is no agreed upon precise statement of the conjecture within the mathematical literature, but there is a general statement that is expected to be close to the correct formulation of the conjecture, which is presented here.Gross, M., Huybrechts, D. and Joyce, D., 2012. Calabi-Yau manifolds and related geometries: lectures at a summer school in Nordfjordeid, Norway, June 2001. Springer Science & Business Media. This statement emphasizes the topological picture of mirror symmetry, but does not precisely characterise the relationship between the complex and symplectic structures of the mirror pairs, or make reference to the associated
Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...
s involved.
SYZ Conjecture: Every 6-dimensional Calabi–Yau manifold X has a mirror 6-dimensional Calabi–Yau manifold \hat such that there are continuous surjections f: X\to B, \hat:\hat \to B to a compact topological manifold B of dimension 3, such that # There exists a dense open subset B_\subset B on which the maps f,\hat are
fibration The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics. Fibrations are used, for example, in postnikov-systems or obstruction theory. In this article, all map ...
s by nonsingular special Lagrangian 3-tori. Furthermore for every point b\in B_, the torus fibres f^(b) and \hat^(b) should be dual to each other in some sense, analogous to duality of Abelian varieties. # For each b\in B\backslash B_, the fibres f^(b) and \hat f^(b) should be singular 3-dimensional special Lagrangian submanifolds of X and \hat respectively.
The situation in which B_ = B so that there is no singular locus is called the ''semi-flat limit'' of the SYZ conjecture, and is often used as a model situation to describe torus fibrations. The SYZ conjecture can be shown to hold in some simple cases of semi-flat limits, for example given by
Abelian varieties In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular function ...
and
K3 surface In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with trivial canonical bundle and irregularity zero. An (algebraic) K3 surface over any field means a smooth proper geometrically connected alg ...
s which are fibred by
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
s. It is expected that the correct formulation of the SYZ conjecture will differ somewhat from the statement above. For example the possible behaviour of the singular set B\backslash B_ is not well understood, and this set could be quite large in comparison to B. Mirror symmetry is also often phrased in terms of degenerating families of Calabi–Yau manifolds instead of for a single Calabi–Yau, and one might expect the SYZ conjecture to reformulated more precisely in this language.


Relation to homological mirror symmetry conjecture

The SYZ mirror symmetry conjecture is one possible refinement of the original mirror symmetry conjecture relating Hodge numbers of mirror Calabi–Yau manifolds. The other is Kontsevich's homological mirror symmetry conjecture (HMS conjecture). These two conjectures encode the predictions of mirror symmetry in different ways: homological mirror symmetry in an ''algebraic'' way, and the SYZ conjecture in a ''geometric'' way. There should be a relationship between these three interpretations of mirror symmetry, but it is not yet known whether they should be equivalent or one proposal is stronger than the other. It is known that under certain assumptions homological mirror symmetry implies Hodge theoretic mirror symmetry.Ganatra, S., Perutz, T. and Sheridan, N., 2015. Mirror symmetry: from categories to curve counts. ''arXiv preprint arXiv:1510.03839''. Nevertheless, in simple settings there are clear ways of relating the SYZ and HMS conjectures. The key feature of HMS is that the conjecture relates objects (either submanifolds or sheaves) on mirror geometric spaces, so the required input to try to understand or prove the HMS conjecture includes a mirror pair of geometric spaces. The SYZ conjecture predicts how these mirror pairs should arise, and so whenever an SYZ mirror pair is found, it is a good candidate to try and prove the HMS conjecture on this pair. To relate the SYZ and HMS conjectures, it is convenient to work in the semi-flat limit. The important geometric feature of a pair of Lagrangian torus fibrations X,\hat X \to B which encodes mirror symmetry is the ''dual torus fibres'' of the fibration. Given a Lagrangian torus T\subset X, the dual torus is given by the
Jacobian variety In mathematics, the Jacobian variety ''J''(''C'') of a non-singular algebraic curve ''C'' of genus ''g'' is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of ''C'', hence an abelian vari ...
of T, denoted \hat T = \mathrm(T). This is again a torus of the same dimension, and the duality is encoded in the fact that \mathrm(\mathrm(T)) = T so T and \hat T are indeed dual under this construction. The Jacobian variety \hat T has the important interpretation as the ''moduli space of line bundles'' on T. This duality and the interpretation of the dual torus as a moduli space of sheaves on the original torus is what allows one to interchange the data of submanifolds and subsheaves. There are two simple examples of this phenomenon: * If p\in X is a point which lies inside some fibre p\in T\subset X of the special Lagrangian torus fibration, then since T = \mathrm(\hat T), the point p corresponds to a line bundle supported on \hat T \subset \hat X. If one chooses a Lagrangian section s: B \to X such that s(B)=L is a Lagrangian submanifold of X, then precisely since s chooses one point in each torus fibre of the SYZ fibration, this Lagrangian section is mirror dual to a choice of line bundle structure supported on each torus fibre of the mirror manifold \hat X, and consequently a line bundle on the total space of \hat X, the simplest example of a
coherent sheaf In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refe ...
appearing in the derived category of the mirror manifold. If the mirror torus fibrations are not in the semi-flat limit, then special care must be taken when crossing over singular set of the base B. * Another example of a Lagrangian submanifold is the torus fibre itself, and one sees that if the entire torus is taken as the Lagrangian T\subset X, with the added data of a flat unitary
line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organisin ...
over it, as is often necessary in homological mirror symmetry, then in the dual torus \hat T \subset \hat X this corresponds to a single point which represents that line bundle over the torus. If one takes the
skyscraper sheaf In mathematics, a sheaf is a tool for systematically tracking data (such as Set (mathematics) , sets, abelian groups, Ring (mathematics) , rings) attached to the open sets of a topological space and defined locally with regard to them. For exam ...
supported on that point in the dual torus, then we see ''torus fibres of the SYZ fibration get sent to skyscraper sheaves supported on points in the mirror torus fibre''. These two examples produce the most extreme kinds of
coherent sheaf In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refe ...
,
locally free sheaves In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with ref ...
(of rank 1) and torsion sheaves supported on points. By more careful construction one can build up more complicated examples of coherent sheaves, analogous to building a coherent sheaf using the torsion filtration. As a simple example, a Lagrangian ''multisection'' (a union of ''k'' Lagrangian sections) should be mirror dual to a rank ''k'' vector bundle on the mirror manifold, but one must take care to account for ''instanton corrections'' by counting ''holomorphic discs'' which are bounded by the multisection, in the sense of Gromov-Witten theory. In this way
enumerative geometry In mathematics, enumerative geometry is the branch of algebraic geometry concerned with counting numbers of solutions to geometric questions, mainly by means of intersection theory. History The problem of Apollonius is one of the earliest examp ...
becomes important for understanding how mirror symmetry interchanges dual objects. By combining the geometry of mirror fibrations in the SYZ conjecture with a detailed understanding of enumerative invariants and the structure of the singular set of the base B, it is possible to use the geometry of the fibration to build the isomorphism of categories from the Lagrangian submanifolds of X to the coherent sheaves of \hat X, the map \mathrm(X) \to \mathrm^b \mathrm(\hat X). By repeating this same discussion in reverse using the duality of the torus fibrations, one similarly can understand coherent sheaves on X in terms of Lagrangian submanifolds of \hat X, and hope to get a complete understanding of how the HMS conjecture relates to the SYZ conjecture.


References

String theory Symmetry Duality theories Conjectures {{applied-math-stub