Mirror symmetry
Encyclopedia
In physics
and mathematics
, mirror symmetry is a relation that can exist between two Calabi-Yau manifold
s. It happens, usually for two such six-dimensional manifolds, that the shapes may look very different geometrically, but nevertheless they are equivalent if they are employed as hidden dimensions of string theory
. The classical formulation of mirror symmetry relates two Calabi-Yau threefolds M and W whose Hodge numbers h1,1 and h1,2 are swapped; string theory compactified on these two manifolds lead to identical effective field theories.
, Nicholas Warner, Brian Greene
, Ronen Plesser, Philip Candelas, Monika Lynker, Rolf Schimmrigk and others. Andrew Strominger
, Shing-Tung Yau
, and Eric Zaslow have showed that mirror symmetry is a special example of T-duality
: the Calabi-Yau manifold may be written as a fiber bundle
whose fiber is a three-dimensional torus
. The simultaneous action of T-duality on all three dimensions of this torus is equivalent to mirror symmetry.
Mathematicians became interested in mirror symmetry in 1990, after Candelas-de la Ossa-Green-Parkes gave predictions for numbers of rational curves in a quintic threefold
via data coming from variation of Hodge structure on the mirror family. These predictions were mathematically proven a few years later by Alexander Givental
and Lian-Liu-Yau.
Most of the physical examples can be conceptualized by the Batyrev-Borisov mirror construction, which uses the duality of reflexive polytopes and nef partitions. In their construction the mirror partners appear as anticanonically embedded hypersurface
s or certain complete intersection
s in Fano
toric varieties. The Gross-Siebert mirror construction generalizes this to non-embedded cases by looking at degenerating families of Calabi-Yau manifolds. This point of view also includes T-duality. Another mathematical framework is provided by the homological mirror symmetry
conjecture.
This follows from the fact that Calabi-Yau hodge numbers satisfy h1,1
1 but h2,1
0. If the Hodge numbers of M are such that h2,1=0 then by definition its mirror dual W is not Calabi-Yau. As a result mirror symmetry allows for the definition of an extended space of compact spaces, which are defined by the W of the above two mirror symmetries.
Mirror symmetry has also been generalized to a duality between supersymmetric
gauge theories
in various numbers of dimensions. In this generalized context the original mirror symmetry, which relates pairs of toric
Calabi-Yau manifolds, relates the moduli space
s of 2-dimensional abelian supersymmetric gauge theories when the sums of the electric charges of the matter are equal to zero.
In all manifestations of mirror symmetry found so far a central role is played by the fact that in a d-dimensional quantum field theory a differential p-form
potential admits a dual formulation as a (d-p-2)-form potential. In 4-dimensions this relates the electric and magnetic vector potential
s and is called electric-magnetic duality. In 3-dimensions this duality relates a vector
and a scalar
, which in an abelian gauge theory correspond to a photon
and a squark. In 2-dimensions it relates two scalars, but while one carries an electric charge, the dual scalar is an uncharged Fayet-Iliopoulos term. In the process of this duality topological solitons called Abrikosov-Nielsen-Oleson vortices
are intercharged with elementary quark
fields in the 3-dimensional case and play the role in instantons in the 2-dimensional case.
The derivations of 2-dimensional mirror symmetry and 3-dimensional mirror symmetry are both inspired by Alexander Polyakov's instanton
calculation in non-supersymmetric quantum electrodynamics
with a scalar Higgs field. In a 1977 article he demonstrated that instanton effects give the photon a mass, where the instanton is a 't Hooft-Polyakov monopole
embedded in an ultraviolet
nonabelian gauge group.
and type II superstring theory. Generalizations to N=(2,0) supersymmetry have also appeared.
The matter content of N=(2,2) gauged linear sigma models consists of three kinds of supermultiplet
. The gauge bosons occur in vector multiplets, the charged matter occurs in chiral multiplet
s and the Fayet-Ilipolous (FI) terms of the various abelian gauge symmetries occur in twisted chiral multiplets. Mirror symmetry exchanges chiral and twisted chiral multiplets.
Mirror symmetry, in a class of models of toric varieties with zero first Chern class
Calabi-Yau manifolds and positive first Chern class (Fano varieties
) was proven by Kentaro Hori and Cumrun Vafa
. Their approach is as follows. A sigma model whose target space is a toric variety may be described by an abelian gauge theory with charged chiral multiplets. Mirror symmetry then replaces these charged chiral multiplets with uncharged twisted chiral multiplets whose vacuum expectation value
s are FI terms. Instantons in the dual theory are now vortices whose action is given by the exponential of the FI term. These vortices each have precisely 2 fermion zeromodes, and so the sole correction to the superpotential
is given by a single vortex. The nonperturbative corrections to the dual superpotential may then be found by simply summing the exponentials of the FI terms. Therefore mirror symmetry allows one to find the full nonperturbative solutions to the theory.
In addition to finding many new dualities, this allowed them to demonstrate many dualities that had been conjectured in the literature. For example, beginning with a sigma model whose target space is the 2-sphere they found an exactly solvable Sine-Gordon model
. More generally, when the original sigma model's target space is the n-complex dimensional projective space they found that the dual theory is the exactly solvable affine Toda model
.
in their 1996 paper as a relation between pairs of 3-dimensional gauge theories such that the Coulomb branch of the moduli space of one is the Higgs branch of the moduli space of the other. It was demonstrated using D-brane
cartoons by Amihay Hanany and Edward Witten
4 months later, where they found that it is a consequence of S-duality
in type IIB string theory.
Four months later it was extended to N=2 gauge theories resulting from supersymmetry breaking
in N=4 theories. Here it was given a physical interpretation in terms of vortices. Vortices in 3-dimensional gauge theories are particles. BPS
vortices, which are those vortices that preserve some supersymmetry, have masses which are given by the FI term of the gauge theory. In particular, on the Higgs branch, where the squarks are massless and condense yielding nontrivial vacuum expectation values (VEVs), the vortices are massive. On the other hand they interpret the Coulomb branch of the gauge theory, where the scalar in the vector multiplet has a VEV, as being the regime where massless vortices condense. Thus the duality between the Coulumb branch in one theory and the Higgs branch in the dual theory is the duality between squarks and vortices.
In this theory the instantons are 't Hooft-Polyakov magnetic monopoles, whose actions are proportional to the VEV of the scalar in the vector multiplet. In this case instanton calculations again reproduce the nonperturbative superpotential. In particular, in the N=4 case with SU(2) gauge symmetry, the metric on the moduli space was found by Nathan Seiberg and Edward Witten using holomorphy and supersymmetric nonrenormalization theorems
several days before Intriligator and Seiberg's 3-dimensional mirror symmetry paper appeared. Their results were reproduced using standard instanton techniques.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
and mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, mirror symmetry is a relation that can exist between two Calabi-Yau manifold
Calabi-Yau manifold
A Calabi-Yau manifold is a special type of manifold that shows up in certain branches of mathematics such as algebraic geometry, as well as in theoretical physics...
s. It happens, usually for two such six-dimensional manifolds, that the shapes may look very different geometrically, but nevertheless they are equivalent if they are employed as hidden dimensions of string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
. The classical formulation of mirror symmetry relates two Calabi-Yau threefolds M and W whose Hodge numbers h1,1 and h1,2 are swapped; string theory compactified on these two manifolds lead to identical effective field theories.
History
The discovery of mirror symmetry is connected with names such as Lance Dixon, Wolfgang Lerche, Cumrun VafaCumrun Vafa
Cumrun Vafa is an Iranian-American leading string theorist from Harvard University where he started as a Harvard Junior Fellow. He is a recipient of the 2008 Dirac Medal.-Birth and education:...
, Nicholas Warner, Brian Greene
Brian Greene
Brian Greene is an American theoretical physicist and string theorist. He has been a professor at Columbia University since 1996. Greene has worked on mirror symmetry, relating two different Calabi-Yau manifolds...
, Ronen Plesser, Philip Candelas, Monika Lynker, Rolf Schimmrigk and others. Andrew Strominger
Andrew Strominger
Andrew Eben Strominger is an American theoretical physicist who works on string theory and son of Jack L. Strominger. He is currently a professor at Harvard University and a senior fellow at the Society of Fellows...
, Shing-Tung Yau
Shing-Tung Yau
Shing-Tung Yau is a Chinese American mathematician working in differential geometry. He was born in Shantou, Guangdong Province, China into a family of scholars from Jiaoling, Guangdong Province....
, and Eric Zaslow have showed that mirror symmetry is a special example of T-duality
T-duality
T-duality is a symmetry of quantum field theories with differing classical descriptions, of which the relationship between small and large distances in various string theories is a special case. Discussion of the subject originated in a paper by T. S. Buscher and was further developed by Martin...
: the Calabi-Yau manifold may be written as a fiber bundle
Fiber bundle
In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...
whose fiber is a three-dimensional torus
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
. The simultaneous action of T-duality on all three dimensions of this torus is equivalent to mirror symmetry.
Mathematicians became interested in mirror symmetry in 1990, after Candelas-de la Ossa-Green-Parkes gave predictions for numbers of rational curves in a quintic threefold
Quintic threefold
In mathematics, a quintic threefold is a degree 5 dimension 3 hypersurface in 4-dimensional projective space. Non-singular quintic threefolds are Calabi-Yau manifolds.The Hodge diamond of a non-singular quintic 3-fold is-Rational curves:...
via data coming from variation of Hodge structure on the mirror family. These predictions were mathematically proven a few years later by Alexander Givental
Alexander Givental
Alexander Givental is a Russian American mathematician working in the area of symplectic topology, singularity theory and their relations to topological string theories. He got his Ph. D. under the supervision of V. I. Arnold. He, first, proved the mirror conjecture for toric Calabi-Yau manifolds,...
and Lian-Liu-Yau.
Applications
Mirror symmetry allowed the physicists to calculate many quantities that seemed virtually incalculable before, by invoking the "mirror" description of a given physical situation, which can be often much easier. Mirror symmetry has also become a tool in mathematics, and although mathematicians have proved theorems based on the physicists' intuition, a full mathematical understanding of the phenomenon of mirror symmetry is still being developed.Most of the physical examples can be conceptualized by the Batyrev-Borisov mirror construction, which uses the duality of reflexive polytopes and nef partitions. In their construction the mirror partners appear as anticanonically embedded hypersurface
Hypersurface
In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface...
s or certain complete intersection
Complete intersection
In mathematics, an algebraic variety V in projective space is a complete intersection if it can be defined by the vanishing of the number of homogeneous polynomials indicated by its codimension...
s in Fano
Fano variety
In algebraic geometry, a Fano variety, introduced by , is a non-singular complete variety whose anticanonical bundle is ample.Fano varieties are quite rare, compared to other families, like Calabi–Yau manifolds and general type surfaces....
toric varieties. The Gross-Siebert mirror construction generalizes this to non-embedded cases by looking at degenerating families of Calabi-Yau manifolds. This point of view also includes T-duality. Another mathematical framework is provided by the homological mirror symmetry
Homological mirror symmetry
Homological mirror symmetry is a mathematical conjecture made by Maxim Kontsevich. It seeks a systematic mathematical explanation for a phenomenon called mirror symmetry first observed by physicists studying string theory.-History:...
conjecture.
Generalizations
There are two different, but closely related, string theory statements of mirror symmetry.- 1. Type IIA string theory on a Calabi-Yau M is mirror dual to Type IIB on W.
- 2. Type IIB string theory on a Calabi-Yau M is mirror dual to Type IIA on W.
This follows from the fact that Calabi-Yau hodge numbers satisfy h1,1
1 but h2,1 0. If the Hodge numbers of M are such that h2,1=0 then by definition its mirror dual W is not Calabi-Yau. As a result mirror symmetry allows for the definition of an extended space of compact spaces, which are defined by the W of the above two mirror symmetries.Mirror symmetry has also been generalized to a duality between supersymmetric
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
gauge theories
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
in various numbers of dimensions. In this generalized context the original mirror symmetry, which relates pairs of toric
Toric geometry
In algebraic geometry, a toric variety or torus embedding is a normal variety containing an algebraic torus as a dense subset, such that the action of the torus on itself extends to the whole variety.-The toric variety of a fan:...
Calabi-Yau manifolds, relates the moduli space
Moduli space
In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...
s of 2-dimensional abelian supersymmetric gauge theories when the sums of the electric charges of the matter are equal to zero.
In all manifestations of mirror symmetry found so far a central role is played by the fact that in a d-dimensional quantum field theory a differential p-form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
potential admits a dual formulation as a (d-p-2)-form potential. In 4-dimensions this relates the electric and magnetic vector potential
Vector potential
In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a scalar potential, which is a scalar field whose negative gradient is a given vector field....
s and is called electric-magnetic duality. In 3-dimensions this duality relates a vector
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
and a scalar
Scalar field
In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the...
, which in an abelian gauge theory correspond to a photon
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
and a squark. In 2-dimensions it relates two scalars, but while one carries an electric charge, the dual scalar is an uncharged Fayet-Iliopoulos term. In the process of this duality topological solitons called Abrikosov-Nielsen-Oleson vortices
Nielsen-Olesen vortex
In theoretical physics, a Nielsen–Olesen vortex is a point-like object localized in two spatial dimensions or, equivalently, a classical solution of field theory with the same property. This particular solution occurs if the configuration space of scalar fields contains non-contractible circles...
are intercharged with elementary quark
Quark
A quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never directly...
fields in the 3-dimensional case and play the role in instantons in the 2-dimensional case.
The derivations of 2-dimensional mirror symmetry and 3-dimensional mirror symmetry are both inspired by Alexander Polyakov's instanton
Instanton
An instanton is a notion appearing in theoretical and mathematical physics. Mathematically, a Yang–Mills instanton is a self-dual or anti-self-dual connection in a principal bundle over a four-dimensional Riemannian manifold that plays the role of physical space-time in non-abelian gauge theory...
calculation in non-supersymmetric quantum electrodynamics
Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...
with a scalar Higgs field. In a 1977 article he demonstrated that instanton effects give the photon a mass, where the instanton is a 't Hooft-Polyakov monopole
't Hooft-Polyakov monopole
In theoretical physics, the t Hooft–Polyakov monopole is a topological soliton similar to the Dirac monopole but without any singularities. It arises in the case of a Yang–Mills theory with a gauge group G, coupled to a Higgs field which spontaneously breaks it down to a smaller group H via the...
embedded in an ultraviolet
Ultraviolet
Ultraviolet light is electromagnetic radiation with a wavelength shorter than that of visible light, but longer than X-rays, in the range 10 nm to 400 nm, and energies from 3 eV to 124 eV...
nonabelian gauge group.
Mirror symmetry in 2-dimensional gauged sigma models
Mirror symmetries in 2-dimensional sigma models are usually considered in cases with N=(2,2) supersymmetry, which means that the fermionic supersymmetry generators are the four real components of a single Dirac spinor. This is the case which is relevant, for example, to topological string theoriesTopological string theory
In theoretical physics, topological string theory is a simplified version of string theory. The operators in topological string theory represent the algebra of operators in the full string theory that preserve a certain amount of supersymmetry...
and type II superstring theory. Generalizations to N=(2,0) supersymmetry have also appeared.
The matter content of N=(2,2) gauged linear sigma models consists of three kinds of supermultiplet
Supermultiplet
In theoretical physics, a supermultiplet is formally a group representation of a supersymmetry algebra. It consists of a collection of particles, called superpartners, corresponding to operators in a quantum field theory which in superspace are represented by superfields.Superfields were...
. The gauge bosons occur in vector multiplets, the charged matter occurs in chiral multiplet
Chiral superfield
In theoretical physics, one often analyzes theories with supersymmetry in which chiral superfields play an important role. In four dimensions, the minimal N=1 supersymmetry may be written using the notion of superspace...
s and the Fayet-Ilipolous (FI) terms of the various abelian gauge symmetries occur in twisted chiral multiplets. Mirror symmetry exchanges chiral and twisted chiral multiplets.
Mirror symmetry, in a class of models of toric varieties with zero first Chern class
Chern class
In mathematics, in particular in algebraic topology and differential geometry, the Chern classes are characteristic classes associated to complex vector bundles.Chern classes were introduced by .-Basic idea and motivation:...
Calabi-Yau manifolds and positive first Chern class (Fano varieties
Fano variety
In algebraic geometry, a Fano variety, introduced by , is a non-singular complete variety whose anticanonical bundle is ample.Fano varieties are quite rare, compared to other families, like Calabi–Yau manifolds and general type surfaces....
) was proven by Kentaro Hori and Cumrun Vafa
Cumrun Vafa
Cumrun Vafa is an Iranian-American leading string theorist from Harvard University where he started as a Harvard Junior Fellow. He is a recipient of the 2008 Dirac Medal.-Birth and education:...
. Their approach is as follows. A sigma model whose target space is a toric variety may be described by an abelian gauge theory with charged chiral multiplets. Mirror symmetry then replaces these charged chiral multiplets with uncharged twisted chiral multiplets whose vacuum expectation value
Vacuum expectation value
In quantum field theory the vacuum expectation value of an operator is its average, expected value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle...
s are FI terms. Instantons in the dual theory are now vortices whose action is given by the exponential of the FI term. These vortices each have precisely 2 fermion zeromodes, and so the sole correction to the superpotential
Superpotential
Superpotential is a concept from particle physics' supersymmetry.-Example of superpotentiality:Let's look at the example of a one dimensional nonrelativistic particle with a 2D internal degree of freedom called "spin"...
is given by a single vortex. The nonperturbative corrections to the dual superpotential may then be found by simply summing the exponentials of the FI terms. Therefore mirror symmetry allows one to find the full nonperturbative solutions to the theory.
In addition to finding many new dualities, this allowed them to demonstrate many dualities that had been conjectured in the literature. For example, beginning with a sigma model whose target space is the 2-sphere they found an exactly solvable Sine-Gordon model
Sine-Gordon equation
The sine–Gordon equation is a nonlinear hyperbolic partial differential equation in 1 + 1 dimensions involving the d'Alembert operator and the sine of the unknown function. It was originally considered in the nineteenth century in the course of study of surfaces of constant negative...
. More generally, when the original sigma model's target space is the n-complex dimensional projective space they found that the dual theory is the exactly solvable affine Toda model
Toda field theory
In the study of field theory and partial differential equations, a Toda field theory is derived from the following Lagrangian:...
.
Mirror symmetry in 3-dimensional gauge theories
Mirror symmetry in 3-dimensional gauge theories with N=4 supersymmetry, or 8 supercharges, was first proposed by Kenneth Intriligator and Nathan SeibergNathan Seiberg
Nathan "Nati" Seiberg is an Israeli American theoretical physicist who works on string theory. He was recipient of a 1996 MacArthur Fellowship and the Dannie Heineman Prize for Mathematical Physics in 1998. He is currently a professor at the Institute for Advanced Study in Princeton, New Jersey, USA...
in their 1996 paper as a relation between pairs of 3-dimensional gauge theories such that the Coulomb branch of the moduli space of one is the Higgs branch of the moduli space of the other. It was demonstrated using D-brane
D-brane
In string theory, D-branes are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named. D-branes were discovered by Dai, Leigh and Polchinski, and independently by Hořava in 1989...
cartoons by Amihay Hanany and Edward Witten
Edward Witten
Edward Witten is an American theoretical physicist with a focus on mathematical physics who is currently a professor of Mathematical Physics at the Institute for Advanced Study....
4 months later, where they found that it is a consequence of S-duality
S-duality
In theoretical physics, S-duality is an equivalence of two quantum field theories or string theories. An S-duality transformation maps states and vacua with coupling constant g in one theory to states and vacua with coupling constant 1/g in the dual theory...
in type IIB string theory.
Four months later it was extended to N=2 gauge theories resulting from supersymmetry breaking
Supersymmetry breaking
In particle physics, supersymmetry breaking is the process to obtain a seemingly non-supersymmetric physics from a supersymmetric theory which is a necessary step to reconcile supersymmetry with actual experiments. It is an example of spontaneous symmetry breaking...
in N=4 theories. Here it was given a physical interpretation in terms of vortices. Vortices in 3-dimensional gauge theories are particles. BPS
BPS
BPS can stand for:In science, mathematics and technology*Base pairs, a measure of length of DNA using the number of nucleotide pairs*Basis points, equal to one one-hundredth of a percentage point...
vortices, which are those vortices that preserve some supersymmetry, have masses which are given by the FI term of the gauge theory. In particular, on the Higgs branch, where the squarks are massless and condense yielding nontrivial vacuum expectation values (VEVs), the vortices are massive. On the other hand they interpret the Coulomb branch of the gauge theory, where the scalar in the vector multiplet has a VEV, as being the regime where massless vortices condense. Thus the duality between the Coulumb branch in one theory and the Higgs branch in the dual theory is the duality between squarks and vortices.
In this theory the instantons are 't Hooft-Polyakov magnetic monopoles, whose actions are proportional to the VEV of the scalar in the vector multiplet. In this case instanton calculations again reproduce the nonperturbative superpotential. In particular, in the N=4 case with SU(2) gauge symmetry, the metric on the moduli space was found by Nathan Seiberg and Edward Witten using holomorphy and supersymmetric nonrenormalization theorems
Supersymmetry nonrenormalization theorems
In theoretical physics a nonrenormalization theorem is a limitation on how a certain quantity in the classical description of a quantum field theory may be modified by renormalization in the full quantum theory...
several days before Intriligator and Seiberg's 3-dimensional mirror symmetry paper appeared. Their results were reproduced using standard instanton techniques.