pccg- jm ashfaque
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String GeometryPostgraduate Conference in Complex Geometry
Cambridge, 2015
Johar M. Ashfaque
University of Liverpool
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Aim of the Talk
To show how geometry has played a key role
To highlight some of the various connections or links
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Outline
Strings Attached
The Role of Geometry
Calabi Yaus
Orbifolds
Extra Dimensions
Coulomb Branch & Higgs Branch in 3D N = 4Supersymmetric Gauge Theories
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
The Fundamental Forces of Nature
Weak Nuclear Force
Strong Nuclear Force
Electromagnetism
Gravity
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Bosons & Fermions
Matter: Fermions
Interactions: Bosons
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Standard Model of Particle Physics
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Then Why String Theory?
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Then Why String Theory?
To incorporate GRAVITY with the Standard Model gaugegroup SU(3)︸ ︷︷ ︸
Strong
×SU(2)× U(1)︸ ︷︷ ︸Electroweak︸ ︷︷ ︸
Rank=4︸ ︷︷ ︸SO(10)
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Why (Bosonic) String Theory Is Not The Complete Story?
Two major setbacks
The ground state of the spectrum always contains a tachyon.As a consequence, the vacuum is unstable.
Does not contain fermions. Where is the matter?
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Why (Bosonic) String Theory Is Not The Complete Story?
Two major setbacks
The ground state of the spectrum always contains a tachyon.As a consequence, the vacuum is unstable.
Does not contain fermions. Where is the matter?
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Why (Bosonic) String Theory Is Not The Complete Story?
Two major setbacks
The ground state of the spectrum always contains a tachyon.As a consequence, the vacuum is unstable.
Does not contain fermions. Where is the matter?
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Z2 Graded Lie Algebra
Let g be a Lie algebra. Then g decomposes as
g = g0 ⊕ g1
where g0 represents even part and g1 represents the odd part.For the linear map
[ , ] : g× g→ g
we have
g0 × g0 → g0
g0 × g1 → g1
g1 × g0 → g1
g1 × g1 → g0
where it can be seen that the linear map on g0 acts as acommutator but on g1 acts as an anti-commutator.
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Why Superstrings?
Supersymmetry is the symmetry that interchanges bosons andfermions.
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Superstring Theories
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Branes
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Dualities
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
T -Duality
T -duality relates a theory with a small compact dimension to atheory where that same dimension is large.
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Modular Invariance
The symmetry group of the torus is SL(2,Z ) which is much biggerthan the U(1) symmetry of the circle.
A string model is consistent whenever all physical quantities areinvariant under these symmetries.
This needs to be checked by looking at the simplest quantity: Theintegrand of the 1-loop vacuum-to-vacuum amplitude known asthe partition function.
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
The Partition Function
Partition function is used to include all physical states
Taking the one-loop partition function transforms theworldsheet into a torus.
A torus has two non-contractible loops often referred toas the ”toroidal” and ”poloidal” directions.
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
The Role of Geometry
Useful insight
Model Building
Low Energy Effective Models of String theory
Models with
4 flat space-time dimensions3 generations of matterN = 1 SUSY
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Calabi-Yaus: Motivation
Superstrings conjectured to exist in 10D: M4 × CY 3(CY 3 is 3 complex dims or 6 real dims)
Compactification of extra dimensions on CY manifolds ispopular as it leaves some of the original SUSY unbroken
Several other motivations for studying these: F-theorycompactifications on CY 4 allow to find many classicalsolutions in the string theory landscape
First attempts at obtaining standard model from string theoryused the now standard compactification of E8 × E8 heteroticstring theory. In such compactifications gens = 1
2 |χ| where χis the Euler characteristic. For 3 generation models, χ = ±6.
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Calabi-Yaus: Motivation
Superstrings conjectured to exist in 10D: M4 × CY 3(CY 3 is 3 complex dims or 6 real dims)
Compactification of extra dimensions on CY manifolds ispopular as it leaves some of the original SUSY unbroken
Several other motivations for studying these: F-theorycompactifications on CY 4 allow to find many classicalsolutions in the string theory landscape
First attempts at obtaining standard model from string theoryused the now standard compactification of E8 × E8 heteroticstring theory. In such compactifications gens = 1
2 |χ| where χis the Euler characteristic. For 3 generation models, χ = ±6.
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Calabi-Yaus: Motivation
Superstrings conjectured to exist in 10D: M4 × CY 3(CY 3 is 3 complex dims or 6 real dims)
Compactification of extra dimensions on CY manifolds ispopular as it leaves some of the original SUSY unbroken
Several other motivations for studying these: F-theorycompactifications on CY 4 allow to find many classicalsolutions in the string theory landscape
First attempts at obtaining standard model from string theoryused the now standard compactification of E8 × E8 heteroticstring theory. In such compactifications gens = 1
2 |χ| where χis the Euler characteristic. For 3 generation models, χ = ±6.
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Complex Manifolds (1)
Note. Essentially, holomorphic transition functions ⇒ complexmanifold.
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Lightening Review: Vector Bundles
A section s of a vector bundle is a map s : B → E such thatπs(x) = x for all x ∈ B.
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
The Various Types
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Complex Manifolds (2)
If (M, J) is an almost complex 2n-fold with N = 0 then J iscalled a complex structure and M a complex n-fold.
This condition of integrability of J being satisfied allows M tobe covered by complex coordinates.
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Kahler Manifolds
Kahler manifolds are a subclass of complex manifolds and, as such,are naturally oriented.In addition to J, Kahler manifolds have a Hermitian metric g (+associated connection) and can thus be denoted by the triplet(M, g , J).
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
de Rham Cohomology (1)
A p-form ω is called closed if dω = 0.
Denote the set of closed p-forms by Zp(M,R). A p-form ω iscalled exact if ω = dη for some (p − 1)-form η.
Denote the set of exact p-forms by Bp(M,R).
Since d2 = 0, exact p-forms are closed. So, the set of exactp-forms is a subset of the set of closed p-forms, that isBp(M,R) ⊂ Zp(M,R), but closed p-forms are not necessarilyexact.
A closed differential form ω on a manifold M is locally exact whena neighbourhood exists around each point in M in which ω = dη.
(Poincare Lemma) Any closed form on a manifoldM is locallyexact.
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
de Rham Cohomology (2)
The de Rham cohomology class of M is defined as
HpdR(M,R) =
Zp(M,R)
Bp(M,R).
The dimension of the de Rham cohomology is given by the p-thBetti number
bp(M) = dimHpdR(M,R).
Poincare duality states
Hk(M) ' Hn−k(M)
and thusbk = bn−k
for an n-dimensional manifold M.
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Chern Classes
Chern classes encode topological information about bundle.
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Calabi-Yaus
CY manifold of real dimension 2m is a compact Kahler manifold(M; J; g) with
Kahler Metric has vanishing Ricci curvature
First Chern class vanishes
a globally defined, nowhere vanishing holomorphic m-form
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Hodge Diamond: Calabi-Yau 3-folds
Interested in CY 3 Hodge numbers hp,q run over p, q = 0, ..., 3.These can’t exceed the top form on the manifold - in this case a(3, 3)-form. This gives Hodge Diamond.
1
0 0
0 h1,1 0
1 h2,1 h1,2 1
0 h1,1 0
0 0
1
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Remarks
Unfixed Hodge numbers in CY 3 Hodge diamond areh1,1, h1,2, h2,1, h2,2
Not Independent as h1,1 = h2,2 (Hodge Dual)h1,2 = h2,1 (Complex Conjugation)
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
K3 Surface
K3 surfaces are examples of Calabi-Yau two-folds.K3 serves as the simplest non-trivial example of Calabi-Yaucompactification. It has also played a crucial role in string dualitiessince the mid 1990s.The Hodge diamond is given by
1 10 0 0
22 1 20 10 0 01 1
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
The Quintic Hypersurface in CP4
We have that the total Chern class for Q is given by
c(Q) = 1 + 10x2 − 40x3
The Euler characteristic is given by the integral over M of the topChern class of M which in the case of the Calabi-Yau 3-fold is
χ =
∫M
c3(M)
The Euler characteristic for the quintic is
χ(Q) = −200
h1,1 = 1 always ⇒ h2,1 = 101
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
The Role of Hodge Diamond
Euler characteristic for CY 3
χ = 2(h1,1 − h2,1)
Interested in CY 3 for 3 generation models with χ = ±6 wecan further restrict to only CY 3 with
h1,1 − h2,1 = ±3
It may be tricky to compute h1,1, h2,1 for certain CY 3.However, there are many ways of computing χ. Often it’seasier to find χ and one of the Hodge numbers. This thenfixes the other and all topological info is fixed.
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Mirror Symmetry
There is a fascinating symmetry of CY manifolds, calledmirror symmetry, that can be seen on Hodge Diamond.
Given a CY manifold M, ∃ another CY manifold M ′ of samedimension h(p,q)(M) = h(3−p,q)(M ′)
This mirror symmetry exchanges h1,1 and h2,1 on Hodgediamond
Although two CY manifolds M, M ′ may look very differentgeometrically, string theory compactification on thesemanifolds leads to identical effective field theories
Means that CY manifolds exist in mirror pairs (M,M ′)
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
The Diophantine Equation
(n + r + 1−
r∑α=1
να
)= 0
By requiring that να ≥ 2, for any fixed value of n there is a finitenumber of solutions. This can be immediately seen as
r∑α=1
να = 1 + n + r ≥ 2r ⇒ 1 + n ≥ r
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Final Remarks: Calabi-Yau 3-Manifolds
For r = 1, ν1 = 5 and N = r + n = 4 we have
CP4[5], χ(CP4[5]) = −200
For r = 2, ν1 + ν2 = 6 and N = r + n = 5 we find
CP5[2, 4], χ(CP5[2, 4]) = −176
CP5[3, 3], χ(CP5[3, 3]) = −144
For r = 3, ν1 + ν2 + ν3 = 7 and N = r + n = 6
CP6[2, 2, 3], χ(CP6[2, 2, 3]) = −144
For r = 4, ν1 + ν2 + ν3 + ν4 = 8 and N = r + n = 7
CP7[2, 2, 2, 2], χ(CP7[2, 2, 2, 2]) = −128
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Summary
Connect String Theory To Low Energy Effective Field Theory
Internal geometry determines 4D theory
Examples
Heterotic Strings on CY 3F-theory on elliptic CY 4
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
A Few Words On Orbifolds
Orbifolds are simply quotient spaces of a manifold modulo somediscrete group.
If G is freely acting, M having no fixed points under G action thenthe orbifold is smooth.
However, if G was to have fixed points then the orbifold hassingularities.
One-dimensional orbifolds are very simple. There are only two ofthem, the circle and the interval.
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Example: The Real Line
Consider the real line R. It has a Z2 symmetry.
This symmetry has one fixed point (a singularity) at x = 0
The orbifoldRZ2
is a half line.
The real line has another infinite symmetry group namely thetranslations
x → x + 2πλ
The resulting orbifold is smooth which is a circle of radius λ.
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Strings Once More
Heterotic strings are hybrid strings with either
left-moving sector being supersymmetric and right-movingsector being bosonic or
left-moving sector being bosonic and right-moving sectorbeing supersymmetric.
There are two heterotic string theories, one associated to thegauge group
E8 × E8
and the other toSO(32)
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Strings Once More
Heterotic strings are hybrid strings with either
left-moving sector being supersymmetric and right-movingsector being bosonic or
left-moving sector being bosonic and right-moving sectorbeing supersymmetric.
There are two heterotic string theories, one associated to thegauge group
E8 × E8
and the other toSO(32)
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Strings Once More
Heterotic strings are hybrid strings with either
left-moving sector being supersymmetric and right-movingsector being bosonic or
left-moving sector being bosonic and right-moving sectorbeing supersymmetric.
There are two heterotic string theories, one associated to thegauge group
E8 × E8
and the other toSO(32)
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
The Free Fermionic Construction
A general boundary condition basis vector is of the form
α ={ψ1,2, χi , y i , ωi |y i , ωi , ψ
1,...,5, η1,2,3, φ
1,...,8}
where i = 1, ..., 6
ψ1,...,5
- SO(10) gauge group
φ1,...,8
- SO(16) gauge group
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
The Very Simple Rules
The ABK Rules[Antoniadis,Bachas,Kounnas, 1987]
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
An Example: The NAHE Set
The NAHE set is the set of basis vectors
B = {1,S,b1,b2,b3}
where
1 = {ψ1,2µ , χ1,...,6, y1,...,6, ω1,...,6|y1,...,6, ω1,...,6, ψ1,...,5, η1,2,3, φ1,...,8},
S = {ψ1,2µ , χ1,...,6},
b1 = {ψ1,2µ , χ1,2, y3,...,6|y3,...,6, ψ1,...,5, η1},
b2 = {ψ1,2µ , χ3,4, y1,2, ω5,6|y1,2, ω5,6, ψ1,...,5, η2},
b3 = {ψ1,2µ , χ5,6, ω1,...,4|ω1,...,4, ψ1,...,5, η3}.
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
The NAHE: The Gauge Group
SO(44)
��SO(10)× E8 × SO(6)3
withN = 4
��N = 2
��N = 1
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
The Various SO(10) Breakings
SO(10)
α
��
α+β // SU(5)× U(1)
SO(6)× SO(4)
�
SU(3)C × U(1)C × SU(2)L × U(1)L
SO(10)
α+β+γ��
SU(3)C × U(1)C × SU(2)L × SU(2)R
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Extra Dimensions
In models with extra dimensions the usual (3 + 1)-dimensionalspace-time xµ = (x0, x1, x2, x3) is extended to include additionalspatial dimensions parametrized by coordinates x4, x5, ..., x3+N
where N is the number of extra dimensions. String theoryarguments would suggest that N can be as large as 6 or 7.
Depending on the type of metric in the bulk, ED models fall intoone of the following two categories:
Flat, also known as universal ED models (UED).
Warped ED models.
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
UED Models
The metric on the extra dimensions is chosen to be flat. However,to implement the chiral fermions of the SM in UED models onemust use an orbifold S1/Z2. The size of the extra dimension issimply parametrized by the radius of the circle R.
In the case of N = 2, one of the many is known as the chiralsquare corresponding to T 2/Z4. The two extra dimensions haveequal size and the boundary conditions are such that adjacent sidesof the chiral square are identified.
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Supersymmetric Gauge Theories
Supersymmetric Gauge Theories in 3D N = 4 are subject to astrange duality: Mirror Symmetry
3D mirror symmetry exchanges Coulomb branch and Higgsbranch of two dual theories.
Mirror Symmetry←−−−−−−−−→
Coulomb branch: moduli space parametrised by scalars in theV-plet
Higgs branch: moduli space parametrised by scalars in theH-plet
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
The Idea
Study moduli space of instantons ⇒ calculate Hilbert Series forHiggs Branch.
What is the Hilbert series?
It is the partition function that counts chiral gauge invariantoperators
Why Bother?
The chiral gauge invariant operators parametrize the modulispace
Hilbert Series encodes all the information: dimension of themoduli space, generators and relations
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Example: Hilbert Series for C2/Z2
C2 with action of Z2: (z1, z2)↔ (−z1,−z2)
Holomorphic functions invariant under this action:z21 , z
22 , z1z2, z
41 , ...
All polynomials constructed from 3 generators subject to 1relation
X = z21 ,Y = z22 ,Z = z1z2
XY = Z 2
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
Example Contd.: Hilbert Series for C2/Z2
Isometry group of C2 = U(2)Cartan Subalgebra: U(1)2
Choose counterst1 is the U(1) charge of z1t2 is the U(1) charge of z2
HS(t1, t2, ;C2) = 1+t21 +t22 +t1t2+... =∞∑
i ,j=0
t i1tj2 =
∏i
1
1− ti
Unrefine
HS(t) =∞∑
i ,j ,...
t i+j = 1 + 3t2 + 5t4 + ... =∑k=0
(2k + 1)t2k
Dimension of the moduli space
=
pole of the unrefined Hilbert series
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge
THANK YOU!!!
Johar M. Ashfaque Postgraduate Conference in Complex Geometry, Cambridge