Prof. Vishnu Jejjala (Witwatersrand) TITLE: "The Geometry of Generations"

1. The Geometry of
Generations
Vishnu Jejjala
University of the witwatersrand
23 September 2014

2. The Collaboration
James Gray Yang-Hui He Cyril Matti Brent Nelson Mike Stillman
“The geometry of generations”
“Veronese geometry and the electroweak moduli space”
“Exploring the vacuum geometry of N=1 gauge theories”
“Vacuum geometry and the search for new physics” Gray, He, VJ, Nelson, hep-th/0511062
He, VJ, Matti, Nelson, Stillman,1408.6841
He, VJ, Matti, Nelson, 1402.3312
Gray, He, VJ, Nelson, hep-th/0604208

3. The Real World
• The objective is to obtain the real world from a string compactification
• We would happily settle for a modestly unreal world
supersymmetry
N = 1
G = SU(3)C ⇥ SU(2)L ⇥ U(1)Y
Matter in chiral representations of :
Superpotential
Three copies of matter such that not identical
Consistent with cosmology
G
W !ij ¯ L R
ij
(3, 2)16
, (3, 1)23
, (3, 1) 1
3
, (1, 2)±12
, (1, 1)1

4. Vacuum Selection Problem
• Program initiated by Candelas, Horowitz, Strominger, Witten (1985)
• Often said that since then, realistic constructions have been obtained – this
is only half true
• Often said that the goal has not been achieved – this is only half false
• No consensus: heterotic compactifications, free fermionic constructions,
intersecting brane models, orbifold constructions, etc.
isolated Δ27
orbifold singularity Berenstein, VJ, Leigh (2001)

5. Phenomenological Motivation
• Unexplained structure exists within the superpotential of phenomenological
theories that is forced on us by experimental and theoretical considerations
• Usually seen as problems: gauge hierarchy and fine tuning – cosmological
constant problem – μ problem – FCNCs – where are the gravitinos and
axions? – what is tan b? – origin of Yukawa couplings – why three
generations? – how come there are discrete symmetries? – whence CP
violation? – is there a GUT? – how is supersymmetry broken? – etc., etc.
• Are these opportunities instead?
μ term: Higgs bilinear coupling must exist, but must be near TeV scale
This hints of new physics, Giudice–Masiero mechanism, for example
R-parity: operators allowed by gauge invariance suppressed to retain
stability of proton, LSP provides a candidate for cold dark matter

6. The Search for Structure
• We propose to search for structure in the geometry of the vacuum spaces of
supersymmetric gauge theories
• Supersymmetric QFTs have scalars with a vacuum space of possible field
vacuum expectation values (vevs) hii
• Phenomenologists have tried to understand how flat directions lift
• The vacuum moduli space may have structure that correlates with
phenomenology of the Standard Model M
• May also tell us about string realizations of these theories
Example: orbifold by ADE – put D3-brane at fixed point – worldvolume
theory then has vacuum moduli space with geometry that recapitulates
resolution of singularity – field theory sees the transverse directions
Douglas, Moore (1996)

7. N=1 Gauge Theory
S =
Z
d4x
Z
d4✓ †i
eV i +
✓
1
4g2
Z
d2✓ trW↵W↵ +
Z
d2✓ W(i) + h.c.
◆%
• i is chiral matter superfield that transforms in representation R i of G
• V
is vector superfield that transforms in Lie algebra
• 2
eV D↵eV
is chiral spinor superfield, which encodes the
gauge field strength W↵ = iD
• W(i)
is superpotential, which is holomorphic in the fields, and receives
no perturbative corrections; it does not renormalize
g

8. Constructing the Vacuum
• Integrating over superspace, the scalar potential is
V (!i, ¯!i) =
X
i
@W
@!i
2
+
g2
4
X
i
qi|!i|2
!2
• Vacuum minimizes the potential
V (i, ¯i) = 0
scalar component of i
DA =
X
i
†i
0TAi0 = 0
F-terms
D-terms
generators of G in adjoint
Fi =
@W()
@i
i=i0
= 0
S =
Z
d4x
Z
d4✓ †i
eV i +
✓
1
4g2
Z
d2✓ trW↵W↵ +
Z
d2✓ W(i) + h.c.
◆%
with charge qi
work in Wess–Zumino gauge

9. How to Solve the Theory
S =
Z
d4x
Z
d4✓ †i
eV i +
✓
1
4g2
Z
d2✓ trW↵W↵ +
Z
d2✓ W(i) + h.c.
• Action has enormous gauge redundancy
• Work with , the complexification of gauge group
◆%
GC
• F-flatness conditions are holomorphic and invariant under GC
• D-flatness fixes the gauge

10. Solving the F-terms D-terms
• Theorem: For every solution to the F-terms, there exists a solution to the
D-terms in the completion of the orbit of the complexified gauge group.
• The set of gauge invariant operators (GIOs) provides a basis for the D-orbits
Luty, Taylor (1996)
• Vacuum moduli space is the symplectic quotient of the master space, which
is the manifold of scalar field vevs that satisfy F-term equations
M= F//G = F/GC
• Open question: given the vacuum moduli space of an N=1 theory, can we
geometrically engineer a string construction?

11. A Machine
W({i}) , i = 1, . . . ,n
rj({i}) , j = 1, . . . ,k
M
Vacuum moduli space
is an affine variety in
S = C[yj=1,...,k]
1. We have quotient ring:
F = C[!1, . . . ,!n]/
⌧
@W
@!i
GIOs are coordinates
Relations among them
describe moduli space
2. We take the image under ring map defined by GIOs:
M= Im(F
D={rj ({i})}
−−−−−−−−−−! S)

12. Rephrasing the Machine
W({i}) , i = 1, . . . ,n
rj({i}) , j = 1, . . . ,k
MVacuum moduli space
is an affine variety in
S = C[yj=1,...,k]
Obtained through the
elimination algorithm
1. Define polynomial ring R = C[i=1,...,n, yj=1,...,k]
2. Define ideal I =
⌧
@W
@i
, yj rj({i})
3. Eliminate variables i from I ⇢ R
4. This gives ideal M in terms of y j variables
Gray (2009)
Hauenstein, He, Mehta (2012)

13. An Example
SU(N) ⇥ SU(N) ('1,'2) (⇤,⇤)
2
, 1
x , x1
2
• theory with fields charged as
(1, 2) (⇤,⇤)
and charged as
• Superpotential W = tr ('11'22 '12'21)
• For simplicity, take N = 1 , so W = 0 (i.e., no F-terms)
• GIOs are
{r1, r2, r3, r4} = {'11,'12,'21,'22}
• There is one relation:
• The vacuum moduli space is the conifold; D3-brane probes on conifold
see this geometry
N
Klebanov, Witten (1998)
M= {r1r4 − r2r3 = 0} ⇢ P4

14. superfields are the gauge field strength and are given
Field Content of MSSM
of !i0, the vacuum expectation values of the scalar
provide a simultaneous solution to the F-term equations
W(!)
@!i
!!!!
i=i0
= 0 (2.2)
X
i
!†i
0 TA !i0 = 0 , (2.3)
group in the adjoint representation, and we have chosen
G = SU(3)C ⇥ SU(2)L ⇥ U(1)Y . We will adopt the
and the field content of the theory. For the moment,
neutrinos, which are gauge singlets.
family) indices
color indices
indices
FIELDS
Qi
a,↵ SU(2)L doublet quarks
ui
a SU(2)L singlet up-quarks
di
a SU(2)L singlet down-quarks
Li
↵ SU(2)L doublet leptons
ei SU(2)L singlet leptons
H↵ up-type Higgs
H↵ down-type Higgs
field content conventions for the MSSM.
in the Lie algebra g. The chiral spinor superfields are the by W↵ = iD2eV D↵eV .
The vacuum of the theory consists of !i0, the vacuum components of the superfields i that provide a simultaneous @W(!)
= 0 @!i
and the D-term equations
!!!!
i=i0
DA =
i
†!TA 0 !i0 = 0 where TA are generators of the gauge group in the adjoint representation, the Wess–Zumino gauge.
X
i
The MSSM fixes the gauge group G = SU(3)C ⇥ SU(notation given in Table 1 for the indices and the field content we do not consider right-organize handed neutrinos, into which two
are gauge Higgs doublets
INDICES
i, j, k, l = 1, 2, . . . ,Nf Flavor (family) indices
a, b, c, d = 1, 2, 3 SU(3)C color indices
↵, $, %, = 1, 2 SU(2)L indices
FIELDS
Qi
a,↵ ui
di
Li
ei H↵ H↵ 15Nf + 4 component fields
Table 1. Indices and field content conventions The corresponding minimal renormalizable superpotential X
X
X
49 MSSM F-terms if
No ⌫ R to start
Nf = 3
⇤
⇤ ⇤
+1/6
2/3
+1/3
1/2
+1
+1/2
1/2

15. Minimal Interactions
Wminimal = C0
X
↵,
H↵H✏↵ +
X
i,j
C1
ij
X
↵,,a
Qi
a,↵uj
aH✏↵
+
X
i,j
C2
ij
X
↵,,a
a,↵dj
aH✏↵ +
Qi
X
i,j
C3
ijei
X
↵,
Lj
↵H✏↵
• μ term plus mass terms for Standard Model particles
• Superpotential is gauge invariant, has terms at renormalizable order only
• All the terms respect R-parity,
(1)3(BL)+2s
• We are agnostic about the couplings in the superpotential and treat them as
generic – i.e., we don’t put mass hierarchies in by hand

16. A GaugGe Invaeriannt Opeeratrorsain ttheoMSSrMs of GIOs in MSSM
Type Explicit Sum Index Number
LH Li
↵H✏↵ i = 1, 2, 3 3
HH H↵H✏↵ 1
udd ui
adj
bdkc
✏abc i, j = 1, 2, 3; k = 1, . . . , j − 1 9
↵Lj
LLe Li
ek✏↵ i, k = 1, 2, 3; j = 1, . . . , j − 1 9
a,↵dj
QdL Qi
aLk
✏↵ i, j, k = 1, 2, 3 27
a,↵uj
QuH Qi
aH✏↵ i, j = 1, 2, 3 9
a,↵dj
QdH Qi
aH✏↵ i, j = 1, 2, 3 9
LHe Li
↵H✏↵ej i, j = 1, 2, 3 9
a,Qj
QQQL Qi
b,#Qk
$✏abc✏#✏↵$ i, j, k, l = 1, 2, 3; i6= k, j6= k,
c,↵Ll
j  i, (i, j, k)6= (3, 2, 1)
24
a,↵uj
QuQd Qi
aQk
b,dl
b✏↵ i, j, k, l = 1, 2, 3 81
a,↵uj
QuLe Qi
aLk
el✏↵ i, j, k, l = 1, 2, 3 81
auj
uude ui
bdkc
el✏abc i, j, k, l = 1, 2, 3;j i 27
a,Qj
QQQH Qi
c,↵H$✏abc✏#✏↵$ i, j, k = 1, 2, 3; i6= k, j6= k,
b,#Qk
j  i, (i, j, k)6= (3, 2, 1)
8
a,↵uj
QuHe Qi
aHek✏↵ i, j, k = 1, 2, 3 27
adj
dddLL di
bdkc
Lm↵
Ln
✏abc✏ijk✏↵ m, n = 1, 2, 3, n m 3
auj
uuuee ui
bukc
emen✏abc✏ijk m, n = 1, 2, 3, n  m 6
a,↵uj
QuQue Qi
aQk
b,umb
en✏↵ i, j, k, m, n = 1, 2, 3;
antisymmetric{(i, j), (k,m)}
108
a,Qj
QQQQu Qi
b,#Qk
c,↵Qm
f,$unf
✏abc✏#✏↵$ i, j, k, m, n = 1, 2, 3; i6= k, j6= k,
j  i, (i, j, k)6= (3, 2, 1)
72
adj
dddLH di
bdkc
Lm↵H✏abc✏ijk✏↵ m = 1, 2, 3 3
auj
uudQdH ui
bdkcQm
f,↵dnf
H✏abc✏↵ i, j, k, m, n = 1, 2, 3;j i 81
(QQQ)4LLH (QQQ)↵#
4 Lm↵
Ln
H# m, n = 1, 2, 3, n  m 6
(QQQ)4LHH (QQQ)↵#
4 Lm↵
HH# m = 1, 2, 3 3
(QQQ)4HHH (QQQ)↵#
4 H↵HH# 1
(QQQ)4LLLe (QQQ)↵#
4 Lm↵
Ln
Lp
#eq m, n, p, q = 1, 2, 3, n  m, p  n 30
auj
uudQdQd ui
bdkc
Qm
f,↵dnf
Qp
g✏abc✏↵ i, j, k, m, n, p, q = 1, 2, 3;
g,dq
j i, antisymmetric{(m, n), (p, q)}
324
(QQQ)4LLHe (QQQ)↵#
4 Lm↵
Ln
H#ep m, n, p = 1, 2, 3, n  m 18
(QQQ)4LHHe (QQQ)↵#
4 Lm↵
HH#en m, n = 1, 2, 3 9
(QQQ)4HHHe (QQQ)↵#
4 H↵HH#em m = 1, 2, 3 3
a,↵Qj
[(QQQ)4]↵# := Qi
Nf = 3
c,#✏abc✏ijk Gherghetta, Kolda, Martin (1995)
b,Qk
Table 7. The set D = {ri} of generators of gauge invariant operators for the MSSM.
i, j, k, l, m, n, p, q = 1, 2, 3 (flavor)
a, b, c, f, g = 1, 2, 3 (color)
↵, , #, $ = 1,2 (SU(2)L)
There are 991 GIOs
that capture orbits
associated to 12 D-terms

17. MSSM Vacuum Geometry
• In principle, since we have a superpotential and know the generators of the
GIOs of MSSM, we can use the machine to calculate vacuum moduli space
• In practice, solving 49 F-terms in terms of 991 GIOs is too hard for desktop
computers using existing algorithms in computational algebraic geometry
• Computation scales with number of GIOs and the dimension of the moduli
space
• We can however restrict to subsectors of the MSSM
• We are (slowly) making progress on the full problem

18. Electroweak Sector
• Subsector of theory where hQi
a,↵i = hui
ai = hdi
ai = 0
• S U ( 3 ) C unbroken as in Nature
Wminimal = C0H↵H✏↵ +
X
i,j
C3
ijeiLj
↵H✏↵
• Minimal superpotential is
• List of generators of GIOs is
will be explicit about flavor only
Type Explicit Sum Index Number
LH Li
↵H✏↵ i = 1, 2, . . . ,Nf Nf
HH H↵H✏↵ 1
LLe Li
↵Lj
ek✏↵ i, k = 1, 2, . . . ,Nf ; j = 1, . . . , i 1 Nf ·
!Nf
2
LHe Li
↵H✏↵ej i, j = 1, 2, . . . ,Nf Nf
2

19. F-terms
Table 2. Minimal generating set of the GIOs for the electroweak sector.
we are explicit about the sums on flavor indices i, j but leave sums on SU(2)L
implicit. The corresponding F-terms are
@Wminimal
@H↵
=) = C0H✏↵ (3.3)
@Wminimal
@H
= C0H↵✏↵ +
X
i,j
C3
ijeiLj
↵✏↵ (3.4)
@Wminimal
@Lj
↵
=
X
i
C3
ijeiH✏↵ (3.5)
@Wminimal
@ei =
X
j
C3
ijLj
↵H✏↵ (3.6)
this yields the following F-term equations for the Higgs fields:
H = 0 , (3.7)
C0H↵ +
X
i,j
C3
ijeiLj
↵ = 0 , (3.8)
and FH
terms, respectively. The other two F-term equations (for the e and L
lead to extra constraints as the vanishing of H renders them trivial.
the {ri}, the only non-trivial GIOs that remain are the LH and LLe operators.
and LHe vanishes by virtue of (3.7). Furthermore, (3.8) specifies the value of
operators in terms of the LLe operators. Multiplying (3.8) by Li
✏↵ and summing
C0Li
↵H✏↵ +
X
j,k
↵Lj
C3
jkLi
ek✏↵ = 0 . (3.9)
H = 0
=) C0H↵ +
X
i,j
C3
ijeiLj
↵ = 0
=) Automatic
• Hit non-trivial F-term with and sum:
C0Li
↵H✏↵ +
X
j,k
C3
jkLi
↵Lj
ek✏↵ = 0
Li
!✏↵!
Nf equations
• This lets us eliminate L H in favor of L L e , of which there are Nf
✓
Nf
2
◆
• So we have an affine variety in C[y1, . . . , yk]
= k

20. Relations among GIOs
• There are non-trivial relations native to GIOs
For example:
(Li
↵Lj
ek✏↵)(Lm#
Ln$
ep✏#$) = (Lm↵
Ln
#Lj
ek✏↵)(Li
$ep✏#$)
• In fact, the number of syzygies is (Nf 1)
✓✓
Nf
2
◆
1
◆
+
✓
Nf 2
2
◆
• The dimension of the vacuum moduli space works out to be
Nf ·
✓
Nf
2
◆
(Nf 1)
✓✓
Nf
2
◆
1
◆
✓
Nf 2
2
◆
= 3Nf 4

21. Thus, the dimension always increases by three when we add another generation of matter
fields to the electroweak sector.
Growth of Dimension
It is a remarkable fact that the dimension increases by the same increment as the number
of fields, despite the number of GIOs growing much faster.
Nf 1 2 3 4 5 6 . . .
number of fields 5 8 11 14 17 21 . . .
number of LLe generators 0 2 9 24 50 90 . . .
vacuum dimension 0 2 5 8 11 14 . . .
• dimMEW Table 3. = Vacuum 3Nf geometry 4 grows dimension much according slower to the than number number generations of GIOs
Nf .
In the following subsections, we will study in greater detail the geometry for the cases
• We next want to characterize the geometry of the vacuum space (especially
Nf = 2, 3, 4, 5.
in the three generation case)
• This is Grassmannian Gr(Nf , – 2) 10 –
⇥ PNf1
2-planes in CNf freely indexed ek
Li
↵Lj
✏↵
dim : 2(Nf 2) + Nf = 3Nf 4

22. Projective Space
• Coordinates on Pn
are written as ; this means
[z1 : z2 : . . . : zn+1] , zi 2 C
(z1, z2, . . . , zn+1) ⇠ (z1, z2, . . . , zn+1) 2 Cn+1 , 2 C
• The Riemann sphere is the complex projective line,
P1
zj = rjei'j , j = 1, 2
|z1|2 + |z2|2 = r2
1 + r2
2 = R2
=
ei'2
R
(Re(ez1))2 + (Im(ez1))2 + ez2
2 = 1
Let
Now,
Define
Take (ez1, ez2) = (z1, z2)
This gives
This is just S2 ⇢ R3
Alternatively, = z1
2 , P1 = C [ {1} ' S2

23. Hilbert Series
• The Hilbert series is a generating function for dimension of graded pieces of
manifold
• d i m M i is number of independent degree i polynomials onM
H(t;M) =
1X
i=−1
(dimMi) ti =
P(t)
(1 t)dimM
• The polynomial P ( t ) in the numerator has integer coefficients
• Dimension of the variety is order of the pole at t = 1
• P ( 1 ) is the degree of M
number of times a generic line intersects variety

24. $
MNfNc
⇥
= 2NcNf (N2
Counts dim
GIOs and Relations
Example: SQCD with Nf = Nc
c 1) . (3.7)
describe the light degrees of freedom in a gauge invariant way by the following basic
Mi
j = Qi
a
⌃Q
aj
(mesons)
a1 . ..QiNc
Bi1...iNc = Qi1
aNc a1...aNc (baryons)
⌃ Bi1...iNc = ⌃Q
a1
i1 . . . ⌃Q
aNc
iNc
a1...aNc (antibaryons)
GIOs
(3.8)
N2 f + 2
Observation 3.3. For Nf ⇤ Nc, under the global SU(Nf )L⇥SU(Nf )R, the mesons M trans-form
There is one relation among GIOs: detM = (⇤B)(⇤ e B)
bifundamental [1, 0, . . . ; 0, . . . , 1] representation, the baryons B and antibaryons ⌃ B
respectively in [0, 0, . . . , 1Nc;L, 0, . . . , 0; 0, . . . , 0] and [0, . . . , 0; 0, . . . , 1Nc;R, 0 . . . , 0].
The gauge group S U ( N c ) is fully broken at generic points
above, 1j;L denotes a 1 in the j-th position from the left, and 1j;R denotes a 1
position from the right.
dimM= 2NcNf (N2
c 1) = N2
Hilbert series encodes all this: one constraint of weight
2Nf
total number of basic generators for the GIOs, coming from the three contributions
therefore
N2
f +
⇤
Nf
Nc
⌅
+
⇤
Nf
Nf Nc
⌅
= N2
⇤
Nf
Nc
f + 2
⌅
. (3.9)
mesons baryons
f + 1
H(t;M) =
1 t2Nf
(1 t2)N2
f (1 tNf )2
Gray, Hanany, He, VJ, Mekareeya (2008)

25. Palindromes
• Theorem: If the numerator of a Hilbert series of a graded Cohen–
Macaulay domain R is palindromic iff R is Gorenstein.
Stanley (1978)
• A polynomial is palindromic if P(t) =
XN
k=0
aktk , ak = aNk
XN
Example: ( 1 + t ) N = is palindromic
k=0
✓
N
k
◆
tk
• Cohen–Macaulay is a technical condition; it means that Krull dimension =
depth; we check this with software
• Gorenstein means that the canonical sheaf is a line bundle (it needn’t be
trivial in general)
• But for affine varieties:
Gorenstein = Calabi–Yau
• Vacuum moduli space of SQCD is Calabi–Yau
Gray, Hanany, He, VJ, Mekareeya (2008)

26. Calabi–Yau
There is a nowhere vanishing holomorphic n -form
The canonical bundle is trivial
There is a Kähler metric with global holonomy in SU(n)

27. Electroweak Geometries
• For , we Nf = 2 have two L L e operators, no relations
Vacuum moduli space is C2
• For N f = 3 , we have a non-trivial vacuum moduli space
MEW = (8|5, 6|29)
embeds in P8
dimension
degree
expressed as 9
polynomials of
degree 2
The Hilbert series is H(t;M) =
1 + 4t + t2
(1 t)5
degree: put t = 1
dimension
Hilbert series has palindromic numerator, so M E W is Calabi–Yau

28. Which Calabi–Yau?
P2 ⇥ P2 −! P8
[x0 : x1 : x2] [z0 : z1 : z2]7! [x0z0 : x0z1 : x0z2 : x1z0 : x1z1
: x1z2 : x2z0 : x2z1 : x2z2]
• This is Grassmannian Gr(3, 2) ⇥ P2
• It has a name: Segrè variety and it is toric
what does
what is so special toric mean?
about this?

29. Toric Varieties
• Consider: |z1|2 + |z2|2 + |z3|2 = 1
2 C3
• Identify ( z , z , z ) ⇠ e i 1 2 3 ( z 1 , z 2 , z 3 ) to define
• Define
(x, y, z) = (|z1|2, |z2|2, |z3|2)
• The original geometry is x + y + z = 1 or z = 1 x y
This is a triangle
• Use U ( 1 ) to choose the phase of z3
• The phases of z 1 , z 2 define a torus over the base
P2 2 C3
B

30. Toric Base
P2 ⇥ P2 −! P8
[x0 : x1 : x2] [z0 : z1 : z2]7! [x0z0 : x0z1 : x0z2 : x1z0 : x1z1
: x1z2 : x2z0 : x2z1 : x2z2]
• It has a name: Segrè variety and it is toric
• We can calculate the topological structure of the base; this is given by the
hp,q(B) =
h0,0
h0,1 h0,1
h0,2 h1,1 h0,2
h0,3 h1,2 h1,2 h0,3
h0,4 h1,3 h2,2 h1,3 h0,4
h0,3 h1,2 h1,2 h0,3
h0,2 h1,1 h0,2
h0,1 h0,1
h0,0
=
1
0 0
0 2 0
0 0 0 0
0 0 3 0 0
0 0 0 0
0 2 0
0 0
1
Hodge diamond

31. Segrè Variety
P2 ⇥ P2 −! P8
[x0 : x1 : x2] [z0 : z1 : z2]7! [x0z0 : x0z1 : x0z2 : x1z0 : x1z1
: x1z2 : x2z0 : x2z1 : x2z2]
• It has a name: Segrè variety and it is toric
• It is one of four Severi varieties
• Theorem: Any smooth non-degenerate algebraic variety of (complex)
n
X
Pm m 32
dimension embedded into with n + 2
has the property that
its secant variety is equal to itself.
Sec(X) Pm Hartshorne, Zak (1984)
• Severi variety: m = 32
n + 2 , Sec(X)6= Pn

32. n = 16: The Cartan variety of the orbit of the highest weight vector of a certain
non-trivial representation of E6.
Severi Varieties
Of these, only two are isomorphic to (a product of) projective space, namely n = 2, 4.
Remarkably, these are the two that show up as the vacuum geometry of the electroweak
sector when Nf = 3.
• There are four division algebras: R , C , H , O
• Consider the projective plane formed out of each of them
• Complexification of these spaces is homeomorphic to Severi varieties
• They are homogeneous spaces
The connection with Severi varieties could be profound. Indeed, it was discussed in [19]
that these four spaces are fundamental to mathematics in the following way. It is well-known
that there are four division algebras: the real numbers R, the complex numbers C, the quater-nions
H, and the octonions O, of, respectively, real dimension 1, 2, 4, 8. Consider the projective
planes formed out of them, viz., RP2, CP2, HP2 and OP2, of real dimension 2, 4, 8, 16. We
have, of course, encountered CP2 repeatedly in our above discussions. The complexification
of these four spaces, of complex dimension 2, 4, 8, 16 are precisely homeomorphic to the four
Severi varieties. Amazingly, they are also homogeneous spaces, being quotients of Lie groups.
In summary, we can tabulate the four Severi varieties
Projective Planes Severi Varieties Homogeneous Spaces
RP2 CP2 SU(3)/S( U(1) ⇥ U(2) )
CP2 CP2 ⇥ CP2 SU(3)2/S( U(1) ⇥ U(2) )2
HP2 Gr(6, 2) SU(6)/S( U(2) ⇥ U(4) )
OP2 S E6/Spin(10) ⇥ U(1)
(3.34)
Returning to our present case of n = 4, the embedding (3.33) can be understood in terms

33. More Flavors
• For N f = 4 , we find M = ( 2 3 | 8 , 7 0 | 2 1 0 0 ) with Hilbert series
H(t;M) =
1 + 16t + 36t2 + 16t3 + t4
(1 t)8
• For N f = 5 , we find M = ( 4 9 | 1 1 , 1 0 5 0 | 2 5 2 5 ) with Hilbert series
H(t;M) =
1 + 39t + 255t2 + 460t3 + 255t4 + 39t5 + t6
(1 t)11
• Both of these theories have Calabi–Yau vacuum geometries since the
numerators of their Hilbert series are palindromic

34. Multiple Higgs Fields
• New GIOs:
Type Explicit Sum Index Number
LH Li
↵Hj
✏↵ i = 1, . . . ,Nf ; j = 1, . . . ,Nh Nf · Nh
HH Hi↵
Hj
✏↵ i, j = 1, . . . ,Nh N2
h
↵Lj
LLe Li
ek✏↵ i, k = 1, . . . ,Nf ; j = 1, . . . , i − 1 Nf ·
!Nf
2
HHe Hi
↵Hj
ek✏↵ i = 1, . . . ,Nh; j = 1, . . . , i − 1; k = 1, . . . ,Nf Nf ·
!Nh
2
↵Hk
✏↵ej i, j = 1, . . . ,Nf ; k = 1, . . . ,Nh Nf
LHe Li
2 · Nh
Table 4. Minimal generating set of the GIOs for the electroweak sector, for number of Higgs doublets
Nh 1.
large rates for μ ! e processes, etc.). But our interest here is to ask whether such a model,
a priori possible, or even natural, from the point of view of an underlying string theory,10 has
a geometry that is significantly di↵erent from that which arises in the one generation case.
When Nh6= 1, we expect a larger set of GIOs and thus, at least na¨ıvely, we might
expect the vacuum moduli space to be of larger dimension than the Nh = 1 case. Indeed,
the operator types LH and LHe from Table 2 now represent Nf · Nh objects, while the
bilinear HH now represents N2
h terms. Since the lepton doublet L and the down-type Higgs
H have the same SU(2)L ⇥ U(1)Y quantum numbers, we can extend the list of GIOs in a
straightforward manner. A new operator type in the electroweak sector is HHe. It is the
• New minimal superpotential:
Wminimal =
X
i,j
C0
ijHi↵
H
j
✏↵ +
X
i,j,k
C3
ij,keiLj
↵H
k
✏↵
• Using machine, we find that the vacuum moduli spaces remain the same
for N h  N f ; they are Calabi–Yau
• N h = 3 not special – this is slightly surprising – cf. trinification

35. Right-handed Neutrinos
• Right-handed neutrinos are gauge singlets ⌫i
• The fields are themselves GIOs
• The superpotential is
Wminimal = C0H↵H✏↵ +
X
i,j
C3
ijeiLj
↵H✏↵ +
X
i,j
C4
ij⌫i⌫j +
X
i,j
C5
ij⌫iLj
↵H✏↵
Yukawa couplings
for Dirac masses
Majorana couplings
• All the interactions respect R-parity; no neutrino tadpoles

36. F-terms and Syzygies
• F-terms tell us that:
⌫ = 0 , LH = 0 , HH = 0 , LHe = 0
X
i,j
C3
ijeiLj
↵Lk
✏↵ = 0
• The vacuum geometry is the solution to the syzygies of operators
intersected with the solutions to the F-term equations
LLe
• To analyze this system, we do a field redefinition: eej :=
X
i
C3
ijei
• If N f = 2 , we find y1 = ee1L1
↵L2
✏↵ = 0 , y2 = ee2L1
↵L2
✏↵ = 0
The vacuum moduli space is a point in C2
,

37. Electroweak Sector of World?
• If N f = 3 , define yI+C(Nf ,2)·(k1) := (1)k1Li
↵Lj
eek✏↵
• Vacuum moduli space works out to be the ideal
h y1y5 − y2y4, y1y6 − y3y4, y2y6 − y3y5, y1y8 − y2y7,
y1y9 − y3y7, y2y9 − y3y8, y4y8 − y5y7, y4y9 − y6y7,
y5y9 − y6y8, y1 − y9, y2 − y6, y4 − y8 i
• Can eliminate linear relations and end up with six quadratic relations
• If only binomials in ideal, it is toric
• This is M = ( 5 | 3 , 4 | 2 6 ) with Hilbert series H(t;M) =
1 + 3t
(1 t)3
• Since numerator is not palindromic, this is not Calabi–Yau

38. and, thus, reduce the ideal as a set of 6 quadratic polynomials. We have,
Veronese Surface
M= (5|3, 4|26) , (5.25)
and the corresponding Hilbert series,
1 + 3t
(1 t)3 . (5.26)
It should be noted that the Hilbert series is not palindromic and therefore the geometry is
not Calabi–Yau.
The Veronese surface is an embedding of P2 into P5. It is in fact the only Severi variety on
projective dimension two, and it is remarkable that two of the four Severi varieties appear as
vacuum geometry for supersymmetric models with three flavor generations. The embedding
is explicitly given by:
P2 ! P5
[x0 : x1 : x2]7! [x0
2 : x0x1 : x1
2 : x0x2 : x1x2 : x2
2]
(5.27)
y1 ! x0x2 , y2 ! x0x1 , y3 ! x20
– 24 –
,
y4 ! x1x2 , y5 ! x21
, y6 ! x1x0 ,
y7 ! x22
, y8 ! x2x1 , y9 ! x2x0
Also a
Severi variety

39. ! ! ! that the e↵ect of (5.11) is therefore to identify the two projective
Grassmannian Gr(3, 2) and P2. Imposing the identification relation
z2] onto (3.33) lead to the vacuum geometry in the presence of
the binomial nature of the polynomial ideal (5.24) that the Veronese
same notation as previously, the corresponding diagram is given
=) (5.29)
Theorem: The toric variety is Calabi–Yau iff the
endpoints of the extremal rays lie on an affine
hyperplane of the form
pictorial representation of the toric cone, as it sits within three
of the affine cone, we can compute its Hodge diamond
=
h0,0
h0,1 h0,1
h0,2 h1,1 h0,2
h0,1 h0,1
h0,0
=
1
0 0
0 1 0
0 0
1
. (5.30)
identification of the Veronese geometry.
– 25 –
Toric Diagram
hp,q(B) =
Endpoints coplanar at height 2 not height 1
h0,0
h0,1 h0,1
h0,2 h1,1 h0,2
h0,1 h0,1
h0,0
Toric diagram is planar, so
why is it not Calabi–Yau?
=
1
0 0
0 1 0
0 0
1
Xd
i=1
aixi = 1 , ai 2 Z .
Hodge diamond confirms Veronese identification:

40. Role of Majorana Masses
• What happens if we get rid of the Majorana masses in superpotential?
• The F-terms become:
LH = 0 , HH = 0 , LHe = 0 ,
X
i,j
C3
ijeiLj
↵Lk
✏↵ = 0 ,
X
i,j
C5
ij⌫iLj
↵Lk
el✏↵ = 0
• , Hilbert series become palindromic again
• For , we have Nf = 3 the vacuum geometry M = ( 8 | 4 , 7 | 2 1 4 ) with
H(t;M) =
1 + 5t + t2
(1 t)4
• Can identify geometry as
P2 ⇥ C −! P8
[x0 : x1 : x2] []7! [x0
2 : x0x1 : x1
2 : x0x2 : x1x2 : x2
2 : x0 : x1 : x2]
dimM= 4Nf 8

41. the cases of Nf = 2, which give points or C2. The table lists the GIOs that are non-vanishing
in the vacuum. The toric property The Calabi–Yau property is checked Summary
refers to whether the ideals are explicitly in a toric form.
by the palindromicity of the numerator of the Hilbert
series associated to the geometry M.
W Vacuum GIOs Nf dimension degree Toric Calabi–Yau
HH + LHe LLe, LH 3 ? 5 6 X X
4 8 70 X
5 11 1050 X
HH + LHe + LH⌫ + ⌫⌫ LLe 3 † 3 4 X
4 6 40
HH + LHe + LH⌫ LLe, ⌫ 3 4 7 X X
4 8 71 X
? = Segrè † = Veronese
Table 6. Summary of algebraic geometries encountered as the vacuum moduli space of supersymmetric
electroweak Only theories. for three Here generations W is the superpotential; is the vacuum vacuum moduli GIOs are the space GIOs toric
after imposing the F-terms,
and thus furnish explicit coordinates of the moduli space, of ane dimension and degree as indicated; Nf
is the • number of generations. We also mark with “X” if the vacuum moduli space is toric or Calabi–Yau.
For three generations we get Severi varieties with MSSM, MSSMn minimal
Furthermore, the corresponds to the cone over the Veronese surface and the ?, the Segr`e variety. These
superpotentials
† two are Severi varieties, in fact, the only two which are isomorphic to (products of) projective spaces.
The observations that can be drawn from this table are the following. First, for the
• minimal superpotential, that is W = HH + LHe, the dimension increases by three when

42. Why Special Geometries?
• Suppose we include higher dimension terms, for example,
that lift Higgs directions but preserve R-parity
(HH)2 , (LH)2
• The vacuum geometry of MSSMn remains Veronese
• However, if we include R-parity violating terms in the superpotential, for
example, the vacuum geometry LLe , (LH) trivializes to point or line
• Remember that we want to preserve R-parity in order to maintain stability
of proton; this also leaves the lightest supersymmetic particle (LSP) as a
stable candidate for cold dark matter
• Observation: Phenomenology and geometry go together!
• Open Question: Can we adopt this idea for model building?

43. A Proposal
• Conjecture: Geometric structure in vacuum moduli space should be
regarded as fundamental.
• If special geometry exists with the minimal superpotential, perhaps we
should add only those deformations at higher mass level that preserve this
structure
• Preserving vacuum geometry predicts that certain operators allowed by
gauge invariance at higher mass level are nevertheless dropped
• Potentially explicates structure of low-energy effective Lagrangian
• Supplies a geometric toolkit for bottom up model building

44. Prospectus
• We are working on a scan of all possible superpotentials for electroweak
sector at renormalizable order for the MSSM, MSSMn, and NMSSM
Work with Daleo, Hauenstein, Mehta uses complementary numerical
algebraic geometry technology
• We intend to test the principle for model building that we have enunciated
• We aim to find the vacuum moduli space of the MSSM
This variety is an intersection of three prime ideals; we have preliminary
results regarding two of these
• The hope is that knowing the vacuum moduli space of the MSSM will
facilitate more compelling string constructions of particle physics

45. Thank You!