FROM F-THEORY TO DYNAMIC GLSM Physics and Geometry of F-theory - - PowerPoint PPT Presentation
FABIO APRUZZI FROM F-THEORY TO DYNAMIC GLSM Physics and Geometry of F-theory (2017), ICTP, Trieste Based on 1602.04221 & 1610.00718 in collaboration with Falk Hassler, Jonathan Heckman and Ilarion Melnikov and see also 1601.02015 by
FABIO APRUZZI
Physics and Geometry of F-theory (2017), ICTP, Trieste
Based on 1602.04221 & 1610.00718 in collaboration with Falk Hassler, Jonathan Heckman and Ilarion Melnikov and see also 1601.02015 by Sakura Schäfer-Nameki and Timo Weigand
A RECENT HISTORY REVIEW
A RECENT HISTORY REVIEW
4D Theories F on CY4:
A RECENT HISTORY REVIEW
4D Theories
… almost everybody in the audience
F on CY4:
A RECENT HISTORY REVIEW
4D Theories 6D (SCFTs) Theories
… almost everybody in the audience
F on CY4: F on CY3:
A RECENT HISTORY REVIEW
4D Theories 6D (SCFTs) Theories
… almost everybody in the audience … a large subset of you
F on CY4: F on CY3:
A RECENT HISTORY REVIEW
2D (0,2) Theories 4D Theories 6D (SCFTs) Theories
… almost everybody in the audience … a large subset of you
F on CY4: F on CY3: F on CY5:
A RECENT HISTORY REVIEW
2D (0,2) Theories 4D Theories 6D (SCFTs) Theories
… almost everybody in the audience … a large subset of you … a smaller subset of you
F on CY4: F on CY3: F on CY5:
MAIN IDEA
CY3
CY5
2D (0,2) Theories
Kähler 4-Manifold + Twist
6D (SCFTs) Theories
▸ Generating novel 2D SCFTs ▸ 2D Theories are Closely Related to String Theories ▸ UV Completion of Non-Critical Strings
WHY?
▸ Generating novel 2D SCFTs ▸ 2D Theories are Closely Related to String Theories ▸ UV Completion of Non-Critical Strings
WHY?
Energy IR UV
Deff >> 10
2D GLSM
+
Extra
+
Gravity
MKK Mstring
▸ Relevant for Time-Dependent and de Sitter Backgrounds ▸ Generating novel 2D SCFTs ▸ 2D Theories are Closely Related to String Theories ▸ UV Completion of Non-Critical Strings
Hellerman; Maloney, Silverstein; + Strominger
WHY?
Energy IR UV
Deff >> 10
2D GLSM
+
Extra
+
Gravity
MKK Mstring
WHAT’S THE PLAN?
CY5
CY5 B4 y2 = x3 + f(B4)x + g(B4)
Elliptically Fibered
B4
CY5 B4 y2 = x3 + f(B4)x + g(B4)
Elliptically Fibered
O(−4KB4)
O(−6KB4)
B4
Section: Homogeneous polynomial of a certain degree
CY5 B4 y2 = x3 + f(B4)x + g(B4)
Elliptically Fibered
O(−4KB4)
O(−6KB4)
B4
Section: Homogeneous polynomial of a certain degree
Non-Compact, Gravity decoupled
CY5 B4 y2 = x3 + f(B4)x + g(B4)
Elliptically Fibered
O(−4KB4)
O(−6KB4)
B4 X3 ∆ = 4f 3 + 27g2 = 0
2D Spacetime + Kähler Threefold X3
Section: Homogeneous polynomial of a certain degree
Non-Compact, Gravity decoupled
CY5 B4 y2 = x3 + f(B4)x + g(B4)
Elliptically Fibered
O(−4KB4)
O(−6KB4)
B4 X3 ∆ = 4f 3 + 27g2 = 0
2D Spacetime + Kähler Threefold X3
Section: Homogeneous polynomial of a certain degree Vanishing Degrees of : Kodaira Classification
(f, g, ∆)
Gauge theory on 7 Branes (Compact)
Non-Compact, Gravity decoupled
10 SYM:
L10D = 1 4g2
Y M
Z d10x
SO(1, 9) → SO(1, 7) × U(1)R → SO(1, 1) × U(3) × U(1)R
X3
(4D) Beasley, Heckman, Vafa; Donagi-Wijnholt
10 SYM:
L10D = 1 4g2
Y M
Z d10x
SO(1, 9) → SO(1, 7) × U(1)R → SO(1, 1) × U(3) × U(1)R
X3
SO(1, 1) × U(3) × U(1)R → SO(1, 1) × SU(3) × U(1)X × U(1)R
(4D) Beasley, Heckman, Vafa; Donagi-Wijnholt
10 SYM:
L10D = 1 4g2
Y M
Z d10x
SO(1, 9) → SO(1, 7) × U(1)R → SO(1, 1) × U(3) × U(1)R
X3
SO(1, 1) × U(3) × U(1)R → SO(1, 1) × SU(3) × U(1)X × U(1)R
Jtop = JX + 3 2JR
(4D) Beasley, Heckman, Vafa; Donagi-Wijnholt
10 SYM:
L10D = 1 4g2
Y M
Z d10x
SO(1, 9) → SO(1, 7) × U(1)R → SO(1, 1) × U(3) × U(1)R
X3
SO(1, 1) × U(3) × U(1)R → SO(1, 1) × SU(3) × U(1)X × U(1)R
Jtop = JX + 3 2JR
(4D) Beasley, Heckman, Vafa; Donagi-Wijnholt
A(10)
I
χ(10)
Chiral Superfield (CS) Multiplets: Vector Multiplet:
V(0,0) v+ D(0,1) ¯ ∂A Φ(3,0)
φ(3,0) ψ+,(0,1) χ+,(3,0) λ−,(0,2) Λ−,(0,2)
Fermi Multiplet:
µ−
10 SYM:
L10D = 1 4g2
Y M
Z d10x
SO(1, 9) → SO(1, 7) × U(1)R → SO(1, 1) × U(3) × U(1)R
X3
SO(1, 1) × U(3) × U(1)R → SO(1, 1) × SU(3) × U(1)X × U(1)R
Jtop = JX + 3 2JR
(4D) Beasley, Heckman, Vafa; Donagi-Wijnholt
Form degree on the 3fold
A(10)
I
χ(10)
Chiral Superfield (CS) Multiplets: Vector Multiplet:
V(0,0) v+ D(0,1) ¯ ∂A Φ(3,0)
φ(3,0) ψ+,(0,1) χ+,(3,0) λ−,(0,2) Λ−,(0,2)
Fermi Multiplet:
µ−
E(0,2) = ∂Wtop ∂Λ3,1 = F(0,2)
J(3,1) = ∂Wtop ∂Λ0,2 = D(0,1)Φ(3,0)
F(0,2) = [D(0,1), D(0,1)]
Wtop = Z
X3
Tr
Z
X3
Tr
Z
X3
¯ ∂Aφ(3,0) = 0
F(0,2) = 0
ωX3 ∧ ωX3 ∧ F(1,1) + [φ(3,0), φ(0,3)] = 0
E(0,2) = ∂Wtop ∂Λ3,1 = F(0,2)
J(3,1) = ∂Wtop ∂Λ0,2 = D(0,1)Φ(3,0)
E.o.M:
F(0,2) = [D(0,1), D(0,1)]
Wtop = Z
X3
Tr
Z
X3
Tr
Z
X3
¯ ∂Aφ(3,0) = 0
F(0,2) = 0
ωX3 ∧ ωX3 ∧ F(1,1) + [φ(3,0), φ(0,3)] = 0
E(0,2) = ∂Wtop ∂Λ3,1 = F(0,2)
J(3,1) = ∂Wtop ∂Λ0,2 = D(0,1)Φ(3,0)
E.o.M:
Susy variations match the Action & E.o.M.
F(0,2) = [D(0,1), D(0,1)]
Wtop = Z
X3
Tr
Z
X3
Tr
Z
X3
¯ ∂Aφ(3,0) = 0
F(0,2) = 0
ωX3 ∧ ωX3 ∧ F(1,1) + [φ(3,0), φ(0,3)] = 0
E(0,2) = ∂Wtop ∂Λ3,1 = F(0,2)
J(3,1) = ∂Wtop ∂Λ0,2 = D(0,1)Φ(3,0)
E.o.M:
Susy variations match the Action & E.o.M.
F(0,2) = [D(0,1), D(0,1)]
Wtop = Z
X3
Tr
Z
X3
Tr
Z
X3
Note: we presented here an action with all the KK modes, but at low energy only 0-modes appear, counted by bundle cohomologies on X3
¯ ∂Aφ(3,0) = 0
F(0,2) = 0
ωX3 ∧ ωX3 ∧ F(1,1) + [φ(3,0), φ(0,3)] = 0
E.o.M.
E(0,2) = ∂Wtop ∂Λ(odd)
(0,2)
= F(even)
(0,2)
J(0,2) = ∂Wtop ∂Λ(even)
(0,2)
= F(odd)
(0,2)
F(0,2) → F(even)
(0,2)
+ F(odd)
(0,2)
Relation to F-theory
Z2 : Ω(4,0) 7! Ω(4,0)
Wtop = Z
Y4
Ω(4,0) ∧ Tr
Z
Y4
Y4 ∼ = O(KX3) → X3
F(0,2) = ωY4xF(1,1) = 0
B4 X1
3
X2
3
Q ∈ K1/2
S
⊗ R1 ⊗ R2 Qc ∈ K1/2
S
⊗ R∨
1 ⊗ R∨ 2
Ψ ∈ Ω(0,1)
S
(K1/2
S
⊗ R1 ⊗ R2) Ψc ∈ Ω(0,1)
S
(K1/2
S
⊗ R∨
1 ⊗ R∨ 2 )
S
Surface
Wtop,S = Z
S
hQc, Λ(0,2)Qi + ⌦ Ψc,
↵ + ⌦ Qc,
↵
Wtop,Σ = Z
Σ
fαβγ δΨ(α)δQ(β)δQ(γ)
Wtop,p = hαβγδ ⇣ δΨ(α)δQ(β)δQ(γ)δQ(δ)⌘ |p
some bundles on X1
3 X2 3
X3 = P2 × P1 y2 = x3 + f(v, X3)x + g(v, x3) B4 = Nv → X3
f(v, X3) ∼ v4f(X3)
g(v, X3) ∼ v5g(X3)
X3 = P2 × P1
Kodaira
Assume No non-trivial bundle
y2 = x3 + f(v, X3)x + g(v, x3) B4 = Nv → X3
f(v, X3) ∼ v4f(X3)
E8 Gauge Symmetry in 2D
g(v, X3) ∼ v5g(X3)
V(0,0), D(0,1), Λ(0,2), Φ(3,0)
X3 = P2 × P1
Kodaira
Assume No non-trivial bundle
y2 = x3 + f(v, X3)x + g(v, x3) B4 = Nv → X3
f(v, X3) ∼ v4f(X3)
E8 Gauge Symmetry in 2D
In cohomology, we have only non-trivial 0 and 2 forms on X3 = P2 × P1
g(v, X3) ∼ v5g(X3)
V(0,0), D(0,1), Λ(0,2), Φ(3,0)
X3 = P2 × P1
Kodaira
Assume No non-trivial bundle
y2 = x3 + f(v, X3)x + g(v, x3) B4 = Nv → X3
f(v, X3) ∼ v4f(X3)
E8 Gauge Symmetry in 2D
Only zero modes that survive are the one associated to gaugino & Fermi In cohomology, we have only non-trivial 0 and 2 forms on X3 = P2 × P1
6= 0
g(v, X3) ∼ v5g(X3)
Igauge = −Ind(Adj(E8)) ×
chirality of fermions in the multiplets
V(0,0), D(0,1), Λ(0,2), Φ(3,0)
X3 = P2 × P1
Kodaira
Assume No non-trivial bundle
y2 = x3 + f(v, X3)x + g(v, x3) B4 = Nv → X3
f(v, X3) ∼ v4f(X3)
E8 Gauge Symmetry in 2D
Only zero modes that survive are the one associated to gaugino & Fermi In cohomology, we have only non-trivial 0 and 2 forms on X3 = P2 × P1 Anomalous!
6= 0
g(v, X3) ∼ v5g(X3)
Igauge = −Ind(Adj(E8)) ×
chirality of fermions in the multiplets
V(0,0), D(0,1), Λ(0,2), Φ(3,0)
Igauge = −Ind(Adj(E8)) ×
SGreen−Schwarz ∝ Z B2 ∧ X8(F, R)
Tadpole Cancellation
Extra Sectors! Igauge = −Ind(Adj(E8)) ×
SGreen−Schwarz ∝ Z B2 ∧ X8(F, R)
Tadpole Cancellation
Extra Sectors! In the previous example, spacetime D3 branes wrapping compact curves Q ⊕ (Qc)† ∈ H0(Σ, RΣ)
Ψ ⊕ (Ψc)† ∈ H1(Σ, RΣ)
some bundle on compact curve
Igauge = −Ind(Adj(E8)) ×
Fermi mult.:
SGreen−Schwarz ∝ Z B2 ∧ X8(F, R)
Tadpole Cancellation
Extra Sectors! In the previous example, spacetime D3 branes wrapping compact curves Q ⊕ (Qc)† ∈ H0(Σ, RΣ)
Ψ ⊕ (Ψc)† ∈ H1(Σ, RΣ)
some bundle on compact curve sum of k trivial bundles on the curve
Igauge = −Ind(Adj(E8)) ×
RΣ = O⊕k
Σ
CS Mult.: Fermi mult.:
SGreen−Schwarz ∝ Z B2 ∧ X8(F, R)
Tadpole Cancellation
Extra Sectors! In the previous example, spacetime D3 branes wrapping compact curves Q ⊕ (Qc)† ∈ H0(Σ, RΣ)
Ψ ⊕ (Ψc)† ∈ H1(Σ, RΣ)
some bundle on compact curve sum of k trivial bundles on the curve
Igauge = −Ind(Adj(E8)) ×
Extra
+Ind(Rep(E8)) × k × (#(Q) + #(Qc)) = 0 RΣ = O⊕k
Σ
CS Mult.: Fermi mult.:
SGreen−Schwarz ∝ Z B2 ∧ X8(F, R)
Tadpole Cancellation
Extra Sectors! In the previous example, spacetime D3 branes wrapping compact curves Q ⊕ (Qc)† ∈ H0(Σ, RΣ)
Ψ ⊕ (Ψc)† ∈ H1(Σ, RΣ)
some bundle on compact curve sum of k trivial bundles on the curve
Igauge = −Ind(Adj(E8)) ×
Fermi multiplets
2D GLSM Extra
+Ind(Rep(E8)) × k × (#(Q) + #(Qc)) = 0 RΣ = O⊕k
Σ
CS Mult.: Fermi mult.:
F-theory on elliptically fibered CY3 on B2 = O(−n) → P1
3 ≤ n ≤ 12
Single gauge group, G, 6D (1,0) theories F-theory on elliptically fibered CY3 on B2 = O(−n) → P1
Morrison-Taylor; Heckman-Morrison-Vafa+Del Zotto+Tomasiello+Rudelius;
3 ≤ n ≤ 12
Single gauge group, G, 6D (1,0) theories F-theory on elliptically fibered CY3 on B2 = O(−n) → P1 Multiplets: (bosonic components)
6D Vector Multiplet:
A(6D)
µ
Tensor multiplet:
(t(6D), B(6D)
µν
)
Morrison-Taylor; Heckman-Morrison-Vafa+Del Zotto+Tomasiello+Rudelius;
3 ≤ n ≤ 12
Single gauge group, G, 6D (1,0) theories F-theory on elliptically fibered CY3 on B2 = O(−n) → P1 Multiplets: (bosonic components)
6D Vector Multiplet:
A(6D)
µ
Tensor multiplet:
(t(6D), B(6D)
µν
) SCFT at Singularity
Morrison-Taylor; Heckman-Morrison-Vafa+Del Zotto+Tomasiello+Rudelius;
3 ≤ n ≤ 12
Single gauge group, G, 6D (1,0) theories F-theory on elliptically fibered CY3 on B2 = O(−n) → P1 Multiplets: (bosonic components)
6D Vector Multiplet:
A(6D)
µ
Tensor multiplet:
(t(6D), B(6D)
µν
)
Coupling, gauge kinetic term
L ⊃ t(6D)Tr(F ∧ ∗F) SCFT at Singularity Far out in Tensor Branch
P1
Morrison-Taylor; Heckman-Morrison-Vafa+Del Zotto+Tomasiello+Rudelius;
3 ≤ n ≤ 12
Single gauge group, G, 6D (1,0) theories F-theory on elliptically fibered CY3 on B2 = O(−n) → P1 Multiplets: (bosonic components)
6D Vector Multiplet:
A(6D)
µ
Tensor multiplet:
(t(6D), B(6D)
µν
)
Coupling, gauge kinetic term
L ⊃ t(6D)Tr(F ∧ ∗F) SCFT at Singularity Far out in Tensor Branch
P1
Compactify on M4
Morrison-Taylor; Heckman-Morrison-Vafa+Del Zotto+Tomasiello+Rudelius;
3 ≤ n ≤ 12
Single gauge group, G, 6D (1,0) theories F-theory on elliptically fibered CY3 on B2 = O(−n) → P1 Multiplets: (bosonic components)
6D Vector Multiplet:
A(6D)
µ
Tensor multiplet:
(t(6D), B(6D)
µν
)
Coupling, gauge kinetic term
L ⊃ t(6D)Tr(F ∧ ∗F) SCFT at Singularity Far out in Tensor Branch
P1
Compactify on M4
*
t(2D) ∼ t(6D)Vol(M4)
Morrison-Taylor; Heckman-Morrison-Vafa+Del Zotto+Tomasiello+Rudelius;
3 ≤ n ≤ 12
Single gauge group, G, 6D (1,0) theories F-theory on elliptically fibered CY3 on B2 = O(−n) → P1 Multiplets: (bosonic components)
6D Vector Multiplet:
A(6D)
µ
Tensor multiplet:
(t(6D), B(6D)
µν
)
Coupling, gauge kinetic term
L ⊃ t(6D)Tr(F ∧ ∗F) SCFT at Singularity Far out in Tensor Branch
P1
Compactify on M4
*
t(2D) ∼ t(6D)Vol(M4)
Dynamic GLSM t(2D)Tr(F+−F +−)
Morrison-Taylor; Heckman-Morrison-Vafa+Del Zotto+Tomasiello+Rudelius;
3 ≤ n ≤ 12
6D Vector Multiplet:
A(6D)
µ
Tensor multiplet:
(t(6D), B(6D)
µν
)
Compactify on M4
1) suppose the 4-Manifold is Kähler, then apply the following Topological Twist
SO(1, 5) × SU(2)R6D → SO(1, 1) × SU(2)M4 × U(1)M4 × SU(2)R6D
U(1)R2D : JU(1)R2D = JU(1)M4 + J3R6D
2) Apply what done for F-theory on CY5, where the 7 branes wrap
X3 = M4 × P1 ⊂ B4
6D Vector Multiplet:
A(6D)
µ
Tensor multiplet:
(t(6D), B(6D)
µν
)
Compactify on M4
1) suppose the 4-Manifold is Kähler, then apply the following Topological Twist
SO(1, 5) × SU(2)R6D → SO(1, 1) × SU(2)M4 × U(1)M4 × SU(2)R6D
U(1)R2D : JU(1)R2D = JU(1)M4 + J3R6D
2) Apply what done for F-theory on CY5, where the 7 branes wrap
X3 = M4 × P1 ⊂ B4
V(0,0) ∈ H0
¯ ∂(M4, OM4)
D(0,1) ∈ H1
¯ ∂(M4, OM4)
Λ(0,2) ∈ H2
¯ ∂(M4, OM4)
CS Multiplet: Vector Multiplet: Fermi Multiplet:
We do not switch on any bundle on the 4-manifold
Right Moving Chiral Bosons: CS Multiplet:
∗H3 = −H3
B(2,0)
(t(2D)R, B
¯ i i)
RS Multiplet:
(t(2D)L, B(1,1))
Left Moving Chiral Bosons:
+ eventual Abelian Vectors
Ganor
2h2,0 + 2 Right moving chiral currents
Left moving chiral currents Chiral Bosons Sector, NLSM GLSM Sector
#(V ) − #(D) + #(Λ) = = dim(G)χ(M4, OM4)
2h2,0 + 2 Right moving chiral currents
Left moving chiral currents Chiral Bosons Sector, NLSM GLSM Sector
#(V ) − #(D) + #(Λ) = = dim(G)χ(M4, OM4)
L(2D) ⊃ µGS Z
M4
B(2,0) ∧ Tr(F(2D) ∧ [D(0,1), D(0,1)]) + µGS Z
M4
B(1,1) ∧ Tr(F(2D) ∧ [D(0,1), D†
(1,0)])
µGS Z
6D
B ∧ X 6D Green-Schwarz:
2h2,0 + 2 Right moving chiral currents
Left moving chiral currents Chiral Bosons Sector, NLSM GLSM Sector
#(V ) − #(D) + #(Λ) = = dim(G)χ(M4, OM4)
L(2D) ⊃ µGS Z
M4
B(2,0) ∧ Tr(F(2D) ∧ [D(0,1), D(0,1)]) + µGS Z
M4
B(1,1) ∧ Tr(F(2D) ∧ [D(0,1), D†
(1,0)])
Gadde,Haghighat, Kim, Kim, Lockhart, Vafa; Kim, Kim , Park; Shimizu, Tachikawa; Del Zotto, Lockhart; Lawrie, Schäfer-Nameki, Weigand
Tensionless strings at the fixed point µGS Z
6D
B ∧ X 6D Green-Schwarz:
Extra Sectors: spacetime filling strings (D3 branes wrapping inside 7 branes)
L2D ⊃ Z
2D
B(2D) Z
M4
− ✓1 4TrF 2 + h∨
G
n ✓ c2(R) + p1(M4) 12 ◆◆
Reduction of 4D N=2 S-W/A-D/M-N
P1
= Nstring Z
2D
B(2D)
2h2,0 + 2 Right moving chiral currents
Left moving chiral currents Chiral Bosons Sector, NLSM GLSM Sector
#(V ) − #(D) + #(Λ) = = dim(G)χ(M4, OM4)
L(2D) ⊃ µGS Z
M4
B(2,0) ∧ Tr(F(2D) ∧ [D(0,1), D(0,1)]) + µGS Z
M4
B(1,1) ∧ Tr(F(2D) ∧ [D(0,1), D†
(1,0)])
Effective Strings
GLSM
Gadde,Haghighat, Kim, Kim, Lockhart, Vafa; Kim, Kim , Park; Shimizu, Tachikawa; Del Zotto, Lockhart; Lawrie, Schäfer-Nameki, Weigand
Tensionless strings at the fixed point µGS Z
6D
B ∧ X 6D Green-Schwarz:
Extra Sectors: spacetime filling strings (D3 branes wrapping inside 7 branes)
L2D ⊃ Z
2D
B(2D) Z
M4
− ✓1 4TrF 2 + h∨
G
n ✓ c2(R) + p1(M4) 12 ◆◆
Reduction of 4D N=2 S-W/A-D/M-N
P1
= Nstring Z
2D
B(2D)
Ohmori, Shimizu, Tachikawa + Yonekura; Heckman, Morrison, Rudelius, Vafa; Intriligator.
I6D = αc2(R)2 + βc2(R)p1(T) + γp1(T)2 + δp2(T)
I2D = Z
M4
I6D(twist(0,2)) = kRR 2 c1(R(2D))2 − kgrav 24 p1(T)
computation
kRR = (3p1(M4) + 6χ(M4))α − 2p1(M4)β kgrav = (6p1(M4) + 12χ(M4))β − 24p1(M4)(2γ + δ)
Ohmori, Shimizu, Tachikawa + Yonekura; Heckman, Morrison, Rudelius, Vafa; Intriligator.
I6D = αc2(R)2 + βc2(R)p1(T) + γp1(T)2 + δp2(T)
I2D = Z
M4
I6D(twist(0,2)) = kRR 2 c1(R(2D))2 − kgrav 24 p1(T)
computation
2D chiral gamma operator
kRR = (3p1(M4) + 6χ(M4))α − 2p1(M4)β kgrav = (6p1(M4) + 12χ(M4))β − 24p1(M4)(2γ + δ)
2 .DGLSM computation
kRR = γ3R2
ferm(GLSM) × #(GLSM) + γ3R2 ferm(chiral bosons) × #(chiral bosons)
+ Nstringh∨
G
kgrav = γ3#(GLSM) + γ3#(chiral bosons) + Nstringh∨
G
Ohmori, Shimizu, Tachikawa + Yonekura; Heckman, Morrison, Rudelius, Vafa; Intriligator.
I6D = αc2(R)2 + βc2(R)p1(T) + γp1(T)2 + δp2(T)
I2D = Z
M4
I6D(twist(0,2)) = kRR 2 c1(R(2D))2 − kgrav 24 p1(T)
computation
R-charges constrained by superpotential couplings 2D chiral gamma operator
kRR = (3p1(M4) + 6χ(M4))α − 2p1(M4)β kgrav = (6p1(M4) + 12χ(M4))β − 24p1(M4)(2γ + δ)
2 .DGLSM computation
kRR = γ3R2
ferm(GLSM) × #(GLSM) + γ3R2 ferm(chiral bosons) × #(chiral bosons)
+ Nstringh∨
G
kgrav = γ3#(GLSM) + γ3#(chiral bosons) + Nstringh∨
G
Ohmori, Shimizu, Tachikawa + Yonekura; Heckman, Morrison, Rudelius, Vafa; Intriligator.
I6D = αc2(R)2 + βc2(R)p1(T) + γp1(T)2 + δp2(T)
I2D = Z
M4
I6D(twist(0,2)) = kRR 2 c1(R(2D))2 − kgrav 24 p1(T)
computation
R-charges constrained by superpotential couplings 2D chiral gamma operator
kRR = (3p1(M4) + 6χ(M4))α − 2p1(M4)β kgrav = (6p1(M4) + 12χ(M4))β − 24p1(M4)(2γ + δ)
Aharony, Tachikawa; Shapere, Tachikawa
2 .DGLSM computation
kRR = γ3R2
ferm(GLSM) × #(GLSM) + γ3R2 ferm(chiral bosons) × #(chiral bosons)
+ Nstringh∨
G
kgrav = γ3#(GLSM) + γ3#(chiral bosons) + Nstringh∨
G
Reduction of the 4D SW/AD/MN N=2 theory on P1
Ohmori, Shimizu, Tachikawa + Yonekura; Heckman, Morrison, Rudelius, Vafa; Intriligator.
I6D = αc2(R)2 + βc2(R)p1(T) + γp1(T)2 + δp2(T)
I2D = Z
M4
I6D(twist(0,2)) = kRR 2 c1(R(2D))2 − kgrav 24 p1(T)
computation
R-charges constrained by superpotential couplings 2D chiral gamma operator
Computations Match: Evidence that the two Theories are at Finite Distance in the Target Space Metric
kRR = (3p1(M4) + 6χ(M4))α − 2p1(M4)β kgrav = (6p1(M4) + 12χ(M4))β − 24p1(M4)(2γ + δ)
Aharony, Tachikawa; Shapere, Tachikawa
2 .DGLSM computation
kRR = γ3R2
ferm(GLSM) × #(GLSM) + γ3R2 ferm(chiral bosons) × #(chiral bosons)
+ Nstringh∨
G
kgrav = γ3#(GLSM) + γ3#(chiral bosons) + Nstringh∨
G
Reduction of the 4D SW/AD/MN N=2 theory on P1
WHAT WE HAVE DONE AND WHAT’S NEXT
▸ Tools for constructing 2D (0,2) Theories; ▸ Novel example of 2D SCFTs with dynamic couplings; ▸ UV completion of super critical string theories; ▸ Extending this method to a larger class of 6D SCFTs; ▸ Add defects on the four-manifold.
AND FINALLY…