SCFTs, Compact CY 3-folds, and TopologicalStrings
Patrick Jefferson(to appear) in collaboration with: Hirotaka Hayashi, Hee-Cheol Kim,
Kantaro Ohmori, and Cumrun Vafa

This subject of this talk is SCFTs and their relationship with SUGRA. In
particular, we would like to use what is known about 6d and 5d SCFTs tobetter understand 6d and 5d SUGRA.
Since 6d/5d theories can be engineered by compactifying F/M theory on
singular 3-folds X , the basic insight is that the relationship between
SCFTs and SUGRA can be interpreted as a relationship between 3-folds:

SCFT

SUGRA

Xnon-compact

XcompactThis subject of this talk is SCFTs and their relationship with SUGRA. In
particular, we would like to use what is known about 6d and 5d SCFTs tobetter understand 6d and 5d SUGRA.
Since 6d/5d theories can be engineered by compactifying F/M theory on
singular 3-folds X , the basic insight is that the relationship between
SCFTs and SUGRA can be interpreted as a relationship between 3-folds:

SCFT,−→SUGRA

Xnon-compact,−→Xcompact

First, 6d SCFTs. 6d (1, 0) SCFTs are associated to singular elliptic
3-folds X → B.
6d SCFTs called (G , G 0 ) conformal matter theories (with global
symmetry subgroup G × G 0 ) have an orbifold realization in terms of
non-compact 3-folds T 2 × C2 /Zm × Zn :
GG0

=

n1

n2···

nk−1nkFirst, 6d SCFTs. 6d (1, 0) SCFTs are associated to singular elliptic
3-folds X → B.
6d SCFTs called (G , G 0 ) conformal matter theories (with global
symmetry subgroup G × G 0 ) have an orbifold realization in terms of
non-compact 3-folds T 2 × C2 /Zm × Zn :GG0=n1n2···

nk−1nkCompact 3-folds X = T 6 /Zm × Zn contain 6d (G , G 0 ) conformal mattertheories as local singularities. However, the compactness of the global
structure means that the symmetries G , G 0 are gauged.
Example: B = (T 2 )2 /Z2 × Z2 , which contains 16 copies of the (D4 , D4 )conformal matter theory. The non-compact curves C carrying D4 global
symmetry are compactified into P1 ’s with self-intersection −4:444411111111111111114444

Compact 3-folds X = T 6 /Zm × Zn contain 6d (G , G 0 ) conformal mattertheories as local singularities. However, the compactness of the global
structure means that the symmetries G , G 0 are gauged.
Example: B = (T 2 )2 /Z2 × Z2 , which contains 16 copies of the (D4 , D4 )conformal matter theory. The non-compact curves C carrying D4 global
symmetry are compactified into P1 ’s with self-intersection −4:4444111111111111111144
44

Compact 3-folds X = T 6 /Zm × Zn contain 6d (G , G 0 ) conformal mattertheories as local singularities. However, the compactness of the global
structure means that the symmetries G , G 0 are gauged.
Example: T 6 /Z2 × Z2 contains 16 copies of the (D4 , D4 ) conformal
matter theory. The non-compact curves C carrying D4 global symmetry
are compactified into P1 ’s with self-intersection −4:4444

11111111111111114444

Next, let’s consider 5d theories. 5d SCFTs are associated non-elliptic
3-folds, such as toric 3-folds. The 5d T5 theory, associated to thenon-compact 3-fold C3 /Z5 × Z5 , can be represented (in a particularCoulomb phase) as↔

Note T5 theory has global symmetry SU(5)3 .

Toric singularities naturally appear in mirror Fermat hypersurfaces
P4w~ [d]/G = {5Xi=1

xipi = 0}/G , pi ≡d.
wi

An example we study is the mirror quintic 3-fold P4 [5]/Z35 , whose
singularities consist of 10 lines of SU(5) singularities meeting triple-wisein 10 singular points Tijk with normal geometry C3 /Z25 .
Locally, these points are T5 theories, but their arrangement in the mirror
quintic means their global SU(5) symmetries are gauged:

Toric singularities naturally appear in mirror Fermat hypersurfaces
P4w~ [d]/G = {5Xi=1xipi = 0}/G , pi ≡d.
wi

An example we study is the mirror quintic 3-fold P4 [5]/Z35 , whose
singularities consist of 10 lines of SU(5) singularities meeting triple-wisein 10 singular points Tijk with normal geometry C3 /Z25 .
Locally, these points are T5 theories, but their arrangement in the mirror
quintic means their global SU(5) symmetries are gauged:

Toric singularities naturally appear in mirror Fermat hypersurfaces
P4w~ [d]/G = {5Xi=1xipi = 0}/G , pi ≡d.
wiAn example we study is the mirror quintic 3-fold P4 [5]/Z35 , whose
singularities consist of 10 lines of SU(5) singularities meeting triple-wisein 10 singular points Tijk with normal geometry C3 /Z25 .
Locally, these points are T5 theories, but their arrangement in the mirror
quintic means their global SU(5) symmetries are gauged:SU(5)
SU(5)

SU(5)

The advantage of these geometric pictures is that they tell us how
SCFTs are coupled consistently in SUGRA, in particular the correct way
to gauge the global symmetries. So what precise information can welearn from this?
1. Holography. We use our description of T 6 /Z22 to propose a 2d
N = (0, 4) quiver holographically dual to type IIB on
AdS3 × S 3 × T 4 /Z22 .
2. Topological string partition function. Observing thatX5dZtop (τ, t, λ) = ZBPS= ZBH = Z0 (τ, λ)ZC (τ, λ)e −t·CC

we use the elliptic genus for O(−1), O(−4) strings to propose
topological string partition function on T 6 /Z22 .
3. We make some progress towards generalizing the topological vertex
to SU(5) gaugings, which gives topological string amplitudes forthe mirror quintic.

The advantage of these geometric pictures is that they tell us howSCFTs are coupled consistently in SUGRA, in particular the correct wayto gauge the global symmetries. So what precise information can welearn from this?
1. Holography. We use our description of T 6 /Z22 to propose a 2d
N = (0, 4) quiver holographically dual to type IIB on
AdS3 × S 3 × T 4 /Z22 .
2. Topological string partition function. Observing thatX
5d
Ztop (τ, t, λ) = ZBPS= ZBH = Z0 (τ, λ)ZC (τ, λ)e −t·CC

we use the elliptic genus for O(−1), O(−4) strings to propose
topological string partition function on T 6 /Z22 .
3. We make some progress towards generalizing the topological vertex
to SU(5) gaugings, which gives topological string amplitudes forthe mirror quintic.

The advantage of these geometric pictures is that they tell us how
SCFTs are coupled consistently in SUGRA, in particular the correct way
to gauge the global symmetries. So what precise information can welearn from this?
1. Holography. We use our description of T 6 /Z22 to propose a 2d
N = (0, 4) quiver holographically dual to type IIB on
AdS3 × S 3 × T 4 /Z22 .
2. Topological string partition function. Observing thatX5d
Ztop (τ, t, λ) = ZBPS= ZBH = Z0 (τ, λ)ZC (τ, λ)e −t·CC

we use the elliptic genus for O(−1), O(−4) strings to propose
topological string partition function on T 6 /Z22 .
3. We make some progress towards generalizing the topological vertex
to SU(5) gaugings, which gives topological string amplitudes forthe mirror quintic.

The advantage of these geometric pictures is that they tell us howSCFTs are coupled consistently in SUGRA, in particular the correct way
to gauge the global symmetries. So what precise information can welearn from this?
1. Holography. We use our description of T 6 /Z22 to propose a 2d
N = (0, 4) quiver holographically dual to type IIB on
AdS3 × S 3 × T 4 /Z22 .
2. Topological string partition function. Observing thatX
5d
Ztop (τ, t, λ) = ZBPS= ZBH = Z0 (τ, λ)ZC (τ, λ)e −t·CC

we use the elliptic genus for O(−1), O(−4) strings to propose
topological string partition function on T 6 /Z22 .
3. We make some progress towards generalizing the topological vertexto SU(5) gaugings, which in principle permits computation of
topological string amplitudes for the mirror quintic.

Thank you!