Discovering T-Dualities of Little String Theories

We describe a general method for deducing T-dualities of little string theories, which are dualities between these theories that arise when they are compactified on circle. The method works for both untwisted and twisted circle compactifications of little string theories and is based on surface geometries associated to these circle compactifications. The surface geometries describe information about Calabi-Yau threefolds on which M-theory can be compactified to construct the corresponding circle compactified little string theories. Using this method, we deduce at least one T-dual, and in some cases multiple T-duals, for untwisted and twisted circle compactifications of most of the little string theories that can be described on their tensor branches in terms of a 6d supersymmetric gauge theory with a simple non-abelian gauge group, which are also known as rank-0 little string theories. This includes little string theories carrying N=(1,1) and N=(1,0) supersymmetries. For many, but not all, circle compactifications of N=(1,1) little string theories, we have T-dualities that realize Langlands dualities between affine Lie algebras. Along the way, we find another discrete theta angle distinct from 0 and $\pi$ for an E-string node.


Introduction
Little string theories (LSTs) are fascinating quantum-mechanical theories.They are expected to be UV complete, but are neither local quantum field theories (QFTs) nor fully-fledged string theories, sitting somewhere in between the two.Unlike string theories, gravity is not dynamical in these theories, and as a consequence they behave very similarly to local QFTs.The key property that differentiates LSTs from local QFTs is that LSTs admit T-dualities [1][2][3], which is the topic of this work.
All the known LSTs exist in six spacetime dimensions, and carry at least N = (1, 0) supersymmetry.Some very special LSTs carry N = (2, 0) or N = (1, 1) supersymmetries.A uniform definition for all the known LSTs is provided as compactifications of F-theory on (a special class of) elliptically fibered Calabi-Yau threefolds.This includes both the standard Ftheory compactifications [4], and also the recently explored frozen F-theory compactifications [5,6].These theories can also be classified using anomaly cancellation [7][8][9].
To study T-dualities1 of such LSTs, we need to compactify the F-theory setups on circle, which are dual to M-theory compactified on the same elliptically fibered Calabi-Yau threefolds 2 .T-dualities thus express themselves as geometric isomorphisms between the corresponding threefolds.
In this paper, we study such geometric isomorphisms in terms of graphical structures that we call surface geometries, which capture the information associated to holomorphic curves and surfaces inside the threefold.Surface geometries have been immensely useful in the recent studies on 5d and 6d superconformal field theories (SCFTs) .They were proposed as useful tools for these studies in [17], while the essential graphical calculus for these purposes was developed in [18][19][20][21][22]25,27,28].The present work is thus part of the vast progress [5, made recently in understanding 5d and 6d supersymmetric UV complete theories -which so far has included SCFTs and KK theories; the present paper adds LSTs to this list.
The strategy for using surface geometries to determine T-dualities of an LST T can be summarized as follows: 1. Use the tensor branch description of T. This can be packaged in terms of a graph with nodes and edges.This graph is used in the following operations for studying T-duals of the untwisted compactification of T. For studying T-duals of a twisted compactification of T, one uses information regarding the twist to convert the graph for T into a new graph, which is then used in the following operations.Let us denote the circle compactified theory being studied as T ρ S 1 , where ρ denotes the twist involved, which takes trivial value if the circle compactification being studied is untwisted.
2. To each node in the graph associated to T ρ S 1 , we can associate a collection of Hirzebruch surfaces (i.e.compact Kahler surfaces with a choice of P 1 fibration) glued to each other.
3. If two nodes are connected by an edge, we glue together the collections of Hirzebruch surfaces associated to the two nodes.By this point, we have described a surface geometry associated to T ρ S 1 .
4. Some Hirzebruch surfaces contain multiple P 1 fibers.Changing the choice of P 1 fibers judiciously lets us recognize the resulting surface geometry as a surface geometry associated to T ρ S 1 , i.e. a ρ (which can take a trivial value for untwisted compactification) twisted circle compactification of an LST T. This means that we have a T-duality between LSTs T and T, where T is compactified with twist ρ and T is compactified with twist ρ.
We illustrate this procedure by finding a T-dual for twisted and untwisted compactifications of many rank-0 LSTs describable on their tensor branches as 6d N = (1, 0) gauge theories with simple gauge algebras.More concretely, we study T-duals of the following systems: • 6d theory with any simple gauge algebra g and a hyper in adjoint representation A. This theory has N = (1, 1) supersymmetry.For g = su(n), we can turn on a rational theta angle θ = 2π p q [91], and for g = sp(n), we can turn on a Z 2 valued theta angle θ = 0, π.We provide a T-dual for each untwisted compactification of these theories for g ̸ = su(n).
On the other hand, for g = su(n), we provide a T-dual for untwisted compactification only of θ = 0 theory3 .
For g = su(n), so(2n), e 6 we have the possibility of twisting the compactification by the Z 2 outer-automorphism of such a Lie algebra g.For g = su(n), we must have θ = 0, π for the twist to be allowed.We provide a T-dual for each such twisted compactification.
For g = so (8), we have an additional possibility of twisting the compactification by the Z 3 outer-automorphism of so (8), and we provide a T-dual for this twisted compactification.
• 6d theory with su(n ≥ 3) gauge algebra, 2 hypers in antisymmetric irrep Λ 2 and 16 hypers in fundamental irrep F. We have the possibility of twisting the compactification by the Z 2 outer-automorphism of su(n).We provide a T-dual for each such compactification.
• 6d theory with su(2) gauge algebra and 16 hypers in fundamental irrep F. We provide a T-dual for the untwisted compactification.
• 6d theory with su(n ≥ 3) gauge algebra, 1 hyper in antisymmetric irrep Λ 2 and 1 hyper in symmetric irrep S 2 .We have the possibility of twisting the compactification by the Z 2 outer-automorphism of su(n).We provide a T-dual only for the untwisted compactification.
• 6d theory with su(6) gauge algebra, 1 hyper in three-index antisymmetric irrep Λ 3 and 18 hypers in fundamental irrep F. We have the possibility of twisting the compactification by the Z 2 outer-automorphism of su (6).We provide a T-dual for each such compactification.
Note that in the T-dual for the untwisted compactification, we find a hitherto unknown discrete theta angle for an E-string sp(0) node, that is distinct from θ = 0, π.We call this as a new theta angle.See equation (3.72).The distinction between the various theta angles is field-theoretically described in terms of different embeddings of the gauge symmetries into the full e 8 flavor symmetry of the E-string theory, leaving behind different residual flavor symmetries.In this case, while the θ = 0, π have residual flavor symmetries su(2) and u(1) respectively, the new theta angle has residual flavor symmetry su (3).The residual flavor symmetries can be determined using the respective surface geometries by finding (−2)-curves inside the compact surfaces that have zero intersection with all the gluing curves, which means that they can be attached to P 1 fibers of non-compact Hirzebruch surfaces describing (non-abelian) flavor symmetries [27,28].
The intersection pattern of these (−2)-curves determines the intersection pattern of the non-compact Hirzebruch surfaces, thus determining the form of the flavor symmetry algebra.
• 6d theory with su(6) gauge algebra, 1 half-hyper in three-index antisymmetric irrep Λ 3 , 1 hyper in antisymmetric irrep Λ 2 and 17 hypers in fundamental irrep F. We have the possibility of twisting the compactification by the Z 2 outer-automorphism of su (6).We provide a T-dual for each such compactification.
• 6d theory with so(8) gauge algebra, 4 hypers in spinor irrep S, 4 hypers in cospinor irrep C and 4 hypers in vector irrep F. We have the possibility of twisting the compactification by either the Z 2 outer-automorphism of so (8) or by the Z 3 outer-automorphism of so (8).
We provide a T-dual for each such compactification.
• 6d theory with so(10) gauge algebra, 4 hypers in spinor irrep S and 6 hypers in vector irrep F. We have the possibility of twisting the compactification by the Z 2 outerautomorphism of so (10).We provide a T-dual for each such compactification.
• 6d theory with so(12) gauge algebra, 2 hypers in spinor irrep S and 8 hypers in vector irrep F. We provide a T-dual for the untwisted compactification.
• 6d theory with so(12) gauge algebra, 1 hyper in spinor irrep S, 1 hyper in cospinor irrep C and 8 hypers in vector irrep F. We have the possibility of twisting the compactification by the Z 2 outer-automorphism of so (12).We provide a T-dual for each such compactification.
• 6d theory with so(14) gauge algebra, 1 hyper in spinor irrep S and 10 hypers in vector irrep F. We have the possibility of twisting the compactification by the Z 2 outerautomorphism of so (14).We provide a T-dual for each such compactification.
• 6d theory with sp(3) gauge algebra, 1 half-hyper in three-index antisymmetric irrep Λ 3 and 35 half-hypers in fundamental irrep F. We provide a T-dual for the untwisted compactification.
• 6d theory with g 2 gauge algebra and 10 hypers in 7-dimensional irrep F. We provide a T-dual for the untwisted compactification.

Surface Geometries for Circle Compactifications of LSTs
In this paper, we will use the conventions and notations for surfaces that were introduced in [22].

Graphs Associated to Circle Compactifications of LSTs
Consider, as in the introduction, a circle compactification T ρ S 1 of a LST T twisted by ρ.To such a circle compactification, one can associate a graph which essentially encodes the information regarding the twist and the low-energy theory arising on the tensor branch of the LST.The construction of the graph was explained in [22] for circle compactifications of 6d SCFTs, and carries over in a similar fashion to circle compactifications of LSTs.
We will denote the nodes of the graph by i.Each node carries a non-negative integer k i and a simple affine Lie algebra g where g i is a simple finite Lie algebra.The affine algebra can be trivial, and there are two possible choices denoted by sp(0) (1) and su(1) (1) .See [22] for more details.
An edge between two nodes can be directional or non-directional, and carries a positive integer label.Using the information about k i and the edges, we can extract a matrix [Ω ij ] as described in [22].For a circle compactification of a 6d SCFT [Ω ij ] is a positive-definite matrix, while for a circle compactification of an LST, [Ω ij ] is a positive semi-definite matrix with a 1-dimensional space of null vectors.
From the structure of the graph (including some additional decorations of nodes and edges), we can also extract a representation R ρ T of a finite Lie algebra where h i is the subalgebra (which is tabulated in [22]) of g i left invariant by the order q i outer-automorphism used in the twist.The physical interpretation of R ρ T is that the circle compactification T ρ S 1 of the 6d LST T admits a Coulomb branch phase in which at extremely low energies we find a 5d gauge theory with h gauge algebra and hypers transforming in representation R ρ T of h.

Surface Geometries from Graphs
The information about the graph associated to T ρ S 1 can be converted into surface geometries describing various 5d N = 1 supersymmetric Coulomb branch phases of T ρ S 1 .Such a phase P of T ρ S 1 can be constructed by compactifying M-theory on a smooth Calabi-Yau threefold X P .A 5d N = 1 abelian gauge theory T P with massive matter content describes the low-energy theory at a point in the interior of the phase P.This gauge theory T P can be identified as the low-energy theory obtained upon compactification of M-theory on X P .Each X P corresponds to a different surface geometry G P .Surface geometries G P and G P ′ for two different phases P and P ′ are related by a sequence of flop transitions.In this work, we will study phases for which the corresponding threefold X P is a genus-one fibration.We will call such phases as genus-one phases.There always exists at least one genus-one phase for each T ρ S 1 .The surface geometry for any genus-one phase takes the following form.For each i, we have a collection of surfaces S i,α where 0 ≤ α ≤ r i and r i is the rank of h i .Consider first the case when h i is non-trivial, i.e. when r i > 0. In this case, each S i,α is a Hirzebruch surface with (typically non-generic) blowups.Let f i,α be the P 1 fiber of the Hirzebruch surface4 S i,α .These Hirzebruch surfaces must intersect such that we have where δ ij is the Kronecker delta and C i,αβ is the Cartan matrix for g Thus each S i,α corresponds to a node in the Dynkin diagram of g (q i ) i , and our convention is that S i,0 corresponds to the affine node, i.e. the nodes S i,α for α ̸ = 0 describe the Dynkin diagram of h i .

The curve
where d i,α is the row null vector of the Cartan matrix of g with minimal positive integer values, is a degenerate genus-one fiber.
For trivial h i , we have a single surface S i,0 which arises only for the following cases: • Ω ii = 1 without loop, in which case the surface S i,0 is a Hirzebruch surface with 8 blowups, which admits an elliptic fibration.There are various ways of representing such a surface.The common representations used in this paper are as follows: -If we represent it as F 8 0 , then the elliptic fiber is -If we represent it as F 8 1 , then the elliptic fiber is -Finally, if we represent it as F 8 2 , then the elliptic fiber is In these cases, we define a curve E i,0 = f − x 1 to be used later.The (trivial) algebra attached to such a node is g = sp(0) (1) .
• Ω ii = 2 without loop, in which case the surface S i,0 is a self-glued Hirzebruch surface which admits an elliptic fibration.In this paper, we typically use the following ways of representing this surface: -F 2 0 with the two blowups x and y glued to each other.The elliptic fiber is -F 2 0 with e − x and e − y glued to each other.The elliptic fiber is In both of these cases, we define a curve E i,0 = e to be used later.The (trivial) algebra attached to such a node is g (1) .
• Ω ii = 2 with a loop, in which case the surface S i,0 is a self-glued Hirzebruch surface which in this paper, we represent this surface as F 2 1 with the two blowups x and y glued to each other.It admits a special genus-one fiber f i = 2h + f − 2x − 2y.We define a curve E i,0 = e to be used later.The (trivial) algebra attached to such a node is (1) .
• It is also possible to have Ω ii = 1 with a loop, which we do not discuss in this paper.
The (trivial) algebra attached to such a node is g (1) .
The curve f i satisfies the property that for all i, j, β.
We have the property for all i, α, j where for h j non-trivial and d ∨ j,β is a null column vector of the Cartan matrix of g (q j ) j with minimal positive integer values, S i = S i,0 for trivial h i , and f i,0 := f i (i.e. the elliptic fiber, not the P 1 fiber of the Hirzebruch surface) for trivial h i .
Additionally we have the property where where E j,0 = e j,0 is the base curve of the Hirzebruch surface S j,0 if h j is non-trivial, and E j,0 for h j trivial was defined above.
The blowups living in the surfaces S i,α for non-trivial h i describe hypermultiplets charged under the gauge algebra h = ⊕ i h i .The map between the blowups and representations of hypers is as follows.Pick a blowup x of a surface S i,α such that h i is non-trivial.It lives in a family F of blowups, where some other blowup y of some other surface S j,β (such that h j is non-trivial) is in the same family F if and only if one of the following conditions hold: • x is glued to y.
• x is glued to f − y.
• f − x is glued to y.
• If (i, α) = (j, β), i.e. if x and y live in the same surface, and for some other surface S k,γ .
• If (i, α) = (j, β), i.e. if x and y live in the same surface, and x − y is glued to f k,γ for some other surface S k,γ .
Such a family F of blowups describes a hypermultiplet living in a representation R F of h.
The information about this representation can be described in terms of its weights which are obtained as follows.For every blowup x ∈ F compute its intersections with all surfaces S k,γ with γ ̸ = 0 for non-trivial h k .These intersection numbers describe the Dynkin coefficients for a weight w x of R F .Collect all weights w x for all blowups x ∈ F.
Then R F is recovered as the representation with minimal number of weights that contain the weights w x for all x ∈ F. The full hypermultiplet content carried by the 5d gauge theory associated to T ρ S 1 is recovered by combining the representations R F for all families F. The intersection number (2.9) for a blowup x living in surface S i,α with non-trivial h i , and for all k, is constrained by the property that Finally, the gluings of curves are constrained such that taking combinations of the gluing rules we find that a representative of a multiple of f i is glued to a representative of a multiple of f j if there is an edge between nodes i and j.More precisely, • If there is an undirected edge carrying a label n between i and j, then the gluing rules imply the gluing but not mf i ∼ mf j for any m < n.
• If there is a directed edge from i to j carrying label n, then the gluing rules imply the gluing

T-dualities from local S-dualities
Consider a circle compactification T ρ 1 1,S 1 of an LST T 1 and a circle compactification T ρ 2 2,S 1 of an LST T 2 .We say that T ρ 1  1,S 1 and T ρ 2 2,S 1 are T-dual if we find a surface geometry G that describes both T ρ 1  1,S 1 and T ρ 2 2,S 1 .In this paper, we will only consider a subset of such T-dualities which can be described as follows.Let G 1 be a surface geometry for a genus-one phase of T ρ 1 1,S 1 , which means that G 1 comes with a preferred choice of P 1 fibers for the Hirzebruch surfaces constituting G 1 .
Similarly, let G 2 be a surface geometry for a genus-one phase of T ρ 2 2,S 1 , which means that G 2 comes with a preferred choice of P 1 fibers for the Hirzebruch surfaces constituting G 2 .Now exchange e and f curves of a subset of degree-0 Hirzebruch surfaces in G 1 , which can be thought of as performing local S-dualities at the location of these surfaces [20].Assume G 1 is converted to G 2 after this exchange5 .Then, G 1 and G 2 are isomorphic as surface geometries, that is they describe the same surface geometry once we forget the special choice of P 1 fibers, and we have found that

Examples of T-dualities 3.1 Gauge Rank 1
At gauge rank 1 in 6d, we have three LSTs • su(2) with 16 hypers in fundamental.We represent this theory by the graph 0 su • su(2) with a hyper in adjoint and discrete theta angle 0. We represent this theory by the graph where the subscript 1 is placed to distinguish it from the graph associated to the previous LST, and indicates that the base curve used in F-theory construction of the LST has genus 1 (while the base curve for previous LST has genus 0).This theory actually has N = (1, 1) supersymmetry.
• su(2) with a hyper in adjoint and discrete theta angle π.We represent this theory by the graph This theory also has N = (1, 1) supersymmetry.
Only untwisted circle compactification is possible for all three LSTs.
For the untwisted circle compactification of the first LST, a surface geometry is Each surface can also be interpreted as a surface associated to a node with Ω = 1 and trivial gauge algebra.Thus, we have the T-duality = 1 sp(0) (1)  1 For the untwisted circle compactification of the second LST, a surface geometry is Each surface can also be interpreted as a surface associated to a node with Ω = 2 and trivial gauge algebra.Thus, we have the T-duality For the untwisted circle compactification of the third LST, a surface geometry is The surface S 0 can also be interpreted as a surface associated to a node with Ω = 2 and trivial gauge algebra, while the surface S 1 can also be interpreted as a surface associated to a node with value 2 (but Ω = 1) and a loop along with trivial gauge algebra.Thus, we have the where we have a directed edge with 2 in the middle because the gluing rules imply that twice the elliptic fiber 2e + 2f − 2x − 2y of S 0 is glued to the genus-one fiber 2h + f − 2x − 2y of S 1 .
On the right hand side of the above equality, we have a twisted circle compactification of the D 5 N = (2, 0) LST.The twist is performed using a Z 4 0-form symmetry of the D 5 N = (2, 0) LST whose action on the nodes of the affine so(10) (1) algebra is such that the affine node goes to spinor node, the spinor node goes to vector node, the vector node goes to cospinor node, the cospinor node goes to affine node, and the other two remaining (trivalent) nodes are exchanged with each other.

Gauge Rank 2
At gauge rank 2 in 6d, we have the following LSTs and their possible circle compactifications This theory can have twisted compactification by the Z 2 outer-automorphism but we do not discuss its T-dual.
• su(3) with a hyper in adjoint and rational theta angle 2πp/q.We represent this theory by the graph where the subscript 1 indicates that the base curve used in F-theory construction of the LST has genus 1.This theory actually has N = (1, 1) supersymmetry.
We only discuss T-duals for untwisted compactification of the θ = 0 LST in the general rank subsection.A compactification twisted by Z 2 outer-automorphism is possible only for θ = 0, π, and we discuss both of these cases in this subsection.
• sp(2) with a hyper in antisymmetric and 16 hypers in fundamental.We represent this theory by the graph 0 sp(2) We will discuss two T-duals for the untwisted compactification of this LST.We discuss one in this subsection, and the other in the subsection on general gauge rank.
• g 2 with 10 hypers in irrep of dimension 7. We represent this theory by the graph 0 First of all, consider the surface geometry Exchanging the P 1 fibers e and f in the surfaces S 1 and S 0 , we can rewrite the surface geometry as The surfaces S 0 and S 1 now describe an sp(1) (1) and the surface S 2 can be recognized to be describing sp(0) (1) , leading to the T-duality = 1 sp(0) (1)  1 sp(1) (1) 0 su(3) (1)   (3.17Now exchanging e and f P 1 fibers in surfaces S 0 and S 1 leads to the surface geometry The three surfaces now describe an sp(2) (1) , leading to the T-duality 0 su(3) (1)  = 0 sp(2) (1)   (3.20) where the right hand side is the untwisted compactification of the sp(2) + Λ 2 + 16F LST.
Moving onto twisted compactifications of the su(3) + 18F LST, we have the following surface geometry The surface S 0 can be recognized as a surface for su(1) (1) and the surface S 1 can be recognized as surface for sp(0) (1) .From the gluing of the elliptic fibers, we read the precise edge between the two nodes, leading to the T-duality = 2 su(1) (1)  1 sp(0) (1) For the twisted compactification of the su(3) 0 + A N = (1, 1) LST, we have the following surface geometry We can interpret both S 0 and S 1 as describing su(2) (1) , leading to the T-duality = 2 su(1) (1)  2 The right hand side is a circle compactification of D 4 N = (2, 0) LST twisted by the Z 4 symmetry rotating the affine, spinor, cospinor and vector nodes of so(8) (1) Dynkin diagram into each other.
For the twisted compactification of the su(3) π + A N = (1, 1) LST, we have the following surface geometry In a similar way as above, we are led to the T-duality = 2 su(1) (1)  2 su(1) (1)  2 Finally, consider the surface geometry As written, the three surfaces describe untwisted circle compactification of the LST g 2 + 10F.
Exchanging e and f fibers in S 1 converts it into a surface geometry such that the three surfaces describe compactification of the LST su(4) + 2Λ 2 + 16F twisted by the Z 2 outer automorphism, leading to the T-duality 0 g = 0 su(4) (2)   (3.30)

Gauge Rank 3
Now we discuss some T-dualities that appear at gauge rank 3, but that do not fall into the general cases discussed later.
The surface geometry describes compactification of LST so(12) + 8F + S + C with a Z 2 outer-automorphism twist.
The surface geometry describes compactification of LST so(14) + 10F + S with a Z 2 outer-automorphism twist.
The surface geometry su(2) (1)  1 where su(6) (2) is formed by the surfaces S 3 , S 4 , S 5 and S 6 , and su(2) (1) is formed by the surfaces S 0 and S 1 .The subscript on sp(0) denotes that theta angle is 0, which means that the flavor symmetry attached to the node is u(2).

LSTs with su(6) gauge algebra
In this subsection, we discuss T-dualities of LSTs with su(6) gauge algebra which do not fall into general classes discussed later.
The surface geometry sp(4) (1)  1 where the theta angle for sp(0) is 0, because it allows coupling to an su(2) flavor symmetry which is shown in square brackets.The gluing curves in S 0 to the P 1 fibers of the P 1 -fibered non-compact surfaces generating the su(2) (1) corresponding to this flavor symmetry are h − The surface geometry describes untwisted compactification of LST su( 6) . Exchanging e and f on S 5 leads to the T-duality = 1 sp(4) (1)  1 sp(0) (1) π u(1) (1) 0 su( 6) (1)   (3.70) where the theta angle for sp(0) is π, because it allows coupling to a u(1) flavor symmetry which is shown in square brackets.One can understand the above as first coupling sp(0) π to su(8) ⊕ u(1) and then decomposing su(8) to sp(4).
The surface geometry sp(4) (1)  1 where the theta angle for sp(0) is neither 0 nor π and hence is denoted by the subscript'new'.
It allows coupling to an su(3) flavor symmetry which is shown in square brackets.The gluing curves in S 0 to the P 1 fibers of the P 1 -fibered non-compact surfaces generating the su(3) (1)   corresponding to this flavor symmetry are One can understand the above as first coupling sp(0) to e 6 ⊕ su(3) and then decomposing e 6 to sp(4).
The T-duals for compactifications twisted by Z 2 outer-automorphisms of the above three theories are as follows.The surface geometry describes twisted compactification of LST su(6) + 16F + 2Λ 2 .Exchanging e and f on S 0 , S 1 and S 2 leads to the T-duality where su(2) (1) is formed by the surfaces S 0 and S 1 , and sp(1) (1) is formed by the surfaces S 2 and S 3 .
The surface geometry describes twisted compactification of LST su(6) . Exchanging e and f on S 0 and S 1 leads to the T-duality where su(2) (1) is formed by the surfaces S 0 and S 1 .

Other T-dualities between
we have a double edge because the gluing rules imply that 2e + 2f − 2x − 2y in S 0 is glued to 2e + 2f − 2x − 2y in S 1 , both of which are twice the elliptic fibers e + f − x − y.The right hand side denotes the untwisted circle compactification of the A 1 N = (2, 0) LST.

•
su(3) with 18 hypers in fundamental.We represent this theory by the graph 0 su(3) (3.10)This theory can have twisted compactification by the Z 2 outer-automorphism.• su(3) with a hyper in 2-index symmetric and a hyper in fundamental.We represent this theory by the graph 0 su( 3) (3.11) i e+2fx i (3.15)which describes the untwisted compactification of the su(3) theory with 18 fundamentals.

)
Now consider another surface geometry for the untwisted compactification of the su(3

2 ( 3 . 26 )
The right hand side is a circle compactification of A 2 N = (2, 0) LST twisted by the Z 2 symmetry exchanging two adjacent nodes of the su(3)(1) Dynkin diagram.Now, consider the following surface geometry (m + 1) describes untwisted circle compactification of su(3) + S 2 + F LST. Exchanging e and f P 1 fibers of surfaces S 0 and S 1 leads to the T-duality = 2

2 e
of so(2m) N = (1, 1) LST, we have the surface geometry , f -x e-x, f -y e-x, f -y e-y, f -x e-y, f -x

114 )For Z 2
outer-automorphism twisted compactification of so(2m + 2) N = (1, 1) LST, we have the surface geometry N = (1, 1) and N = (2, 0) In a similar way, the reader can derive remaining T-dualities of compactifications of N = (1, 1) LSTs, for which we quote the results below: • The untwisted compactification of e n N = (1, 1) LST is T-dual to untwisted compactification of E n N = (2, 0) LST, for each 6 ≤ n ≤ 8. • The Z 2 outer-automorphism twisted compactification of e 6 N = (1, 1) LST is T-dual to compactification of E 6 N = (2, 0) LST twisted by a Z 2 symmetry exchanging two of the three single valent nodes.The su(1) (1) involved in the compactification of the N = (2, 0) theory arrange themselves in the form of Dynkin diagram of f (1) 4 which is Langlands dual to the affine algebra e the compactification of the T-dual N = (1, 1) LST.• The untwisted compactification of f 4 N = (1, 1) LST is T-dual to compactification of E 7 N = (2, 0) LST twisted by a Z 2 symmetry exchanging two of the three single valent nodes.The su(1) (1) involved in the compactification of the N = (2, 0) theory arrange themselves in the form of Dynkin diagram of e (2) 6 which is Langlands dual to the affine algebra f the compactification of the T-dual N = (1, 1) LST.
compactification of the T-dual N = (1, 1) LST.• The Z 3 outer-automorphism twisted compactification of so(8) N = (1, 1) LST is T-dual to compactification of D 4 N = (2, 0) LST twisted by a Z 3 symmetry rotating three out of four single valent nodes.The su(1) (1) involved in the compactification of the N = (2, 0) theory arrange themselves in the form of Dynkin diagram of g (1)