Punctures and dynamical systems

Falk Hassler; Jonathan J. Heckman

doi:10.1007/s11005-018-1118-4

Letters in Mathematical Physics

March 2019, Volume 109, Issue 3, pp 449–495 | Cite as

Punctures and dynamical systems

Authors
Authors and affiliations

Falk Hassler
Jonathan J. Heckman

Article

First Online: 03 August 2018

Abstract

With the aim of better understanding the class of 4D theories generated by compactifications of 6D superconformal field theories (SCFTs), we study the structure of $\mathcal {N}=1$ supersymmetric punctures for class $\mathcal {S}_{\varGamma }$ theories, namely the 6D SCFTs obtained from M5-branes probing an ADE singularity. For M5-branes probing a $\mathbb {C}^2 / \mathbb {Z}_{k}$ singularity, the punctures are governed by a dynamical system in which evolution in time corresponds to motion to a neighboring node in an affine A-type quiver. Classification of punctures reduces to determining consistent initial conditions which produce periodic orbits. The study of this system is particularly tractable in the case of a single M5-brane. Even in this “simple” case, the solutions exhibit a remarkable level of complexity: Only specific rational values for the initial momenta lead to periodic orbits and small perturbations in these values lead to vastly different late-time behavior. Another difference from half BPS punctures of class $\mathcal {S}$ theories includes the appearance of a continuous complex “zero mode” modulus in some puncture solutions. The construction of punctures with higher-order poles involves a related set of recursion relations. The resulting structures also generalize to systems with multiple M5-branes as well as probes of D- and E-type orbifold singularities.

Keywords

6D SCFTs Punctures Dynamical systems 4D SCFTs

Mathematics Subject Classification

37N20 17B08

This is a preview of subscription content, to check access.

Notes

Acknowledgements

We thank F. Apruzzi, J. Marincel, N. Miller and T. Rudelius for helpful discussions. JJH thanks the 2017 Summer Workshop at the Simons Center for Geometry and Physics as well as the Aspen Center for Physics Winter Conference in 2017 on Superconformal Field Theories in $d\ge 4$, NSF Grant PHY-1066293, for hospitality during part of this work. The work of FH and JJH is supported by NSF CAREER Grant PHY-1756996. The work of FH is also supported by NSF Grant PHY-1620311.

A Periodic orbit proofs

With notation as in Sect. 3.4, in this Appendix we establish that for an eternal puncture with initial momentum $p(1) = a/b$ with a, b coprime, when $b \ne 0$ mod 4, we always obtain a periodic orbit.

We first establish some important properties of the leading-order contribution to the dynamical system, as given in Eq. 3.56.

Lemma 1

Let $p=a/b$ where $a>0$ and $b>0$ are coprime. If $b\,\text {mod} \, 4 \ne 0$, there exists a periodic orbit for the leading contribution $\varDelta ^{(m)}_\mathrm {L}$ which is independent of the initial value $\varDelta ^{(1)}_\mathrm {L}$. The number of extrema in this orbit is given by

$$\begin{aligned} k' = {\left\{ \begin{array}{ll} b &{} b \text { is even} \\ 2 b &{} b \text { is odd} \end{array}\right. } \end{aligned}$$

(A.1)

for which we have

$$\begin{aligned} \varDelta ^{(m+k')}_\mathrm {L} = \varDelta ^{(m)}_\mathrm {L} . \end{aligned}$$

(A.2)

Proof of Lemma 1

In order to proof this lemma, we have to show that

$$\begin{aligned} 2 k' p + \sum \limits _{l=0}^{k'-1} (-1)^l \, \text {Ceil}(\varDelta ^{(m)}_\mathrm {L} + 2 l p) = 0 \end{aligned}$$

(A.3)

holds under the assumptions stated in the lemma. If this is the case, (A.2) follows directly from (3.56). Most of the terms in the sum cancel due to the alternating sign. If we only keep the non-vanishing contributions, the equation we have to check can be rewritten as

$$\begin{aligned} 2 k' p + \sum \limits _{l=0}^{k'-1} (-1)^l \, \text {Ceil}(\varDelta ^{(m)}_\mathrm {L} + 2 l p) = \sum \limits _{l=0}^{2 k' p - 1} \text {sgn}_- \left( 2 p - ( l \, \mathrm {mod}\, 4 p ) \right) = 0 \end{aligned}$$

(A.4)

with

$$\begin{aligned} \text {sgn}_- (x) = {\left\{ \begin{array}{ll} \phantom {-} 1 &{} x>0 \\ -1 &{} x \le 0 . \end{array}\right. } \end{aligned}$$

(A.5)

The argument of the $\text {sgn}_-$ function has the point symmetry

$$\begin{aligned} 2 p - ( ( l_* + l ) \, \mathrm {mod} \, 4 p ) = - 2 p + ( ( l_* - l ) \, \mathrm {mod} \, 4 p ) \quad \forall \, l \in \mathbb {N} , \, 0< l < 2 k' p - l_* \end{aligned}$$

(A.6)

around the point

$$\begin{aligned} l_* = {\left\{ \begin{array}{ll} a &{} \text {for } b \, \mathrm {mod} \, 2 = 0 \\ 2 a &{} \text {for } b \, \mathrm {mod} \, 2 \ne 0 . \end{array}\right. } \end{aligned}$$

(A.7)

Hence, there are even more terms in the sum (A.4) which cancel, too. Finally, one is left with

$$\begin{aligned} \sum \limits _{l=0}^{2 k' p - 1} \text {sgn}_- \left( 2 p - ( l \, \mathrm {mod}\, 4 p ) \right)&= \sum \limits _{l\in \{0,\,l_*\}} \text {sgn}_- \left( 2 p - ( l \, \mathrm {mod}\, 4 p ) \right) \nonumber \\&= \text {sgn}_- ( 2 p ) - \text {sgn}_- (0) = 1 - 1 = 0 . \end{aligned}$$

(A.8)

Lemma 2

Let $p=a/b$ where $a>0$ and $b>0$ are coprime. If $b\,\text {mod}\, 4 = 0$, $\varDelta ^{(m)}_\mathrm {L}$ is shifted by 1 after b / 2 time steps

$$\begin{aligned} \varDelta ^{(m+b/2)}_\mathrm {L} = \varDelta ^{(m)}_\mathrm {L} + 1 . \end{aligned}$$

(A.9)

Proof of Lemma 2

The proof for this lemma requires only a very slight modification of the previous proof. Instead of (A.3), we now want to show that

$$\begin{aligned} b p + \sum \limits _{l=0}^{b/2-1} (-1)^l \, \text {Ceil}(\varDelta ^{(m)}_\mathrm {L} + 2 l p) = 1 \end{aligned}$$

(A.10)

holds. Again, we rewrite this sum as

$$\begin{aligned} b p + \sum \limits _{l=0}^{b/2-1} (-1)^l \, \text {Ceil}(\varDelta ^{(m)}_\mathrm {L} + 2 l p) = \sum \limits _{l=0}^{b p - 1} \text {sgn}_- \left( 2 p - ( l \, \mathrm {mod}\, 4 p ) \right) = 1 \end{aligned}$$

(A.11)

and take advantage of the point symmetry of the argument of the $\text {sgn}_-$ function. Now, $l_*=a/2$ and therewith not an integer. Thus, instead of two contributions to the same which cancel each other, we find

$$\begin{aligned} b p + \sum \limits _{l=0}^{b/2-1} (-1)^l \, \text {Ceil}(\varDelta ^{(m)}_\mathrm {L} + 2 l p) = \text {sgn}_- (2p) = 1 . \end{aligned}$$

(A.12)

Now consider a periodic orbit $\{\varDelta ^{(1)}_\mathrm {L}, \dots , \varDelta ^{(k')}_\mathrm {L}\}$ which arises from Lemma 1 and take instead of $\varDelta ^{(1)}_\mathrm {L}$, $\varDelta ^{(2)}_\mathrm {L}$ as the initial condition. In this case, we obtain another periodic orbit $\{\varDelta ^{(2)}_\mathrm {L}, \ldots , \varDelta ^{(k')}_\mathrm {L}, \varDelta ^{(1)}_\mathrm {L}\}$. The only difference between the two of them is that all elements are shifted to the right by one position. Periodic orbits which arise from such shifts are members of the same equivalence class. All their relevant properties do not change within the class. Hence, it is sufficient to study only one representative of each equivalence class.

Lemma 3

A unique representative of each equivalence class for periodic orbits $\varDelta ^{(m)}_\mathrm {L}$ which arise from Lemma 1 is given by the initial conditions

$$\begin{aligned} 0< |\varDelta ^{(1)}_\mathrm {L}| < \frac{1}{k'} \end{aligned}$$

(A.13)

for $p=a/b$, $a>0$ and $b>0$. For all other remaining $\varDelta ^{(m)}_\mathrm {L}$ in this orbit, we instead have:

$$\begin{aligned} \left| \left( \varDelta ^{(m)}_\mathrm {L} + \frac{1}{2}\right) \,\text {mod}\, 1 - \frac{1}{2} \right| > \frac{1}{k'} \quad \forall \, m \in \{2, \ldots ,k'/2,k'/2+2, \ldots , k'\} \end{aligned}$$

(A.14)

and

$$\begin{aligned} \left| \left( \varDelta ^{(k'/2+1)}_\mathrm {L} + \frac{1}{2}\right) \,\text {mod}\, 1 - \frac{1}{2} \right| < \frac{1}{k'} . \end{aligned}$$

(A.15)

Proof of Lemma 3

First, we prove that from the representatives (A.13) all other orbits of the same equivalence class can be obtained by shifting the time evolution. More specifically, we shift by n steps to the left

$$\begin{aligned} \varDelta '^{(m)}_\mathrm {L} = \varDelta ^{(m+2n)}_\mathrm {L} - 2 \, \text {Floor}\left( \frac{\varDelta ^{(1+2 n)}_\mathrm {L} + 1}{2} \right) \end{aligned}$$

(A.16)

and perform an additional integer shift to keep $\varDelta '^{(1)}$ in the fixed interval $(-1,1)$. According to (3.54), this interval covers all possible initial conditions. A shift by an integer leaves the iteration (3.53) for the leading contributions $\varDelta ^{(m)}_\mathrm {L}$ invariant and therefore does not change the equivalence class of the orbit. After this transformation, we obtain

$$\begin{aligned} \varDelta '^{(1)}_\mathrm {L} = ( \varDelta ^{(1)}_\mathrm {L} + 2 n p + 1) \, \text {mod} \, 2 - 1 . \end{aligned}$$

(A.17)

as new initial condition. Sweeping n from 1 to $k'$, we find that all initial conditions fill the lattice

$$\begin{aligned} \varDelta '^{(1)}_\mathrm {L} \in \left\{ -1 + 1/k' + \varDelta ^{(1)}_\mathrm {L}, \, -1/2 + 3/k' + \varDelta ^{(1)}_\mathrm {L}, \dots , 1 - 1/k' + \varDelta ^{(1)}_\mathrm {L} \right\} . \end{aligned}$$

(A.18)

Varying also $\varDelta ^{(1)}_\mathrm {L}$ in the bounds given by (A.13), we eventually see that

$$\begin{aligned} \forall \, \varDelta '^{(1)}_\mathrm {L} \in ( -1 , 1) \quad \exists \, n \in \{1, \ldots , k'\} \text { and } \varDelta ^{(1)}_\mathrm {L} \in \left( -\frac{1}{k'}, \frac{1}{k'}\right) \text { s.t. (A.17) holds.} \end{aligned}$$

(A.19)

This proves the first part of the lemma.

In order to prove the second part, we first note that

$$\begin{aligned} \left( \varDelta ^{(m)}_\mathrm {L} + \frac{1}{2}\right) \,\text {mod}\, 1 = \left( \varDelta ^{(1)}_\mathrm {L} + 2 (m - 1) p + \frac{1}{2}\right) \,\text {mod}\, 1 \quad \forall \, m \in \mathbb {Z}, \end{aligned}$$

(A.20)

which follows from (3.53) after stripping off all integer contributions. Because

$$\begin{aligned} k' p \in \mathbb {N} , \end{aligned}$$

(A.21)

we immediately find that

$$\begin{aligned} \left( \varDelta ^{(m + k'/2)}_\mathrm {L} + \frac{1}{2}\right) \,\text {mod}\, 1 = \left( \varDelta ^{(m)}_\mathrm {L} + \frac{1}{2}\right) \,\text {mod}\, 1 \end{aligned}$$

(A.22)

holds. Under the assumption (A.13), this relation directly implies (A.15). More generally,

$$\begin{aligned} \left( \varDelta ^{(1)}_\mathrm {L} + 2 (m - 1) p + \frac{1}{2}\right) \,\text {mod}\, 1 - \frac{1}{2} \in \varLambda _{k'}(\varDelta ^{(1)}_\mathrm {L}) \end{aligned}$$

(A.23)

only takes values on the lattice

$$\begin{aligned} \varLambda _{k'}(\varDelta ) = \{ -1/2 + 1/k' + \varDelta ,\, -1/2 + 3/k' + \varDelta , \ldots , 1/2 - 1/k' + \varDelta \} , \end{aligned}$$

(A.24)

with $|\varDelta |<1/k'$. This lattice has exactly $k'/2$ elements, and for $m \in \{1, \ldots ,k'/2 \}$, every one occurs exactly one times. There is only one element in this lattice whose absolute value is small that $1/k'$. This element is $\varDelta $. It arises for $m=0$ from (A.23). All other elements have an absolute value which is larger that $1/k'$. This proves (A.14) for $m\in \{2, \ldots , k'/2\}$. The remaining conditions follow immediately from (A.22). $\square $

We call this canonical representative an aligned periodic orbit:

Definition 1

A periodic orbit ($p=a/b$, a, b coprime and $b\, \text {mod} \, 4 \ne 0$) with

$$\begin{aligned} 0< |\varDelta ^{(1)}_\mathrm {L}| < \frac{1}{k'} \end{aligned}$$

(A.25)

is called aligned.

A direct consequence of Lemma 3 is that every orbit can be aligned by a suitable shift.

Lemma 4

Let $\varDelta ^{(m)}_\mathrm {L}$ capture a periodic orbit which arises from Lemma 1. All $\varDelta ^{(m)}_\mathrm {L}$ are bounded by

$$\begin{aligned} | \varDelta ^{(m)}_\mathrm {L} | < k' p + 2 = {\left\{ \begin{array}{ll} a + 1 &{} \text {for } b \text { even} \\ 2 a + 1 &{} \text {for } b \text { odd} . \end{array}\right. } \end{aligned}$$

(A.26)

Proof of Lemma 4

First, we deduce for (3.53) that

$$\begin{aligned} | \varDelta ^{(m+1)} | < |\varDelta ^{(m)} | + 1 \end{aligned}$$

(A.27)

holds. In order to prove the lemma, we now show that

$$\begin{aligned} | \varDelta ^{(1+2n)} | < k' p + 1 \quad \forall \, n \in \mathbb {Z} . \end{aligned}$$

(A.28)

As in the proofs of the Lemmas 1 and 2, we write the expression for $\varDelta ^{(1+2n)}_\mathrm {L}$ in (3.56) as

$$\begin{aligned} \varDelta ^{(1+2 n)}_\mathrm {L} =( \varDelta ^{(1)}_\mathrm {L} + 4 n p ) \, \text {mod} \, 1 + \sum \limits _{l=0}^{\text {Floor}( 4 n p ) - 1} \text {sgn}_- \, ( 2 p - ( l \, \text {mod} \, 4p ) ) . \end{aligned}$$

(A.29)

For $n=k'/2$, the sum on the right-hand side vanishes according to Lemma 1. Moreover, the $\text {sgn}_-$ function can only take values of 1 or $-1$. Hence, the sum is bounded by

$$\begin{aligned} \left| \sum \limits _{l=0}^{\text {Floor}( 4 n p ) - 1} \text {sgn}_- \, ( 2 p - ( l \, \text {mod} \, 4p ) ) \right| < k' p \quad \forall n \in \mathbb {Z} . \end{aligned}$$

(A.30)

Taking into account the first part of (A.29) too, we see that (A.28) holds. This completes the proof. $\square $

Let us now start to analyze the effect of the corrections $\delta ^{(m)}$ in (3.51) for the case where the leading contributions $\varDelta ^{(m)}_\mathrm {L}$ describe an aligned, periodic orbit defined in definition 1. In this discussion, the functions $\delta _1(\varDelta , y)$, $\delta _2(\varDelta , y)$ and $\tilde{\delta }_2(\varDelta , y)$, which we defined in (3.45), (3.47) and (3.50), play an important role. Under the assumption that $\varDelta $ is bound by

$$\begin{aligned} |\varDelta | < \varDelta _\mathrm {B} , \end{aligned}$$

(A.31)

we can bound them from above by

$$\begin{aligned}&|\delta _1(\varDelta , y)|< \frac{(2 \varDelta _\mathrm {B} + 2)^2}{2 (l - 1)} , \quad |\delta _2(\varDelta , y)|< \frac{(2 \varDelta _\mathrm {B} + 2)^2}{2 (l - 1)}\nonumber \\&\quad \text {and} \quad |\tilde{\delta }_2(\varDelta , y)| < \frac{(2 \varDelta _\mathrm {B} + 2)^2}{2 (l - 1)} . \end{aligned}$$

(A.32)

We want to guarantee that the subleading part $|\delta ^{(m)}|$ is small and does not change the qualitative behavior of the time evolution. If this is the case, the time evolution is governed by the leading contributions $\varDelta ^{(m)}_\mathrm {L}$ for which we already proved an analytic expression in (3.56). More specifically, $|\delta ^{(m)}|$ has to be smaller than a certain boundary $\delta _\mathrm {B}<1$, at least for the first $k'/2$ steps. Thus, we require

$$\begin{aligned} |\delta ^{(m)}|< \delta _\mathrm {B} \quad \text {and} \quad |\varDelta ^{(m)}| < \varDelta _\mathrm {B} \quad \forall \, m \in \{ 1, \dots , k'/2 + 1\} \end{aligned}$$

(A.33)

and fix $\delta _\mathrm {B}$ such that

$$\begin{aligned} \text {Ceil}( \varDelta ^{(m)}_\mathrm {L} ) = \text {Ceil}\big ( \varDelta ^{(m)} + \delta _1( \varDelta ^{(m)}, y^{(m)} ) \big ) \quad \forall \, m \in \{ 1, \ldots , k'/2 \} \end{aligned}$$

(A.34)

is not violated. If it would be violated, then $|\delta ^{(m)}|$ would be immediately bigger than one and our analysis would break down. For $m=1$, we have

$$\begin{aligned} \text {Ceil}( \varDelta ^{(1)}_\mathrm {L} ) = \text {Ceil}\big ( \varDelta ^{(1)}_\mathrm {L} - \delta _1( \varDelta ^{(1)}, y^{(1)} ) + \delta _1( \varDelta ^{(1)}, y^{(1)} ) \big ) = \text {Ceil}( \varDelta ^{(1)}_\mathrm {L} ) \end{aligned}$$

(A.35)

automatically. For $m=2$, we obtain the constraint

$$\begin{aligned} \text {Ceil}( \varDelta ^{(2)}_\mathrm {L} ) = \text {Ceil}\big ( \varDelta ^{(2)}_\mathrm {L} + \delta ^{(2)} + \delta _1( \varDelta ^{(2)}, y^{(2)} ) \big ) = \text {Ceil}( \varDelta ^{(2)}_\mathrm {L} ), \end{aligned}$$

(A.36)

where

$$\begin{aligned} |\delta ^{(2)}| \le | \delta ^{(1)} + \delta _2(\varDelta ^{(1)}, y^{(1)}) | < 2 \frac{(2\varDelta _\mathrm {B} + 2)^2}{2(l-1)} \end{aligned}$$

(A.37)

after taking into account the bounds (A.32). This constraint only holds, if we restrict the domain of allowed values for $\varDelta ^{(2)}_\mathrm {L}$ to

$$\begin{aligned} \left| \left( \varDelta ^{(2)}_\mathrm {L} + \frac{1}{2}\right) \,\text {mod}\, 1 - \frac{1}{2} \right| \ge 3 \frac{(2 \varDelta _\mathrm {B} + 2)^2}{2(l-1)} . \end{aligned}$$

(A.38)

Repeating this reasoning step for step, we eventually find

$$\begin{aligned} \bigwedge \limits _{m=2}^{k'/2} \left| \left( \varDelta ^{(m)}_\mathrm {L} + \frac{1}{2}\right) \,\text {mod}\, 1 - \frac{1}{2} \right| \ge (m+1) \frac{(2 \varDelta _\mathrm {B} + 2)^2}{2(l-1)} , \end{aligned}$$

(A.39)

which implies

$$\begin{aligned} | \delta ^{(k'/2+1)} | < k' \frac{(\varDelta _\mathrm {B} + 1)^2}{l-1} = \delta _\mathrm {B} . \end{aligned}$$

(A.40)

For the following discussion, it is sufficient to use the weaker, but simpler version

$$\begin{aligned} \left| \left( \varDelta ^{(m)}_\mathrm {L} + \frac{1}{2}\right) \,\text {mod}\, 1 - \frac{1}{2} \right| \ge (k' + 2) \frac{( \varDelta _\mathrm {B} + 1)^2}{l-1} = \delta _\mathrm {B} \quad \forall \, m \in \{ 2, \ldots , k'/2 \} . \end{aligned}$$

(A.41)

If it holds, the subleading part $\delta ^{(m)}$ is given by

$$\begin{aligned} \delta ^{(m)} = \delta ^{(1)} + \sum \limits _{n=1}^{m-1} \delta _2\big ( \varDelta _\mathrm {L}^{(n)}, y^{(n)} \big ) \quad \text {for} \quad \forall \, m \in \{ 2, \ldots , k'/2 + 1\} . \end{aligned}$$

(A.42)

Taking into account Lemma 3, we know that:

$$\begin{aligned} \left| \left( \varDelta ^{(m)}_\mathrm {L} - \frac{1}{2}\right) \,\text {mod}\, 1 + \frac{1}{2} \right| > \frac{1}{k'}. \end{aligned}$$

(A.43)

On the other hand, the constraint (A.41) also tells us that:

$$\begin{aligned} \delta _\mathrm {B} \le \left| \left( \varDelta ^{(m)}_\mathrm {L} - \frac{1}{2}\right) \,\text {mod}\, 1 + \frac{1}{2} \right| . \end{aligned}$$

(A.44)

Scanning over the admissible values of $\delta _{B}$, we see that the largest $\delta _{B}$ which will satisfy line A.43 satisfies the inequality:

$$\begin{aligned} \delta _\mathrm {B} \le \frac{1}{k'} . \end{aligned}$$

(A.45)

If we furthermore combine Lemma 4 with $|\delta ^{(m)}|< \delta _\mathrm {B} < 1$ for $m\in \{1, \ldots ,k'/2\}$, we see that there are two additional inequalities we can write:

$$\begin{aligned} |\varDelta ^{(m)}|&< \varDelta _\mathrm {B} \end{aligned}$$

(A.46)

$$\begin{aligned} |\varDelta ^{(m)}|&\le |\varDelta ^{(m)}_\mathrm {L}| + |\delta ^{(m)}| < k' p + 3 \quad \forall \, m \in \{ 2, \ldots , k'/2 \} \end{aligned}$$

(A.47)

hold. Again, we seek out the largest value of $\varDelta _{B}$ compatible with these two conditions. This is fixed by taking:

$$\begin{aligned} \varDelta _\mathrm {B} = k' p + 3 . \end{aligned}$$

(A.48)

Hence, we can express both boundaries $\delta _\mathrm {B}$ and $\varDelta _\mathrm {B}$ in (A.33) in terms of $k'$ and p.

So far we just discussed the subleading part $\delta ^{(m)}$ for the first half of the periodic orbit. To go beyond $\varDelta ^{(k'/2 + 1)}$ is more involved, because according to Lemma 3

$$\begin{aligned} \left| \left( \varDelta ^{(k'/2+1)}_\mathrm {L} + \frac{1}{2}\right) \,\text {mod}\, 1 - \frac{1}{2} \right| < \frac{1}{k'} \end{aligned}$$

(A.49)

holds and enables us to apply the bounds discussed above. There are now two ways one could go. First one could improve the analysis of the contributions from $\delta _1(\varDelta ,y)$ and $\delta _2(\varDelta ,y)$. We only used very crude bounds. They worked well for the analysis of the first $k'/2$ corrections $\delta ^{(m)}$ but are insufficient as soon as one want to go beyond the special point $k'/2 + 1$. Alternatively, we can avoid this point completely by approaching the second part of the orbit with the inverse iteration prescription (3.49). In analogy with (A.33), one now requires

$$\begin{aligned} | \delta ^{(m)} |< \delta _B \quad \text {and} \quad | \varDelta ^{(m)} | < \varDelta _B \quad \forall \, m \in \{- k'/2 + 1 , \ldots , 0 \} . \end{aligned}$$

(A.50)

As before, we again require

$$\begin{aligned} \text {Ceil}( \varDelta ^{(m)}_\mathrm {L} ) = \text {Ceil}\big ( \varDelta ^{(m)} + \delta _1( \varDelta ^{(m)}, y^{(m)} ) \big ) \quad \forall \, m \in \{ - k'/2 + 1, \ldots , 0 \} \end{aligned}$$

(A.51)

in order to keep $\delta ^{(m)}$ smaller than one. Following the same reasoning as above, we find that this constraint is fulfilled when:

$$\begin{aligned} \delta _\mathrm {B} = (k' + 2) \frac{(\varDelta _\mathrm {B} + 1)^2}{l - 1} \le \frac{1}{k'} \end{aligned}$$

(A.52)

with the $\varDelta _\mathrm {B}$ in (A.48). Alternatively, we can also state this inequation for l, resulting in

$$\begin{aligned} 1 + k' ( k' + 2 ) ( k' p + 4 )^2 \le l . \end{aligned}$$

(A.53)

If it is satisfied, the corrections for the second part of the orbits are given by

$$\begin{aligned} \delta ^{(m)} = \delta ^{(1)} + \sum \limits _{n=m+1}^{1} \tilde{\delta }_2\big ( \varDelta _\mathrm {L}^{(n)}, y^{(n)} \big ) \quad \text {for} \quad \forall \, m \in \{ - k'/2 + 1, \ldots , 0\} . \end{aligned}$$

(A.54)

We call the corresponding periodic orbits dominant orbits.

Definition 2

Orbits with $p=a/b$ where a, b are coprime and $b \, \text {mod}\, 4 \ne 0$ are called dominant if they fulfill

$$\begin{aligned} l \ge 9 + 2 k' p + k' ( k' + 2 )( k' p + 4 )^2 = {\left\{ \begin{array}{ll} 9 + 2 a + b ( b + 2 ) ( a + 4 )^2 &{} \text {for } b \text { even} \\ 9 + 4 a + 4 b ( b + 1 ) ( 2 a + 4 )^2 &{} \text {for } b \text { odd} . \end{array}\right. } \end{aligned}$$

(A.55)

Note that this definition also takes orbits into account which are not aligned. To this end, we require that not only $l^{(1)}=l$ satisfies the bound (A.53) but all $l^{(m)}$. For dominant orbits, we can now establish

Lemma 5

For dominant orbits, the subleading part is given by

$$\begin{aligned} \delta ^{(m)} = \delta ^{(1)} + \sum \limits _{n=1}^{m-1} \delta _2\big ( \varDelta _\mathrm {L}^{(n)}, y^{(n)} \big ) \quad \text {for} \quad \forall \, m \in \{ 2, \ldots , k'/2 + 1\} \end{aligned}$$

(A.56)

and

$$\begin{aligned} \delta ^{(m)} = \delta ^{(1)} + \sum \limits _{n=m+1}^{1} \tilde{\delta }_2\big ( \varDelta _\mathrm {L}^{(n)}, y^{(n)} \big ) \quad \text {for} \quad \forall \, m \in \{ - k'/2 + 1, \ldots , 0\} . \end{aligned}$$

(A.57)

After all this preparation, we can finally state the important theorems for dominant periodic orbits.

Theorem 1

Dominant orbits are periodic and contain $k'$ (given by (A.1)) extrema.

Proof

In order to prove that we indeed have a periodic orbit, we have to show that

$$\begin{aligned} \varDelta ^{(-k'/2 + 1)} = \varDelta ^{(k'/2 + 1)} \end{aligned}$$

(A.58)

holds. This relation is equivalent to

$$\begin{aligned} \varDelta ^{(-k'/2 + 1)}_\mathrm {L} + \delta ^{(-k'/2 + 1)} = \varDelta ^{(k'/2 + 1)}_\mathrm {L} + \delta ^{(k'/2 + 1)} . \end{aligned}$$

(A.59)

For Lemma 1, we know that $\varDelta ^{(-k'/2 + 1)}_\mathrm {L} = \varDelta ^{(k'/2 + 1)}_\mathrm {L}$ holds. Therefore, we only have to show $\delta ^{(-k'/2 + 1)} = \delta ^{(k'/2 + 1)}$ which is equivalent to

$$\begin{aligned} \sum \limits _{n = -k'/2 + 2}^1 \tilde{\delta }_2 (\varDelta _\mathrm {L}^{(n)}, y^{(n)}) = \sum \limits _{n = 1}^{k'/2} \delta _2 (\varDelta _\mathrm {L}^{(n)}, y^{(n)}) . \end{aligned}$$

(A.60)

First, we note that Lemma 1 implies

$$\begin{aligned} \sum \limits _{n = -k'/2 + 2}^1 \tilde{\delta }_2 (\varDelta _\mathrm {L}^{(n)}, y^{(n)}) = \sum \limits _{n = k'/2 + 2}^{k'+1} \tilde{\delta }_2 (\varDelta _\mathrm {L}^{(n)}, y^{(n)}) \end{aligned}$$

(A.61)

and furthermore

$$\begin{aligned} \tilde{\delta }_2 (\varDelta _\mathrm {L}^{(n)}, y^{(n)}) = - \delta _2 (\varDelta _\mathrm {L}^{(n-1)}, y^{(n-1)}) \end{aligned}$$

(A.62)

follows form the definitions (3.47), (3.50) and the iteration prescription (3.53) for the leading contribution. Therefore (A.60) is equivalent to

$$\begin{aligned} 0&=\sum \limits _{n=1}^{k'} \delta _2(\varDelta _\mathrm {L}^{(n)}, y^{(n)})\nonumber \\&= \frac{2}{l-1} \sum \limits _{n=0}^{k'/2 - 1} \left[ 3 \text {Ceil}(\varDelta _\mathrm {L}^{(2n+1)}) - \text {Ceil}(\varDelta _\mathrm {L}^{(2n+1)}+2p) - 1 \right] \nonumber \\&\quad \times \left[ 2 p + \text {Ceil}(\varDelta _\mathrm {L}^{(2n+1)}) - \text {Ceil}(\varDelta _\mathrm {L}^{(2n+1)}+2p) \right] . \end{aligned}$$

(A.63)

After introducing

$$\begin{aligned} s^{(n)} = \text {Ceil}(\varDelta _\mathrm {L}^{(1)} + 4 p n) - \text {Ceil}(\varDelta _\mathrm {L}^{(1)} + 4 p n + 2 p ) , \end{aligned}$$

(A.64)

this equation is equivalent to

$$\begin{aligned} 0 = \sum \limits _{n=0}^{k'/2 - 1} 2 \text {Ceil}(\varDelta _\mathrm {L}^{(2n+1)}) (2 p + s^{(n)}) + \sum \limits _{n=0}^{k'/2 - 1} (2 p + s^{(n)})(s^{(n)} - 1) . \end{aligned}$$

(A.65)

Writing (3.56) as

$$\begin{aligned} \varDelta _\mathrm {L}^{(1+2n)} = \varDelta _\mathrm {L}^{(1)} + 2 \sum \limits _{l=0}^{n-1} s^{(l)} , \end{aligned}$$

(A.66)

Lemma 1 implies

$$\begin{aligned} \sum \limits _{l=0}^{k'/2+1} s^{(l)} = -\, k' p . \end{aligned}$$

(A.67)

Moreover, we take into account that $(s^{(l)})^2 = - s^{(l)}$. Thus, (A.60) is equivalent to

$$\begin{aligned} \sum \limits _{n=0}^{k'/2-1} \text {Ceil}(\varDelta _\mathrm {L}^{(2n+1)})(2 p + s^{(n)})&= \sum \limits _{n=0}^{k'/2-1} \left( \text {Ceil}(\varDelta _\mathrm {L}^{(1)} + 4 p n) + 2 \sum \limits _l^{n-1} s^{(l)} \right) \left( 2 p + s^{(n)} \right) \nonumber \\&= \frac{k' p}{2} ( 2 p - 1) . \end{aligned}$$

(A.68)

All what remains is to calculate the four remaining sums on the right-hand side in the first line. Let us start with

$$\begin{aligned} 2 p \sum \limits _{n=0}^{k'/2-1} \text {Ceil}(\varDelta _\mathrm {L}^{(1)}+4pn)= & {} \frac{k'p}{2} (2 k'p + 1) -2 k' p^2 + p \, \text {sgn }\varDelta _\mathrm {L}^{(1)} \quad \text {for} \quad 0< |\varDelta _\mathrm {L}^{(1)}| < \frac{1}{k'}.\nonumber \\ \end{aligned}$$

(A.69)

In order to better understand this result, assume for a moment that $\varDelta _\mathrm {L}^{(1)}=0$. Now

$$\begin{aligned} \text {Ceil}(4 n p) + \text {Ceil}\left( 4 ( k'/2 - n) p \right) = 2 k' p + 1 \end{aligned}$$

(A.70)

holds. We find this contribution exactly $k'/4-1/2$ times in the sum in (A.69). If $|\varDelta _\mathrm {L}^{(1)}|<1/k'$ instead of being zero, it only affects the summand for $n=0$ and therefore gives rise to the $\text {sgn}$ function in (A.69). Second, we evaluate the sum

$$\begin{aligned} 2 \sum \limits _{n=0}^{k'/2-1} \sum \limits _{l=0}^{n-1} s^{(n)} s^{(l)} = 2 \sum \limits _{n=1}^{k' p} (n-1) = k'p(k'p - 1) . \end{aligned}$$

(A.71)

Thus, we finally need to show that

$$\begin{aligned} \sum \limits _{n=0}^{k'/2-1} \left( \text {Ceil}(\varDelta _\mathrm {L}^{(1)}+4pn) s^{(n)} + 4 p \sum \limits _{l=0}^{n-1} s^{(l)}\right) = - k' p ( 2 k' p - 3 p) - p \text {sgn }(\varDelta _\mathrm {L}^{(1)}) \end{aligned}$$

(A.72)

holds. To do, let us consider the following problem: Choose $i=1, \dots , k'p$ integers $0 \le m_i \le 2 k' p$ such that they fulfill

$$\begin{aligned} \exists \, n_i \in \mathbb {N} \text { s.t. } \varDelta _\mathrm {L}^{(1)} + 4 p n_i< m_i < \varDelta _\mathrm {L}^{(1)} + 4 p n_i + 2 p . \end{aligned}$$

(A.73)

In terms of these integers, we can write (A.72) as

$$\begin{aligned} \sum \limits _{i=1}^{k'p} \left( 4 p\,\text {Ceil} \left( \frac{m_i - \varDelta _\mathrm {L}^{(1)}}{4p}\right) - m_i - 2 k' p \right) = - k' p ( 2 k' p - 3 p) - p \, \text {sgn }(\varDelta _\mathrm {L}^{(1)}) \end{aligned}$$

(A.74)

which simplifies to

$$\begin{aligned} 4 \sum \limits _{i=1}^{k'p} \left( \frac{m_i - \varDelta _\mathrm {L}^{(1)}}{4 p} \text { mod } 1 \right) = k'p + \text {sgn } \varDelta _\mathrm {L}^{(1)} - \varDelta _\mathrm {L}^{(1)} k' . \end{aligned}$$

(A.75)

To see that this relations is indeed fulfilled, we again check first the case for $\varDelta _\mathrm {L}^{(1)} = 0$. In this case, it is easy to check that if an integer m satisfies (A.73), there is another integer

$$\begin{aligned} m' = ( k' p - m ) \text { mod } 2 k' p \end{aligned}$$

(A.76)

which does so, too. Let us add up this two contributions to the sum on the left-hand side of (A.75):

$$\begin{aligned} \frac{m}{4 p} \text { mod } 1 + \left( \frac{k'}{4} - \frac{m}{4 p} \right) \text { mod } 1 = \frac{1}{2} . \end{aligned}$$

(A.77)

Here, we have used that

$$\begin{aligned} \frac{k'}{4} \text { mod } 1 = \frac{1}{2} \quad \text {and} \quad 0 \le \frac{m}{4p} \text { mod } < \frac{1}{2} , \end{aligned}$$

(A.78)

which follows directly from (A.73). This situation occurs $k' p/2$ times, and we reproduces (A.75) for $\varDelta _\mathrm {L}^{(1)}=0$. For $|\varDelta _\mathrm {L}^{(1)}|<\frac{1}{k'}$, the deviations from this argumentation are minor. It is straightforward to check that they reproduce exactly the remaining terms on the right-hand side of (A.75). $\square $

Corollary 1

All orbits with $p = a/b$ with a, b coprime and $b \, \text {mod} \, 4 \ne 0$ are periodic.

Proof

We prove this statement by contradiction. Assume we start from a $y^{(1)}$ which does not give rise to a periodic orbit. Of course this $y^{(1)}$ cannot be the initial condition of a dominant orbit because these are periodic. Thus,

$$\begin{aligned} |y^{(1)}| < y_\mathrm {max} \end{aligned}$$

(A.79)

is bounded from above (with increasing $|y^{(1)}|$ also l increases gradually and at some point would fulfill the requirement in definition 2 for a dominant orbit). Moreover, we note that $y^{(m)}$ can only change in multiples of 1 / b according to (3.35). As the sequence $y^{(m)}$ does not describes a periodic orbit, all $y^{(m)}$ are unique and cannot repeat. These two observations give rise to

$$\begin{aligned} \exists \, y \in \{ y^{(2)}, \ldots , y^{(N)} \} \quad \text {with} \quad | y - y^{(1)} | \ge \frac{N}{2 b}, \end{aligned}$$

(A.80)

which implies

$$\begin{aligned} | y | \ge \frac{N}{2 b} - y_\mathrm {max} . \end{aligned}$$

(A.81)

For $N=\text {Ceil}( 4 b y_\mathrm {max})$, there exists an y in the first N elements of the time evolution which is bigger than $y_\mathrm {max}$. This is a contradiction. Such a y implies a periodic, dominant orbit, but we started with the assumption that the orbit is not periodic. $\square $

Theorem 2

Let $x^{(i)}$ describe a dominant, aligned, periodic orbit. Its length, which is defined as

$$\begin{aligned} x(i) = x(i+k) , \end{aligned}$$

(A.82)

is

$$\begin{aligned} k = k' ( l + 2 p ) = {\left\{ \begin{array}{ll} l b + 2 a &{} \text {for } b \text { even} \\ 2 l b + 4 a &{} \text {for } b \text { odd} . \end{array}\right. } \end{aligned}$$

(A.83)

Proof

The length of the orbit is given by

$$\begin{aligned} k = \sum \limits _{l=1}^{k'} l^{(m)} = \sum \limits _{l=0}^{k'/2-1} \left[ l^{(1+2 l)} + l^{(2 + 2l)} \right] . \end{aligned}$$

(A.84)

It is straightforward to check that

$$\begin{aligned} l^{(m)} + l^{(m+1)} = 2 l - 2 \text {Ceil} ( \varDelta ^{(m)}_\mathrm {L} ) + 2 \text {Ceil} ( \varDelta ^{(m)}_\mathrm {L} + 2p ) \end{aligned}$$

(A.85)

follows from (3.44). It allows us to write

$$\begin{aligned} k = l k' - \sum \limits _{l=0}^{k'-1} (-1)^l \, \text {Ceil} ( \varDelta ^{(1)}_\mathrm {L} + 2 l p ) = l k' + \varDelta ^{(1)}_\mathrm {L} - \varDelta _{\mathrm {L}}^{(1+k')} + 2 k' p \end{aligned}$$

(A.86)

after taking into account (3.56). According to Lemma 1, the part $\varDelta ^{(1)}_\mathrm {L} - \varDelta _{\mathrm {L}}^{(1+k')}$ vanishes. $\square $

References

1.
Witten, E.: String theory dynamics in various dimensions. Nucl. Phys. B 443, 85–126 (1995). arXiv:hep-th/9503124 ADSMathSciNetCrossRefzbMATHGoogle Scholar
2.
Witten, E.: Some comments on string dynamics. In: Future Perspectives in String Theory. Proceedings, Conference, Strings’95, Los Angeles, USA, March 13–18, 1995 (1995). arXiv:hep-th/9507121
3.
Strominger, A.: Open p-branes. Phys. Lett. B 383, 44–47 (1996). arXiv:hep-th/9512059 ADSMathSciNetCrossRefzbMATHGoogle Scholar
4.
Seiberg, N.: Nontrivial fixed points of the renormalization group in six-dimensions. Phys. Lett. B 390, 169–171 (1997). arXiv:hep-th/9609161 ADSMathSciNetCrossRefGoogle Scholar
5.
Witten, E.: Small instantons in string theory. Nucl. Phys. B 460, 541–559 (1996). arXiv:hep-th/9511030 ADSMathSciNetCrossRefzbMATHGoogle Scholar
6.
Ganor, O.J., Hanany, A.: Small $E_8$ instantons and tensionless noncritical strings. Nucl. Phys. B 474, 122–140 (1996). arXiv:hep-th/9602120 ADSCrossRefzbMATHGoogle Scholar
7.
Morrison, D.R., Vafa, C.: Compactifications of F-theory on Calabi–Yau threefolds—II. Nucl. Phys. B 476, 437–469 (1996). arXiv:hep-th/9603161 ADSMathSciNetCrossRefzbMATHGoogle Scholar
8.
Seiberg, N., Witten, E.: Comments on string dynamics in six-dimensions. Nucl. Phys. B 471, 121–134 (1996). arXiv:hep-th/9603003 ADSMathSciNetCrossRefzbMATHGoogle Scholar
9.
Bershadsky, M., Johansen, A.: Colliding Singularities in F-theory and Phase Transitions. Nucl. Phys. B 489, 122–138 (1997). arXiv:hep-th/9610111 ADSMathSciNetCrossRefzbMATHGoogle Scholar
10.
Brunner, I., Karch, A.: Branes at orbifolds versus Hanany Witten in six-dimensions. JHEP 03, 003 (1998). arXiv:hep-th/9712143 ADSMathSciNetCrossRefzbMATHGoogle Scholar
11.
Blum, J.D., Intriligator, K.A.: Consistency conditions for branes at orbifold singularities. Nucl. Phys. B 506, 223–235 (1997). arXiv:hep-th/9705030 ADSMathSciNetCrossRefzbMATHGoogle Scholar
12.
Aspinwall, P.S., Morrison, D.R.: Point-like instantons on K3 orbifolds. Nucl. Phys. B 503, 533–564 (1997). arXiv:hep-th/9705104 ADSMathSciNetCrossRefzbMATHGoogle Scholar
13.
Intriligator, K.A.: New string theories in six-dimensions via branes at orbifold singularities. Adv. Theor. Math. Phys. 1, 271–282 (1998). arXiv:hep-th/9708117 MathSciNetCrossRefzbMATHGoogle Scholar
14.
Hanany, A., Zaffaroni, A.: Branes and six-dimensional supersymmetric theories. Nucl. Phys. B 529, 180–206 (1998). arXiv:hep-th/9712145 ADSMathSciNetCrossRefzbMATHGoogle Scholar
15.
Heckman, J.J., Morrison, D.R., Vafa, C.: On the Classification of 6D SCFTs and Generalized ADE Orbifolds. JHEP 05, 028 (2014). arXiv:1312.5746 [hep-th]. [Erratum: JHEP 06 (2015) 017]ADSCrossRefGoogle Scholar
16.
Gaiotto, D., Tomasiello, A.: Holography for (1,0) theories in six dimensions. JHEP 12, 003 (2014). arXiv:1404.0711 [hep-th]ADSCrossRefGoogle Scholar
17.
Del Zotto, M., Heckman, J.J., Tomasiello, A., Vafa, C.: 6D conformal matter. JHEP 02, 054 (2015). arXiv:1407.6359 [hep-th]MathSciNetCrossRefzbMATHGoogle Scholar
18.
Del Zotto, M., Heckman, J.J., Morrison, D.R., Park, D.S.: 6D SCFTs and gravity. JHEP 06, 158 (2015). arXiv:1412.6526 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
19.
Heckman, J.J., Morrison, D.R., Rudelius, T., Vafa, C.: Atomic classification of 6D SCFTs. Fortsch. Phys. 63, 468–530 (2015). arXiv:1502.05405 [hep-th]ADSCrossRefzbMATHGoogle Scholar
20.
Bhardwaj, L.: Classification of 6D $ \cal{N}=\left(1,0\right) $ Gauge theories. JHEP 11, 002 (2015). arXiv:1502.06594 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
21.
Chang, C.-M., Lin, Y.-H.: Carving out the end of the world or (superconformal bootstrap in six dimensions). JHEP 08, 128 (2017). arXiv:1705.05392 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
22.
Apruzzi, F., Fazzi, M., Rosa, D., Tomasiello, A.: All $AdS_7$ solutions of type II supergravity. JHEP 04, 064 (2014). arXiv:1309.2949 [hep-th]ADSCrossRefGoogle Scholar
23.
Heckman, J.J.: More on the matter of 6D SCFTs. Phys. Lett. B 747, 73–75 (2015). arXiv:1408.0006 [hep-th]ADSCrossRefzbMATHGoogle Scholar
24.
Del Zotto, M., Heckman, J.J., Park, D.S., Rudelius, T.: On the defect group of a 6D SCFT. Lett. Math. Phys. 106(6), 765–786 (2016). arXiv:1503.04806 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
25.
Gaiotto, D., Razamat, S.S.: $ \cal{N}=1 $ theories of class $ {\cal{S}}_k $. JHEP 07, 073 (2015). arXiv:1503.05159 [hep-th]ADSCrossRefGoogle Scholar
26.
Ohmori, K., Shimizu, H., Tachikawa, Y., Yonekura, K.: 6d ${\cal{N}}=(1,0)$ theories on $T^2$ and class S theories: Part I. JHEP 07, 014 (2015). arXiv:1503.06217 [hep-th]ADSCrossRefzbMATHGoogle Scholar
27.
Franco, S., Hayashi, H., Uranga, A.: Charting class $\cal{S}_k$ territory. Phys. Rev. D 92(4), 045004 (2015). arXiv:1504.05988 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
28.
Del Zotto, M., Vafa, C., Xie, D.: Geometric engineering, mirror symmetry and $ 6{\text{ d }_{\left(1,0\right)}\rightarrow 4\text{ d }}_{\left(\cal{N}=2\right)} $. JHEP 11, 123 (2015). arXiv:1504.08348 [hep-th]ADSCrossRefGoogle Scholar
29.
Heckman, J.J., Morrison, D.R., Rudelius, T., Vafa, C.: Geometry of 6D RG flows. JHEP 09, 052 (2015). arXiv:1505.00009 [hep-th]MathSciNetCrossRefzbMATHGoogle Scholar
30.
Cordova, C., Dumitrescu, T.T., Intriligator, K.: Anomalies, renormalization group flows, and the a-theorem in six-dimensional (1, 0) theories. JHEP 10, 080 (2016). arXiv:1506.03807 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
31.
Hanany, A., Maruyoshi, K.: Chiral theories of class $ \cal{S} $. JHEP 12, 080 (2015). arXiv:1505.05053 [hep-th]ADSMathSciNetzbMATHGoogle Scholar
32.
Aganagic, M., Haouzi, N.: ADE little string theory on a Riemann surface (and triality). arXiv:1506.04183 [hep-th]
33.
Louis, J., Lüst, S.: Supersymmetric AdS$_{7}$ backgrounds in half-maximal supergravity and marginal operators of (1, 0) SCFTs. JHEP 10, 120 (2015). arXiv:1506.08040 [hep-th]ADSCrossRefzbMATHGoogle Scholar
34.
Ohmori, K., Shimizu, H., Tachikawa, Y., Yonekura, K.: 6D $\cal{N}=\left(1,\;0\right) $ theories on S$^{1}$ /T$^{2}$ and class S theories: part II. JHEP 12, 131 (2015). arXiv:1508.00915 [hep-th]ADSMathSciNetzbMATHGoogle Scholar
35.
Coman, I., Pomoni, E., Taki, M., Yagi, F.: Spectral curves of $\cal{N}=1$ theories of class $\cal{S}_k$. arXiv:1512.06079 [hep-th]
36.
Cremonesi, S., Tomasiello, A.: 6D holographic anomaly match as a continuum limit. JHEP 05, 031 (2016). arXiv:1512.02225 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
37.
Heckman, J.J., Rudelius, T., Tomasiello, A.: 6D RG flows and nilpotent hierarchies. JHEP 07, 082 (2016). arXiv:1601.04078 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
38.
Cordova, C., Dumitrescu, T.T., Intriligator, K.: Deformations of superconformal theories. JHEP 11, 135 (2016). arXiv:1602.01217 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
39.
Morrison, D.R., Vafa, C.: F-theory and $\cal{N} = 1$ SCFTs in four dimensions. arXiv:1604.03560 [hep-th]
40.
Heckman, J.J., Jefferson, P., Rudelius, T., Vafa, C.: Punctures for theories of class $\cal{S}_\Gamma $. JHEP 03, 171 (2017). arXiv:1609.01281 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
41.
Cordova, C., Dumitrescu, T.T., Intriligator, K.: Multiplets of superconformal symmetry in diverse dimensions. arXiv:1612.00809 [hep-th]
42.
Kim, H.-C., Kim, S., Park, J.: 6D strings from new chiral gauge theories. arXiv:1608.03919 [hep-th]
43.
Shimizu, H., Tachikawa, Y.: Anomaly of strings of 6D $ \cal{N}=\left(1,0\right) $ theories. JHEP 11, 165 (2016). arXiv:1608.05894 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
44.
Mekareeya, N., Rudelius, T., Tomasiello, A.: T-branes, anomalies and moduli spaces in 6D SCFTs. arXiv:1612.06399 [hep-th]
45.
Del Zotto, M., Lockhart, G.: On exceptional instanton strings. JHEP 09, 081 (2017). arXiv:1609.00310 [hep-th]MathSciNetCrossRefzbMATHGoogle Scholar
46.
Apruzzi, F., Hassler, F., Heckman, J.J., Melnikov, I.V.: From 6D SCFTs to dynamic GLSMs. arXiv:1610.00718 [hep-th]
47.
Razamat, S.S., Vafa, C., Zafrir, G.: 4D $ \cal{N}=1 $ from 6D (1, 0). JHEP 04, 064 (2017). arXiv:1610.09178 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
48.
Bah, I., Hanany, A., Maruyoshi, K., Razamat, S.S., Tachikawa, Y., Zafrir, G.: 4D $ \cal{N}=1 $ from 6D $ \cal{N}=\left(1,0\right) $ on a torus with fluxes. JHEP 06, 022 (2017). arXiv:1702.04740 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
49.
Mitev, V., Pomoni, E.: 2D CFT blocks for the 4D class $\cal{S}_k$ theories. JHEP 08, 009 (2017). arXiv:1703.00736 [hep-th]ADSCrossRefzbMATHGoogle Scholar
50.
Bah, I., Passias, A., Tomasiello, A.: $AdS_5$ compactifications with punctures in massive IIA supergravity. arXiv:1704.07389 [hep-th]
51.
Del Zotto, M., Heckman, J.J., Morrison, D.R.: 6D SCFTs and phases of 5D theories. arXiv:1703.02981 [hep-th]
52.
Apruzzi, F., Heckman, J.J., Rudelius, T.: Green–Schwarz automorphisms and 6D SCFTs. arXiv:1707.06242 [hep-th]
53.
Heckman, J.J., Tizzano, L.: 6D fractional quantum hall effect. arXiv:1708.02250 [hep-th]
54.
Kim, H.-C., Razamat, S.S., Vafa, C., Zafrir, G.: E-string theory on Riemann surfaces. arXiv:1709.02496 [hep-th]
55.
Razamat, S.S., Zafrir, G.: $E_8$ orbits of IR dualities. arXiv:1709.06106 [hep-th]
56.
Gaiotto, D.: $\cal{N} = 2$ dualities. JHEP 08, 034 (2012). arXiv:0904.2715 [hep-th]ADSCrossRefGoogle Scholar
57.
Douglas, M.R., Moore, G.W.: D-branes, quivers, and ALE instantons. arXiv:hep-th/9603167
58.
Xie, D.: M5 brane and four dimensional $\cal{N} = 1$ theories I. JHEP 04, 154 (2014). arXiv:1307.5877 [hep-th]ADSCrossRefGoogle Scholar
59.
Gaiotto, D., Maldacena, J.: The gravity duals of $N=2$ superconformal field theories. JHEP 10, 189 (2012). arXiv:0904.4466 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
60.
Apruzzi, F., Fazzi, M., Passias, A., Rota, A., Tomasiello, A.: Six-dimensional superconformal theories and their compactifications from type IIA supergravity. Phys. Rev. Lett. 115(6), 061601 (2015). arXiv:1502.06616 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
61.
Denef, F.: TASI lectures on complex structures. In: Proceedings, Theoretical Advanced Study Institute in Elementary Particle Physics (TASI 2010). String Theory and Its Applications: From meV to the Planck Scale: Boulder, Colorado, USA, June 1–25, 2010, pp. 407–512 (2011). arXiv:1104.0254 [hep-th]

Copyright information

Authors and Affiliations

Falk Hassler
- 1
- 2
Email authorView author's OrcID profile
Jonathan J. Heckman
- 2

1.Department of Physics and AstronomyUniversity of PennsylvaniaPhiladelphiaUSA
2.Department of PhysicsUniversity of North CarolinaChapel HillUSA

Punctures and dynamical systems

Abstract

Keywords

Mathematics Subject Classification

Notes

Acknowledgements

References

Copyright information

Authors and Affiliations

Personalised recommendations

Cite article