BPS states in the Minahan-Nemeschansky $E_7$ theory

TL;DR

This paper employs spectral networks to compute BPS degeneracies in the Minahan-Nemeschansky $E_7$ theory, revealing patterns in BPS indices related to charge multiples and extending previous calculations in the $E_6$ theory.

Contribution

It introduces a spectral network method to calculate BPS states in the $E_7$ theory and uncovers a recurring pattern in BPS indices across different charges.

Findings

BPS degeneracies follow a specific pattern related to charge multiples.

Extended calculations of BPS states in the $E_6$ theory for larger charges.

Confirmed the pattern of BPS indices in the $E_7$ theory.

Abstract

We use the method of spectral networks to calculate BPS degeneracies in the Minahan-Nemeschansky $E_7$ theory, as representations of the $E_7$ flavor symmetry. Our results provide another example of a pattern noticed earlier in the Minahan-Nemeschansky $E_6$ theory: when the electromagnetic charge is $n$ times a primitive charge, the BPS index is a positive integer multiple of $(-1)^{n+1} n$. We also calculate BPS degeneracies in the Minahan-Nemeschansky $E_6$ theory for larger charges than were previously computed.

Figures

Tables

Table 1. Table 1: Reduced indices for charges n ​ γ 1 𝑛 subscript 𝛾 1 n\gamma_{1} in the E 7 subscript 𝐸 7 E_{7} Minahan-Nemeschansky theory, with flavor information included.

$n$	${𝛀}_{\mathrm{red}}\left(n{\gamma }_1\right)$
1	$1\times \mathrm{𝟓𝟔}$
2	$1\times \mathrm{𝟏𝟑𝟑}+2\times \mathrm{𝟏}$
3	$1\times \mathrm{𝟗𝟏𝟐}+2\times \mathrm{𝟓𝟔}$
4	$1\times \mathrm{𝟖𝟔𝟒𝟓}+2\times \mathrm{𝟏𝟓𝟑𝟗}+6\times \mathrm{𝟏𝟑𝟑}+7\times \mathrm{𝟏}$
5	$1\times \mathrm{𝟖𝟔𝟏𝟖𝟒}+2\times \mathrm{𝟐𝟕𝟔𝟔𝟒}+6\times \mathrm{𝟔𝟒𝟖𝟎}+13\times \mathrm{𝟗𝟏𝟐}+23\times \mathrm{𝟓𝟔}$
6	$1\times \mathrm{𝟓𝟕𝟑𝟒𝟒𝟎}+2\times \mathrm{𝟑𝟔𝟓𝟕𝟓𝟎}+1\times \mathrm{𝟐𝟓𝟑𝟗𝟑𝟓}+6\times \mathrm{𝟏𝟓𝟐𝟏𝟓𝟐}+13\times \mathrm{𝟒𝟎𝟕𝟓𝟓}+29\times \mathrm{𝟖𝟔𝟒𝟓}+12\times \mathrm{𝟕𝟑𝟕𝟏}$
	$+51\times \mathrm{𝟏𝟓𝟑𝟗}+16\times \mathrm{𝟏𝟒𝟔𝟑}+93\times \mathrm{𝟏𝟑𝟑}+79\times \mathrm{𝟏}$
7	$3\times \mathrm{𝟑𝟕𝟗𝟐𝟎𝟗𝟔}+1\times \mathrm{𝟑𝟔𝟑𝟓𝟖𝟒𝟎}+6\times \mathrm{𝟐𝟐𝟖𝟐𝟐𝟖𝟎}+13\times \mathrm{𝟖𝟔𝟏𝟖𝟒𝟎}+29\times \mathrm{𝟑𝟔𝟐𝟖𝟖𝟎}+12\times \mathrm{𝟑𝟐𝟎𝟏𝟏𝟐}$
	$+78\times \mathrm{𝟖𝟔𝟏𝟖𝟒}+44\times \mathrm{𝟓𝟏𝟎𝟕𝟐}+107\times \mathrm{𝟐𝟕𝟔𝟔𝟒}+256\times \mathrm{𝟔𝟒𝟖𝟎}+320\times \mathrm{𝟗𝟏𝟐}+448\times \mathrm{𝟓𝟔}$
8	$7\times \mathrm{𝟐𝟒𝟑𝟖𝟔𝟔𝟕𝟎}+1\times \mathrm{𝟏𝟗𝟎𝟒𝟔𝟔𝟔𝟒}+3\times \mathrm{𝟏𝟖𝟑𝟕𝟐𝟑𝟓𝟒}+2\times \mathrm{𝟏𝟑𝟕𝟐𝟖𝟕𝟗𝟐}+13\times \mathrm{𝟏𝟏𝟑𝟏𝟔𝟑𝟎𝟓}$
	$+29\times \mathrm{𝟕𝟏𝟒𝟐𝟒𝟗𝟗}+12\times \mathrm{𝟔𝟔𝟏𝟗𝟐𝟑𝟗}+78\times \mathrm{𝟑𝟒𝟐𝟒𝟐𝟓𝟔}+100\times \mathrm{𝟗𝟖𝟎𝟑𝟒𝟑}+28\times \mathrm{𝟔𝟏𝟕𝟐𝟓𝟑}$
	$+146\times \mathrm{𝟓𝟕𝟑𝟒𝟒𝟎}+235\times \mathrm{𝟑𝟔𝟓𝟕𝟓𝟎}+97\times \mathrm{𝟐𝟓𝟑𝟗𝟑𝟓}+21\times \mathrm{𝟐𝟑𝟖𝟔𝟎𝟐}+537\times \mathrm{𝟏𝟓𝟐𝟏𝟓𝟐}$
	$+163\times \mathrm{𝟏𝟓𝟎𝟖𝟐𝟐}+852\times \mathrm{𝟒𝟎𝟕𝟓𝟓}+1205\times \mathrm{𝟖𝟔𝟒𝟓}+589\times \mathrm{𝟕𝟑𝟕𝟏}+1726\times \mathrm{𝟏𝟓𝟑𝟗}$
	$+745\times \mathrm{𝟏𝟒𝟔𝟑}+2272\times \mathrm{𝟏𝟑𝟑}+1398\times \mathrm{𝟏}$
9	$9\times \mathrm{𝟏𝟗𝟓𝟏𝟎𝟐𝟑𝟑𝟔}+14\times \mathrm{𝟏𝟎𝟎𝟔𝟕𝟕𝟖𝟎𝟖}+1\times \mathrm{𝟗𝟒𝟎𝟓𝟕𝟔𝟎𝟎}+13\times \mathrm{𝟖𝟔𝟏𝟖𝟒𝟎𝟎𝟎}+29\times \mathrm{𝟖𝟔𝟏𝟒𝟏𝟒𝟒𝟎}$
	$+78\times \mathrm{𝟔𝟑𝟒𝟑𝟏𝟒𝟐𝟒}+2\times \mathrm{𝟑𝟐𝟗𝟗𝟓𝟐𝟒𝟖}+146\times \mathrm{𝟐𝟏𝟔𝟑𝟑𝟐𝟒𝟖}+84\times \mathrm{𝟏𝟒𝟗𝟏𝟎𝟖𝟗𝟔}+228\times \mathrm{𝟏𝟑𝟎𝟔𝟗𝟎𝟓𝟔}$
	$+97\times \mathrm{𝟗𝟒𝟖𝟎𝟐𝟒𝟎}+21\times \mathrm{𝟗𝟒𝟎𝟓𝟕𝟔𝟎}+444\times \mathrm{𝟒𝟓𝟐𝟐𝟎𝟎𝟎}+576\times \mathrm{𝟑𝟕𝟗𝟐𝟎𝟗𝟔}+287\times \mathrm{𝟑𝟔𝟑𝟓𝟖𝟒𝟎}$
	$+1055\times \mathrm{𝟐𝟐𝟖𝟐𝟐𝟖𝟎}+532\times \mathrm{𝟖𝟖𝟓𝟐𝟒𝟖}+1707\times \mathrm{𝟖𝟔𝟏𝟖𝟒𝟎}+3110\times \mathrm{𝟑𝟔𝟐𝟖𝟖𝟎}+1551\times \mathrm{𝟑𝟐𝟎𝟏𝟏𝟐}$
	$+5082\times \mathrm{𝟖𝟔𝟏𝟖𝟒}+3662\times \mathrm{𝟓𝟏𝟎𝟕𝟐}+5985\times \mathrm{𝟐𝟕𝟔𝟔𝟒}+376\times \mathrm{𝟐𝟒𝟑𝟐𝟎}+11595\times \mathrm{𝟔𝟒𝟖𝟎}$
	$+11009\times \mathrm{𝟗𝟏𝟐}+12397\times \mathrm{𝟓𝟔}$
10	$15\times \mathrm{𝟕𝟖𝟓𝟔𝟕𝟒𝟖𝟗𝟎}+79\times \mathrm{𝟕𝟏𝟓𝟑𝟎𝟗𝟎𝟓𝟔}+30\times \mathrm{𝟔𝟖𝟖𝟒𝟎𝟎𝟖𝟓𝟔}+7\times \mathrm{𝟔𝟐𝟐𝟑𝟗𝟔𝟕𝟕𝟓}+15\times \mathrm{𝟔𝟎𝟐𝟑𝟓𝟎𝟕𝟒𝟗}$
	$+6\times \mathrm{𝟓𝟔𝟏𝟔𝟑𝟐𝟒𝟎𝟎}+1\times \mathrm{𝟒𝟏𝟐𝟕𝟕𝟖𝟒𝟗𝟔}+146\times \mathrm{𝟑𝟕𝟖𝟐𝟐𝟒𝟔𝟒𝟎}+212\times \mathrm{𝟐𝟎𝟗𝟖𝟔𝟖𝟖𝟏𝟑}+22\times \mathrm{𝟏𝟕𝟓𝟎𝟔𝟏𝟐𝟓𝟎}$
	$+97\times \mathrm{𝟏𝟔𝟑𝟔𝟎𝟏𝟒𝟑𝟖}+569\times \mathrm{𝟏𝟑𝟐𝟕𝟗𝟑𝟑𝟕𝟓}+287\times \mathrm{𝟏𝟑𝟎𝟗𝟒𝟓𝟖𝟏𝟓}+56\times \mathrm{𝟏𝟎𝟗𝟏𝟐𝟎𝟔𝟒𝟖}+962\times \mathrm{𝟕𝟐𝟖𝟒𝟕𝟎𝟐𝟔}$
	$+281\times \mathrm{𝟒𝟖𝟑𝟏𝟔𝟗𝟎𝟓}+2114\times \mathrm{𝟐𝟒𝟑𝟖𝟔𝟔𝟕𝟎}+1387\times \mathrm{𝟐𝟑𝟗𝟔𝟗𝟕𝟗𝟐}+484\times \mathrm{𝟏𝟗𝟎𝟒𝟔𝟔𝟔𝟒}+939\times \mathrm{𝟏𝟖𝟑𝟕𝟐𝟑𝟓𝟒}$
	$+673\times \mathrm{𝟏𝟑𝟕𝟐𝟖𝟕𝟗𝟐}+3122\times \mathrm{𝟏𝟏𝟑𝟏𝟔𝟑𝟎𝟓}+1905\times \mathrm{𝟕𝟒𝟖𝟐𝟔𝟏𝟖}+6053\times \mathrm{𝟕𝟏𝟒𝟐𝟒𝟗𝟗}+962\times \mathrm{𝟔𝟕𝟔𝟎𝟑𝟗𝟎}$
	$+3135\times \mathrm{𝟔𝟔𝟏𝟗𝟐𝟑𝟗}+33\times \mathrm{𝟓𝟐𝟒𝟖𝟕𝟓𝟎}+12854\times \mathrm{𝟑𝟒𝟐𝟒𝟐𝟓𝟔}+13764\times \mathrm{𝟗𝟖𝟎𝟑𝟒𝟑}+2015\times \mathrm{𝟗𝟏𝟓𝟕𝟎𝟓}$
	$+4489\times \mathrm{𝟔𝟏𝟕𝟐𝟓𝟑}+14710\times \mathrm{𝟓𝟕𝟑𝟒𝟒𝟎}+20735\times \mathrm{𝟑𝟔𝟓𝟕𝟓𝟎}+8745\times \mathrm{𝟐𝟓𝟑𝟗𝟑𝟓}+2676\times \mathrm{𝟐𝟑𝟖𝟔𝟎𝟐}$
	$+40812\times \mathrm{𝟏𝟓𝟐𝟏𝟓𝟐}+17974\times \mathrm{𝟏𝟓𝟎𝟖𝟐𝟐}+53180\times \mathrm{𝟒𝟎𝟕𝟓𝟓}+59182\times \mathrm{𝟖𝟔𝟒𝟓}+30938\times \mathrm{𝟕𝟑𝟕𝟏}$
	$+72422\times \mathrm{𝟏𝟓𝟑𝟗}+36278\times \mathrm{𝟏𝟒𝟔𝟑}+75217\times \mathrm{𝟏𝟑𝟑}+34869\times \mathrm{𝟏}$
11	$81\times \mathrm{𝟓𝟑𝟏𝟏𝟕𝟑𝟓𝟎𝟎𝟎}+19\times \mathrm{𝟓𝟐𝟓𝟔𝟖𝟕𝟗𝟐𝟔𝟒}+147\times \mathrm{𝟑𝟗𝟗𝟑𝟖𝟑𝟎𝟒𝟎𝟎}+29\times \mathrm{𝟑𝟓𝟏𝟔𝟑𝟎𝟕𝟐𝟎𝟎}$
	$+288\times \mathrm{𝟐𝟏𝟕𝟔𝟕𝟔𝟏𝟔𝟎𝟎}+553\times \mathrm{𝟐𝟏𝟐𝟎𝟎𝟓𝟖𝟑𝟎𝟒}+24\times \mathrm{𝟐𝟎𝟑𝟐𝟑𝟏𝟔𝟎𝟎𝟎}+13\times \mathrm{𝟏𝟗𝟐𝟒𝟕𝟐𝟐𝟖𝟎𝟎}$
	$+185\times \mathrm{𝟏𝟗𝟎𝟑𝟕𝟐𝟓𝟖𝟐𝟒}+1\times \mathrm{𝟏𝟕𝟏𝟒𝟏𝟗𝟗𝟕𝟔𝟎}+97\times \mathrm{𝟏𝟕𝟎𝟎𝟕𝟓𝟓𝟎𝟓𝟔}+799\times \mathrm{𝟗𝟖𝟓𝟗𝟒𝟒𝟗𝟔𝟎}$
	$+2021\times \mathrm{𝟕𝟖𝟗𝟕𝟎𝟑𝟗𝟗𝟐}+484\times \mathrm{𝟔𝟓𝟔𝟓𝟗𝟒𝟒𝟎𝟎}+932\times \mathrm{𝟔𝟏𝟗𝟕𝟑𝟔𝟖𝟑𝟐}+666\times \mathrm{𝟒𝟔𝟐𝟏𝟒𝟑𝟐𝟑𝟐}$
	$+2802\times \mathrm{𝟑𝟑𝟗𝟎𝟔𝟔𝟎𝟎𝟎}+855\times \mathrm{𝟐𝟑𝟔𝟖𝟖𝟖𝟗𝟔𝟎}+4440\times \mathrm{𝟏𝟗𝟓𝟏𝟎𝟐𝟑𝟑𝟔}+34\times \mathrm{𝟏𝟗𝟎𝟒𝟔𝟔𝟔𝟒𝟎}$
	$+4848\times \mathrm{𝟏𝟖𝟖𝟗𝟕𝟐𝟕𝟖𝟒}+2546\times \mathrm{𝟏𝟕𝟗𝟐𝟔𝟐𝟕𝟐𝟎}+5517\times \mathrm{𝟏𝟎𝟎𝟔𝟕𝟕𝟖𝟎𝟖}+826\times \mathrm{𝟗𝟒𝟎𝟓𝟕𝟔𝟎𝟎}$
	$+5914\times \mathrm{𝟖𝟔𝟏𝟖𝟒𝟎𝟎𝟎}+10638\times \mathrm{𝟖𝟔𝟏𝟒𝟏𝟒𝟒𝟎}+7772\times \mathrm{𝟔𝟕𝟑𝟗𝟓𝟖𝟖𝟖}+24528\times \mathrm{𝟔𝟑𝟒𝟑𝟏𝟒𝟐𝟒}$
	$+829\times \mathrm{𝟑𝟐𝟗𝟗𝟓𝟐𝟒𝟖}+36469\times \mathrm{𝟐𝟏𝟔𝟑𝟑𝟐𝟒𝟖}+7858\times \mathrm{𝟏𝟕𝟗𝟐𝟔𝟐𝟕𝟐}+21827\times \mathrm{𝟏𝟒𝟗𝟏𝟎𝟖𝟗𝟔}$
	$+3211\times \mathrm{𝟏𝟒𝟐𝟐𝟎𝟑𝟔𝟎}+49219\times \mathrm{𝟏𝟑𝟎𝟔𝟗𝟎𝟓𝟔}+21586\times \mathrm{𝟗𝟒𝟖𝟎𝟐𝟒𝟎}+6833\times \mathrm{𝟗𝟒𝟎𝟓𝟕𝟔𝟎}$
	$+81797\times \mathrm{𝟒𝟓𝟐𝟐𝟎𝟎𝟎}+75935\times \mathrm{𝟑𝟕𝟗𝟐𝟎𝟗𝟔}+42709\times \mathrm{𝟑𝟔𝟑𝟓𝟖𝟒𝟎}+126875\times \mathrm{𝟐𝟐𝟖𝟐𝟐𝟖𝟎}$
	$+8651\times \mathrm{𝟐𝟐𝟕𝟑𝟗𝟐𝟎}+79317\times \mathrm{𝟖𝟖𝟓𝟐𝟒𝟖}+174663\times \mathrm{𝟖𝟔𝟏𝟖𝟒𝟎}+274960\times \mathrm{𝟑𝟔𝟐𝟖𝟖𝟎}$
	$+147738\times \mathrm{𝟑𝟐𝟎𝟏𝟏𝟐}+345964\times \mathrm{𝟖𝟔𝟏𝟖𝟒}+271269\times \mathrm{𝟓𝟏𝟎𝟕𝟐}+367084\times \mathrm{𝟐𝟕𝟔𝟔𝟒}$
	$+42715\times \mathrm{𝟐𝟒𝟑𝟐𝟎}+601036\times \mathrm{𝟔𝟒𝟖𝟎}+475863\times \mathrm{𝟗𝟏𝟐}+444394\times \mathrm{𝟓𝟔}$

Table 2. Table 2: Numerical reduced indices for charges n ​ γ 1 𝑛 subscript 𝛾 1 n\gamma_{1} in the E 7 subscript 𝐸 7 E_{7} Minahan-Nemeschansky theory.

$n$	${\Omega }_{\mathrm{red}}\left(n{\gamma }_1\right)$
1	56
2	135
3	1024
4	12528
5	193536
6	3455104
7	68179968
8	1447549920
9	32495488000
10	762222261888
11	18524656253952

Table 3. Table 3: Reduced indices for charges n ​ ( γ 1 + γ 2 ) 𝑛 subscript 𝛾 1 subscript 𝛾 2 n(\gamma_{1}+\gamma_{2}) in the E 7 subscript 𝐸 7 E_{7} Minahan-Nemeschansky theory, with flavor information included.

$n$	${𝛀}_{\mathrm{red}}\left(n{\gamma }_1\right)$
1	$1\times \mathrm{𝟏𝟑𝟑}+3\times \mathrm{𝟏}$
2	$1\times \mathrm{𝟏𝟓𝟑𝟗}+2\times \mathrm{𝟏𝟑𝟑}+4\times \mathrm{𝟏}$
3	$1\times \mathrm{𝟒𝟎𝟕𝟓𝟓}+2\times \mathrm{𝟖𝟔𝟒𝟓}+6\times \mathrm{𝟏𝟓𝟑𝟗}+2\times \mathrm{𝟏𝟒𝟔𝟑}+11\times \mathrm{𝟏𝟑𝟑}+12\times \mathrm{𝟏}$
4	$1\times \mathrm{𝟗𝟖𝟎𝟑𝟒𝟑}+2\times \mathrm{𝟑𝟔𝟓𝟕𝟓𝟎}+1\times \mathrm{𝟐𝟓𝟑𝟗𝟑𝟓}+6\times \mathrm{𝟏𝟓𝟐𝟏𝟓𝟐}+2\times \mathrm{𝟏𝟓𝟎𝟖𝟐𝟐}+19\times \mathrm{𝟒𝟎𝟕𝟓𝟓}$
	$+27\times \mathrm{𝟖𝟔𝟒𝟓}+7\times \mathrm{𝟕𝟑𝟕𝟏}+57\times \mathrm{𝟏𝟓𝟑𝟗}+27\times \mathrm{𝟏𝟒𝟔𝟑}+82\times \mathrm{𝟏𝟑𝟑}+67\times \mathrm{𝟏}$
5	$1\times \mathrm{𝟐𝟑𝟗𝟔𝟗𝟕𝟗𝟐}+3\times \mathrm{𝟏𝟏𝟑𝟏𝟔𝟑𝟎𝟓}+2\times \mathrm{𝟕𝟒𝟖𝟐𝟔𝟏𝟖}+6\times \mathrm{𝟕𝟏𝟒𝟐𝟒𝟗𝟗}+19\times \mathrm{𝟑𝟒𝟐𝟒𝟐𝟓𝟔}+42\times \mathrm{𝟗𝟖𝟎𝟑𝟒𝟑}$
	$+6\times \mathrm{𝟗𝟏𝟓𝟕𝟎𝟓}+12\times \mathrm{𝟔𝟏𝟕𝟐𝟓𝟑}+23\times \mathrm{𝟓𝟕𝟑𝟒𝟒𝟎}+67\times \mathrm{𝟑𝟔𝟓𝟕𝟓𝟎}+29\times \mathrm{𝟐𝟓𝟑𝟗𝟑𝟓}+166\times \mathrm{𝟏𝟓𝟐𝟏𝟓𝟐}$
	$+78\times \mathrm{𝟏𝟓𝟎𝟖𝟐𝟐}+330\times \mathrm{𝟒𝟎𝟕𝟓𝟓}+386\times \mathrm{𝟖𝟔𝟒𝟓}+149\times \mathrm{𝟕𝟑𝟕𝟏}+664\times \mathrm{𝟏𝟓𝟑𝟗}+349\times \mathrm{𝟏𝟒𝟔𝟑}$
	$+778\times \mathrm{𝟏𝟑𝟑}+498\times \mathrm{𝟏}$

Table 4. Table 4: Reduced BPS indices for the Minahan-Nemeschansky E 6 subscript 𝐸 6 E_{6} theory.

$n$	${𝛀}_{\mathrm{red}}\left(n{\gamma }_1\right)$
1	$\overline{\mathrm{𝟐𝟕}}$
2	$\mathrm{𝟐𝟕}$
3	$\mathrm{𝟕𝟖}+2\times \mathrm{𝟏}$
4	$\overline{\mathrm{𝟑𝟓𝟏}}+2\times \overline{\mathrm{𝟐𝟕}}$
5	$\mathrm{𝟏𝟕𝟐𝟖}+2\times \mathrm{𝟑𝟓𝟏}+6\times \mathrm{𝟐𝟕}$
6	$\mathrm{𝟓𝟖𝟐𝟒}+2\times \mathrm{𝟐𝟗𝟐𝟓}+\mathrm{𝟐𝟒𝟑𝟎}+6\times \mathrm{𝟔𝟓𝟎}+13\times \mathrm{𝟕𝟖}+16\times \mathrm{𝟏}$
7	$\overline{\mathrm{𝟏𝟗𝟑𝟎𝟓}}+3\times \overline{\mathrm{𝟏𝟕𝟓𝟓𝟎}}+6\times \overline{\mathrm{𝟕𝟑𝟕𝟏}}+13\times \overline{\mathrm{𝟏𝟕𝟐𝟖}}+12\times \overline{{\mathrm{𝟑𝟓𝟏}}^{\prime }}+29\times \overline{\mathrm{𝟑𝟓𝟏}}+44\times \overline{\mathrm{𝟐𝟕}}$
8	$1\times \mathrm{𝟓𝟒𝟎𝟓𝟒}+7\times \mathrm{𝟓𝟏𝟗𝟕𝟓}+3\times \mathrm{𝟒𝟔𝟑𝟑𝟐}+2\times \mathrm{𝟑𝟒𝟑𝟗𝟖}+13\times \mathrm{𝟏𝟕𝟓𝟓𝟎}+12\times \mathrm{𝟕𝟕𝟐𝟐}+29\times \mathrm{𝟕𝟑𝟕𝟏}$
	$+78\times \mathrm{𝟏𝟕𝟐𝟖}+28\times {\mathrm{𝟑𝟓𝟏}}^{\prime }+100\times \mathrm{𝟑𝟓𝟏}+163\times \mathrm{𝟐𝟕}$
9	$9\times \mathrm{𝟐𝟓𝟐𝟐𝟓𝟐}+1\times \mathrm{𝟏𝟒𝟔𝟒𝟑𝟐}+14\times \mathrm{𝟏𝟎𝟓𝟔𝟎𝟎}+13\times \mathrm{𝟕𝟖𝟗𝟕𝟓}+29\times \mathrm{𝟕𝟎𝟎𝟕𝟎}+2\times \mathrm{𝟒𝟑𝟕𝟓𝟖}$
	$+78\times \mathrm{𝟑𝟒𝟕𝟒𝟗}+84\times \overline{\mathrm{𝟓𝟖𝟐𝟒}}+146\times \mathrm{𝟓𝟖𝟐𝟒}+21\times \mathrm{𝟑𝟎𝟎𝟑}+228\times \mathrm{𝟐𝟗𝟐𝟓}+97\times \mathrm{𝟐𝟒𝟑𝟎}$
	$+444\times \mathrm{𝟔𝟓𝟎}+532\times \mathrm{𝟕𝟖}+376\times \mathrm{𝟏}$
10	$15\times \overline{\mathrm{𝟒𝟗𝟒𝟐𝟎𝟖}}+7\times \overline{\mathrm{𝟒𝟓𝟗𝟒𝟓𝟗}}+6\times \overline{\mathrm{𝟒𝟏𝟐𝟕𝟕𝟔}}+15\times \overline{\mathrm{𝟑𝟗𝟑𝟖𝟐𝟐}}+30\times \overline{\mathrm{𝟑𝟖𝟔𝟏𝟎𝟎}}+1\times \overline{{\mathrm{𝟑𝟓𝟗𝟒𝟐𝟒}}^{\prime }}$
	$+79\times \overline{\mathrm{𝟑𝟏𝟒𝟒𝟗𝟔}}+146\times \overline{\mathrm{𝟏𝟏𝟐𝟑𝟐𝟎}}+22\times \overline{\mathrm{𝟔𝟏𝟒𝟐𝟓}}+212\times \overline{\mathrm{𝟓𝟏𝟗𝟕𝟓}}+97\times \overline{\mathrm{𝟒𝟔𝟑𝟑𝟐}}+56\times \overline{\mathrm{𝟑𝟒𝟑𝟗𝟖}}$
	$+287\times \overline{\mathrm{𝟏𝟗𝟑𝟎𝟓}}+569\times \overline{\mathrm{𝟏𝟕𝟓𝟓𝟎}}+281\times \overline{\mathrm{𝟕𝟕𝟐𝟐}}+962\times \overline{\mathrm{𝟕𝟑𝟕𝟏}}+1387\times \overline{\mathrm{𝟏𝟕𝟐𝟖}}+1905\times \overline{\mathrm{𝟑𝟓𝟏}}$
	$+962\times \overline{{\mathrm{𝟑𝟓𝟏}}^{\prime }}+2015\times \overline{\mathrm{𝟐𝟕}}$
11	$19\times \mathrm{𝟐𝟎𝟖𝟖𝟒𝟓𝟎}+81\times \mathrm{𝟏𝟔𝟒𝟎𝟗𝟐𝟓}+29\times \mathrm{𝟏𝟐𝟓𝟑𝟎𝟕𝟎}+147\times \mathrm{𝟗𝟔𝟕𝟔𝟖𝟎}+1\times \mathrm{𝟖𝟓𝟑𝟐𝟖𝟏}+13\times \mathrm{𝟕𝟒𝟏𝟑𝟏𝟐}$
	$24\times \mathrm{𝟓𝟕𝟗𝟏𝟓𝟎}+97\times \mathrm{𝟑𝟗𝟑𝟖𝟐𝟐}+185\times \mathrm{𝟑𝟖𝟔𝟏𝟎𝟎}+288\times \mathrm{𝟑𝟓𝟗𝟒𝟐𝟒}+553\times \mathrm{𝟑𝟏𝟒𝟒𝟗𝟔}+799\times \mathrm{𝟏𝟏𝟐𝟑𝟐𝟎}$
	$+484\times \mathrm{𝟓𝟒𝟎𝟓𝟒}+2021\times \mathrm{𝟓𝟏𝟗𝟕𝟓}+932\times \mathrm{𝟒𝟔𝟑𝟑𝟐}+666\times \mathrm{𝟑𝟒𝟑𝟗𝟖}+855\times \mathrm{𝟏𝟗𝟑𝟎𝟓}+34\times {\mathrm{𝟏𝟗𝟑𝟎𝟓}}^{\prime }$
	$+2802\times \mathrm{𝟏𝟕𝟓𝟓𝟎}+2546\times \mathrm{𝟕𝟕𝟐𝟐}+4848\times \mathrm{𝟕𝟑𝟕𝟏}+7772\times \mathrm{𝟏𝟕𝟐𝟖}+3211\times {\mathrm{𝟑𝟓𝟏}}^{\prime }+7858\times \mathrm{𝟑𝟓𝟏}$
	$+8651\times \mathrm{𝟐𝟕}$
12	$78\times \mathrm{𝟓𝟎𝟓𝟒𝟒𝟎𝟎}+147\times \mathrm{𝟒𝟕𝟓𝟐𝟑𝟖𝟒}+28\times \mathrm{𝟑𝟑𝟎𝟗𝟔𝟗𝟔}+13\times \mathrm{𝟑𝟏𝟔𝟐𝟏𝟓𝟗}+12\times \mathrm{𝟑𝟎𝟎𝟕𝟑𝟔𝟖}$
	$+290\times \mathrm{𝟐𝟗𝟕𝟕𝟗𝟕𝟓}+526\times \mathrm{𝟐𝟒𝟓𝟑𝟖𝟏𝟒}+97\times \mathrm{𝟏𝟗𝟏𝟏𝟏𝟗𝟓}+1\times \mathrm{𝟏𝟖𝟗𝟔𝟏𝟖𝟎}+16\times \mathrm{𝟏𝟓𝟓𝟗𝟑𝟕𝟔}$
	$130\times \mathrm{𝟏𝟑𝟑𝟕𝟎𝟓𝟎}+485\times \mathrm{𝟗𝟕𝟐𝟗𝟕𝟐}+1859\times \mathrm{𝟖𝟓𝟐𝟗𝟑𝟎}+916\times \mathrm{𝟖𝟏𝟐𝟏𝟕𝟓}+650\times \overline{\mathrm{𝟔𝟎𝟎𝟔𝟎𝟎}}$
	$519\times \mathrm{𝟔𝟎𝟎𝟔𝟎𝟎}+7\times \mathrm{𝟓𝟑𝟕𝟗𝟔𝟔}+36\times \mathrm{𝟑𝟕𝟏𝟖𝟎𝟎}+2270\times \overline{\mathrm{𝟐𝟓𝟐𝟐𝟓𝟐}}+4340\times \mathrm{𝟐𝟓𝟐𝟐𝟓𝟐}$
	$827\times \mathrm{𝟏𝟒𝟔𝟒𝟑𝟐}+5197\times \mathrm{𝟏𝟎𝟓𝟔𝟎𝟎}+1584\times \mathrm{𝟖𝟓𝟐𝟗𝟑}+2943\times \overline{\mathrm{𝟕𝟖𝟗𝟕𝟓}}+5325\times \mathrm{𝟕𝟖𝟗𝟕𝟓}$
	$+9433\times \mathrm{𝟕𝟎𝟎𝟕𝟎}+822\times \mathrm{𝟒𝟑𝟕𝟓𝟖}+19446\times \mathrm{𝟑𝟒𝟕𝟒𝟗}+16290\times \overline{\mathrm{𝟓𝟖𝟐𝟒}}+21759\times \mathrm{𝟓𝟖𝟐𝟒}$
	$1572\times \overline{\mathrm{𝟑𝟎𝟎𝟑}}+4157\times \mathrm{𝟑𝟎𝟎𝟑}+28563\times \mathrm{𝟐𝟗𝟐𝟓}+12841\times \mathrm{𝟐𝟒𝟑𝟎}+43257\times \mathrm{𝟔𝟓𝟎}$
	$+37146\times \mathrm{𝟕𝟖}+17436\times \mathrm{𝟏}$
13	$288\times \overline{\mathrm{𝟏𝟒𝟎𝟏𝟕𝟓𝟑𝟔}}+133\times \overline{\mathrm{𝟏𝟑𝟒𝟕𝟖𝟒𝟎𝟎}}+34\times \overline{\mathrm{𝟏𝟐𝟔𝟒𝟖𝟔𝟑𝟔}}+469\times \overline{\mathrm{𝟏𝟎𝟑𝟕𝟖𝟑𝟔𝟖}}+485\times \overline{\mathrm{𝟕𝟕𝟓𝟕𝟏𝟎𝟎}}$
	$+49\times \overline{\mathrm{𝟔𝟔𝟕𝟓𝟔𝟔𝟗}}+888\times \overline{\mathrm{𝟔𝟐𝟒𝟑𝟐𝟑𝟕}}+1580\times \overline{\mathrm{𝟓𝟕𝟕𝟔𝟎𝟓𝟔}}+91\times \overline{\mathrm{𝟓𝟓𝟎𝟏𝟗𝟐𝟓}}+623\times \overline{\mathrm{𝟒𝟓𝟖𝟐𝟔𝟓𝟔}}$
	$+4162\times \overline{\mathrm{𝟒𝟐𝟎𝟎𝟕𝟔𝟖}}+1\times \overline{\mathrm{𝟒𝟎𝟖𝟖𝟒𝟒𝟖}}+40\times \overline{\mathrm{𝟑𝟐𝟖𝟏𝟖𝟓𝟎}}+829\times \overline{\mathrm{𝟐𝟓𝟓𝟗𝟖𝟒𝟑}}+4665\times \overline{\mathrm{𝟏𝟔𝟒𝟎𝟗𝟐𝟓}}$
	$+1416\times \overline{\mathrm{𝟏𝟐𝟓𝟑𝟎𝟕𝟎}}+4364\times \overline{\mathrm{𝟏𝟏𝟐𝟑𝟐𝟎𝟎}}+7529\times \overline{\mathrm{𝟗𝟔𝟕𝟔𝟖𝟎}}+806\times \overline{\mathrm{𝟕𝟒𝟏𝟑𝟏𝟐}}+10516\times \overline{\mathrm{𝟒𝟗𝟒𝟐𝟎𝟖}}$
	$5099\times \overline{\mathrm{𝟒𝟓𝟗𝟒𝟓𝟗}}+4462\times \overline{\mathrm{𝟒𝟏𝟐𝟕𝟕𝟔}}+8824\times \overline{\mathrm{𝟑𝟗𝟑𝟖𝟐𝟐}}+16388\times \overline{\mathrm{𝟑𝟖𝟔𝟏𝟎𝟎}}+11674\times \overline{\mathrm{𝟑𝟓𝟗𝟒𝟐𝟒}}$
	$+1296\times \overline{{\mathrm{𝟑𝟓𝟗𝟒𝟐𝟒}}^{\prime }}+37016\times \overline{\mathrm{𝟑𝟏𝟒𝟒𝟗𝟔}}+53396\times \overline{\mathrm{𝟏𝟏𝟐𝟑𝟐𝟎}}+52\times \overline{\mathrm{𝟏𝟎𝟎𝟑𝟖𝟔}}+10794\times \overline{\mathrm{𝟔𝟏𝟒𝟐𝟓}}$
	$+8808\times \overline{\mathrm{𝟓𝟒𝟎𝟓𝟒}}+64712\times \overline{\mathrm{𝟓𝟏𝟗𝟕𝟓}}+31256\times \overline{\mathrm{𝟒𝟔𝟑𝟑𝟐}}+20191\times \overline{\mathrm{𝟑𝟒𝟑𝟗𝟖}}+61577\times \overline{\mathrm{𝟏𝟗𝟑𝟎𝟓}}$
	$+105595\times \overline{\mathrm{𝟏𝟕𝟓𝟓𝟎}}+65758\times \overline{\mathrm{𝟕𝟕𝟐𝟐}}+153851\times \overline{\mathrm{𝟕𝟑𝟕𝟏}}+174359\times \overline{\mathrm{𝟏𝟕𝟐𝟖}}+182898\times \overline{\mathrm{𝟑𝟓𝟏}}$
	$+101325\times \overline{{\mathrm{𝟑𝟓𝟏}}^{\prime }}+147690\times \overline{\mathrm{𝟐𝟕}}$
14	$258\times \mathrm{𝟑𝟖𝟏𝟒𝟔𝟔𝟖𝟎}+472\times {\mathrm{𝟑𝟒𝟗𝟎𝟔𝟗𝟓𝟎}}^{\prime }+3853\times \mathrm{𝟑𝟎𝟏𝟏𝟓𝟖𝟎𝟎}+826\times \mathrm{𝟐𝟔𝟕𝟎𝟐𝟔𝟕𝟔}+328\times \mathrm{𝟐𝟐𝟎𝟎𝟕𝟕𝟎𝟎}$
	$827\times \mathrm{𝟏𝟗𝟕𝟔𝟖𝟑𝟐𝟎}+566\times \mathrm{𝟏𝟗𝟐𝟗𝟕𝟐𝟕𝟖}+1059\times \mathrm{𝟏𝟕𝟗𝟏𝟖𝟗𝟎𝟏}+46\times \mathrm{𝟏𝟕𝟒𝟓𝟑𝟒𝟕𝟓}+22\times \mathrm{𝟏𝟔𝟗𝟗𝟐𝟔𝟏𝟐}$
	$+21\times \mathrm{𝟏𝟔𝟓𝟒𝟎𝟓𝟐𝟒}+68\times \mathrm{𝟏𝟒𝟖𝟎𝟓𝟓𝟎𝟒}+3810\times \mathrm{𝟏𝟎𝟑𝟕𝟖𝟑𝟔𝟖}+1\times \mathrm{𝟖𝟒𝟎𝟏𝟓𝟑𝟔}+68\times \mathrm{𝟖𝟐𝟖𝟏𝟖𝟒𝟓}$
	$+9540\times \mathrm{𝟕𝟔𝟎𝟏𝟗𝟓𝟖}+4936\times \mathrm{𝟕𝟓𝟐𝟖𝟗𝟓𝟎}+4300\times \mathrm{𝟔𝟕𝟒𝟕𝟑𝟎𝟎}+8276\times \mathrm{𝟔𝟐𝟒𝟑𝟐𝟑𝟕}+1296\times \mathrm{𝟔𝟏𝟏𝟎𝟐𝟎𝟖}$
	$+14484\times \mathrm{𝟓𝟕𝟕𝟔𝟎𝟓𝟔}+2780\times \mathrm{𝟓𝟓𝟓𝟑𝟗𝟎𝟎}+778\times \mathrm{𝟓𝟓𝟎𝟏𝟗𝟐𝟓}+4588\times \mathrm{𝟒𝟓𝟖𝟐𝟔𝟓𝟔}+29245\times \mathrm{𝟒𝟐𝟎𝟎𝟕𝟔𝟖}$
	$+20562\times \mathrm{𝟐𝟎𝟖𝟖𝟒𝟓𝟎}+56\times \mathrm{𝟏𝟖𝟑𝟕𝟖𝟑𝟔}+62331\times \mathrm{𝟏𝟔𝟒𝟎𝟗𝟐𝟓}+24493\times \mathrm{𝟏𝟐𝟓𝟑𝟎𝟕𝟎}+31638\times \mathrm{𝟏𝟏𝟐𝟑𝟐𝟎𝟎}$
	$+98816\times \mathrm{𝟗𝟔𝟕𝟔𝟖𝟎}+2033\times \mathrm{𝟖𝟓𝟑𝟐𝟖𝟏}+10941\times \mathrm{𝟕𝟒𝟏𝟑𝟏𝟐}+6637\times \mathrm{𝟔𝟑𝟖𝟖𝟐𝟎}+21892\times \mathrm{𝟓𝟕𝟗𝟏𝟓𝟎}$
	$+36525\times \mathrm{𝟒𝟗𝟒𝟐𝟎𝟖}+18415\times \mathrm{𝟒𝟓𝟗𝟒𝟓𝟗}+14349\times \mathrm{𝟒𝟏𝟐𝟕𝟕𝟔}+57206\times \mathrm{𝟑𝟗𝟑𝟖𝟐𝟐}+103179\times \mathrm{𝟑𝟖𝟔𝟏𝟎𝟎}$
	$+148984\times \mathrm{𝟑𝟓𝟗𝟒𝟐𝟒}+246082\times \mathrm{𝟑𝟏𝟒𝟒𝟗𝟔}+304906\times \mathrm{𝟏𝟏𝟐𝟑𝟐𝟎}+31152\times \mathrm{𝟔𝟏𝟒𝟐𝟓}+147492\times \mathrm{𝟓𝟒𝟎𝟓𝟒}$
	$490011\times \mathrm{𝟓𝟏𝟗𝟕𝟓}+239499\times \mathrm{𝟒𝟔𝟑𝟑𝟐}+174995\times \mathrm{𝟑𝟒𝟑𝟗𝟖}+253940\times \mathrm{𝟏𝟗𝟑𝟎𝟓}+14701\times {\mathrm{𝟏𝟗𝟑𝟎𝟓}}^{\prime }$
	$564835\times \mathrm{𝟏𝟕𝟓𝟓𝟎}+479896\times \mathrm{𝟕𝟕𝟐𝟐}+826668\times \mathrm{𝟕𝟑𝟕𝟏}+995856\times \mathrm{𝟏𝟕𝟐𝟖}+413309\times {\mathrm{𝟑𝟓𝟏}}^{\prime }$
	$+853638\times \mathrm{𝟑𝟓𝟏}+703835\times \mathrm{𝟐𝟕}$

Equations

$$\Omega (9{\gamma }_1)=292459392000.$$

$$|\Omega (n{\gamma }_1)|\sim c{n}^{-\frac{5}{2}}(17+12\sqrt{2}{)}^n.$$

$$\mathbf{\Omega }(3{\gamma }_1)=3\times 912+6\times 56.$$

$$\mathbf{\Omega }(2({\gamma }_1+{\gamma }_2))=-2\times 1539-4\times 133-8\times 1.$$

$$\text{det}(\lambda -\Phi (z))=0,$$

$${\lambda }^4+{\varphi }_2(z){\lambda }^2+{\varphi }_3(z)\lambda +{\varphi }_4(z)=0,$$

$${\varphi }_2=0,{\varphi }_3=0,{\varphi }_4=-\frac{u\mathrm{d}{z}^4}{(z-1{)}^3(z-\omega {)}^3(z-{\omega }^2{)}^2}.$$

$$\Sigma =\left\{x^4-\frac{1}{(z-1{)}^3(z-\omega {)}^3(z-{\omega }^2{)}^2}=0\right\}.$$

$${\Gamma }_g=H_1(\overline{\Sigma },\mathbb{Z}).$$

$$Z_{{\gamma }_1}=\frac{1}{\pi }{\int }_1^{\omega }{\lambda }_1+\frac{1}{\pi }{\int }_{\omega }^1{\lambda }_4=\frac{4\sqrt{\frac{2}{3}}}{{\pi }^{3/2}}\Gamma \left[\frac{5}{4}\right]\Gamma \left[\frac{1}{4}\right]{\mathrm{e}}^{-\frac{7\pi }{12}\mathrm{i}}\approx 1.92749{\mathrm{e}}^{-\frac{7\pi }{12}\mathrm{i}},$$

$$Z_{{\gamma }_2}=-\mathrm{i}{Z}_{{\gamma }_1}=\frac{1}{\pi }{\int }_{\omega }^{{\omega }^2}{\lambda }_4+\frac{1}{\pi }{\int }_{{\omega }^2}^{\omega }{\lambda }_2=\frac{4\sqrt{\frac{2}{3}}}{{\pi }^{3/2}}\Gamma \left[\frac{5}{4}\right]\Gamma \left[\frac{1}{4}\right]{\mathrm{e}}^{\frac{11\pi }{12}\mathrm{i}}\approx 1.92749{\mathrm{e}}^{\frac{11\pi }{12}\mathrm{i}}.$$

$$z^{\prime }(t)=-(x_1(t)-x_2(t){)}^{-1}{\mathrm{e}}^{\mathrm{i}\vartheta },x_1^{\prime }(t)=\frac{\mathrm{d}x}{\mathrm{d}z}{z}^{\prime }(t),x_2^{\prime }(t)=\frac{\mathrm{d}x}{\mathrm{d}z}{z}^{\prime }(t),$$

$$\frac{\mathrm{d}x}{\mathrm{d}z}=\frac{\frac{\partial F}{\partial z}}{\frac{\partial F}{\partial x}},F(x,z)=x^4+{\varphi }_4(z).$$

$${\mathbf{M}}_1$$

$${\mathbf{M}}_2$$

$${\mathbf{M}}_3$$

$$\det \left(\begin{array}{cccc}{m}_1-t& 0& 0& 0\\ 0& m_2-t& 0& 0\\ 0& 0& m_3-t& 0\\ 0& 0& 0& m_4-t\end{array}\right)=1-4t+6t^2-\bar{4}{t}^3+t^4,$$

$$\det ({\mathbf{M}}_l-t)=1-\bar{4}t+6t^2-4t^3+t^4.$$

$${\mathbf{M}}_3^2-2{\mathbf{M}}_3+{Id}_{4\times 4}=0,$$

$$f_{14}(x)$$

$$f_{41}(x)$$

$$Q_p(x)=1+{\tau }_p{\nu }_p.$$

$$Q_{ij}(x)=1+x{f}_{ij}(x)f_{ji}(x).$$

$$Q_p(x)={\wedge }^{\ast }({\mathbf{\alpha }}_1(p)x)\otimes {\text{Sym}}^{\ast }(-{\mathbf{\alpha }}_2(p)x^2)\otimes \cdots$$

$$Q_{41}$$

$$+2\times (6,6,1)+(\overline{10},6,1)+(4,4,2)+(15,1,1))x^2+\cdots$$

$$Q_{41}$$

$$+2\times (6,6,1)+(4,4,2)+(15,1,1))x^2)\otimes \cdots$$

$${\mathbf{\alpha }}_1(p_{41})$$

$$-{\mathbf{\alpha }}_2(p_{41})$$

$${\mathbf{\alpha }}_1(p_{41})$$

$${\mathbf{\alpha }}_1(p_{32})$$

$${\mathbf{\alpha }}_1(p_{13})$$

$${\mathbf{\alpha }}_1(p_{24})$$

$$\mathbf{L}(n{\gamma }_1)=\sum _p{\mathbf{\alpha }}_n(p)p_{\Sigma }.$$

Full text

††preprint:

UTTG–02–19

\gitAuthorIsoDate\gitAbbrevHash

1 Introduction

Minahan and Nemeschansky discovered $\mathcal{N}=2$ superconformal theories in four dimensions with flavor symmetry $E_{6}$, $E_{7}$ and $E_{8}$ Minahan:1996fg ; Minahan:1996cj . These remarkable theories have been studied extensively since then. In this paper we study the BPS spectrum of the $E_{7}$ theory.

We consider the class $S$ construction Gaiotto:2009we ; Gaiotto:2009hg , applied with Lie algebra $A_{3}$ and Riemann surface $C=\mathbb{CP}^{1}\setminus\{z_{1},z_{2},z_{3}\}$, where $z_{1}$, $z_{2}$ are full punctures and $z_{3}$ is a puncture of type $[2,2]$. This construction produces a superconformal ${\mathcal{N}}=2$ theory with manifest flavor symmetry $SU(4)\times SU(4)\times SU(2)$.111$E_{7}$ does not have a subgroup isomorphic to $SU(4)\times SU(4)\times SU(2)$, but has subgroup isomorphic to $SU(4)\times SU(4)\times SU(2)/\mathbb{Z}_{4}$. The $\mathbb{Z}_{4}$ is the subgroup $\{(1,1,1)$, $(\mu,\mu,\mu^{2})$, $(\mu^{2},\mu^{2},1)$, $(\mu^{3},\mu^{3},\mu^{2})\}$, where $\mu$ is a primitive fourth root of unity. Thus the enhancement of symmetry requires that this $\mathbb{Z}_{4}$ acts trivially. Already in Gaiotto:2009we , Gaiotto proposed that in this class $S$ theory the manifest $SU(4)\times SU(4)\times SU(2)$ should actually be enhanced to an $E_{7}$ flavor symmetry, and indeed the theory should be the Minahan-Nemeschansky $E_{7}$ theory. Some checks of this proposal are given by Benini, Benvenuti and Tachikawa in Benini:2009gi and by Tachikawa in Tachikawa:2013kta .

Having this class $S$ realization of the Minahan-Nemeschansky $E_{7}$ theory allows us to study its BPS states using the method of spectral networks Gaiotto:2012rg , and this is what we do in this paper. Our approach is mostly parallel to what was done for the $E_{6}$ theory in Hollands:2016kgm , and thus we are rather brief, focusing mainly on those points which are different for the $E_{7}$ theory; see Hollands:2016kgm for more background and explanations of the method.

On the Coulomb branch, the Hilbert space is graded by electromagnetic charge corresponding to the $U(1)$ gauge symmetry. The electromagnetic charge lattice has rank $2$, and can be identified with the homology $H_{1}(\overline{\Sigma},{\mathbb{Z}})$ where $\overline{\Sigma}$ is the Seiberg-Witten curve, given below as (8). We introduce a basis $\{\gamma_{1},\gamma_{2}\}$ for this charge lattice, where $\langle\gamma_{1},\gamma_{2}\rangle=1$; we call $\gamma_{1}$ the primitive electric charge, and $\gamma_{2}$ the primitive magnetic charge. A general charge can be written as $\gamma=p\gamma_{1}+q\gamma_{2}$. The theory has a $\mathbb{Z}_{2}$ symmetry, induced from the symmetry of $C$ which exchanges the two full punctures; this symmetry swaps the charges $\gamma_{1}\leftrightarrow\gamma_{2}$.

The main new result in this paper is the computation of BPS indices for various charges, of the form $n\gamma_{1}$ and $n(\gamma_{1}+\gamma_{2})$, as we now describe.

The spectral network relevant for computing BPS indices for particles with charges $n\gamma_{1}$ looks like a circle; it is shown in Figure 3 below. Using this spectral network we have computed the indexed counts $\Omega(n\gamma_{1})$ of 4d BPS states for $1\leq n\leq 200$; for $1\leq n\leq 11$ the results are given in Table 2. For example, we find

[TABLE]

We also show that the BPS index has asymptotic exponential growth

[TABLE]

Our computation has manifest $SU(4)\times SU(4)\times SU(2)$ flavor symmetry, so the integers $\Omega(\gamma)$ admit an “upgrade” to characters $\mathbf{\Omega}(\gamma)$ of representations of $SU(4)\times SU(4)\times SU(2)$. Since the flavor symmetry is predicted to be enhanced to $E_{7}$, these characters should a posteriori assemble into characters of representations of $E_{7}$. We compute $\mathbf{\Omega}(n\gamma_{1})$ for $1\leq n\leq 11$, and find that indeed they are characters of $E_{7}$, as predicted. For example, we find

[TABLE]

The full results are given in LABEL:table:electric-indices below.

We also calculate the BPS index for the charges $n(\gamma_{1}+\gamma_{2})$ with $1\leq n\leq 5$; this involves a different spectral network, shown in Figure 7. We obtain, for example,

[TABLE]

The full results are in LABEL:table:11-indices below.

Our results exhibit the same pattern observed in the $E_{6}$ Minahan-Nemeschansky theory Hollands:2016kgm : BPS states carrying electromagnetic charges which are $n$ times a primitive charge always come with index a positive integer multiple of $(-1)^{n+1}n$. For example, in (3) above, the multiplicities of irreducible representations of $E_{7}$ are $3$ and $6$, which are positive integer multiples of $3$; similarly in (4) above, all the multiplicities are positive integer multiples of $-2$.

BPS states for rank 1 Minahan-Nemeschansky theories were recently studied in Distler:2019eky , using string junctions in the $F$-theory realization. In that paper the precise BPS multiplicities were not computed; rather, what was computed is a list of which representations of the flavor symmetry group can occur in the spectrum, for each electromagnetic charge. Our results for the $E_{7}$ theory are all in agreement with the lists of representations given in Distler:2019eky .

Finally, we revisit the BPS states of the Minahan-Nemeschansky $E_{6}$ theory, extending the results of Hollands:2016kgm to higher charges: see LABEL:table:electric-indices-e6 below.

Acknowledgements

We thank Jacques Distler and Mario Martone for helpful discussions. LH’s work on this paper is supported by a Royal Society Dorothy Hodgkin Fellowship. QH’s and AN’s work on this paper is supported by NSF grant DMS-1711692.

2 Seiberg-Witten curve

The IR $U(1)$ gauge theory of the theory on its Coulomb branch is described by the Seiberg-Witten curve, which takes the form

[TABLE]

where $\Phi(z)$ is the Higgs field in the corresponding Hitchin system. (5) can also be written as

[TABLE]

where $\phi_{d}(z)$ are meromorphic differentials on $C$, degree-$d$ invariant polynomial combinations of the eigenvalues of $\Phi(z)$. As discussed in Gaiotto:2009we , and reviewed e.g. in Chacaltana:2012zy , the form of $\Phi(z)$ around a puncture is $\Phi(z)=(\frac{\Phi_{-1}}{z}+\Phi_{0}+\dots)\,{\mathrm{d}}z$, where $\Phi_{-1}$ is constrained to lie in a specific nilpotent orbit of $\mathfrak{sl}_{4}$, determined by a partition $\rho$ of $4$. $\rho$ determines the Jordan block structure of $\Phi_{-1}$, which in turn determines the form of the meromorphic differentials $\phi_{d}(z)$; $\phi_{d}(z)$ has a pole of order at most $p_{d}(\rho)$ at a puncture with partition $\rho$.

In our case the two full punctures have $\rho=[4]$ and for the third puncture $\rho=[2,2]$; the two full punctures have $(p_{2},p_{3},p_{4})=(1,2,3)$ and the other puncture has $(p_{2},p_{3},p_{4})=(1,1,2)$.

Using these constraints, the only nonzero differential in the Minahan-Nemeschansky $E_{7}$ theory turns out to be $\phi_{4}$. Using the $PSL(2,\mathbb{C})$ symmetry of ${\mathbb{CP}}^{1}$, the three punctures can be fixed to $(z_{1}=1,z_{2}=\omega,z_{3}=\omega^{2})$, where $\omega=e^{2\pi i/3}$. Then the differentials are concretely

[TABLE]

The free parameter $u\in\mathbb{C}$ parameterizes the 1-complex-dimensional Coulomb branch. By the scale invariance and $U(1)_{R}$ symmetry of the theory, all points with $u\neq 0$ are equivalent, so from now on we set $u=1$. We also write $\lambda=x\,{\mathrm{d}}z$. Then the Seiberg-Witten curve (6) becomes

[TABLE]

By filling in the punctures, we get a smooth compact genus 1 curve $\overline{\Sigma}$. The projection $\pi:\overline{\Sigma}\rightarrow C$ is a degree $4$ covering, branched over the $3$ punctures on $C$.

The electromagnetic charge lattice of the IR $U(1)$ theory on the Coulomb branch is

[TABLE]

Two basis charges $\gamma_{1}$ and $\gamma_{2}$ are sketched in Figure 1; by convention we call $\gamma_{1}$ “electric” and $\gamma_{2}$ “magnetic.” The central charge corresponding to the EM charge $\gamma$ is given by the integral $Z_{\gamma}=\frac{1}{\pi}\oint_{\gamma}\lambda$.

For the primitive electric charge this gives

[TABLE]

and for the primitive magnetic charge

[TABLE]

Notice that the curve $\Sigma$ given in (8) has $\mathbb{Z}_{4}$ symmetry, generated by the transformation $\lambda\mapsto{\mathrm{i}}\lambda$. This generator permutes the primitive electric and magnetic charges: it maps $\gamma_{1}\mapsto-\gamma_{2}$ and $\gamma_{2}\mapsto\gamma_{1}$.

3 Computing the BPS states

We use the same spectral network technique for computing BPS states as was used for the Minahan-Nemeschansky $E_{6}$ theory in Hollands:2016kgm . The surface $C$ is a parameter space for surface defects; the spectral network $W(\vartheta)$ consists of the points $z$ such that the surface defect with parameter $z$ supports a BPS soliton of central charge $Z$ with phase $\arg(-Z)=\vartheta$. To study the bulk BPS states of charge $\gamma$, we must choose the phase of the spectral network to be $\vartheta=\vartheta_{\gamma}=\arg(-Z_{\gamma})$.

3.1 Building the spectral network

We comment briefly on how we compute the spectral network $W(\vartheta)$. The network is made up of “$S$-walls” which obey differential equations. In addition to its parameterization $z(t)$, each wall carries a pair of labels $ij$ and two auxiliary functions $x_{1}(t)$ and $x_{2}(t)$: if the wall is labeled $ij$ then $x_{1}(t)$ gives the $i$-th sheet of $\Sigma$ over the wall, and $x_{2}(t)$ the $j$-th sheet. The condition $\arg(-Z)=\vartheta$ translates into differential equations which control the $t$ dependence of each wall:222Alternatively one could say that only $z(t)$ is determined by a differential equation, while $x_{1}(z)$, $x_{2}(z)$ are just solutions of the algebraic equation $F(x,z(t))=0$ which move continuously with $t$. This continuity condition is hard to implement in practice, because of the branch cut of the fourth-root function. The advantage of writing differential equations for $x_{1}(t)$, $x_{2}(t)$ as well as for $z(t)$ is that it automatically enforces the continuity, thus avoiding having to deal with cuts. This method is implemented in swn-plotter .

[TABLE]

where

[TABLE]

Next we need to explain the initial conditions: where do new walls originate? There are a few possibilities: either branch points of the covering $\Sigma\to C$, or from places where existing walls intersect one another. In the example we consider here, we will only have to deal with the case of walls originating from branch points. In this theory the branch points coincide with the punctures. Thus we need to study solutions of (12) originating at a puncture. Integrating (12) from the puncture $z_{i}$ to some nearby point $z_{i}+\delta$ gives the constraint $-({\mathrm{e}}^{-{\mathrm{i}}\vartheta}\int_{z_{i}}^{z_{i}+\delta}(x_{1}(z)-x_{2}(z))\,{\mathrm{d}}z)\in\mathbb{R}_{+}$. For small $\delta$, and a wall of type $ij$, this integral is proportional to $({\mathrm{e}}^{{\mathrm{i}}\pi i/2}-{\mathrm{e}}^{{\mathrm{i}}\pi j/2})\delta^{\frac{1}{4}}$ for a full puncture, or to $({\mathrm{e}}^{{\mathrm{i}}\pi i/2}-{\mathrm{e}}^{{\mathrm{i}}\pi j/2})\delta^{\frac{1}{2}}$ for a type $[2,2]$ puncture; thus, in either case, this constraint singles out distinguished directions $\arg(\delta)$ and sheet labels $ij$. We also impose the additional constraint that $x_{1}=x_{2}$ at the puncture. See Figure 2 for the resulting distinguished directions and sheet labels, at the phase $\vartheta=\vartheta_{\gamma_{1}}=\frac{5\pi}{12}$. (Note that two different walls emerging from a puncture can be exactly degenerate: e.g. emanating from the puncture at $z=z_{1}=1$ there is a wall labeled $14$ which is exactly degenerate with a wall labeled $23$.)

We then choose initial points $z=z_{i}+\delta$ very close to the punctures, and initial values $x_{1},x_{2}$ determined by the sheet labels, and numerically integrate the equations (12)-(13) to determine the full $S$-walls. After finite time, we find that the $S$-walls emerging from one puncture run into a neighboring puncture; at that point we just terminate them.

We make the most conservative possible assumption, that any $S$-wall allowed by this analysis indeed exists. (This assumption will be verified in the next section when we compute the soliton counts on the walls and see that they are nonzero.)

The outcome of this process at the phase $\vartheta=\vartheta_{\gamma_{1}}=\frac{5\pi}{12}$ is the very simple spectral network shown in Figure 3.

3.2 Finding the solitons

In order to determine the bulk BPS indices, following the strategy in Hollands:2016kgm , we first deform $\vartheta$ infinitesimally to get a resolution of the spectral network, as shown in Figure 4.

Next we apply the constraints of homotopy invariance for 2d-4d framed BPS spectra, as described in Section 4.3 of Hollands:2016kgm , to the resolved spectral network. This involves studying the generating functions of 2d-4d framed BPS states associated to three different interfaces, associated to loops around the three punctures. See Figure 5 and Figure 6 for the loops we consider, and Appendix A for more details about the generating functions and formal variables we use below.

The generating functions of 2d-4d framed BPS states associated to these three loops have the form:

[TABLE]

Here in each case $p$ and $p^{\prime}$ denote two semicircles on $C$, making up a circular loop $\wp$ around a puncture; $p$ is the top half and $p^{\prime}$ the bottom half. $p$ does not cross a branch cut, while $p^{\prime}$ does cross a cut, thus going from one sheet to the next according to the permutation attached to the cut. The generating function $\bm{F}_{p}$ is the sum of formal variables $X_{p^{(i)}}$ associated to the four lifts of $p$ to $\Sigma$, and similarly $\bm{F}_{p^{\prime}}$. Finally, $\tau_{i}$ or $\nu_{i}$ denote the soliton generating functions. Each of these functions counts BPS solitons with charges of the form $a+n\gamma_{1}$, $n\geq 0$, where $a$ is a basic soliton charge; thus the function is of the form $f(x)X_{a}$, where $X_{a}$ is the formal variable standing in for soliton charge $a$, and $x=-X_{\widetilde{\gamma}_{1}}$ is the formal variable standing in for 4d particle charge $\gamma_{1}$ (see Appendix A for more details on this tricky sign.) These functions $f(x)$ are the main undetermined quantities which we need to find. As it happens, the spectral network in Figure 4 contains at most one wall of any type $ij$; thus we can distinguish the various $f(x)$ by labeling them $f_{ij}(x)$, and our job is to determine the eight functions $f_{13}$, $f_{31}$, $f_{14}$, $f_{41}$, $f_{23}$, $f_{32}$, $f_{42}$, $f_{24}$.

The $\bm{M}_{l}$, $l=1,2,3$, are naturally viewed as $4\times 4$ matrices, since they contain counts of solitons going from vacuum $i$ to vacuum $j$ with $1\leq i,j\leq 4$. Moreover, since the solitons are charged under the flavor symmetry, the $\bm{M}_{l}$ can be promoted from numbers to characters depending on a flavor parameter $g\in SU(4)\times SU(4)\times SU(2)$. As in Hollands:2016kgm , we impose the constraint that the characteristic polynomial of $\bm{M}_{l}$ equals the characteristic polynomial of $g$ acting in a representation $R_{l}$. $R_{l}$ is the representation of $SU(4)\times SU(4)\times SU(2)$ which is the fundamental for the $l$-th factor, and the trivial representation of the other two factors. Explicitly, for the full punctures, this characteristic polynomial can be written as

[TABLE]

where $m_{i}$ are the eigenvalues of $g$ acting in $R_{l}$, obeying $m_{1}m_{2}m_{3}m_{4}=1$. Combinations of their products give characters of other representations of $SU(4)$, which we denote by bold numbers: e.g. ${\bf 4}=\frac{1}{m_{1}}+\frac{1}{m_{2}}+\frac{1}{m_{3}}+\frac{1}{m_{4}}$. So, explicitly, our constraint at each full puncture is

[TABLE]

For the type $[2,2]$ puncture, we impose a stronger constraint:

[TABLE]

where $\bm{2}=m+1/m$ is the character of the fundamental representation of $SU(2)$. Since the monodromy matrix $\bm{M}_{3}$ has determinant $1$, and (19) requires the diagonal elements to be either $m$ or $1/m$, it implies that the eigenvalues are $\{m,m,1/m,1/m\}$. However, (19) is stronger than just fixing the eigenvalues: it also rules out nontrivial Jordan blocks.

Using the constraints (18), (19) for all three punctures simultaneously we obtain a system of equations for the $f_{ij}(x)$. We have not found a closed solution, but making the assumption that all of the $f_{ij}(x)$ have series expansions in nonnegative powers of $x$, we can solve the equations iteratively in powers of $x$. For example, we find:

[TABLE]

As we will see explicitly below, finding the $f_{ij}(x)$ up to order $x^{n}$ is sufficient to determine the BPS indices up to charge $n\gamma_{1}$.

3.3 The bulk BPS indices

Once the $f_{ij}(x)$ have been determined, the next step in determining the spectrum of bulk BPS states with phase $\vartheta$ is to construct a generating function $Q_{p}(x)$ attached to each double $S$-wall $p$. As explained in gaiotto2013framed ; Gaiotto:2012rg , $Q_{p}(x)$ is a generating function determining the jumping behavior of the framed BPS spectrum attached to an interface between surface defects, when the phase of the interface crosses $\vartheta$. It is given by Gaiotto:2012rg ; Hollands:2016kgm

[TABLE]

In our case there are four double $S$-walls $p_{13}$, $p_{41}$, $p_{23}$, $p_{42}$, and thus four functions $Q_{ij}(x)$, which are explicitly

[TABLE]

The extra factor of $x$ appearing in (23) is the product of the basic soliton factors $X_{a}$ in $\tau$ and $\nu$.

$Q_{p}(x)$ is the character of a representation of $U(1)\times SU(4)\times SU(4)\times SU(2)$, where the $U(1)$ keeps track of the electric charge (exponent of $x$) and the rest keeps track of the flavor charge. This representation is a Fock space built from basic fermionic and bosonic constituents (fermionic for odd electric charge, bosonic for even charge), and what we need to do is to extract those constituents. We define $\bm{\alpha}_{n}(p)$ to be the constituent vector space with electric charge $n$, so that $Q_{p}(x)$ has an expansion of the form

[TABLE]

There is a straightforward algorithm to compute the $\bm{\alpha}_{n}(p)$ order by order in $n$. The coefficient of $x^{1}$ in $Q_{p}(x)$ gives $\bm{\alpha}_{1}(p)$. We then formally divide $Q_{p}(x)$ by the fermionic Fock space generated by $\bm{\alpha}_{1}(p)x$. The remaining terms of order $x^{2}$ give $-\bm{\alpha}_{2}(p)x^{2}$. We then formally divide out by the bosonic Fock space generated by $-\bm{\alpha}_{2}(p)x^{2}$, and so on. For example, the expansion of $Q_{41}(x)$ to order $x^{2}$ is

[TABLE]

Its expansion in terms of constituents is

[TABLE]

From this expansion we now read off the characters of the constituent vector spaces,

[TABLE]

We record here the answers to first order for all of the double walls, obtained by expanding the corresponding $Q_{ij}$ to first order in $x$:

[TABLE]

Next, as in Hollands:2016kgm , we define

[TABLE]

The sum in (33) runs over the $4$ double $S$-walls $p$, each of which lifts to a chain $p_{\Sigma}$ on $\Sigma$. $\bm{L}(n\gamma_{1})$ is a 1-cycle on $\Sigma$ valued in representations of $SU(4)\times SU(4)\times SU(2)$. Its homology class $[\bm{L}(n\gamma_{1})]$ is necessarily a multiple of $\gamma_{1}$.

For example, for $n=1$ we have

[TABLE]

Happily, this is the decomposition of the representation $\bm{56}$ of $E_{7}$. For this spectral network, each $p_{\Sigma}$ is in fact a closed chain, in the class $\gamma_{1}$, so (33) becomes

[TABLE]

Finally, as in Gaiotto:2012rg ; Hollands:2016kgm , the BPS index is computed as the ratio:

[TABLE]

Thus we find

[TABLE]

and by similar computations we can compute $\bm{\Omega}(n\gamma_{1})$ for larger $n$. The fact that these BPS indices turn out to be characters of representations of $E_{7}$, not only $SU(4)\times SU(4)\times SU(2)$, constitutes evidence for the expected enhancement of flavor symmetry in this theory.

4 Results

Up to $n=11$, the multiplicity for each representation in $\bm{\Omega}(n\gamma_{1})$ turns out to be a positive integer multiple of $(-1)^{(n+1)}n$ (and up to $n=200$, the unflavored $\Omega(n\gamma_{1})$ is a positive integer multiple of $(-1)^{(n+1)}n$). This continues the pattern observed in Hollands:2016kgm for the $E_{6}$ theory. It is convenient to define a reduced index by dividing out this common factor:

[TABLE]

Our results for $\bm{\Omega}_{{\mathrm{red}}}(n\gamma_{1})$ for $1\leq n\leq 11$ are shown in LABEL:table:electric-indices.

We make a few comments about these results:

• •

The result can be compared with Huang:2013yta , which gives refined BPS states of a 5d theory with $E_{7}$ flavor symmetry obtained by compactifying M-theory on a CY manifold which is a bundle over a del Pezzo surface. It is natural to suspect that further compactifying this theory on $S^{1}$ to four dimensions would give the Minahan-Nemeschansky $E_{7}$ theory. As far as the BPS states go, the precise relation between the 5d and 4d theories is not clear; but following Hollands:2016kgm we can find a surprisingly close match by the following ad hoc procedure. We sum the 5d results over spins $j_{L}$ and $j_{R}$, i.e. we just count the total number of multiplets, and compare that with the reduced index in 4d. For charges $\gamma_{1}$ and $3\gamma_{1}$, the 4d and 5d results match exactly. However, for charge $2\gamma_{1}$ the 4d result contains one more $\bm{1}$ than the 5d, and for $4\gamma_{1}$ the 4d result has an extra $\bm{133}+\bm{1}$. For charge $5\gamma_{1}$, the results are different by $193536-192568=968$ in size. It would be natural to try to identify this mismatch as coming from the multiplets $\bm{912}+\bm{56}$, but comparing our result with the $E_{7}$ decompositions given in Huang:2013yta , it seems that the mismatch is actually worse: the representations given in Huang:2013yta in 5d are not a subset of the representations we have computed in 4d. For charge $6\gamma_{1}$, we only looked at the difference in sizes: it is $3455104-3451215=3889$. It would be very interesting to understand better what the precise relation between the 5d and 4d results should be.

• •

If we omit the flavor information, replacing representations by their dimensions, then we can actually solve for the generating functions in closed form, assuming the symmetries $f_{13}(x)=f_{42}(x),\,f_{31}(x)=f_{24}(x)$. Building the corresponding $Q$ we obtain:

[TABLE]

[TABLE]

The $Q$’s satisfy algebraic equations:

[TABLE]

The discriminants both vanish at $(17+12\sqrt{2})^{-1}$. Using the technique in Mainiero:2016xaj and Hollands:2016kgm , these algebraic equations determine the asymptotic behavior of the BPS degeneracies as

[TABLE]

for some constant $c$. Using (41)-(42) it becomes possible to compute the $\Omega(n\gamma_{1})$ for much larger $n$, e.g. $1\leq n\leq 200$, and compare with these asymptotics; the agreement is very good.

5 BPS states with charge $\gamma_{1}+\gamma_{2}$

The circular spectral network is the simplest case, but we can use the same method for other charges, at the price of dealing with more complicated spectral networks. In this paper we limit ourselves to briefly considering the next simplest case, the BPS states of charge $\gamma_{1}+\gamma_{2}$. The spectral network for charge $\gamma_{1}+\gamma_{2}$ is shown in Figure 7; it can be computed by the methods we reviewed in subsection 3.1.

In this case there are six double walls, shown in Figure 7, which we label $p^{1},\dots,p^{6}$. Carrying out the computations of soliton degeneracies and plethystic logarithms as in section 3 above, we obtain at leading order

[TABLE]

The lifts of paths $p^{1}$ and $p^{6}$ are cycles in the class $\gamma_{1}+\gamma_{2}$. However, the lifts of paths $p^{2},p^{3},p^{4}$ and $p^{5}$ do not form closed loops individually. Instead, we get closed cycles with charge $\gamma_{1}+\gamma_{2}$ as combinations of these lifts:

[TABLE]

As defined above,

[TABLE]

which gives

[TABLE]

This is the decomposition of the representation $\bm{133}+3\times\bm{1}$ of $E_{7}$, so altogether we have found

[TABLE]

With computer assistance we calculated the BPS index $\bm{\Omega}(n(\gamma_{1}+\gamma_{2}))$, where $1\leq n\leq 5$. The results are given in LABEL:table:11-indices below. As before, the results are consistent with the string-network analysis of Distler:2019eky , and as before, they continue the pattern of being divisible by $(-1)^{n+1}n$: thus, as before, we give the results for the reduced BPS index (38).

6 Minahan-Nemeschansky $E_{6}$ theory revisited

As we have mentioned, the Fock space decomposition method for extracting the $\bm{\alpha}_{n}(p)$ from $Q_{p}$ is a bit more efficient than the method used in Hollands:2016kgm . Thus we revisited the $E_{6}$ theory using the Fock space decomposition method. We were able to compute ${\bf\Omega}(n\gamma_{1})$ for $n\leq 14$. Our results are presented in LABEL:table:electric-indices-e6 below. For $n\leq 7$ the results agree with those in Hollands:2016kgm ; we include them here just for convenience.

Appendix A Sign rules

In this appendix we address a tricky question of signs which arises in the computation of the BPS indices.

We need to recall a few details from Gaiotto:2012rg . In that paper, the generating function of framed 2d-4d BPS states for a given interface $\wp$ is written as an expansion in formal variables, of the form

[TABLE]

Here the index $c$ runs roughly over possible charges for a BPS state of the interface $\wp$, and $\underline{\overline{\Omega}}(\wp,c)$ is roughly the BPS index counting states of charge $c$. However, the precise meaning is a bit subtler, because of ambiguity in defining the fermion number in a system with only two-dimensional Poincare invariance. We parameterize our ignorance by saying $c$ is valued in a ${\mathbb{Z}}_{2}$ extension of the naive space of charges for the interface, and letting $H$ denote the generator of the extension, we have $\underline{\overline{\Omega}}(\wp,c+H)=-\underline{\overline{\Omega}}(\wp,c)$. To compensate this we further define $X_{c+H}=-X_{c}$, so that the product $\underline{\overline{\Omega}}(\wp,c)X_{c}$ appearing in (57) is well defined and independent of how we lift the charge to this extension.

The soliton generating functions $\tau$ and $\nu$ on an $S$-wall of type $ij$-$ji$ are similarly written in terms of formal variables $X_{a}$, which also lie in ${\mathbb{Z}}_{2}$ extensions of the naive space of soliton charges: in $\tau$ the extended charges $a$ which appear are charges of solitons from vacuum $i$ to vacuum $j$, while in $\nu$ the extended charges $b$ are solitons from vacuum $j$ to vacuum $i$. Given such an $a$ and $b$, there is also a charge $\mathrm{cl}(a+b)$, which is an extended 4d charge: it lives in a ${\mathbb{Z}}_{2}$ extension of the lattice of charges for 4d particles. We introduce the notation $\mathrm{cl}(X_{a}X_{b})=X_{\mathrm{cl}(a+b)}$. Then the generating functions

[TABLE]

are functions in the formal variables $X_{\widetilde{\gamma}}$. Once we consider purely 4d particles, there is no fermion number ambiguity, and thus it is possible to choose a canonical extended $\widetilde{\gamma}$ for each ordinary charge $\gamma$.

In Gaiotto:2012rg a specific geometric realization of the extended charges is chosen. The key technical device is to consider paths on the unit tangent bundle $\widetilde{\Sigma}:=UT\Sigma$, instead of on $\Sigma$ itself. Then:

• •

All extended charges are homology classes of paths in $\widetilde{\Sigma}$, considered modulo the relation $2H=0$, where $H$ represents a loop winding once around a fiber of $\widetilde{\Sigma}$.

• •

The extended charges of states of an interface $\wp$ are realized as homology classes of paths on $\widetilde{\Sigma}$, ending on the preimages of the tangent vectors to $\wp$ at its endpoints.

• •

The extended soliton charges $a$ on a wall of the spectral network are realized as homology classes of paths on $\widetilde{\Sigma}$, whose endpoints are tangent vectors pointing in opposite directions along the wall: the initial vector points in the direction of decreasing soliton mass, while the final vector points in the direction of increasing mass.

• •

The extended 4d charges $\widetilde{\gamma}$ are realized as homology classes of closed paths on $\widetilde{\Sigma}$.

• •

A closed path realizing $\mathrm{cl}(a+b)$ is obtained by gluing open paths realizing $a$ and $b$ at their endpoints to make a closed loop on $\widetilde{\Sigma}$.

• •

A canonical lift $\widetilde{\gamma}$ of a homology class $\gamma\in H_{1}(\Sigma)$ is obtained as follows. Represent $\gamma$ by an oriented submanifold $P\subset\Sigma$. The oriented unit tangent vector field to $P$ gives a lift to a submanifold $\widetilde{P}\subset\widetilde{\Sigma}$. Finally $\widetilde{\gamma}=[\widetilde{P}]+n_{P}H$ where $n_{P}$ is the number of connected components of $P$.

Although this realization is canonical and theoretically convenient, keeping track of lifts to the unit tangent bundle can be annoying, so it is sometimes useful to switch to an alternate realization of the extended charges. In this alternate realization, which we call the “untwisted formalism,” instead of $\widetilde{\Sigma}$ we consider $\Sigma^{\prime}=\Sigma\setminus b$, where $b$ is the branch locus of the covering $\pi:\Sigma\to C$. Then:

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Extended charges are represented by homology classes of paths on $\Sigma^{\prime}$ plus multiples of a formal variable $H$, where we impose $2H=0$, and also the following relation: if $L$ is a loop around a branch point with ramification index $n$, then $L=(n-1)H$.

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The extended charges of states of an interface $\wp$ are realized as homology classes of open paths on $\Sigma^{\prime}$, ending on the preimages of the endpoints of $\wp$.

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Given a soliton associated to a wall of a spectral network, its extended charge $a$ is a homology class of paths on $\Sigma^{\prime}$. The charge $a$ depends on a choice of a co-orientation of the wall; if we reverse the co-orientation then the charge $a$ is replaced by $a+H$. In practice, we generally fix once and for all a co-orientation for each wall.

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The extended 4d charges $\widetilde{\gamma}$ are realized as homology classes of closed paths on $\Sigma^{\prime}$.

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A closed path realizing $\mathrm{cl}(a+b)$ is obtained by gluing open paths representing $a$ and $b$. The result of this process is independent of the co-orientation we choose on the wall, since reversing the co-orientation changes both $a\to a+H$ and $b\to b+H$, thus changes $\mathrm{cl}(a+b)$ by $2H=0$.

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If $\gamma\in H_{1}(\Sigma)$ obeys $\pi_{*}\gamma=0$, then a canonical lift $\widetilde{\gamma}$ of $\gamma$ is obtained as follows. Represent $\gamma$ by an oriented submanifold $P\subset\Sigma^{\prime}$, such that $\pi_{*}P$ has only transverse self-intersections. Then $\widetilde{\gamma}=[P]+n^{\prime}_{P}H$, where $n^{\prime}_{P}$ is the number of self-intersections of $\pi_{*}P$. (For $\gamma$ which do not necessarily obey $\pi_{*}\gamma=0$ we cannot get a canonical lift for free, but we can get one after making a choice of a spin structure on $C$.)

The two formalisms are equivalent; however, to construct an explicit equivalence between them, one needs to use a spin structure on $C$.

In this paper, as well as in Hollands:2016kgm , we work in the untwisted formalism. We choose once and for all a co-orientation on each wall. Thus all the formal variables which we use concretely represent homology classes of paths on $\Sigma^{\prime}$, and the product of formal variables is induced from concatenation or addition of homology classes. For concrete computations we make a choice of a basic soliton charge $a$ along each wall, and, as indicated in (23), we define a formal variable $x$ by $\mathrm{cl}(X_{a}X_{b})=x$, where $a$ and $b$ are the two solitons along the wall. After so doing, we have to check carefully whether $x=X_{\widetilde{\gamma}}$ or $x=-X_{\widetilde{\gamma}}$. According to the rules above, this means we have to draw a loop representing $a+b$, and count how many self-intersections its projection to $C$ has: calling this number $k$, we have $x=(-1)^{k}X_{\widetilde{\gamma}}$. In the computations described in section 3 of this paper, as well as in the main example described in Hollands:2016kgm , we chose the basic charges $a$ in such a way that $x=-X_{\widetilde{\gamma}}$. This minus sign is ultimately responsible for the fact that when we decompose $Q_{p}$ we use fermionic constituents for odd charges and bosonic for even charges; if $x=X_{\widetilde{\gamma}}$ instead then we would use bosonic constituents for all charges.