Emergent higher-form symmetry from type IIB superstring theory

We investigate a higher-form symmetry in type IIB superstring theory, which possesses an <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>SL</a:mi> <a:mo stretchy="false">(</a:mo> <a:mn>2</a:mn> <a:mo>,</a:mo> <a:mi mathvariant="double-struck">Z</a:mi> <a:mo stretchy="false">)</a:mo> </a:math> symmetry. From the point of view of the low-energy effective field theory, the <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:mi>SL</f:mi> <f:mo stretchy="false">(</f:mo> <f:mn>2</f:mn> <f:mo>,</f:mo> <f:mi mathvariant="double-struck">Z</f:mi> <f:mo stretchy="false">)</f:mo> </f:math> symmetry is treated as a gauge symmetry. Hence, an 8-form global symmetry <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:msubsup> <k:mi mathvariant="double-struck">Z</k:mi> <k:mn>12</k:mn> <k:mrow> <k:mo stretchy="false">[</k:mo> <k:mn>8</k:mn> <k:mo stretchy="false">]</k:mo> </k:mrow> </k:msubsup> </k:math> emerges as a quantum symmetry. In this paper, we present an explicit construction of the topological operator associated with the <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"> <p:msubsup> <p:mi mathvariant="double-struck">Z</p:mi> <p:mn>12</p:mn> <p:mrow> <p:mo stretchy="false">[</p:mo> <p:mn>8</p:mn> <p:mo stretchy="false">]</p:mo> </p:mrow> </p:msubsup> </p:math> symmetry. In this construction, the discriminant <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:mi mathvariant="normal">Δ</u:mi> <u:mo stretchy="false">(</u:mo> <u:mi>τ</u:mi> <u:mo stretchy="false">)</u:mo> </u:math> plays a central role. As a result, it becomes manifest that <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"> <z:msubsup> <z:mi mathvariant="double-struck">Z</z:mi> <z:mn>12</z:mn> <z:mrow> <z:mo stretchy="false">[</z:mo> <z:mn>8</z:mn> <z:mo stretchy="false">]</z:mo> </z:mrow> </z:msubsup> </z:math> is the solitonic symmetry of 7-branes. Furthermore, taking into account the extensions of the duality group, we also discuss what global symmetries emerge when considering not <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"> <eb:mi>SL</eb:mi> <eb:mo stretchy="false">(</eb:mo> <eb:mn>2</eb:mn> <eb:mo>,</eb:mo> <eb:mi mathvariant="double-struck">Z</eb:mi> <eb:mo stretchy="false">)</eb:mo> </eb:math> but <jb:math xmlns:jb="http://www.w3.org/1998/Math/MathML" display="inline"> <jb:mi>Mp</jb:mi> <jb:mo stretchy="false">(</jb:mo> <jb:mn>2</jb:mn> <jb:mo>,</jb:mo> <jb:mi mathvariant="double-struck">Z</jb:mi> <jb:mo stretchy="false">)</jb:mo> </jb:math> , <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"> <ob:mi>GL</ob:mi> <ob:mo stretchy="false">(</ob:mo> <ob:mn>2</ob:mn> <ob:mo>,</ob:mo> <ob:mi mathvariant="double-struck">Z</ob:mi> <ob:mo stretchy="false">)</ob:mo> </ob:math> , and <tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML" display="inline"> <tb:msup> <tb:mi>Pin</tb:mi> <tb:mo>+</tb:mo> </tb:msup> <tb:mo stretchy="false">(</tb:mo> <tb:mi>GL</tb:mi> <tb:mo stretchy="false">(</tb:mo> <tb:mn>2</tb:mn> <tb:mo>,</tb:mo> <tb:mi mathvariant="double-struck">Z</tb:mi> <tb:mo stretchy="false">)</tb:mo> <tb:mo stretchy="false">)</tb:mo> <tb:mo>=</tb:mo> <tb:msup> <tb:mi>GL</tb:mi> <tb:mo>+</tb:mo> </tb:msup> <tb:mo stretchy="false">(</tb:mo> <tb:mn>2</tb:mn> <tb:mo>,</tb:mo> <tb:mi mathvariant="double-struck">Z</tb:mi> <tb:mo stretchy="false">)</tb:mo> </tb:math> .