Extended AdS spacetime without boundaries and entanglement without holography

We glue together two copies of pure anti–de Sitter (AdS) spacetime along their conformal boundaries creating a manifold without boundaries. The resulting space, which in dimension <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>d</a:mi> <a:mo>+</a:mo> <a:mn>2</a:mn> </a:math> we denote by <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mrow> <c:mi mathvariant="normal">A</c:mi> <c:mi mathvariant="normal">d</c:mi> <c:msubsup> <c:mrow> <c:mi mathvariant="normal">S</c:mi> </c:mrow> <c:mrow> <c:mo>±</c:mo> </c:mrow> <c:mrow> <c:mi>d</c:mi> <c:mo>+</c:mo> <c:mn>2</c:mn> </c:mrow> </c:msubsup> </c:mrow> </c:math> , has the topology of <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:msup> <h:mi>S</h:mi> <h:mn>2</h:mn> </h:msup> <h:mo>×</h:mo> <h:msup> <h:mi mathvariant="normal">Σ</h:mi> <h:mi>d</h:mi> </h:msup> </h:math> , where <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:msup> <k:mi mathvariant="normal">Σ</k:mi> <k:mi>d</k:mi> </k:msup> </k:math> is a <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"> <n:mi>d</n:mi> </n:math> manifold without boundary. Acting with <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"> <p:msub> <p:mi mathvariant="double-struck">Z</p:mi> <p:mi>n</p:mi> </p:msub> </p:math> on the <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:msup> <s:mi>S</s:mi> <s:mn>2</s:mn> </s:msup> </s:math> factor amounts to coupling a pair of membranes at the north and south poles of the two-sphere. Moreover, extending the domain of the two-sphere polar coordinate from <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:mo stretchy="false">[</u:mo> <u:mn>0</u:mn> <u:mo>,</u:mo> <u:mi>π</u:mi> <u:mo stretchy="false">]</u:mo> </u:math> to the interval <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:mo stretchy="false">[</y:mo> <y:mn>0</y:mn> <y:mo>,</y:mo> <y:mo stretchy="false">(</y:mo> <y:mn>2</y:mn> <y:mi>N</y:mi> <y:mo>−</y:mo> <y:mn>1</y:mn> <y:mo stretchy="false">)</y:mo> <y:mi>π</y:mi> <y:mo stretchy="false">]</y:mo> </y:math> , where <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"> <eb:mi>N</eb:mi> <eb:mo>&gt;</eb:mo> <eb:mn>1</eb:mn> </eb:math> , enables the coupling of one stack of <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"> <gb:mi>N</gb:mi> </gb:math> coincident membranes at each pole of the two-sphere ( <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"> <ib:mn>2</ib:mn> <ib:mi>N</ib:mi> </ib:math> membranes in total). Assuming the existence of a quantum gravity theory on the glued spacetime, we compute the classical approximation of the entanglement entropy across an entangling surface consisting of the two antipodal stacks of membranes. We find that the resulting entropy exhibits a boundary cutoff divergence that can be canceled by taking the limit of an infinite number of membranes. This large- <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"> <kb:mi>N</kb:mi> </kb:math> cancelation—possible only in the doubled, extended geometry without boundaries—yields a finite, universal quarter-area law. The calculation does not require details of the quantum theory other than its infrared limit, which we assume to be Einstein gravity.