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Title: Towards bulk metric reconstruction from extremal area variations

Abstract

The Ryu–Takayanagi and Hubeny–Rangamani–Takayanagi formulae suggest that bulk geometry emerges from the entanglement structure of the boundary theory. Using these formulae, we build on a result of Alexakis, Balehowsky, and Nachman to show that in four bulk dimensions, the entanglement entropies of boundary regions of disk topology uniquely fix the bulk metric in any region foliated by the corresponding HRT surfaces. More generally, for a bulk of any dimension , knowledge of the (variations of the) areas of two-dimensional boundary-anchored extremal surfaces of disk topology uniquely fixes the bulk metric wherever these surfaces reach. This result is covariant and not reliant on any symmetry assumptions; its applicability thus includes regions of strong dynamical gravity such as the early-time interior of black holes formed from collapse. While we only show uniqueness of the metric, the approach we present provides a clear path towards an explicit spacetime metric reconstruction.

Authors:
 [1]; ORCiD logo [2]; ORCiD logo [3]; ORCiD logo [4]
  1. Univ. of California, Berkeley, CA (United States); Brookhaven National Lab. (BNL), Upton, NY (United States)
  2. California Inst. of Technology (CalTech), Pasadena, CA (United States); Univ. of Maryland, College Park, MD (United States)
  3. McGill Univ., Montreal, QC (Canada)
  4. Arizona State Univ., Tempe, AZ (United States)
Publication Date:
Research Org.:
Brookhaven National Lab. (BNL), Upton, NY (United States); Arizona State Univ., Tempe, AZ (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Advanced Scientific Computing Research (SC-21); USDOE Office of Science (SC), High Energy Physics (HEP)
OSTI Identifier:
1566869
Alternate Identifier(s):
OSTI ID: 1597022
Report Number(s):
BNL-212128-2019-JAAM
Journal ID: ISSN 0264-9381; TRN: US2001006
Grant/Contract Number:  
SC0012704; SC0019470
Resource Type:
Accepted Manuscript
Journal Name:
Classical and Quantum Gravity
Additional Journal Information:
Journal Volume: 36; Journal Issue: 18; Journal ID: ISSN 0264-9381
Publisher:
IOP Publishing
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; AdS/CFT; bulk reconstruction; holographic entanglement entropy; boundary rigidity; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS

Citation Formats

Bao, Ning, Cao, ChunJun, Fischetti, Sebastian, and Keeler, Cynthia. Towards bulk metric reconstruction from extremal area variations. United States: N. p., 2019. Web. doi:10.1088/1361-6382/ab377f.
Bao, Ning, Cao, ChunJun, Fischetti, Sebastian, & Keeler, Cynthia. Towards bulk metric reconstruction from extremal area variations. United States. doi:10.1088/1361-6382/ab377f.
Bao, Ning, Cao, ChunJun, Fischetti, Sebastian, and Keeler, Cynthia. Tue . "Towards bulk metric reconstruction from extremal area variations". United States. doi:10.1088/1361-6382/ab377f. https://www.osti.gov/servlets/purl/1566869.
@article{osti_1566869,
title = {Towards bulk metric reconstruction from extremal area variations},
author = {Bao, Ning and Cao, ChunJun and Fischetti, Sebastian and Keeler, Cynthia},
abstractNote = {The Ryu–Takayanagi and Hubeny–Rangamani–Takayanagi formulae suggest that bulk geometry emerges from the entanglement structure of the boundary theory. Using these formulae, we build on a result of Alexakis, Balehowsky, and Nachman to show that in four bulk dimensions, the entanglement entropies of boundary regions of disk topology uniquely fix the bulk metric in any region foliated by the corresponding HRT surfaces. More generally, for a bulk of any dimension , knowledge of the (variations of the) areas of two-dimensional boundary-anchored extremal surfaces of disk topology uniquely fixes the bulk metric wherever these surfaces reach. This result is covariant and not reliant on any symmetry assumptions; its applicability thus includes regions of strong dynamical gravity such as the early-time interior of black holes formed from collapse. While we only show uniqueness of the metric, the approach we present provides a clear path towards an explicit spacetime metric reconstruction.},
doi = {10.1088/1361-6382/ab377f},
journal = {Classical and Quantum Gravity},
number = 18,
volume = 36,
place = {United States},
year = {2019},
month = {8}
}

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margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Cao, ChunJun; Carroll, Sean M.</span> </li> <li> Physical Review D, Vol. 97, Issue 8</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevD.97.086003" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevD.97.086003<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevLett.112.011601" target="_blank" rel="noopener noreferrer" class="name">Entanglement Tsunami: Universal Scaling in Holographic Thermalization<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2014-01-01">January 2014</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Liu, Hong; Suh, S. Josephine</span> </li> <li> Physical Review Letters, Vol. 112, Issue 1</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevLett.112.011601" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevLett.112.011601<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevD.86.065007" target="_blank" rel="noopener noreferrer" class="name">Entanglement renormalization and holography<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2012-09-01">September 2012</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Swingle, Brian</span> </li> <li> Physical Review D, Vol. 86, Issue 6</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevD.86.065007" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevD.86.065007<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1007/JHEP09(2015)130" target="_blank" rel="noopener noreferrer" class="name">The holographic entropy cone<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2015-09-01">September 2015</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Bao, Ning; Nezami, Sepehr; Ooguri, Hirosi</span> </li> <li> Journal of High Energy Physics, Vol. 2015, Issue 9</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1007/JHEP09(2015)130" class="text-muted" target="_blank" rel="noopener noreferrer">10.1007/JHEP09(2015)130<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> </div> <div class="pagination-container small"> <a class="pure-button prev page" href="#" rel="prev"><span class="fa fa-angle-left"></span></a><ul class="pagination d-inline-block" style="padding-left:.2em;"></ul><a class="pure-button next page" href="#" rel="next"><span class="fa fa-angle-right"></span></a> </div> </div> </div> <div class="col-sm-3 order-sm-3"> <ul class="nav nav-stacked"> <li class="active"><a href="" class="reference-type-filter tab-nav" data-tab="biblio-references" data-filter="type" data-pattern="*"><span class="fa fa-angle-right"></span> All References</a></li> <li class="small" style="margin-left:.75em; 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list-style-type: none;"> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="1" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/pages/biblio/1764602-more-bulk-from-extremal-area-variations" itemprop="url">More of the bulk from extremal area variations</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Journal Article</small><span class="authors"> <span class="author">Bao, Ning</span> ; <span class="author">Cao, ChunJun</span> ; <span class="author">Fischetti, Sebastian</span> ; <span class="author">...</span> <span class="text-muted pubdata"> - Classical and Quantum Gravity</span> </span> </div> <div class="abstract">It was shown recently in (Bao N et al 2019 Class. Quantum Grav. 36 185002), building on work of Alexakis, Balehowksy, and Nachman (Alexakis S et al 2017 arXiv:1711.09379), that the geometry of (some portion of) a manifold with boundary is uniquely fixed by the areas of a foliation of two-dimensional disk-shaped surfaces anchored to the boundary. In the context of AdS/CFT, this implies that (a portion of) a four-dimensional bulk geometry can be fixed uniquely from the entanglement entropies of disk-shaped boundary regions, subject to several constraints. Here, we loosen some of these constraints, in particular allowing for the<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> bulk foliation of extremal surfaces to be local and removing the constraint of disk topology; these generalizations ensure uniqueness of more of the deep bulk geometry by allowing for e.g. surfaces anchored on disconnected asymptotic boundaries, or HRT surfaces past a phase transition. We also explore in more depth the generality of the local foliation requirement, showing that even in a highly dynamical geometry like AdS-Vaidya it is satisfied.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink">DOI: <a class="misc doi-link " href="https://doi.org/10.1088/1361-6382/abcfd0" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1764602" data-product-type="Journal Article" data-product-subtype="AM" >10.1088/1361-6382/abcfd0</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="2" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/pages/biblio/1434555-new-constraints-holographic-entropy-from-maximin-go-theorem" itemprop="url">New constraints for holographic entropy from maximin: A no-go theorem</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Journal Article</small><span class="authors"> <span class="author">Rota, Massimiliano</span> ; <span class="author">Weinberg, Sean J.</span> <span class="text-muted pubdata"> - Physical Review D</span> </span> </div> <div class="abstract">The Ryu-Takayanagi (RT) formula for static spacetimes arising in the AdS/CFT correspondence satisfies inequalities that are not yet proven in the case of the Rangamani-Hubeny-Takayanagi (HRT) formula, which applies to general dynamical spacetimes. Wall’s maximin construction is the only known technique for extending inequalities of holographic entanglement entropy from the static to dynamical case. We show that this method currently has no further utility when dealing with inequalities for five or fewer regions. Despite this negative result, we propose the validity of one new inequality for covariant holographic entanglement entropy for five regions. This inequality, while not maximin provable, is<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> much weaker than many of the inequalities satisfied by the RT formula and should therefore be easier to prove. If it is valid, then there is strong evidence that holographic entanglement entropy plays a role in general spacetimes including those that arise in cosmology. Our new inequality is obtained by the assumption that the HRT formula satisfies every known balanced inequality obeyed by the Shannon entropies of classical probability distributions. This is a property that the RT formula has been shown to possess and which has been previously conjectured to hold for quantum mechanics in general.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <span class="fa fa-book text-muted" aria-hidden="true"></span> Cited by 3<div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink">DOI: <a class="misc doi-link " href="https://doi.org/10.1103/PhysRevD.97.086013" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1434555" data-product-type="Journal Article" data-product-subtype="PA" >10.1103/PhysRevD.97.086013</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="4" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/pages/biblio/1598352-tensor-networks-adic-fields-algebraic-curves-arithmetic-ads3-cft2-correspondence" itemprop="url">Tensor networks, <em>p</em>-adic fields, and algebraic curves: arithmetic and the AdS<sub>3</sub>/CFT<sub>2</sub> correspondence</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Journal Article</small><span class="authors"> <span class="author">Heydeman, Matthew</span> ; <span class="author">Marcolli, Matilde</span> ; <span class="author">Saberi, Ingmar A.</span> ; <span class="author">...</span> <span class="text-muted pubdata"> - Advances in Theoretical and Mathematical Physics</span> </span> </div> <div class="abstract">One of the many remarkable properties of conformal field theory in two dimensions is its connection to algebraic geometry. Since every compact Riemann surface is a projective algebraic curve, many constructions of interest in physics (which <i>a priori</i> depend on the analytic structure of the spacetime) can be formulated in purely algebraic language. This opens the door to interesting generalizations, obtained by taking another choice of field: for instance, the p-adics. Herein, we generalize the<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> $$\mathrm{AdS} / \mathrm{CFT}$$ correspondence according to this principle; the result is a formulation of holography in which the bulk geometry is discrete—the Bruhat–Tits tree for $$\mathrm{PGL} (2, \mathbb{Q}_p)$$—but the group of bulk isometries nonetheless agrees with that of boundary conformal transformations and is not broken by discretization. We anticipate that this forms the natural geometric setting for tensor networks that have been proposed as models of bulk reconstruction via quantum error correcting codes; in certain cases, geodesics in the Bruhat–Tits tree reproduce those constructed using quantum error correction. Other aspects of holography also hold: Standard holographic results for massive free scalar fields in a fixed background carry over to the tree, whose vertical direction can be interpreted as a renormalization-group scale for modes in the boundary $$\mathrm{CFT}$$. Higher-genus bulk geometries (the $$\mathrm{BTZ}$$ black hole and its generalizations) can be understood straightforwardly in our setting, and the Ryu–Takayanagi formula for the entanglement entropy appears naturally</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <span class="fa fa-book text-muted" aria-hidden="true"></span> Cited by 14<div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink">DOI: <a class="misc doi-link " href="https://doi.org/10.4310/ATMP.2018.v22.n1.a4" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1598352" data-product-type="Journal Article" data-product-subtype="AM" >10.4310/ATMP.2018.v22.n1.a4</a></span></li> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc fulltext-link " href="/pages/servlets/purl/1598352" title="Link to document media" target="_blank" rel="noopener" data-ostiid="1598352" data-product-type="Journal Article" data-product-subtype="AM" >Full Text Available</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="6" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/pages/biblio/1431154-bulk-entanglement-gravity-without-boundary-towards-finding-einsteins-equation-hilbert-space" itemprop="url">Bulk entanglement gravity without a boundary: Towards finding Einstein’s equation in Hilbert space</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Journal Article</small><span class="authors"> <span class="author">Cao, ChunJun</span> ; <span class="author">Carroll, Sean M.</span> <span class="text-muted pubdata"> - Physical Review D</span> </span> </div> <div class="abstract">We consider the emergence from quantum entanglement of spacetime geometry in a bulk region. For certain classes of quantum states in an appropriately factorized Hilbert space, a spatial geometry can be defined by associating areas along codimension-one surfaces with the entanglement entropy between either side. We show how radon transforms can be used to convert these data into a spatial metric. Under a particular set of assumptions, the time evolution of such a state traces out a four-dimensional spacetime geometry, and we argue using a modified version of Jacobson’s “entanglement equilibrium” that the geometry should obey Einstein’s equation in the<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> weak-field limit. Here, we also discuss how entanglement equilibrium is related to a generalization of the Ryu-Takayanagi formula in more general settings, and how quantum error correction can help specify the emergence map between the full quantum-gravity Hilbert space and the semiclassical limit of quantum fields propagating on a classical spacetime.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <span class="fa fa-book text-muted" aria-hidden="true"></span> Cited by 3<div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink">DOI: <a class="misc doi-link " href="https://doi.org/10.1103/PhysRevD.97.086003" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1431154" data-product-type="Journal Article" data-product-subtype="PA" >10.1103/PhysRevD.97.086003</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="7" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/pages/biblio/1434617-quantum-corrections-holographic-mutual-information" itemprop="url">Quantum corrections to holographic mutual information</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Journal Article</small><span class="authors"> <span class="author">Agon, Cesar A.</span> ; <span class="author">Faulkner, Thomas</span> <span class="text-muted pubdata"> - Journal of High Energy Physics (Online)</span> </span> </div> <div class="abstract">We compute the leading contribution to the mutual information (MI) of two disjoint spheres in the large distance regime for arbitrary conformal field theories (CFT) in any dimension. This is achieved by refining the operator product expansion method introduced by Cardy [1]. For CFTs with holographic duals the leading contribution to the MI at long distances comes from bulk quantum corrections to the Ryu-Takayanagi area formula. According to the FLM proposal [2] this equals the bulk MI between the two disjoint regions spanned by the boundary spheres and their corresponding minimal area surfaces. We compute this quantum correction and provide<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> in this way a non-trivial check of the FLM proposal.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <span class="fa fa-book text-muted" aria-hidden="true"></span> Cited by 12<div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink">DOI: <a class="misc doi-link " href="https://doi.org/10.1007/JHEP08(2016)118" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1434617" data-product-type="Journal Article" data-product-subtype="AM" >10.1007/JHEP08(2016)118</a></span></li> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc fulltext-link " href="/pages/servlets/purl/1434617" title="Link to document media" target="_blank" rel="noopener" data-ostiid="1434617" data-product-type="Journal Article" data-product-subtype="AM" >Full Text Available</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> </ul> </aside> </div> </section> </div> <div class="col-sm-3 order-sm-3"> <ul class="nav nav-stacked"> <li class="active"><a class="tab-nav disabled" data-tab="related" style="color: #636c72 !important; opacity: 1;"><span class="fa fa-angle-right"></span> Similar Records</a></li> </ul> </div> </div> </section> </div></div> </div> </div> </section> <footer class="" style="background-color:#f9f9f9; /* padding-top: 0.5rem; */"> <div class="footer-minor"> <div class="container"> <hr class="footer-separator" /> <div class="text-center" style="margin-top:1.25rem;"> <div class="pure-menu pure-menu-horizontal"> <ul class="pure-menu-list" id="footer-org-menu"> <li class="pure-menu-item"> <a href="https://energy.gov" target="_blank" rel="noopener noreferrer"> <img src="data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==" class="sprite sprite-footer-us-doe-min" alt="U.S. Department of Energy" /> </a> </li> <li class="pure-menu-item"> <a href="https://www.energy.gov/science/office-science" target="_blank" rel="noopener noreferrer"> <img src="data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==" class="sprite sprite-footer-office-of-science-min" alt="Office of Science" /> </a> </li> <li class="pure-menu-item"> <a href="/"> <img src="data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==" class="sprite sprite-footer-osti-min" alt="Office of Scientific and Technical Information" /> </a> </li> </ul> </div> </div> <div class="text-center small" style="margin-top:0.5em;margin-bottom:2.0rem;"> <div class="pure-menu pure-menu-horizontal"> <ul class="pure-menu-list"> <li class="pure-menu-item"><a href="/disclaim" class="pure-menu-link"><span class="fa fa-institution"></span> Website Policies <span class="hidden-xs">/ Important Links</span></a></li> <li class="pure-menu-item"><a href="/pages/contact" class="pure-menu-link"><span class="fa fa-comments-o"></span> Contact Us</a></li> <li class="d-block d-md-none"></li> <li class="pure-menu-item"><a href="https://www.facebook.com/ostigov" target="_blank" rel="noopener noreferrer" class="pure-menu-link social"><span class="fa fa-facebook" style=""></span></a></li> <li class="pure-menu-item"><a href="https://twitter.com/OSTIgov" target="_blank" rel="noopener noreferrer" class="pure-menu-link social"><span class="fa fa-twitter" style=""></span></a></li> <li class="pure-menu-item"><a href="https://www.youtube.com/user/ostigov" target="_blank" rel="noopener noreferrer" class="pure-menu-link social"><span class="fa fa-youtube-play" style=""></span></a></li> </ul> </div> </div> </div> </div> </footer> <link href="/pages/css/pages.fonts.200912.1307.css" rel="stylesheet"> <script src="/pages/js/pages.200912.1307.js"></script><noscript></noscript> <script defer src="/pages/js/pages.biblio.200912.1307.js"></script><noscript></noscript> <script defer src="/pages/js/lity.js"></script><noscript></noscript> <script async type="text/javascript" src="/pages/js/Universal-Federated-Analytics-Min.js?agency=DOE" id="_fed_an_ua_tag"></script><noscript></noscript> </body> <!-- DOE PAGES v.200912.1307 --> </html>