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# Stratification of the Generalized Gauge Orbit Space
Christian Fleischhack∗
Mathematisches Institut Institut fir Theoretische Physik Universitat Leipzig Universitat Leipzig Augustusplatz 10/11 Augustusplatz 10/11 04109 Leipzig, Germany 04109 Leipzig, Germany
Max-Planck-Institut fir Mathematik in den Naturwissenschaften Inselstraße 22-26 04103 Leipzig, Germany
January 5, 2000
# Abstract
The action of Ashtekar’s generalized gauge group $\overline{{\mathcal{G}}}$ on the space $\overline{{\mathcal{A}}}$ of generalized connections is investigated for compact structure groups $\mathbf{G}$ .
First a stratum is defined to be the set of all connections of one and the same gauge orbit type, i.e. the conjugacy class of the centralizer of the holonomy group. Then a slice theorem is proven on $\overline{{\mathcal{A}}}$ . This yields the openness of the strata. Afterwards, a denseness theorem is proven for the strata. Hence, $\overline{{\mathcal{A}}}$ is topologically regularly stratified by $\overline{{\mathcal{G}}}$ . These results coincide with those of Kondracki and Rogulski for Sobolev connections. As a by-product, we prove that the set of all gauge orbit types equals the set of all (conjugacy classes of) Howe subgroups of $\mathbf{G}$ . Finally, we show that the set of all gauge orbits with maximal type has the full induced Haar measure 1.
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<html><body>
<h1 data-bbox="83 55 516 79">Stratification of the Generalized Gauge Orbit Space </h1>
<p data-bbox="232 97 371 113">Christian Fleischhack∗ </p>
<p data-bbox="133 129 486 189">Mathematisches Institut Institut fir Theoretische Physik Universitat Leipzig Universitat Leipzig Augustusplatz 10/11 Augustusplatz 10/11 04109 Leipzig, Germany 04109 Leipzig, Germany </p>
<p data-bbox="135 203 466 248">Max-Planck-Institut fir Mathematik in den Naturwissenschaften Inselstraße 22-26 04103 Leipzig, Germany </p>
<p data-bbox="250 259 350 276">January 5, 2000 </p>
<h1 data-bbox="275 318 324 332">Abstract </h1>
<p data-bbox="92 339 507 366">The action of Ashtekar’s generalized gauge group $\overline{{\mathcal{G}}}$ on the space $\overline{{\mathcal{A}}}$ of generalized connections is investigated for compact structure groups $\mathbf{G}$ . </p>
<p data-bbox="92 367 508 474">First a stratum is defined to be the set of all connections of one and the same gauge orbit type, i.e. the conjugacy class of the centralizer of the holonomy group. Then a slice theorem is proven on $\overline{{\mathcal{A}}}$ . This yields the openness of the strata. Afterwards, a denseness theorem is proven for the strata. Hence, $\overline{{\mathcal{A}}}$ is topologically regularly stratified by $\overline{{\mathcal{G}}}$ . These results coincide with those of Kondracki and Rogulski for Sobolev connections. As a by-product, we prove that the set of all gauge orbit types equals the set of all (conjugacy classes of) Howe subgroups of $\mathbf{G}$ . Finally, we show that the set of all gauge orbits with maximal type has the full induced Haar measure 1. </p>
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0001008v1
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# 1 Introduction
For quite a long time the geometric structure of gauge theories has been investigated. A classical (pure) gauge theory consists of three basic objects: First the set $\mathcal{A}$ of smooth connections (”gauge fields”) in a principle fiber bundle, then the set $\mathcal{G}$ of all smooth gauge transforms, i.e. automorphisms of this bundle, and finally the action of $\mathcal{G}$ on $\mathcal{A}$ . Physically, two gauge fields that are related by a gauge transform describe one and the same situation. Thus, the space of all gauge orbits, i.e. elements in $\scriptstyle A/\mathcal G$ , is the configuration space for the gauge theory. Unfortunately, in contrast to $\overline{{\mathcal{A}}}$ , which is an affine space, the space $\scriptstyle A/\mathcal G$ has a very complicated structure: It is non-affin, non-compact and infinite-dimensional and it is not a manifold. This causes enormous problems, in particular, when one wants to quantize a gauge theory. One possible quantization method is the path integral quantization. Here one has to find an appropriate measure on the configuration space of the classical theory, hence a measure on $\mathcal{A}/\mathcal{G}$ . As just indicated, this is very hard to find. Thus, one has hoped for a better understanding of the structure of $\scriptstyle A/\mathcal G$ . However, up to now, results are quite rare.
About 20 years ago, the efforts were focussed on a related problem: The consideration of connections and gauge transforms that are contained in a certain Sobolev class (see, e.g., [16]). Now, $\mathcal{G}$ is a Hilbert-Lie group and acts smoothly on $\mathcal{A}$ . About 15 years ago, Kondracki and Rogulski [12] found lots of fundamental properties of this action. Perhaps, the most remarkable theorem they obtained was a slice theorem on $\mathcal{A}$ . This means, for every orbit $A\circ\mathcal{G}\subseteq A$ there is an equivariant retraction from a (so-called tubular) neighborhood of $A$ onto $A\circ{\mathcal{G}}$ . Using this theorem they could clarify the structure of the so-called strata. A stratum contains all connections that have the same, fixed type, i.e. the same (conjugacy class of the) stabilizer under the action of $\mathcal{G}$ . Using a denseness theorem for the strata, Kondracki and Rogulski proved that the space $\mathcal{A}$ is regularly stratified by the action of $\mathcal{G}$ . In particular, all the strata are smooth submanifolds of $\mathcal{A}$ .
Despite these results the mathematically rigorous construction of a measure on $\scriptstyle A/\mathcal{G}$ has not been achieved. This problem was solved – at least preliminary – by Ashtekar et al. [1, 2], but, however, not for $\mathcal{A}/\mathcal{G}$ itself. Their idea was to drop simply all smoothness conditions for the connections and gauge transforms. In detail, they first used the fact that a connection can always be reconstructed uniquely by its parallel transports. On the other hand, these parallel transports can be identified with an assignment of elements of the structure group $\mathbf{G}$ to the paths in the base manifold $M$ such that the concatenation of paths corresponds to the product of these group elements. It is intuitively clear that for smooth connections the parallel transports additionally depend smoothly on the paths [14]. But now this restriction is removed for the generalized connections. They are only homomorphisms from the groupoid $\mathcal{P}$ of paths to the structure group $\mathbf{G}$ . Analogously, the set $\overline{{g}}$ of generalized gauge transforms collects all functions from $M$ to $\mathbf{G}$ . Now the action of $\mathcal{G}$ to $\overline{{\mathcal{A}}}$ is defined purely algebraically. Given $\overline{{\mathcal{A}}}$ and $\mathcal{G}$ the topologies induced by the topology of $\mathbf{G}$ , one sees that, for compact $\mathbf{G}$ , these spaces are again compact. This guarantees the existence of a natural induced Haar measure on $\overline{{\mathcal{A}}}$ and $\overline{{\mathcal{A}}}/\overline{{\mathcal{G}}}$ , the new configuration space for the path integral quantization.
Both from the mathematical and from the physical point of view it is very interesting how the ”classical” regular gauge theories are related to the generalized formulation in the Ashtekar framework. First of all, it has been proven that $\mathcal{A}$ and $\mathcal{G}$ are dense subsets in $\overline{{\mathcal{A}}}$ and $\overline{{g}}$ , respectively [17]. Furthermore, $\mathcal{A}$ is contained in a set of induced Haar measure zero [15]. These properties coincide exactly with the experiences known from the Wiener or Feynman path integral. Then the Wilson loop expectation values have been determined for the twodimensional pure Yang-Mills theory [5, 11] – in coincidence with the known results in the standard framework. In the present paper we continue the investigations on how the results of Kondracki and Rogulski can be extended to the Ashtekar framework. In a previous paper [9] we have already shown that the gauge orbit type is determined by the centralizer of the holonomy group. This closely related to the observations of Kondracki and Sadowski [13]. In the present paper we are going to prove that there is a slice theorem and a denseness theorem for the space of connections in the Ashtekar framework as well. However, our methods are completely different to those of Kondracki and Rogulski.
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<html><body>
<h1 data-bbox="63 10 200 29">1 Introduction </h1>
<p data-bbox="63 41 538 228">For quite a long time the geometric structure of gauge theories has been investigated. A classical (pure) gauge theory consists of three basic objects: First the set $\mathcal{A}$ of smooth connections (”gauge fields”) in a principle fiber bundle, then the set $\mathcal{G}$ of all smooth gauge transforms, i.e. automorphisms of this bundle, and finally the action of $\mathcal{G}$ on $\mathcal{A}$ . Physically, two gauge fields that are related by a gauge transform describe one and the same situation. Thus, the space of all gauge orbits, i.e. elements in $\scriptstyle A/\mathcal G$ , is the configuration space for the gauge theory. Unfortunately, in contrast to $\overline{{\mathcal{A}}}$ , which is an affine space, the space $\scriptstyle A/\mathcal G$ has a very complicated structure: It is non-affin, non-compact and infinite-dimensional and it is not a manifold. This causes enormous problems, in particular, when one wants to quantize a gauge theory. One possible quantization method is the path integral quantization. Here one has to find an appropriate measure on the configuration space of the classical theory, hence a measure on $\mathcal{A}/\mathcal{G}$ . As just indicated, this is very hard to find. Thus, one has hoped for a better understanding of the structure of $\scriptstyle A/\mathcal G$ . However, up to now, results are quite rare. </p>
<p data-bbox="63 229 538 388">About 20 years ago, the efforts were focussed on a related problem: The consideration of connections and gauge transforms that are contained in a certain Sobolev class (see, e.g., [16]). Now, $\mathcal{G}$ is a Hilbert-Lie group and acts smoothly on $\mathcal{A}$ . About 15 years ago, Kondracki and Rogulski [12] found lots of fundamental properties of this action. Perhaps, the most remarkable theorem they obtained was a slice theorem on $\mathcal{A}$ . This means, for every orbit $A\circ\mathcal{G}\subseteq A$ there is an equivariant retraction from a (so-called tubular) neighborhood of $A$ onto $A\circ{\mathcal{G}}$ . Using this theorem they could clarify the structure of the so-called strata. A stratum contains all connections that have the same, fixed type, i.e. the same (conjugacy class of the) stabilizer under the action of $\mathcal{G}$ . Using a denseness theorem for the strata, Kondracki and Rogulski proved that the space $\mathcal{A}$ is regularly stratified by the action of $\mathcal{G}$ . In particular, all the strata are smooth submanifolds of $\mathcal{A}$ . </p>
<p data-bbox="63 388 537 605">Despite these results the mathematically rigorous construction of a measure on $\scriptstyle A/\mathcal{G}$ has not been achieved. This problem was solved – at least preliminary – by Ashtekar et al. [1, 2], but, however, not for $\mathcal{A}/\mathcal{G}$ itself. Their idea was to drop simply all smoothness conditions for the connections and gauge transforms. In detail, they first used the fact that a connection can always be reconstructed uniquely by its parallel transports. On the other hand, these parallel transports can be identified with an assignment of elements of the structure group $\mathbf{G}$ to the paths in the base manifold $M$ such that the concatenation of paths corresponds to the product of these group elements. It is intuitively clear that for smooth connections the parallel transports additionally depend smoothly on the paths [14]. But now this restriction is removed for the generalized connections. They are only homomorphisms from the groupoid $\mathcal{P}$ of paths to the structure group $\mathbf{G}$ . Analogously, the set $\overline{{g}}$ of generalized gauge transforms collects all functions from $M$ to $\mathbf{G}$ . Now the action of $\mathcal{G}$ to $\overline{{\mathcal{A}}}$ is defined purely algebraically. Given $\overline{{\mathcal{A}}}$ and $\mathcal{G}$ the topologies induced by the topology of $\mathbf{G}$ , one sees that, for compact $\mathbf{G}$ , these spaces are again compact. This guarantees the existence of a natural induced Haar measure on $\overline{{\mathcal{A}}}$ and $\overline{{\mathcal{A}}}/\overline{{\mathcal{G}}}$ , the new configuration space for the path integral quantization. </p>
<p data-bbox="64 605 537 677">Both from the mathematical and from the physical point of view it is very interesting how the ”classical” regular gauge theories are related to the generalized formulation in the Ashtekar framework. First of all, it has been proven that $\mathcal{A}$ and $\mathcal{G}$ are dense subsets in $\overline{{\mathcal{A}}}$ and $\overline{{g}}$ , respectively [17]. Furthermore, $\mathcal{A}$ is contained in a set of induced Haar measure zero [15]. These properties coincide exactly with the experiences known from the Wiener or Feynman path integral. Then the Wilson loop expectation values have been determined for the twodimensional pure Yang-Mills theory [5, 11] – in coincidence with the known results in the standard framework. In the present paper we continue the investigations on how the results of Kondracki and Rogulski can be extended to the Ashtekar framework. In a previous paper [9] we have already shown that the gauge orbit type is determined by the centralizer of the holonomy group. This closely related to the observations of Kondracki and Sadowski [13]. In the present paper we are going to prove that there is a slice theorem and a denseness theorem for the space of connections in the Ashtekar framework as well. However, our methods are completely different to those of Kondracki and Rogulski. </p>
</body></html>
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0001008v1
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This causes enormous problems, in particular, when one wants to quantize a gauge theory. One possible quantization method is the path integral quantization. Here one has to find an appropriate measure on the configuration space of the classical theory, hence a measure on $\\mathcal{A}/\\mathcal{G}$ . As just indicated, this is very hard to find. Thus, one has hoped for a better understanding of the structure of $\\scriptstyle A/\\mathcal G$ . However, up to now, results are quite rare. ", "page_idx": 1}, {"type": "text", "text": "About 20 years ago, the efforts were focussed on a related problem: The consideration of connections and gauge transforms that are contained in a certain Sobolev class (see, e.g., [16]). Now, $\\mathcal{G}$ is a Hilbert-Lie group and acts smoothly on $\\mathcal{A}$ . About 15 years ago, Kondracki and Rogulski [12] found lots of fundamental properties of this action. Perhaps, the most remarkable theorem they obtained was a slice theorem on $\\mathcal{A}$ . This means, for every orbit $A\\circ\\mathcal{G}\\subseteq A$ there is an equivariant retraction from a (so-called tubular) neighborhood of $A$ onto $A\\circ{\\mathcal{G}}$ . Using this theorem they could clarify the structure of the so-called strata. A stratum contains all connections that have the same, fixed type, i.e. the same (conjugacy class of the) stabilizer under the action of $\\mathcal{G}$ . Using a denseness theorem for the strata, Kondracki and Rogulski proved that the space $\\mathcal{A}$ is regularly stratified by the action of $\\mathcal{G}$ . In particular, all the strata are smooth submanifolds of $\\mathcal{A}$ . ", "page_idx": 1}, {"type": "text", "text": "Despite these results the mathematically rigorous construction of a measure on $\\scriptstyle A/\\mathcal{G}$ has not been achieved. This problem was solved – at least preliminary – by Ashtekar et al. [1, 2], but, however, not for $\\mathcal{A}/\\mathcal{G}$ itself. Their idea was to drop simply all smoothness conditions for the connections and gauge transforms. In detail, they first used the fact that a connection can always be reconstructed uniquely by its parallel transports. On the other hand, these parallel transports can be identified with an assignment of elements of the structure group $\\mathbf{G}$ to the paths in the base manifold $M$ such that the concatenation of paths corresponds to the product of these group elements. It is intuitively clear that for smooth connections the parallel transports additionally depend smoothly on the paths [14]. But now this restriction is removed for the generalized connections. They are only homomorphisms from the groupoid $\\mathcal{P}$ of paths to the structure group $\\mathbf{G}$ . Analogously, the set $\\overline{{g}}$ of generalized gauge transforms collects all functions from $M$ to $\\mathbf{G}$ . Now the action of $\\mathcal{G}$ to $\\overline{{\\mathcal{A}}}$ is defined purely algebraically. Given $\\overline{{\\mathcal{A}}}$ and $\\mathcal{G}$ the topologies induced by the topology of $\\mathbf{G}$ , one sees that, for compact $\\mathbf{G}$ , these spaces are again compact. This guarantees the existence of a natural induced Haar measure on $\\overline{{\\mathcal{A}}}$ and $\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}$ , the new configuration space for the path integral quantization. ", "page_idx": 1}, {"type": "text", "text": "Both from the mathematical and from the physical point of view it is very interesting how the ”classical” regular gauge theories are related to the generalized formulation in the Ashtekar framework. First of all, it has been proven that $\\mathcal{A}$ and $\\mathcal{G}$ are dense subsets in $\\overline{{\\mathcal{A}}}$ and $\\overline{{g}}$ , respectively [17]. Furthermore, $\\mathcal{A}$ is contained in a set of induced Haar measure zero [15]. These properties coincide exactly with the experiences known from the Wiener or Feynman path integral. Then the Wilson loop expectation values have been determined for the twodimensional pure Yang-Mills theory [5, 11] – in coincidence with the known results in the standard framework. In the present paper we continue the investigations on how the results of Kondracki and Rogulski can be extended to the Ashtekar framework. In a previous paper [9] we have already shown that the gauge orbit type is determined by the centralizer of the holonomy group. This closely related to the observations of Kondracki and Sadowski [13]. In the present paper we are going to prove that there is a slice theorem and a denseness theorem for the space of connections in the Ashtekar framework as well. However, our methods are completely different to those of Kondracki and Rogulski. ", "page_idx": 1}]
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{"preproc_blocks": [{"type": "title", "bbox": [63, 10, 200, 29], "lines": [{"bbox": [63, 13, 199, 29], "spans": [{"bbox": [63, 15, 73, 28], "score": 1.0, "content": "1", "type": "text"}, {"bbox": [90, 13, 199, 29], "score": 1.0, "content": "Introduction", "type": "text"}], "index": 0}], "index": 0}, {"type": "text", "bbox": [63, 41, 538, 228], "lines": [{"bbox": [62, 43, 537, 58], "spans": [{"bbox": [62, 43, 537, 58], "score": 1.0, "content": "For quite a long time the geometric structure of gauge theories has been investigated. A", "type": "text"}], "index": 1}, {"bbox": [62, 58, 538, 73], "spans": [{"bbox": [62, 58, 445, 73], "score": 1.0, "content": "classical (pure) gauge theory consists of three basic objects: First the set ", "type": "text"}, {"bbox": [445, 60, 455, 68], "score": 0.89, "content": "\\mathcal{A}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [455, 58, 538, 73], "score": 1.0, "content": " of smooth con-", "type": "text"}], "index": 2}, {"bbox": [62, 71, 537, 87], "spans": [{"bbox": [62, 71, 416, 87], "score": 1.0, "content": "nections (”gauge fields”) in a principle fiber bundle, then the set ", "type": "text"}, {"bbox": [417, 74, 425, 84], "score": 0.9, "content": "\\mathcal{G}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [425, 71, 537, 87], "score": 1.0, "content": " of all smooth gauge", "type": "text"}], "index": 3}, {"bbox": [62, 87, 536, 101], "spans": [{"bbox": [62, 87, 434, 101], "score": 1.0, "content": "transforms, i.e. automorphisms of this bundle, and finally the action of ", "type": "text"}, {"bbox": [434, 88, 442, 98], "score": 0.91, "content": "\\mathcal{G}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [443, 87, 462, 101], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [463, 88, 473, 97], "score": 0.9, "content": "\\mathcal{A}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [473, 87, 536, 101], "score": 1.0, "content": ". Physically,", "type": "text"}], "index": 4}, {"bbox": [62, 101, 538, 116], "spans": [{"bbox": [62, 101, 538, 116], "score": 1.0, "content": "two gauge fields that are related by a gauge transform describe one and the same situation.", "type": "text"}], "index": 5}, {"bbox": [63, 116, 537, 129], "spans": [{"bbox": [63, 116, 334, 129], "score": 1.0, "content": "Thus, the space of all gauge orbits, i.e. elements in ", "type": "text"}, {"bbox": [334, 117, 358, 129], "score": 0.94, "content": "\\scriptstyle A/\\mathcal G", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [358, 116, 537, 129], "score": 1.0, "content": ", is the configuration space for the", "type": "text"}], "index": 6}, {"bbox": [61, 130, 538, 145], "spans": [{"bbox": [61, 130, 291, 145], "score": 1.0, "content": "gauge theory. Unfortunately, in contrast to ", "type": "text"}, {"bbox": [292, 130, 302, 141], "score": 0.89, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [302, 130, 491, 145], "score": 1.0, "content": ", which is an affine space, the space ", "type": "text"}, {"bbox": [491, 131, 515, 144], "score": 0.94, "content": "\\scriptstyle A/\\mathcal G", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [515, 130, 538, 145], "score": 1.0, "content": " has", "type": "text"}], "index": 7}, {"bbox": [62, 145, 538, 159], "spans": [{"bbox": [62, 145, 538, 159], "score": 1.0, "content": "a very complicated structure: It is non-affin, non-compact and infinite-dimensional and it is", "type": "text"}], "index": 8}, {"bbox": [61, 158, 538, 174], "spans": [{"bbox": [61, 158, 538, 174], "score": 1.0, "content": "not a manifold. This causes enormous problems, in particular, when one wants to quantize a", "type": "text"}], "index": 9}, {"bbox": [61, 173, 538, 188], "spans": [{"bbox": [61, 173, 538, 188], "score": 1.0, "content": "gauge theory. One possible quantization method is the path integral quantization. Here one", "type": "text"}], "index": 10}, {"bbox": [62, 188, 538, 203], "spans": [{"bbox": [62, 188, 538, 203], "score": 1.0, "content": "has to find an appropriate measure on the configuration space of the classical theory, hence", "type": "text"}], "index": 11}, {"bbox": [61, 202, 538, 217], "spans": [{"bbox": [61, 202, 136, 217], "score": 1.0, "content": "a measure on ", "type": "text"}, {"bbox": [136, 204, 159, 216], "score": 0.95, "content": "\\mathcal{A}/\\mathcal{G}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [160, 202, 538, 217], "score": 1.0, "content": ". As just indicated, this is very hard to find. Thus, one has hoped for a", "type": "text"}], "index": 12}, {"bbox": [62, 216, 523, 231], "spans": [{"bbox": [62, 216, 271, 231], "score": 1.0, "content": "better understanding of the structure of ", "type": "text"}, {"bbox": [272, 218, 295, 230], "score": 0.94, "content": "\\scriptstyle A/\\mathcal G", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [295, 216, 523, 231], "score": 1.0, "content": ". However, up to now, results are quite rare.", "type": "text"}], "index": 13}], "index": 7}, {"type": "text", "bbox": [63, 229, 538, 388], "lines": [{"bbox": [63, 231, 539, 246], "spans": [{"bbox": [63, 231, 539, 246], "score": 1.0, "content": "About 20 years ago, the efforts were focussed on a related problem: The consideration of", "type": "text"}], "index": 14}, {"bbox": [61, 244, 538, 263], "spans": [{"bbox": [61, 244, 538, 263], "score": 1.0, "content": "connections and gauge transforms that are contained in a certain Sobolev class (see, e.g.,", "type": "text"}], "index": 15}, {"bbox": [63, 261, 537, 275], "spans": [{"bbox": [63, 261, 123, 275], "score": 1.0, "content": "[16]). Now, ", "type": "text"}, {"bbox": [124, 262, 131, 272], "score": 0.9, "content": "\\mathcal{G}", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [132, 261, 360, 275], "score": 1.0, "content": " is a Hilbert-Lie group and acts smoothly on ", "type": "text"}, {"bbox": [360, 262, 370, 271], "score": 0.89, "content": "\\mathcal{A}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [370, 261, 537, 275], "score": 1.0, "content": ". About 15 years ago, Kondracki", "type": "text"}], "index": 16}, {"bbox": [63, 275, 538, 289], "spans": [{"bbox": [63, 275, 538, 289], "score": 1.0, "content": "and Rogulski [12] found lots of fundamental properties of this action. Perhaps, the most", "type": "text"}], "index": 17}, {"bbox": [62, 289, 537, 303], "spans": [{"bbox": [62, 289, 371, 303], "score": 1.0, "content": "remarkable theorem they obtained was a slice theorem on ", "type": "text"}, {"bbox": [371, 291, 381, 299], "score": 0.9, "content": "\\mathcal{A}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [382, 289, 537, 303], "score": 1.0, "content": ". This means, for every orbit", "type": "text"}], "index": 18}, {"bbox": [63, 303, 538, 319], "spans": [{"bbox": [63, 305, 113, 316], "score": 0.93, "content": "A\\circ\\mathcal{G}\\subseteq A", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [113, 303, 501, 319], "score": 1.0, "content": " there is an equivariant retraction from a (so-called tubular) neighborhood of ", "type": "text"}, {"bbox": [501, 305, 510, 314], "score": 0.89, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [511, 303, 538, 319], "score": 1.0, "content": " onto", "type": "text"}], "index": 19}, {"bbox": [63, 318, 537, 332], "spans": [{"bbox": [63, 320, 91, 330], "score": 0.92, "content": "A\\circ{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [91, 318, 537, 332], "score": 1.0, "content": ". Using this theorem they could clarify the structure of the so-called strata. A stratum", "type": "text"}], "index": 20}, {"bbox": [61, 332, 538, 348], "spans": [{"bbox": [61, 332, 538, 348], "score": 1.0, "content": "contains all connections that have the same, fixed type, i.e. the same (conjugacy class of the)", "type": "text"}], "index": 21}, {"bbox": [62, 347, 538, 361], "spans": [{"bbox": [62, 347, 218, 361], "score": 1.0, "content": "stabilizer under the action of ", "type": "text"}, {"bbox": [219, 349, 227, 358], "score": 0.91, "content": "\\mathcal{G}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [227, 347, 538, 361], "score": 1.0, "content": ". Using a denseness theorem for the strata, Kondracki and", "type": "text"}], "index": 22}, {"bbox": [61, 360, 538, 376], "spans": [{"bbox": [61, 360, 226, 376], "score": 1.0, "content": "Rogulski proved that the space ", "type": "text"}, {"bbox": [226, 363, 236, 372], "score": 0.91, "content": "\\mathcal{A}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [236, 360, 435, 376], "score": 1.0, "content": " is regularly stratified by the action of ", "type": "text"}, {"bbox": [435, 363, 443, 373], "score": 0.92, "content": "\\mathcal{G}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [443, 360, 538, 376], "score": 1.0, "content": ". In particular, all", "type": "text"}], "index": 23}, {"bbox": [63, 376, 276, 390], "spans": [{"bbox": [63, 376, 262, 390], "score": 1.0, "content": "the strata are smooth submanifolds of ", "type": "text"}, {"bbox": [262, 378, 272, 386], "score": 0.9, "content": "\\mathcal{A}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [272, 376, 276, 390], "score": 1.0, "content": ".", "type": "text"}], "index": 24}], "index": 19}, {"type": "text", "bbox": [63, 388, 537, 605], "lines": [{"bbox": [63, 390, 537, 405], "spans": [{"bbox": [63, 390, 471, 405], "score": 1.0, "content": "Despite these results the mathematically rigorous construction of a measure on ", "type": "text"}, {"bbox": [471, 391, 495, 403], "score": 0.94, "content": "\\scriptstyle A/\\mathcal{G}", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [495, 390, 537, 405], "score": 1.0, "content": " has not", "type": "text"}], "index": 25}, {"bbox": [63, 405, 536, 418], "spans": [{"bbox": [63, 405, 536, 418], "score": 1.0, "content": "been achieved. This problem was solved – at least preliminary – by Ashtekar et al. [1, 2],", "type": "text"}], "index": 26}, {"bbox": [61, 419, 537, 433], "spans": [{"bbox": [61, 419, 173, 433], "score": 1.0, "content": "but, however, not for ", "type": "text"}, {"bbox": [173, 420, 196, 433], "score": 0.94, "content": "\\mathcal{A}/\\mathcal{G}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [197, 419, 537, 433], "score": 1.0, "content": " itself. Their idea was to drop simply all smoothness conditions for", "type": "text"}], "index": 27}, {"bbox": [62, 434, 537, 448], "spans": [{"bbox": [62, 434, 537, 448], "score": 1.0, "content": "the connections and gauge transforms. In detail, they first used the fact that a connection", "type": "text"}], "index": 28}, {"bbox": [63, 449, 537, 462], "spans": [{"bbox": [63, 449, 537, 462], "score": 1.0, "content": "can always be reconstructed uniquely by its parallel transports. On the other hand, these", "type": "text"}], "index": 29}, {"bbox": [62, 462, 537, 477], "spans": [{"bbox": [62, 462, 537, 477], "score": 1.0, "content": "parallel transports can be identified with an assignment of elements of the structure group", "type": "text"}], "index": 30}, {"bbox": [63, 476, 538, 491], "spans": [{"bbox": [63, 479, 74, 487], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [74, 476, 252, 491], "score": 1.0, "content": " to the paths in the base manifold ", "type": "text"}, {"bbox": [253, 479, 265, 487], "score": 0.91, "content": "M", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [266, 476, 538, 491], "score": 1.0, "content": " such that the concatenation of paths corresponds to", "type": "text"}], "index": 31}, {"bbox": [63, 491, 538, 506], "spans": [{"bbox": [63, 491, 538, 506], "score": 1.0, "content": "the product of these group elements. It is intuitively clear that for smooth connections the", "type": "text"}], "index": 32}, {"bbox": [63, 507, 537, 520], "spans": [{"bbox": [63, 507, 537, 520], "score": 1.0, "content": "parallel transports additionally depend smoothly on the paths [14]. But now this restriction", "type": "text"}], "index": 33}, {"bbox": [62, 520, 536, 533], "spans": [{"bbox": [62, 520, 536, 533], "score": 1.0, "content": "is removed for the generalized connections. They are only homomorphisms from the groupoid", "type": "text"}], "index": 34}, {"bbox": [63, 534, 537, 549], "spans": [{"bbox": [63, 537, 72, 545], "score": 0.91, "content": "\\mathcal{P}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [73, 534, 237, 549], "score": 1.0, "content": " of paths to the structure group ", "type": "text"}, {"bbox": [238, 536, 249, 545], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [249, 534, 363, 549], "score": 1.0, "content": ". Analogously, the set ", "type": "text"}, {"bbox": [363, 534, 371, 546], "score": 0.91, "content": "\\overline{{g}}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [372, 534, 537, 549], "score": 1.0, "content": " of generalized gauge transforms", "type": "text"}], "index": 35}, {"bbox": [62, 548, 537, 564], "spans": [{"bbox": [62, 548, 198, 564], "score": 1.0, "content": "collects all functions from ", "type": "text"}, {"bbox": [198, 551, 211, 559], "score": 0.91, "content": "M", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [211, 548, 228, 564], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [228, 551, 239, 560], "score": 0.88, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [240, 548, 342, 564], "score": 1.0, "content": ". Now the action of ", "type": "text"}, {"bbox": [342, 549, 350, 561], "score": 0.91, "content": "\\mathcal{G}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [351, 548, 368, 564], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [368, 549, 378, 560], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [378, 548, 537, 564], "score": 1.0, "content": " is defined purely algebraically.", "type": "text"}], "index": 36}, {"bbox": [62, 563, 537, 578], "spans": [{"bbox": [62, 563, 97, 578], "score": 1.0, "content": "Given ", "type": "text"}, {"bbox": [97, 564, 107, 574], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [107, 563, 133, 578], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [134, 564, 142, 575], "score": 0.91, "content": "\\mathcal{G}", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [142, 563, 364, 578], "score": 1.0, "content": " the topologies induced by the topology of ", "type": "text"}, {"bbox": [364, 565, 375, 574], "score": 0.9, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [375, 563, 522, 578], "score": 1.0, "content": ", one sees that, for compact ", "type": "text"}, {"bbox": [522, 565, 533, 574], "score": 0.88, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [533, 563, 537, 578], "score": 1.0, "content": ",", "type": "text"}], "index": 37}, {"bbox": [62, 578, 537, 592], "spans": [{"bbox": [62, 578, 537, 592], "score": 1.0, "content": "these spaces are again compact. This guarantees the existence of a natural induced Haar", "type": "text"}], "index": 38}, {"bbox": [62, 591, 513, 608], "spans": [{"bbox": [62, 591, 124, 608], "score": 1.0, "content": "measure on ", "type": "text"}, {"bbox": [125, 592, 135, 603], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [135, 591, 161, 608], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [161, 592, 185, 606], "score": 0.94, "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 14, "width": 24}, {"bbox": [185, 591, 513, 608], "score": 1.0, "content": ", the new configuration space for the path integral quantization.", "type": "text"}], "index": 39}], "index": 32}, {"type": "text", "bbox": [64, 605, 537, 677], "lines": [{"bbox": [62, 606, 537, 622], "spans": [{"bbox": [62, 606, 537, 622], "score": 1.0, "content": "Both from the mathematical and from the physical point of view it is very interesting how the", "type": "text"}], "index": 40}, {"bbox": [61, 619, 537, 637], "spans": [{"bbox": [61, 619, 537, 637], "score": 1.0, "content": "”classical” regular gauge theories are related to the generalized formulation in the Ashtekar", "type": "text"}], "index": 41}, {"bbox": [61, 634, 537, 650], "spans": [{"bbox": [61, 634, 324, 650], "score": 1.0, "content": "framework. First of all, it has been proven that ", "type": "text"}, {"bbox": [324, 637, 334, 646], "score": 0.9, "content": "\\mathcal{A}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [334, 634, 362, 650], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [362, 637, 370, 647], "score": 0.91, "content": "\\mathcal{G}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [371, 634, 486, 650], "score": 1.0, "content": " are dense subsets in ", "type": "text"}, {"bbox": [486, 636, 496, 646], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [496, 634, 524, 650], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [524, 636, 533, 647], "score": 0.89, "content": "\\overline{{g}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [533, 634, 537, 650], "score": 1.0, "content": ",", "type": "text"}], "index": 42}, {"bbox": [63, 650, 536, 664], "spans": [{"bbox": [63, 650, 228, 664], "score": 1.0, "content": "respectively [17]. Furthermore, ", "type": "text"}, {"bbox": [229, 652, 239, 660], "score": 0.89, "content": "\\mathcal{A}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [239, 650, 536, 664], "score": 1.0, "content": " is contained in a set of induced Haar measure zero [15].", "type": "text"}], "index": 43}, {"bbox": [63, 664, 538, 680], "spans": [{"bbox": [63, 664, 538, 680], "score": 1.0, "content": "These properties coincide exactly with the experiences known from the Wiener or Feynman", "type": "text"}], "index": 44}], "index": 42}], "layout_bboxes": [], "page_idx": 1, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [295, 704, 303, 715], "lines": [{"bbox": [295, 705, 304, 718], "spans": [{"bbox": [295, 705, 304, 718], "score": 1.0, "content": "2", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "title", "bbox": [63, 10, 200, 29], "lines": [{"bbox": [63, 13, 199, 29], "spans": [{"bbox": [63, 15, 73, 28], "score": 1.0, "content": "1", "type": "text"}, {"bbox": [90, 13, 199, 29], "score": 1.0, "content": "Introduction", "type": "text"}], "index": 0}], "index": 0}, {"type": "text", "bbox": [63, 41, 538, 228], "lines": [{"bbox": [62, 43, 537, 58], "spans": [{"bbox": [62, 43, 537, 58], "score": 1.0, "content": "For quite a long time the geometric structure of gauge theories has been investigated. A", "type": "text"}], "index": 1}, {"bbox": [62, 58, 538, 73], "spans": [{"bbox": [62, 58, 445, 73], "score": 1.0, "content": "classical (pure) gauge theory consists of three basic objects: First the set ", "type": "text"}, {"bbox": [445, 60, 455, 68], "score": 0.89, "content": "\\mathcal{A}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [455, 58, 538, 73], "score": 1.0, "content": " of smooth con-", "type": "text"}], "index": 2}, {"bbox": [62, 71, 537, 87], "spans": [{"bbox": [62, 71, 416, 87], "score": 1.0, "content": "nections (”gauge fields”) in a principle fiber bundle, then the set ", "type": "text"}, {"bbox": [417, 74, 425, 84], "score": 0.9, "content": "\\mathcal{G}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [425, 71, 537, 87], "score": 1.0, "content": " of all smooth gauge", "type": "text"}], "index": 3}, {"bbox": [62, 87, 536, 101], "spans": [{"bbox": [62, 87, 434, 101], "score": 1.0, "content": "transforms, i.e. automorphisms of this bundle, and finally the action of ", "type": "text"}, {"bbox": [434, 88, 442, 98], "score": 0.91, "content": "\\mathcal{G}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [443, 87, 462, 101], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [463, 88, 473, 97], "score": 0.9, "content": "\\mathcal{A}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [473, 87, 536, 101], "score": 1.0, "content": ". Physically,", "type": "text"}], "index": 4}, {"bbox": [62, 101, 538, 116], "spans": [{"bbox": [62, 101, 538, 116], "score": 1.0, "content": "two gauge fields that are related by a gauge transform describe one and the same situation.", "type": "text"}], "index": 5}, {"bbox": [63, 116, 537, 129], "spans": [{"bbox": [63, 116, 334, 129], "score": 1.0, "content": "Thus, the space of all gauge orbits, i.e. elements in ", "type": "text"}, {"bbox": [334, 117, 358, 129], "score": 0.94, "content": "\\scriptstyle A/\\mathcal G", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [358, 116, 537, 129], "score": 1.0, "content": ", is the configuration space for the", "type": "text"}], "index": 6}, {"bbox": [61, 130, 538, 145], "spans": [{"bbox": [61, 130, 291, 145], "score": 1.0, "content": "gauge theory. Unfortunately, in contrast to ", "type": "text"}, {"bbox": [292, 130, 302, 141], "score": 0.89, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [302, 130, 491, 145], "score": 1.0, "content": ", which is an affine space, the space ", "type": "text"}, {"bbox": [491, 131, 515, 144], "score": 0.94, "content": "\\scriptstyle A/\\mathcal G", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [515, 130, 538, 145], "score": 1.0, "content": " has", "type": "text"}], "index": 7}, {"bbox": [62, 145, 538, 159], "spans": [{"bbox": [62, 145, 538, 159], "score": 1.0, "content": "a very complicated structure: It is non-affin, non-compact and infinite-dimensional and it is", "type": "text"}], "index": 8}, {"bbox": [61, 158, 538, 174], "spans": [{"bbox": [61, 158, 538, 174], "score": 1.0, "content": "not a manifold. This causes enormous problems, in particular, when one wants to quantize a", "type": "text"}], "index": 9}, {"bbox": [61, 173, 538, 188], "spans": [{"bbox": [61, 173, 538, 188], "score": 1.0, "content": "gauge theory. One possible quantization method is the path integral quantization. Here one", "type": "text"}], "index": 10}, {"bbox": [62, 188, 538, 203], "spans": [{"bbox": [62, 188, 538, 203], "score": 1.0, "content": "has to find an appropriate measure on the configuration space of the classical theory, hence", "type": "text"}], "index": 11}, {"bbox": [61, 202, 538, 217], "spans": [{"bbox": [61, 202, 136, 217], "score": 1.0, "content": "a measure on ", "type": "text"}, {"bbox": [136, 204, 159, 216], "score": 0.95, "content": "\\mathcal{A}/\\mathcal{G}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [160, 202, 538, 217], "score": 1.0, "content": ". As just indicated, this is very hard to find. Thus, one has hoped for a", "type": "text"}], "index": 12}, {"bbox": [62, 216, 523, 231], "spans": [{"bbox": [62, 216, 271, 231], "score": 1.0, "content": "better understanding of the structure of ", "type": "text"}, {"bbox": [272, 218, 295, 230], "score": 0.94, "content": "\\scriptstyle A/\\mathcal G", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [295, 216, 523, 231], "score": 1.0, "content": ". However, up to now, results are quite rare.", "type": "text"}], "index": 13}], "index": 7, "bbox_fs": [61, 43, 538, 231]}, {"type": "text", "bbox": [63, 229, 538, 388], "lines": [{"bbox": [63, 231, 539, 246], "spans": [{"bbox": [63, 231, 539, 246], "score": 1.0, "content": "About 20 years ago, the efforts were focussed on a related problem: The consideration of", "type": "text"}], "index": 14}, {"bbox": [61, 244, 538, 263], "spans": [{"bbox": [61, 244, 538, 263], "score": 1.0, "content": "connections and gauge transforms that are contained in a certain Sobolev class (see, e.g.,", "type": "text"}], "index": 15}, {"bbox": [63, 261, 537, 275], "spans": [{"bbox": [63, 261, 123, 275], "score": 1.0, "content": "[16]). Now, ", "type": "text"}, {"bbox": [124, 262, 131, 272], "score": 0.9, "content": "\\mathcal{G}", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [132, 261, 360, 275], "score": 1.0, "content": " is a Hilbert-Lie group and acts smoothly on ", "type": "text"}, {"bbox": [360, 262, 370, 271], "score": 0.89, "content": "\\mathcal{A}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [370, 261, 537, 275], "score": 1.0, "content": ". About 15 years ago, Kondracki", "type": "text"}], "index": 16}, {"bbox": [63, 275, 538, 289], "spans": [{"bbox": [63, 275, 538, 289], "score": 1.0, "content": "and Rogulski [12] found lots of fundamental properties of this action. Perhaps, the most", "type": "text"}], "index": 17}, {"bbox": [62, 289, 537, 303], "spans": [{"bbox": [62, 289, 371, 303], "score": 1.0, "content": "remarkable theorem they obtained was a slice theorem on ", "type": "text"}, {"bbox": [371, 291, 381, 299], "score": 0.9, "content": "\\mathcal{A}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [382, 289, 537, 303], "score": 1.0, "content": ". This means, for every orbit", "type": "text"}], "index": 18}, {"bbox": [63, 303, 538, 319], "spans": [{"bbox": [63, 305, 113, 316], "score": 0.93, "content": "A\\circ\\mathcal{G}\\subseteq A", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [113, 303, 501, 319], "score": 1.0, "content": " there is an equivariant retraction from a (so-called tubular) neighborhood of ", "type": "text"}, {"bbox": [501, 305, 510, 314], "score": 0.89, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [511, 303, 538, 319], "score": 1.0, "content": " onto", "type": "text"}], "index": 19}, {"bbox": [63, 318, 537, 332], "spans": [{"bbox": [63, 320, 91, 330], "score": 0.92, "content": "A\\circ{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [91, 318, 537, 332], "score": 1.0, "content": ". Using this theorem they could clarify the structure of the so-called strata. A stratum", "type": "text"}], "index": 20}, {"bbox": [61, 332, 538, 348], "spans": [{"bbox": [61, 332, 538, 348], "score": 1.0, "content": "contains all connections that have the same, fixed type, i.e. the same (conjugacy class of the)", "type": "text"}], "index": 21}, {"bbox": [62, 347, 538, 361], "spans": [{"bbox": [62, 347, 218, 361], "score": 1.0, "content": "stabilizer under the action of ", "type": "text"}, {"bbox": [219, 349, 227, 358], "score": 0.91, "content": "\\mathcal{G}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [227, 347, 538, 361], "score": 1.0, "content": ". Using a denseness theorem for the strata, Kondracki and", "type": "text"}], "index": 22}, {"bbox": [61, 360, 538, 376], "spans": [{"bbox": [61, 360, 226, 376], "score": 1.0, "content": "Rogulski proved that the space ", "type": "text"}, {"bbox": [226, 363, 236, 372], "score": 0.91, "content": "\\mathcal{A}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [236, 360, 435, 376], "score": 1.0, "content": " is regularly stratified by the action of ", "type": "text"}, {"bbox": [435, 363, 443, 373], "score": 0.92, "content": "\\mathcal{G}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [443, 360, 538, 376], "score": 1.0, "content": ". In particular, all", "type": "text"}], "index": 23}, {"bbox": [63, 376, 276, 390], "spans": [{"bbox": [63, 376, 262, 390], "score": 1.0, "content": "the strata are smooth submanifolds of ", "type": "text"}, {"bbox": [262, 378, 272, 386], "score": 0.9, "content": "\\mathcal{A}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [272, 376, 276, 390], "score": 1.0, "content": ".", "type": "text"}], "index": 24}], "index": 19, "bbox_fs": [61, 231, 539, 390]}, {"type": "text", "bbox": [63, 388, 537, 605], "lines": [{"bbox": [63, 390, 537, 405], "spans": [{"bbox": [63, 390, 471, 405], "score": 1.0, "content": "Despite these results the mathematically rigorous construction of a measure on ", "type": "text"}, {"bbox": [471, 391, 495, 403], "score": 0.94, "content": "\\scriptstyle A/\\mathcal{G}", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [495, 390, 537, 405], "score": 1.0, "content": " has not", "type": "text"}], "index": 25}, {"bbox": [63, 405, 536, 418], "spans": [{"bbox": [63, 405, 536, 418], "score": 1.0, "content": "been achieved. This problem was solved – at least preliminary – by Ashtekar et al. [1, 2],", "type": "text"}], "index": 26}, {"bbox": [61, 419, 537, 433], "spans": [{"bbox": [61, 419, 173, 433], "score": 1.0, "content": "but, however, not for ", "type": "text"}, {"bbox": [173, 420, 196, 433], "score": 0.94, "content": "\\mathcal{A}/\\mathcal{G}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [197, 419, 537, 433], "score": 1.0, "content": " itself. Their idea was to drop simply all smoothness conditions for", "type": "text"}], "index": 27}, {"bbox": [62, 434, 537, 448], "spans": [{"bbox": [62, 434, 537, 448], "score": 1.0, "content": "the connections and gauge transforms. In detail, they first used the fact that a connection", "type": "text"}], "index": 28}, {"bbox": [63, 449, 537, 462], "spans": [{"bbox": [63, 449, 537, 462], "score": 1.0, "content": "can always be reconstructed uniquely by its parallel transports. On the other hand, these", "type": "text"}], "index": 29}, {"bbox": [62, 462, 537, 477], "spans": [{"bbox": [62, 462, 537, 477], "score": 1.0, "content": "parallel transports can be identified with an assignment of elements of the structure group", "type": "text"}], "index": 30}, {"bbox": [63, 476, 538, 491], "spans": [{"bbox": [63, 479, 74, 487], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [74, 476, 252, 491], "score": 1.0, "content": " to the paths in the base manifold ", "type": "text"}, {"bbox": [253, 479, 265, 487], "score": 0.91, "content": "M", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [266, 476, 538, 491], "score": 1.0, "content": " such that the concatenation of paths corresponds to", "type": "text"}], "index": 31}, {"bbox": [63, 491, 538, 506], "spans": [{"bbox": [63, 491, 538, 506], "score": 1.0, "content": "the product of these group elements. It is intuitively clear that for smooth connections the", "type": "text"}], "index": 32}, {"bbox": [63, 507, 537, 520], "spans": [{"bbox": [63, 507, 537, 520], "score": 1.0, "content": "parallel transports additionally depend smoothly on the paths [14]. But now this restriction", "type": "text"}], "index": 33}, {"bbox": [62, 520, 536, 533], "spans": [{"bbox": [62, 520, 536, 533], "score": 1.0, "content": "is removed for the generalized connections. They are only homomorphisms from the groupoid", "type": "text"}], "index": 34}, {"bbox": [63, 534, 537, 549], "spans": [{"bbox": [63, 537, 72, 545], "score": 0.91, "content": "\\mathcal{P}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [73, 534, 237, 549], "score": 1.0, "content": " of paths to the structure group ", "type": "text"}, {"bbox": [238, 536, 249, 545], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [249, 534, 363, 549], "score": 1.0, "content": ". Analogously, the set ", "type": "text"}, {"bbox": [363, 534, 371, 546], "score": 0.91, "content": "\\overline{{g}}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [372, 534, 537, 549], "score": 1.0, "content": " of generalized gauge transforms", "type": "text"}], "index": 35}, {"bbox": [62, 548, 537, 564], "spans": [{"bbox": [62, 548, 198, 564], "score": 1.0, "content": "collects all functions from ", "type": "text"}, {"bbox": [198, 551, 211, 559], "score": 0.91, "content": "M", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [211, 548, 228, 564], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [228, 551, 239, 560], "score": 0.88, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [240, 548, 342, 564], "score": 1.0, "content": ". Now the action of ", "type": "text"}, {"bbox": [342, 549, 350, 561], "score": 0.91, "content": "\\mathcal{G}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [351, 548, 368, 564], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [368, 549, 378, 560], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [378, 548, 537, 564], "score": 1.0, "content": " is defined purely algebraically.", "type": "text"}], "index": 36}, {"bbox": [62, 563, 537, 578], "spans": [{"bbox": [62, 563, 97, 578], "score": 1.0, "content": "Given ", "type": "text"}, {"bbox": [97, 564, 107, 574], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [107, 563, 133, 578], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [134, 564, 142, 575], "score": 0.91, "content": "\\mathcal{G}", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [142, 563, 364, 578], "score": 1.0, "content": " the topologies induced by the topology of ", "type": "text"}, {"bbox": [364, 565, 375, 574], "score": 0.9, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [375, 563, 522, 578], "score": 1.0, "content": ", one sees that, for compact ", "type": "text"}, {"bbox": [522, 565, 533, 574], "score": 0.88, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [533, 563, 537, 578], "score": 1.0, "content": ",", "type": "text"}], "index": 37}, {"bbox": [62, 578, 537, 592], "spans": [{"bbox": [62, 578, 537, 592], "score": 1.0, "content": "these spaces are again compact. This guarantees the existence of a natural induced Haar", "type": "text"}], "index": 38}, {"bbox": [62, 591, 513, 608], "spans": [{"bbox": [62, 591, 124, 608], "score": 1.0, "content": "measure on ", "type": "text"}, {"bbox": [125, 592, 135, 603], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [135, 591, 161, 608], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [161, 592, 185, 606], "score": 0.94, "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 14, "width": 24}, {"bbox": [185, 591, 513, 608], "score": 1.0, "content": ", the new configuration space for the path integral quantization.", "type": "text"}], "index": 39}], "index": 32, "bbox_fs": [61, 390, 538, 608]}, {"type": "text", "bbox": [64, 605, 537, 677], "lines": [{"bbox": [62, 606, 537, 622], "spans": [{"bbox": [62, 606, 537, 622], "score": 1.0, "content": "Both from the mathematical and from the physical point of view it is very interesting how the", "type": "text"}], "index": 40}, {"bbox": [61, 619, 537, 637], "spans": [{"bbox": [61, 619, 537, 637], "score": 1.0, "content": "”classical” regular gauge theories are related to the generalized formulation in the Ashtekar", "type": "text"}], "index": 41}, {"bbox": [61, 634, 537, 650], "spans": [{"bbox": [61, 634, 324, 650], "score": 1.0, "content": "framework. First of all, it has been proven that ", "type": "text"}, {"bbox": [324, 637, 334, 646], "score": 0.9, "content": "\\mathcal{A}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [334, 634, 362, 650], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [362, 637, 370, 647], "score": 0.91, "content": "\\mathcal{G}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [371, 634, 486, 650], "score": 1.0, "content": " are dense subsets in ", "type": "text"}, {"bbox": [486, 636, 496, 646], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [496, 634, 524, 650], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [524, 636, 533, 647], "score": 0.89, "content": "\\overline{{g}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [533, 634, 537, 650], "score": 1.0, "content": ",", "type": "text"}], "index": 42}, {"bbox": [63, 650, 536, 664], "spans": [{"bbox": [63, 650, 228, 664], "score": 1.0, "content": "respectively [17]. Furthermore, ", "type": "text"}, {"bbox": [229, 652, 239, 660], "score": 0.89, "content": "\\mathcal{A}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [239, 650, 536, 664], "score": 1.0, "content": " is contained in a set of induced Haar measure zero [15].", "type": "text"}], "index": 43}, {"bbox": [63, 664, 538, 680], "spans": [{"bbox": [63, 664, 538, 680], "score": 1.0, "content": "These properties coincide exactly with the experiences known from the Wiener or Feynman", "type": "text"}], "index": 44}, {"bbox": [63, 18, 536, 31], "spans": [{"bbox": [63, 18, 536, 31], "score": 1.0, "content": "path integral. Then the Wilson loop expectation values have been determined for the two-", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [63, 32, 537, 45], "spans": [{"bbox": [63, 32, 537, 45], "score": 1.0, "content": "dimensional pure Yang-Mills theory [5, 11] – in coincidence with the known results in the", "type": "text", "cross_page": true}], "index": 1}, {"bbox": [61, 45, 537, 60], "spans": [{"bbox": [61, 45, 537, 60], "score": 1.0, "content": "standard framework. In the present paper we continue the investigations on how the results", "type": "text", "cross_page": true}], "index": 2}, {"bbox": [62, 59, 537, 75], "spans": [{"bbox": [62, 59, 537, 75], "score": 1.0, "content": "of Kondracki and Rogulski can be extended to the Ashtekar framework. In a previous paper", "type": "text", "cross_page": true}], "index": 3}, {"bbox": [63, 75, 537, 90], "spans": [{"bbox": [63, 75, 537, 90], "score": 1.0, "content": "[9] we have already shown that the gauge orbit type is determined by the centralizer of the", "type": "text", "cross_page": true}], "index": 4}, {"bbox": [61, 88, 538, 106], "spans": [{"bbox": [61, 88, 538, 106], "score": 1.0, "content": "holonomy group. This closely related to the observations of Kondracki and Sadowski [13]. In", "type": "text", "cross_page": true}], "index": 5}, {"bbox": [61, 104, 537, 119], "spans": [{"bbox": [61, 104, 537, 119], "score": 1.0, "content": "the present paper we are going to prove that there is a slice theorem and a denseness theorem", "type": "text", "cross_page": true}], "index": 6}, {"bbox": [63, 119, 536, 132], "spans": [{"bbox": [63, 119, 536, 132], "score": 1.0, "content": "for the space of connections in the Ashtekar framework as well. However, our methods are", "type": "text", "cross_page": true}], "index": 7}, {"bbox": [64, 133, 352, 146], "spans": [{"bbox": [64, 133, 352, 146], "score": 1.0, "content": "completely different to those of Kondracki and Rogulski.", "type": "text", "cross_page": true}], "index": 8}], "index": 42, "bbox_fs": [61, 606, 538, 680]}]}
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[{"type": "title", "bbox": [63, 10, 200, 29], "content": "1 Introduction", "index": 0}, {"type": "text", "bbox": [63, 41, 538, 228], "content": "For quite a long time the geometric structure of gauge theories has been investigated. A classical (pure) gauge theory consists of three basic objects: First the set of smooth con- nections (”gauge fields”) in a principle fiber bundle, then the set of all smooth gauge transforms, i.e. automorphisms of this bundle, and finally the action of on . Physically, two gauge fields that are related by a gauge transform describe one and the same situation. Thus, the space of all gauge orbits, i.e. elements in , is the configuration space for the gauge theory. Unfortunately, in contrast to , which is an affine space, the space has a very complicated structure: It is non-affin, non-compact and infinite-dimensional and it is not a manifold. This causes enormous problems, in particular, when one wants to quantize a gauge theory. One possible quantization method is the path integral quantization. Here one has to find an appropriate measure on the configuration space of the classical theory, hence a measure on . As just indicated, this is very hard to find. Thus, one has hoped for a better understanding of the structure of . However, up to now, results are quite rare.", "index": 1}, {"type": "text", "bbox": [63, 229, 538, 388], "content": "About 20 years ago, the efforts were focussed on a related problem: The consideration of connections and gauge transforms that are contained in a certain Sobolev class (see, e.g., [16]). Now, is a Hilbert-Lie group and acts smoothly on . About 15 years ago, Kondracki and Rogulski [12] found lots of fundamental properties of this action. Perhaps, the most remarkable theorem they obtained was a slice theorem on . This means, for every orbit there is an equivariant retraction from a (so-called tubular) neighborhood of onto . Using this theorem they could clarify the structure of the so-called strata. A stratum contains all connections that have the same, fixed type, i.e. the same (conjugacy class of the) stabilizer under the action of . Using a denseness theorem for the strata, Kondracki and Rogulski proved that the space is regularly stratified by the action of . In particular, all the strata are smooth submanifolds of .", "index": 2}, {"type": "text", "bbox": [63, 388, 537, 605], "content": "Despite these results the mathematically rigorous construction of a measure on has not been achieved. This problem was solved – at least preliminary – by Ashtekar et al. [1, 2], but, however, not for itself. Their idea was to drop simply all smoothness conditions for the connections and gauge transforms. In detail, they first used the fact that a connection can always be reconstructed uniquely by its parallel transports. On the other hand, these parallel transports can be identified with an assignment of elements of the structure group to the paths in the base manifold such that the concatenation of paths corresponds to the product of these group elements. It is intuitively clear that for smooth connections the parallel transports additionally depend smoothly on the paths [14]. But now this restriction is removed for the generalized connections. They are only homomorphisms from the groupoid of paths to the structure group . Analogously, the set of generalized gauge transforms collects all functions from to . Now the action of to is defined purely algebraically. Given and the topologies induced by the topology of , one sees that, for compact , these spaces are again compact. This guarantees the existence of a natural induced Haar measure on and , the new configuration space for the path integral quantization.", "index": 3}, {"type": "text", "bbox": [64, 605, 537, 677], "content": "Both from the mathematical and from the physical point of view it is very interesting how the ”classical” regular gauge theories are related to the generalized formulation in the Ashtekar framework. First of all, it has been proven that and are dense subsets in and , respectively [17]. Furthermore, is contained in a set of induced Haar measure zero [15]. These properties coincide exactly with the experiences known from the Wiener or Feynman path integral. Then the Wilson loop expectation values have been determined for the two- dimensional pure Yang-Mills theory [5, 11] – in coincidence with the known results in the standard framework. In the present paper we continue the investigations on how the results of Kondracki and Rogulski can be extended to the Ashtekar framework. In a previous paper [9] we have already shown that the gauge orbit type is determined by the centralizer of the holonomy group. This closely related to the observations of Kondracki and Sadowski [13]. In the present paper we are going to prove that there is a slice theorem and a denseness theorem for the space of connections in the Ashtekar framework as well. However, our methods are completely different to those of Kondracki and Rogulski.", "index": 4}]
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[{"bbox": [63, 13, 199, 29], "content": "1 Introduction", "parent_index": 0, "line_index": 0}, {"bbox": [62, 43, 537, 58], "content": "For quite a long time the geometric structure of gauge theories has been investigated. A", "parent_index": 1, "line_index": 0}, {"bbox": [62, 58, 538, 73], "content": "classical (pure) gauge theory consists of three basic objects: First the set of smooth con-", "parent_index": 1, "line_index": 1}, {"bbox": [62, 71, 537, 87], "content": "nections (”gauge fields”) in a principle fiber bundle, then the set of all smooth gauge", "parent_index": 1, "line_index": 2}, {"bbox": [62, 87, 536, 101], "content": "transforms, i.e. automorphisms of this bundle, and finally the action of on . Physically,", "parent_index": 1, "line_index": 3}, {"bbox": [62, 101, 538, 116], "content": "two gauge fields that are related by a gauge transform describe one and the same situation.", "parent_index": 1, "line_index": 4}, {"bbox": [63, 116, 537, 129], "content": "Thus, the space of all gauge orbits, i.e. elements in , is the configuration space for the", "parent_index": 1, "line_index": 5}, {"bbox": [61, 130, 538, 145], "content": "gauge theory. Unfortunately, in contrast to , which is an affine space, the space has", "parent_index": 1, "line_index": 6}, {"bbox": [62, 145, 538, 159], "content": "a very complicated structure: It is non-affin, non-compact and infinite-dimensional and it is", "parent_index": 1, "line_index": 7}, {"bbox": [61, 158, 538, 174], "content": "not a manifold. This causes enormous problems, in particular, when one wants to quantize a", "parent_index": 1, "line_index": 8}, {"bbox": [61, 173, 538, 188], "content": "gauge theory. One possible quantization method is the path integral quantization. Here one", "parent_index": 1, "line_index": 9}, {"bbox": [62, 188, 538, 203], "content": "has to find an appropriate measure on the configuration space of the classical theory, hence", "parent_index": 1, "line_index": 10}, {"bbox": [61, 202, 538, 217], "content": "a measure on . As just indicated, this is very hard to find. Thus, one has hoped for a", "parent_index": 1, "line_index": 11}, {"bbox": [62, 216, 523, 231], "content": "better understanding of the structure of . However, up to now, results are quite rare.", "parent_index": 1, "line_index": 12}, {"bbox": [63, 231, 539, 246], "content": "About 20 years ago, the efforts were focussed on a related problem: The consideration of", "parent_index": 2, "line_index": 0}, {"bbox": [61, 244, 538, 263], "content": "connections and gauge transforms that are contained in a certain Sobolev class (see, e.g.,", "parent_index": 2, "line_index": 1}, {"bbox": [63, 261, 537, 275], "content": "[16]). Now, is a Hilbert-Lie group and acts smoothly on . About 15 years ago, Kondracki", "parent_index": 2, "line_index": 2}, {"bbox": [63, 275, 538, 289], "content": "and Rogulski [12] found lots of fundamental properties of this action. Perhaps, the most", "parent_index": 2, "line_index": 3}, {"bbox": [62, 289, 537, 303], "content": "remarkable theorem they obtained was a slice theorem on . This means, for every orbit", "parent_index": 2, "line_index": 4}, {"bbox": [63, 303, 538, 319], "content": "there is an equivariant retraction from a (so-called tubular) neighborhood of onto", "parent_index": 2, "line_index": 5}, {"bbox": [63, 318, 537, 332], "content": ". Using this theorem they could clarify the structure of the so-called strata. A stratum", "parent_index": 2, "line_index": 6}, {"bbox": [61, 332, 538, 348], "content": "contains all connections that have the same, fixed type, i.e. the same (conjugacy class of the)", "parent_index": 2, "line_index": 7}, {"bbox": [62, 347, 538, 361], "content": "stabilizer under the action of . Using a denseness theorem for the strata, Kondracki and", "parent_index": 2, "line_index": 8}, {"bbox": [61, 360, 538, 376], "content": "Rogulski proved that the space is regularly stratified by the action of . In particular, all", "parent_index": 2, "line_index": 9}, {"bbox": [63, 376, 276, 390], "content": "the strata are smooth submanifolds of .", "parent_index": 2, "line_index": 10}, {"bbox": [63, 390, 537, 405], "content": "Despite these results the mathematically rigorous construction of a measure on has not", "parent_index": 3, "line_index": 0}, {"bbox": [63, 405, 536, 418], "content": "been achieved. This problem was solved – at least preliminary – by Ashtekar et al. [1, 2],", "parent_index": 3, "line_index": 1}, {"bbox": [61, 419, 537, 433], "content": "but, however, not for itself. Their idea was to drop simply all smoothness conditions for", "parent_index": 3, "line_index": 2}, {"bbox": [62, 434, 537, 448], "content": "the connections and gauge transforms. In detail, they first used the fact that a connection", "parent_index": 3, "line_index": 3}, {"bbox": [63, 449, 537, 462], "content": "can always be reconstructed uniquely by its parallel transports. On the other hand, these", "parent_index": 3, "line_index": 4}, {"bbox": [62, 462, 537, 477], "content": "parallel transports can be identified with an assignment of elements of the structure group", "parent_index": 3, "line_index": 5}, {"bbox": [63, 476, 538, 491], "content": "to the paths in the base manifold such that the concatenation of paths corresponds to", "parent_index": 3, "line_index": 6}, {"bbox": [63, 491, 538, 506], "content": "the product of these group elements. It is intuitively clear that for smooth connections the", "parent_index": 3, "line_index": 7}, {"bbox": [63, 507, 537, 520], "content": "parallel transports additionally depend smoothly on the paths [14]. But now this restriction", "parent_index": 3, "line_index": 8}, {"bbox": [62, 520, 536, 533], "content": "is removed for the generalized connections. They are only homomorphisms from the groupoid", "parent_index": 3, "line_index": 9}, {"bbox": [63, 534, 537, 549], "content": "of paths to the structure group . Analogously, the set of generalized gauge transforms", "parent_index": 3, "line_index": 10}, {"bbox": [62, 548, 537, 564], "content": "collects all functions from to . Now the action of to is defined purely algebraically.", "parent_index": 3, "line_index": 11}, {"bbox": [62, 563, 537, 578], "content": "Given and the topologies induced by the topology of , one sees that, for compact ,", "parent_index": 3, "line_index": 12}, {"bbox": [62, 578, 537, 592], "content": "these spaces are again compact. This guarantees the existence of a natural induced Haar", "parent_index": 3, "line_index": 13}, {"bbox": [62, 591, 513, 608], "content": "measure on and , the new configuration space for the path integral quantization.", "parent_index": 3, "line_index": 14}, {"bbox": [62, 606, 537, 622], "content": "Both from the mathematical and from the physical point of view it is very interesting how the", "parent_index": 4, "line_index": 0}, {"bbox": [61, 619, 537, 637], "content": "”classical” regular gauge theories are related to the generalized formulation in the Ashtekar", "parent_index": 4, "line_index": 1}, {"bbox": [61, 634, 537, 650], "content": "framework. First of all, it has been proven that and are dense subsets in and ,", "parent_index": 4, "line_index": 2}, {"bbox": [63, 650, 536, 664], "content": "respectively [17]. Furthermore, is contained in a set of induced Haar measure zero [15].", "parent_index": 4, "line_index": 3}, {"bbox": [63, 664, 538, 680], "content": "These properties coincide exactly with the experiences known from the Wiener or Feynman", "parent_index": 4, "line_index": 4}, {"bbox": [63, 18, 536, 31], "content": "path integral. Then the Wilson loop expectation values have been determined for the two-", "parent_index": 4, "line_index": 5}, {"bbox": [63, 32, 537, 45], "content": "dimensional pure Yang-Mills theory [5, 11] – in coincidence with the known results in the", "parent_index": 4, "line_index": 6}, {"bbox": [61, 45, 537, 60], "content": "standard framework. In the present paper we continue the investigations on how the results", "parent_index": 4, "line_index": 7}, {"bbox": [62, 59, 537, 75], "content": "of Kondracki and Rogulski can be extended to the Ashtekar framework. In a previous paper", "parent_index": 4, "line_index": 8}, {"bbox": [63, 75, 537, 90], "content": "[9] we have already shown that the gauge orbit type is determined by the centralizer of the", "parent_index": 4, "line_index": 9}, {"bbox": [61, 88, 538, 106], "content": "holonomy group. This closely related to the observations of Kondracki and Sadowski [13]. In", "parent_index": 4, "line_index": 10}, {"bbox": [61, 104, 537, 119], "content": "the present paper we are going to prove that there is a slice theorem and a denseness theorem", "parent_index": 4, "line_index": 11}, {"bbox": [63, 119, 536, 132], "content": "for the space of connections in the Ashtekar framework as well. However, our methods are", "parent_index": 4, "line_index": 12}, {"bbox": [64, 133, 352, 146], "content": "completely different to those of Kondracki and Rogulski.", "parent_index": 4, "line_index": 13}]
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"The outline of the paper is as follows:\n\nAfter fixing the notations we prove a very crucial lemma(...TRUNCATED) | "<html><body>\n<p data-bbox=\"63 155 257 169\">The outline of the paper is as follows: </p>\n<p data(...TRUNCATED) |
0001008v1
| 2 | 612 | 792 | 1,275 | 1,650 | "[{\"type\": \"text\", \"text\": \"\", \"page_idx\": 2}, {\"type\": \"text\", \"text\": \"The outlin(...TRUNCATED) | "{\"layout_dets\": [{\"category_id\": 1, \"poly\": [176, 42, 1496, 42, 1496, 403, 176, 403], \"score(...TRUNCATED) | "{\"preproc_blocks\": [{\"type\": \"text\", \"bbox\": [63, 15, 538, 145], \"lines\": [{\"bbox\": [63(...TRUNCATED) | "[{\"type\": \"text\", \"bbox\": [63, 15, 538, 145], \"content\": \"\", \"index\": 0}, {\"type\": \"(...TRUNCATED) | "[{\"bbox\": [63, 157, 257, 171], \"content\": \"The outline of the paper is as follows:\", \"parent(...TRUNCATED) |
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| "[{\"bbox\": [157, 246, 173, 254], \"content\": \"\\\\mathbf{G}^{n}\", \"parent_index\": 3, \"subtyp(...TRUNCATED) |
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"• A generalized connection ${\\overline{{A}}}\\in{\\overline{{A}}}$ is a homomorphism1 $h_{\\over(...TRUNCATED) | "<html><body>\n<p data-bbox=\"65 116 538 173\">• A generalized connection ${\\overline{{A}}}\\in{\(...TRUNCATED) |
0001008v1
| 3 | 612 | 792 | 1,275 | 1,650 | "[{\"type\": \"text\", \"text\": \"\", \"page_idx\": 3}, {\"type\": \"text\", \"text\": \"• A gene(...TRUNCATED) | "{\"layout_dets\": [{\"category_id\": 1, \"poly\": [216, 40, 1495, 40, 1495, 322, 216, 322], \"score(...TRUNCATED) | "{\"preproc_blocks\": [{\"type\": \"text\", \"bbox\": [77, 14, 538, 115], \"lines\": [{\"bbox\": [79(...TRUNCATED) | "[{\"type\": \"text\", \"bbox\": [77, 14, 538, 115], \"content\": \"\", \"index\": 0}, {\"type\": \"(...TRUNCATED) | "[{\"bbox\": [61, 117, 538, 134], \"content\": \"• A generalized connection is a homomorphism1 (...TRUNCATED) |
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| "[{\"bbox\": [213, 118, 248, 129], \"content\": \"{\\\\overline{{A}}}\\\\in{\\\\overline{{A}}}\", \"(...TRUNCATED) |
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"Definition 3.3 Let $t\\in\\mathcal T$ . We define the following expressions:\n\n$$\n\\begin{array}{(...TRUNCATED) | "<html><body>\n<p data-bbox=\"62 14 397 29\">Definition 3.3 Let $t\\in\\mathcal T$ . We define the f(...TRUNCATED) |
0001008v1
| 4 | 612 | 792 | 1,275 | 1,650 | "[{\"type\": \"text\", \"text\": \"Definition 3.3 Let $t\\\\in\\\\mathcal T$ . We define the followi(...TRUNCATED) | "{\"layout_dets\": [{\"category_id\": 8, \"poly\": [714, 89, 1197, 89, 1197, 221, 714, 221], \"score(...TRUNCATED) | "{\"preproc_blocks\": [{\"type\": \"text\", \"bbox\": [62, 14, 397, 29], \"lines\": [{\"bbox\": [62,(...TRUNCATED) | "[{\"type\": \"text\", \"bbox\": [62, 14, 397, 29], \"content\": \"Definition 3.3 Let . We define t(...TRUNCATED) | "[{\"bbox\": [62, 17, 396, 31], \"content\": \"Definition 3.3 Let . We define the following express(...TRUNCATED) |
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| "[{\"bbox\": [174, 19, 203, 28], \"content\": \"t\\\\in\\\\mathcal T\", \"parent_index\": 0, \"subty(...TRUNCATED) |
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"Proof • $\\varphi_{\\alpha}:{\\overline{{\\mathcal{A}}}}\\longrightarrow\\mathbf{G}^{\\#\\alpha}$(...TRUNCATED) | "<html><body>\n<p data-bbox=\"61 12 539 146\">Proof • $\\varphi_{\\alpha}:{\\overline{{\\mathcal{A(...TRUNCATED) |
0001008v1
| 5 | 612 | 792 | 1,275 | 1,650 | "[{\"type\": \"text\", \"text\": \"Proof • $\\\\varphi_{\\\\alpha}:{\\\\overline{{\\\\mathcal{A}}}(...TRUNCATED) | "{\"layout_dets\": [{\"category_id\": 0, \"poly\": [174, 456, 806, 456, 806, 504, 174, 504], \"score(...TRUNCATED) | "{\"preproc_blocks\": [{\"type\": \"text\", \"bbox\": [61, 12, 539, 146], \"lines\": [{\"bbox\": [61(...TRUNCATED) | "[{\"type\": \"text\", \"bbox\": [61, 12, 539, 146], \"content\": \"Proof • is as a map into a pr(...TRUNCATED) | "[{\"bbox\": [61, 14, 536, 35], \"content\": \"Proof • is as a map into a product space continuou(...TRUNCATED) |
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| "[{\"bbox\": [123, 17, 209, 30], \"content\": \"\\\\varphi_{\\\\alpha}:{\\\\overline{{\\\\mathcal{A}(...TRUNCATED) |
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"# 5.1 The Idea\n\nOur proof imitates in a certain sense the proof of the standard slice theorem (se(...TRUNCATED) | "<html><body>\n<h1 data-bbox=\"62 12 165 29\">5.1 The Idea </h1>\n<p data-bbox=\"62 36 538 166\">Our(...TRUNCATED) |
0001008v1
| 6 | 612 | 792 | 1,275 | 1,650 | "[{\"type\": \"text\", \"text\": \"5.1 The Idea \", \"text_level\": 1, \"page_idx\": 6}, {\"type\": (...TRUNCATED) | "{\"layout_dets\": [{\"category_id\": 1, \"poly\": [172, 866, 1496, 866, 1496, 985, 172, 985], \"sco(...TRUNCATED) | "{\"preproc_blocks\": [{\"type\": \"title\", \"bbox\": [62, 12, 165, 29], \"lines\": [{\"bbox\": [61(...TRUNCATED) | "[{\"type\": \"title\", \"bbox\": [62, 12, 165, 29], \"content\": \"5.1 The Idea\", \"index\": 0}, {(...TRUNCATED) | "[{\"bbox\": [61, 15, 164, 30], \"content\": \"5.1 The Idea\", \"parent_index\": 0, \"line_index\": (...TRUNCATED) |
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| "[{\"bbox\": [367, 55, 384, 64], \"content\": \"L i e\", \"parent_index\": 1, \"subtype\": \"inline\(...TRUNCATED) |
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"# 5.2 The Proof\n\nProof 1. Let ${\\overline{{A}}}\\in{\\overline{{A}}}$ . Choose for $\\overline{{(...TRUNCATED) | "<html><body>\n<h1 data-bbox=\"61 12 176 29\">5.2 The Proof </h1>\n<p data-bbox=\"62 35 539 80\">Pro(...TRUNCATED) |
0001008v1
| 7 | 612 | 792 | 1,275 | 1,650 | "[{\"type\": \"text\", \"text\": \"5.2 The Proof \", \"text_level\": 1, \"page_idx\": 7}, {\"type\":(...TRUNCATED) | "{\"layout_dets\": [{\"category_id\": 8, \"poly\": [544, 1186, 1363, 1186, 1363, 1448, 544, 1448], \(...TRUNCATED) | "{\"preproc_blocks\": [{\"type\": \"title\", \"bbox\": [61, 12, 176, 29], \"lines\": [{\"bbox\": [62(...TRUNCATED) | "[{\"type\": \"title\", \"bbox\": [61, 12, 176, 29], \"content\": \"5.2 The Proof\", \"index\": 0}, (...TRUNCATED) | "[{\"bbox\": [62, 15, 174, 29], \"content\": \"5.2 The Proof\", \"parent_index\": 0, \"line_index\":(...TRUNCATED) |
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| "[{\"bbox\": [153, 39, 189, 50], \"content\": \"{\\\\overline{{A}}}\\\\in{\\\\overline{{A}}}\", \"pa(...TRUNCATED) |
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"6. $F$ is equivariant.\n\nLet $\\overline{{A}}^{\\prime\\prime}=\\overline{{A}}^{\\prime}\\circ\\ov(...TRUNCATED) | "<html><body>\n<p data-bbox=\"110 15 218 28\">6. $F$ is equivariant. </p>\n<p data-bbox=\"132 28 309(...TRUNCATED) |
0001008v1
| 8 | 612 | 792 | 1,275 | 1,650 | "[{\"type\": \"text\", \"text\": \"6. $F$ is equivariant. \", \"page_idx\": 8}, {\"type\": \"text\",(...TRUNCATED) | "{\"layout_dets\": [{\"category_id\": 8, \"poly\": [736, 129, 1172, 129, 1172, 348, 736, 348], \"sco(...TRUNCATED) | "{\"preproc_blocks\": [{\"type\": \"text\", \"bbox\": [110, 15, 218, 28], \"lines\": [{\"bbox\": [11(...TRUNCATED) | "[{\"type\": \"text\", \"bbox\": [110, 15, 218, 28], \"content\": \"6. is equivariant.\", \"index\"(...TRUNCATED) | "[{\"bbox\": [111, 17, 217, 30], \"content\": \"6. is equivariant.\", \"parent_index\": 0, \"line_i(...TRUNCATED) |
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| "[{\"bbox\": [132, 19, 141, 28], \"content\": \"F\", \"parent_index\": 0, \"subtype\": \"inline\"}, (...TRUNCATED) |
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"Since $\\varphi$ , $f$ and $\\tau_{\\mathbf G}$ are continuous, the map\n\n$$\nF^{\\prime}:=\\tau_{(...TRUNCATED) | "<html><body>\n<p data-bbox=\"147 15 371 29\">Since $\\varphi$ , $f$ and $\\tau_{\\mathbf G}$ are co(...TRUNCATED) |
0001008v1
| 9 | 612 | 792 | 1,275 | 1,650 | "[{\"type\": \"text\", \"text\": \"Since $\\\\varphi$ , $f$ and $\\\\tau_{\\\\mathbf G}$ are continu(...TRUNCATED) | "{\"layout_dets\": [{\"category_id\": 8, \"poly\": [474, 1358, 1429, 1358, 1429, 1637, 474, 1637], \(...TRUNCATED) | "{\"preproc_blocks\": [{\"type\": \"text\", \"bbox\": [147, 15, 371, 29], \"lines\": [{\"bbox\": [14(...TRUNCATED) | "[{\"type\": \"text\", \"bbox\": [147, 15, 371, 29], \"content\": \"Since , and are continuous,(...TRUNCATED) | "[{\"bbox\": [149, 17, 369, 31], \"content\": \"Since , and are continuous, the map\", \"parent(...TRUNCATED) |
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| "[{\"bbox\": [179, 22, 187, 30], \"content\": \"\\\\varphi\", \"parent_index\": 0, \"subtype\": \"in(...TRUNCATED) |
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