Analytic Continuation Of Chern-Simons Theory

By E. Witten. 2010

576 citations recorded by the source corpus.

The title of this article refers to analytic continuation of three-dimensional ChernSimons gauge theory away from integer values of the usual coupling parameter k, to explore questions such as the volume conjecture, or analytic continuation of threedimensional quantum gravity (to the extent that it can be described by gauge theory) from Lorentzian to Euclidean signature. Such analytic continuation can be carried out by generalizing the usual integration cycle of the Feynman path integral. Morse theory or Picard-Lefschetz theory gives a natural framework for describing the appropriate integration cycles. An important part of the analysis involves flow equations that turn out to have a surprising four-dimensional symmetry. After developing a general framework, we describe some specific examples (involving the trefoil and figure-eight knots in S3). We also find that the space of possible integration cycles for ChernSimons theory can be interpreted as the “physical Hilbert space” of a twisted version of N = 4 super Yang-Mills theory in four dimensions.

Subfields: Geometry and Topology, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/8e5afbd46989e677f41aabf3b3c480a1df104327

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Sergei Gukov
2. C. Vafa
3. Tudor Dimofte
4. E. Witten
5. Mina Aganagic
6. Marcos Mariño
7. D. Gaiotto
8. Sh. Shakirov

In Nuclear and High Energy Physics

1. Sergei Gukov
2. C. Vafa
3. E. Witten
4. Tudor Dimofte
5. Marcos Mariño
6. Mina Aganagic
7. D. Gaiotto
8. Sh. Shakirov

In Statistical and Nonlinear Physics

1. Sergei Gukov
2. E. Witten
3. C. Vafa
4. Marcos Mariño
5. Tudor Dimofte
6. Mina Aganagic
7. D. Gaiotto
8. K. Jensen

Citations in the corpus

• Analytic continuation of Liouville theory (2011, methodology): “escent contour, but in higher dimension, the stationary phase condition is not enough to determine C n and one must use the steepest descent condition. For much more on such matters, see section 2 of [45]. Notice that, in our case, because the function Ihas opposite sign in (C.14) relative to (C.1),”
• R-Twisting and 4d/2d Correspondences (2010, background): “The meaning of such irrational q has recently been clarified from the path-integral viewpoint in [22].”
• Vortex Counting and Lagrangian 3-Manifolds (2010, background): “al CFT, this partition function consists of products of holomorphic and anti-holomorphic pieces, each of which is related to the analytic continuation of Chern-Simons theory with compact gauge group G[1], and takes the following general form Z(M;~) = exp 1 ~ S 0 + X1 n=0 S n+1 ~ n ! : (1.1) Here ~ 2”
• Decoding perturbation theory using resurgence: Stokes phenomena, new saddle points and Lefschetz thimbles (2014, methodology): “In doing so, we divide the complex z-plane into “good” and “bad” regions [52].”
• Lorentzian quantum cosmology (2017, background): “More recently, Witten [27] and others have discussed its use in quantum mechanical path integrals.”
• Knot Homology from Refined Chern-Simons Theory (2011, background): “The dependence of the path integral on the contour of integration is related to the choice of the vacuum, as discussed in [58,16,54].”
• A Note On Complex Spacetime Metrics (2021, unknown): “One can attempt to use gradient flow to construct the integration cycle Γ, as described in detail for three-dimensional Chern-Simons theory in [27].”
• Fivebranes and Knots (2011, methodology): “The relation of the Chern-Simons path integral onW to the N = 4 path integral onW × R+ is one of the main results of [45] (and the basic idea is suggested in the conclusions of [14]).”
• Quantum Riemann Surfaces in Chern-Simons Theory (2011, background): “ocally holomorphic functions of u, rather than functions of m= eu. One way to see that holomorphic blocks should be non-periodic functions of u is to extend Witten’s analysis of analytic continuation [2] from knots in closed three manifolds to knot complements M = MnK. For example, suppose that we c”
• Low-dimensional de Sitter quantum gravity (2019, result): “t higher genus or with more boundaries. Our implicit point of view throughout is that to compute dS observables we must still perform the honest dS computation. This is in accord with the analysis in [62,84]. 5.2 Discussion Let us conclude with an outlook on some directions for future study. One str”
• Exact Kähler potential from gauge theory and mirror symmetry (2012, methodology): “The problem of summing over all worldsheet instantons defines an interesting class of topological invariants, known as the Gromow-Witten invariants [4, 5].”
• Resurgence theory, ghost-instantons, and analytic continuation of path integrals (2013, unknown): “The analytic continuation of path integration by itself has received some interest recently; see for example [9, 23, 24, 44, 45].”
• The finiteness conjecture for skein modules (2019, background): “It seems natural to view Walker’s TFT associated to the ribbon category Repq(G) with q generic in the context of analytically continued Chern-Simons theory as discussed in [95].”
• A new look at the path integral of quantum mechanics (2010, unknown): “) As explained in [2], if W is sufficiently generic, there are no flows between distinct critical orbits.”