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Decoding perturbation theory using resurgence: Stokes phenomena, new saddle points and Lefschetz thimbles
, 2014
"... Resurgence theory implies that the nonperturbative (NP) and perturbative (P) data in a QFT are quantitatively related, and that detailed information about nonperturbative saddle point field configurations of path integrals can be extracted from perturbation theory. Traditionally, only stable NP ..."
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Resurgence theory implies that the nonperturbative (NP) and perturbative (P) data in a QFT are quantitatively related, and that detailed information about nonperturbative saddle point field configurations of path integrals can be extracted from perturbation theory. Traditionally, only stable NP saddle points are considered in QFT, and homotopy group considerations are used to classify them. However, in many QFTs the relevant homotopy groups are trivial, and even when they are nontrivial they leave many NP saddle points undetected. Resurgence provides a refined classification of NPsaddles, going beyond conventional topological considerations. To demonstrate some of these ideas, we study the SU(N) principal chiral model (PCM), a two dimensional asymptotically free matrix field theory which has no instantons, because the relevant homotopy group is trivial. Adiabatic continuity is used to reach a weakly coupled regime where NP effects are calculable. We then use resurgence theory to uncover the existence and role of novel ‘fracton ’ saddle points, which turn out to be the fractionalized constituents of previously observed unstable ‘uniton’ saddle points. The fractons play a crucial role in the physics of the PCM, and are responsible for the dynamically generated mass gap of the theory. Moreover, we show that the fractonantifracton events are the weak coupling realization of ’t Hooft’s renormalons, and argue that the renormalon ambiguities are systematically cancelled in the semiclassical expansion. Along the way, we also observe that the semiclassical expansion of the path integral can be geometrized as a sum over Lefschetz thimbles.
Resurgence and Transseries in Quantum Field Theory: The CP(N1
 Model, JHEP
"... Abstract: This work is a step towards a nonperturbative continuum definition of quantum field theory (QFT), beginning with asymptotically free two dimensional nonlinear sigmamodels, using recent ideas from mathematics and QFT. The ideas from mathematics are resurgence theory, the transseries fra ..."
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Abstract: This work is a step towards a nonperturbative continuum definition of quantum field theory (QFT), beginning with asymptotically free two dimensional nonlinear sigmamodels, using recent ideas from mathematics and QFT. The ideas from mathematics are resurgence theory, the transseries framework, and BorelÉcalle resummation. The ideas from QFT use continuity on R1×S1L, i.e, the absence of any phase transition as N → ∞ or rapidcrossovers for finiteN, and the smallL weak coupling limit to render the semiclassical sector welldefined and calculable. We classify semiclassical configurations with actions 1/N (kinkinstantons), 2/N (bions and bikinks), in units where the 2d instanton action is normalized to one. Perturbation theory possesses the IRrenormalon ambiguity that arises due to nonBorel summability of the largeorders perturbation series (of Gevrey1 type), for which a microscopic cancellation mechanism was unknown. This divergence must be present because the corresponding expansion is on a singular Stokes ray in the complexified coupling constant plane, and the sum exhibits the Stokes phenomenon crossing the ray. We show that there is also a nonperturbative ambiguity inherent to certain neutral topological molecules (neutral bions and bionantibions) in the semiclassical expansion. We find a set
Phases of planar 5dimensional supersymmetric ChernSimons theory
"... In this paper we investigate the largeN behavior of 5dimensional N = 1 super YangMills with a level k ChernSimons term and an adjoint hypermultiplet. As in threedimensional ChernSimons theories, one must choose an integration contour to completely define the theory. Using localization, we red ..."
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In this paper we investigate the largeN behavior of 5dimensional N = 1 super YangMills with a level k ChernSimons term and an adjoint hypermultiplet. As in threedimensional ChernSimons theories, one must choose an integration contour to completely define the theory. Using localization, we reduce the path integral to a matrix model with a cubic action and compute its free energy in various scenarios. In the limit of infinite YangMills coupling and for particular choices of the contours, we find that the freeenergy scales as N5/2 for U(N) gauge groups with large values of the ChernSimons ’t Hooft coupling, λ ̃ ≡ N/k. If we also set the hypermultiplet mass to zero, then this limit is a superconformal fixed point and the N5/2 behavior parallels other fixed points which have known supergravity duals. We also demonstrate that SU(N) gauge groups cannot have this N5/2 scaling for their freeenergy. At finite YangMills coupling we establish the existence of a third order phase transition where the theory crosses over from the Yang Mills phase to the ChernSimons phase. The phase transition exists for any value of λ̃, although the details differ between small and large values of λ̃. For pure ChernSimons theories we present evidence for a chain of phase transitions as λ ̃ is increased. We also find the expectation values for supersymmetric circular Wilson loops in these various scenarios and show that the ChernSimons term leads to different physical properties for fundamental and antifundamental Wilson loops. Different choices of the integration contours also lead to different properties for the loops. ar
Thresholds of large N factorization in CFT4 : Exploring bulk locality
 in AdS5,” arXiv:1403.5281 [hepth
"... Large N factorization ensures that, for lowdimension gaugeinvariant operators in the halfBPS sector of N = 4 SYM, products of holomorphic traces have vanishing correlators with single antiholomorphic traces. This is important in mapping these operators to states in the dual AdS5, with a Fock spa ..."
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Large N factorization ensures that, for lowdimension gaugeinvariant operators in the halfBPS sector of N = 4 SYM, products of holomorphic traces have vanishing correlators with single antiholomorphic traces. This is important in mapping these operators to states in the dual AdS5, with a Fock space structure of gravitons based on trace structures in CFT4. We investigate the factorization thresholds where the vanishing correlators become order one in the large N limit. Using the UV/IR relation, we can map the operator dimensions to the radial AdS positions of the gravitons, and interpret the factorization threshold as giving information on the limitations of bulk locality in quantum gravity in AdS. Quite generally, we find the threshold to be when the product of the two holomorphic operator dimensions is of order N logN. Our discussion includes nonextremal correlators and LLM backgrounds, and we observe intriguing similarities between the the energydependent running coupling of nonabelian gauge theories and some of our threshold equations.
Contents
, 2006
"... Abstract: We consider pure spinor strings that propagate in the background generated by a sequence of TsT transformations. We use the fact that U(1) isometry variables of TsTtransformed background are related to the isometry variables of the initial background in the universal way that is independe ..."
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Abstract: We consider pure spinor strings that propagate in the background generated by a sequence of TsT transformations. We use the fact that U(1) isometry variables of TsTtransformed background are related to the isometry variables of the initial background in the universal way that is independent of the details of the background. We will argue that after redefinitions of pure spinors and the fermionic variables we can construct pure spinor action with manifest U(1) isometry. This fact implies that the pure spinor string in TsTtransformed background is described by pure spinor string in the original background where worldvolume modes are subject to twisted boundary conditions. We will argue that these twisted boundary conditions generally prevent to prove the quantum conformal invariance of the pure spinor string in AdS5 × S 5 background. We determine the conditions under which this quantum conformal invariance can be proved. We also determine the Lax pair for pure spinor strings in the TsTtransformed background.
Physical mathematics and the future
, 2014
"... These are some thoughts meant to accompany one of the summary talks at Strings2014, Princeton, June 27, 2014. This is a snapshot of a personal and perhaps heterodox view of the relation of Physics and Mathematics, together with some guesses about some of the directions forward in the field of Physic ..."
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These are some thoughts meant to accompany one of the summary talks at Strings2014, Princeton, June 27, 2014. This is a snapshot of a personal and perhaps heterodox view of the relation of Physics and Mathematics, together with some guesses about some of the directions forward in the field of Physical Mathematics. At least, this is my view as of July 21, 2014.
Preprint typeset in JHEP style PAPER VERSION The spectral problem of the ABJ Fermi gas
"... Abstract: The partition function on the threesphere of ABJ theory can be rewritten into a partition function of a noninteracting Fermi gas, with an accompanying oneparticle Hamiltonian. We study the spectral problem defined by this Hamiltonian. We determine the exact WKB quantization condition, ..."
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Abstract: The partition function on the threesphere of ABJ theory can be rewritten into a partition function of a noninteracting Fermi gas, with an accompanying oneparticle Hamiltonian. We study the spectral problem defined by this Hamiltonian. We determine the exact WKB quantization condition, which involves quantities from refined topological string theory, and test it successfully against numerical calculations of the spectrum. ar X iv