<p>We study nonperturbative effects of torus partition function of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>T</mi><mover accent="true"><mi>T</mi><mo stretchy="false">¯</mo></mover></math>-deformed 2D conformal field theory (CFT) by resurgence in this Letter and a companion paper. The deformed partition function can be written as an infinite series of the deformation parameter <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>λ</mi></math>. We develop highly efficient methods to compute perturbative coefficients in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>λ</mi></math> expansion. To exemplify, the first 600 coefficients for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>T</mi><mover accent="true"><mi>T</mi><mo stretchy="false">¯</mo></mover></math>-deformed free boson and free fermion are computed. The method can be generalized to any rational CFTs; we present the key idea in this Letter while relegating details and results to the companion paper. Equipped with the large order perturbative data, we provide convincing numerical evidence that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>λ</mi></math> series is asymptotic and not Borel resummable. We extract the nonperturbative contribution by resurgence and propose that they originate from new complex saddle points after analytically continuing the modular parameters in the integral representation of the partition function. The proposal is checked by comparing the predicted asymptotic behavior of the coefficients and large order perturbative data, which match nicely. The implications of these nonperturbative contributions for the Stokes phenomenon, which relates the positive and negative signs of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>λ</mi></math>, as well as possible comparisons with holography, is also discussed.</p>

China

Jie Gu

Yunfeng Jiang

Huajia Wang

Resurgence of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>T</mi><mover accent="true"><mrow><mi>T</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow></math>-Deformed Partition Function

<p>We elaborate on the resurgence analysis on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>T</mi><mover accent="true"><mi>T</mi><mo stretchy="false">¯</mo></mover></math>-deformed 2D conformal field theory (CFT). Writing the deformed partition function as an infinite series in the deformation parameter <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>λ</mi></math>, we develop efficient analytical methods to compute high-order terms of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>λ</mi></math>-series. Based on the asymptotic behavior of the large-order perturbative series, we show that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>λ</mi></math>-series is asymptotic and extract nonperturbative contributions by resurgence. This paper extends a previous Letter of the same authors to interacting rational CFTs, with Lee-Yang CFT as a concrete example. We also discuss possible implications of the resurgence results on holography.</p>

Resurgent properties of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>T</mi><mover accent="true"><mi>T</mi><mo stretchy="false">¯</mo></mover></math>-deformed conformal field theories

We study the resurgent structures of Wilson loops in refined topological string theory. We argue that the Borel singularities should be integral periods, and that the associated Stokes constants are refined Donaldson-Thomas invariants, just like the free energies, except that the Borel singularities cannot be local flat coordinates. We also solve the non-perturbative series in closed form from the holomorphic anomaly equations for the refined Wilson loops. We illustrate these results with the examples of local $\textit{ℙ}$ and local $\textit{ℙ}$ × $\textit{ℙ}$ .

Gengbei Guo

Resurgent Wilson loops in refined topological string

We study the transverse momentum ( $p_T$ ) differential anisotropic flow and flow fluctuation in Pb+Pb collisions at $\sqrt{s_{_{NN}}}$ =5.02 TeV at the LHC. A (3+1)-dimensional CLVisc hydrodynamics framework with fluctuating TRENTO (or AMPT) initial conditions is utilized to simulate the space-time evolution of the quark-gluon plasma (QGP) medium. The effects of shear viscosity and the sub-nucleon structure on anisotropic flow and flow fluctuation are analyzed. Our result shows that shear viscosity tends to suppress both flow coefficients ( ${v_2\{2}\}$ , ${v_2\{4\}}$ , and ${\langle v_2\rangle}$ ) and flow fluctuation ( ${\sigma_{v_2}}$ ) owing to its smearing effect on local density fluctuation. The flow coefficients appear to be insensitive to the sub-nucleon structure, whereas the flow fluctuation ${\sigma_{v_2}}$ tends to be suppressed by the sub-nucleon structure in central collisions but enhanced in peripheral collisions. After taking into account the sub-nucleon structure effect, our numerical result can qualitatively describe the relative flow fluctuations ( ${v_2\{4\}/v_2\{2\}}$ , $F({v_2})$ ) measured by the ALICE Collaboration at the LHC. We further investigate the effects of shear viscosity, sub-nucleon structure, and initial condition model on the flow angle and flow magnitude decorrelations ( ${A_2^f}$ , ${M_2^f}$ ) using the four-particle correlation method. We find that the flow decorrelation effect is typically stronger in central collisions than in peripheral collisions. The flow angle decorrelation is found to be insensitive to the shear viscosity and sub-nucleon structure, whereas the flow magnitude decorrelation shows quite different behaviors when using the TRENTO and AMPT initial condition models. Our study sheds light on the anisotropic flow, transport properties, and initial structure of the QGP created in high-energy nuclear collisions.

Germany

Jie Zhu

Canada

Xiang-Yu Wu

Guang-You Qin

Anisotropic flow, flow fluctuation, and flow decorrelation in relativistic heavy-ion collisions: the roles of sub-nucleon structure and shear viscosity * Supported in part by the Natural Science Foundation of China (12225503, 11890710, 11890711, 11935007). Zhu Jie is supported in part by the China Scholarship Council (202306770009). Wu Xiang-Yu is supported in part by the National Science Foundation within the framework of the JETSCAPE collaboration (OAC-2004571) and in part by the Natural Sciences and Engineering Research Council of Canada. Some of the calculations were performed in the Nuclear Science Computing Center at Central China Normal University (NSC$^{3}$), Wuhan, Hubei, China.

The trans-series completion of perturbative series of a wide class of quantum mechanical systems can be determined by combining the resurgence program with extra input coming from exact WKB analysis. In this paper, we reexamine the Harper-Hofstadter model and its spectrum, Hofstadter’s butterfly in light of recent developments. We demonstrate the connection between the perturbative energy series of the Harper-Hofstadter model and the vev of 1/2-BPS Wilson loop of 5d SYM and clarify the differences between their non-perturbative corrections. Taking insights from the cosine potential model, we construct the full energy trans-series for flux $\textit{ϕ}$ = 2$\textit{π}$/$\textit{Q}$ and provide numerical evidence with remarkably high precision. Finally, we revisit the problem of self-similarity of the butterfly and discuss the possibility of a completed version of the Rammal-Wilkinson formula.

Zhaojie Xu

Towards full instanton trans-series in Hofstadter’s butterfly

In this paper we demonstrate that the Stokes constants of unrefined free energies and the Stokes constants of Nekrasov-Shatashvili free energies of topological string on a non-compact Calabi-Yau threefold are identical, possibly up to a sign, for any Borel singularity which is not associated to a compact two-cycle that intersects only with non-compact four-cycles. Since the Stokes constants of Nekrasov-Shatashvili free energies are conjectured to coincide with those of quantum periods and therefore have the interpretation of BPS invariants, our results give strong support that the Stokes constants of unrefined free energies may also be identified with BPS invariants.

Relations between Stokes constants of unrefined and Nekrasov-Shatashvili topological strings

In this note we study the resurgent structure of sl(2, ℂ) Chern-Simons state integral model on knot complements S 3\4 1 , S 3\5 2 with generic discrete level k ≥ 1 and with small boundary holonomy deformation. The coefficients of the saddle point expansions are in the trace field of the knot extended by the holonomy parameter. Despite increasing complication of the asymptotic series as the level k increases, the resurgent structure of the asymptotic series is universal: both the distribution of Borel plane singularities and the associated Stokes constants are independent of the level k.

South Korea

Zhihao Duan

Resurgence in complex Chern-Simons theory at generic levels

Given the recent geometrical classification of 6d (1, 0) SCFTs, a major question is how to compute for this large class their elliptic genera. The latter encode the refined BPS spectrum of the SCFTs, which determines geometric invariants of the associated elliptic non-compact Calabi-Yau threefolds. In this paper we establish for all 6d (1, 0) SCFTs in the atomic classification blowup equations that fix these elliptic genera to large extent. The latter fall into two types: the unity and the vanishing blowup equations. For almost all rank one theories, we find unity blowup equations which determine the elliptic genera completely. We develop several techniques to compute elliptic genera and BPS invariants from the blowup equations, including a recursion formula with respect to the number of strings, a Weyl orbit expansion, a refined BPS expansion and an ϵ 1 , ϵ 2 expansion. For higher-rank theories, we propose a gluing rule to obtain all their blowup equations based on those of rank one theories. For example, we explicitly give the elliptic blowup equations for the three higher-rank non-Higgsable clusters, ADE chain of −2 curves and conformal matter theories. We also give the toric construction for many elliptic non-compact Calabi- Yau threefolds which engineer 6d (1, 0) SCFTs with various matter representations.

Switzerland

Babak Haghighat

Albrecht Klemm

Italy

Kaiwen Sun

Xin Wang

Elliptic blowup equations for 6d SCFTs. Part IV. Matters

We establish the elliptic blowup equations for E-strings and M-strings and solve elliptic genera and refined BPS invariants from them. Such elliptic blowup equations can be derived from a path integral interpretation. We provide toric hypersurface construction for the Calabi-Yau geometries of M-strings and those of E-strings with up to three mass parameters turned on, as well as an approach to derive the perturbative prepotential directly from the local description of the Calabi-Yau threefolds. We also demonstrate how to systematically obtain blowup equations for all rank one 5d SCFTs from E-string by blow-down operations. Finally, we present blowup equations for E–M and M string chains.

Elliptic blowup equations for 6d SCFTs. Part III. E-strings, M-strings and chains

We study various non-perturbative approaches to the quantization of the Seiberg-Witten curve of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2, SU(2) super Yang-Mills theory, which is closely related to the modified Mathieu operator. The first approach is based on the quantum WKB periods and their resurgent properties. We show that these properties are encoded in the TBA equations of Gaiotto-Moore-Neitzke determined by the BPS spectrum of the theory, and we relate the Borel-resummed quantum periods to instanton calculus. In addition, we use the TS/ST correspondence to obtain a closed formula for the Fredholm determinant of the modified Mathieu operator. Finally, by using blowup equations, we explain the connection between this operator and the τ function of Painlevé III.