Gopakumar-Vafa invariants via vanishing cycles

In this paper, we propose an ansatz for defining Gopakumar-Vafa invariants of Calabi-Yau threefolds, using perverse sheaves of vanishing cycles. Our proposal is a modification of a recent approach of Kiem-Li, which is itself based on earlier ideas of Hosono-Saito-Takahashi. We conjecture that these invariants are equivalent to other curve-counting theories such as Gromov-Witten theory and Pandharipande-Thomas theory. Our main theorem is that, for local surfaces, our invariants agree with PT invariants for irreducible one-cycles. We also give a counter-example to the Kiem-Li conjectures, where our invariants match the predicted answer. Finally, we give examples where our invariant matches the expected answer in cases where the cycle is non-reduced, non-planar, or non-primitive.Comment: 63 pages, many improvements of the exposition following referee comments, final version to appear in Inventione

Enumerative geometry of Calabi-Yau 4-folds

Gromov-Witten theory is used to define an enumerative geometry of curves in Calabi-Yau 4-folds. The main technique is to find exact solutions to moving multiple cover integrals. The resulting invariants are analogous to the BPS counts of Gopakumar and Vafa for Calabi-Yau 3-folds. We conjecture the 4-fold invariants to be integers and expect a sheaf theoretic explanation. Several local Calabi-Yau 4-folds are solved exactly. Compact cases, including the sextic Calabi-Yau in CP5, are also studied. A complete solution of the Gromov-Witten theory of the sextic is conjecturally obtained by the holomorphic anomaly equation.Comment: 44 page

Gauge theory, topological strings, and S-duality

We offer a derivation of the duality between the topological U(1) gauge theory on a Calabi-Yau 3-fold and the topological A-model on the same manifold. This duality was conjectured recently by Iqbal, Nekrasov, Okounkov, and Vafa. We deduce it from the S-duality of the IIB superstring. We also argue that the mirror version of this duality relates the topological B-model on a Calabi-Yau 3-fold and a topological sector of the Type IIA Little String Theory on the same manifold.Comment: 9 pages, latex. v2: a footnote has been added. The footnote corrects an inaccuracy in the original argument; the results are unchanged. v3: exposition improve

Comparative Analysis of a Transition Region Bright Point with a Blinker and Coronal Bright Point Using Multiple EIS Emission Lines

Since their discovery twenty year ago, transition region bright points (TRBPs) have never been observed spectroscopically. Bright point properties have not been compared with similar transition region and coronal structures. In this work we have investigated three transient quiet Sun brightenings including a TRBP, a coronal BP (CBP) and a blinker. We use time-series observations of the extreme ultraviolet emission lines of a wide range of temperature T (log T = 5.3 - 6.4) from the EUV imaging spectrometer (EIS) onboard the Hinode satellite. We present the EIS temperature maps and Doppler maps, which are compared with magnetograms from the Michelson Doppler Imager (MDI) onboard the SOHO satellite. Doppler velocities of the TR BP and blinker are <,25 km s$^{-1}$, which is typical of transient TR phenomena. The Dopper velocities of the CBP were found to be < 20 km s^{-1} with exception of those measured at log T = 6.2 where a distinct bi-directional jet is observed. From an EM loci analysis we find evidence of single and double isothermal components in the TRBP and CBP, respectively. TRBP and CBP loci curves are characterized by broad distributions suggesting the existence of unresolved structure. By comparing and contrasting the physical characteristics of the events we find the BP phenomena are an indication of multi-scaled self similarity, given similarities in both their underlying magnetic field configuration and evolution in relation to EUV flux changes. In contrast, the blinker phenomena and the TRBP are sufficiently dissimilar in their observed properties as to constitute different event classes. Our work indicates that the measurement of similar characteristics across multiple event types holds class-predictive power, and is a significant step towards automated solar atmospheric multi-class classification of unresolved transient EUV sources.Comment: 38 pages, 16 figure

Direct Integration of the Topological String

We present a new method to solve the holomorphic anomaly equations governing the free energies of type B topological strings. The method is based on direct integration with respect to the non-holomorphic dependence of the amplitudes, and relies on the interplay between non-holomorphicity and modularity properties of the topological string amplitudes. We develop a formalism valid for any Calabi-Yau manifold and we study in detail two examples, providing closed expressions for the amplitudes at low genus, as well as a discussion of the boundary conditions that fix the holomorphic ambiguity. The first example is the non-compact Calabi-Yau underlying Seiberg-Witten theory and its gravitational corrections. The second example is the Enriques Calabi-Yau, which we solve in full generality up to genus six. We discuss various aspects of this model: we obtain a new method to generate holomorphic automorphic forms on the Enriques moduli space, we write down a new product formula for the fiber amplitudes at all genus, and we analyze in detail the field theory limit. This allows us to uncover the modularity properties of SU(2), N=2 super Yang-Mills theory with four massless hypermultiplets.Comment: 75 pages, 3 figure

Drug use in acute otitis media: a prospective study at a tertiary care teaching hospital

Background: Drug use study identifies the problems that arise from prescription and highlights the current approaches to the rational use of drugs. The objective of the study was to assess drug use pattern in patients diagnosed of acute otitis media in tertiary care teaching hospital.Methods: This prospective observational study was carried in the Otorhinolaryngology department of a tertiary care teaching hospital over a period of twelve months. The data collected for patients with acute otitis media included the patient's demographic details and the drugs prescribed. Data were analysed for drug use pattern and cost per prescription and assessment of rationality of prescription.Results: Total 153 patients were analysed, 100 (65.35%) belonged to male patients and 53 (34.65%) belonged to female patients. Children less than 2years age were the most diagnosed with AOM 47.71%, the major diagnostic symptoms were earache (58.16%) and fever (54.90%) and signs were congestion (52.94%) and discharge (43.13%). In a total 153 prescriptions (469 drugs), 33.68% were antimicrobials, followed by mineral supplements (23.67%). Average number of drugs per prescription was found to be 3.0. Most common antibiotic prescribed was amoxicillin (with or without clavulanate) in 142 (92.81%) patients. Paracetamol alone or in fixed dose combination with antihistaminics were prescribed in 131 patients. Average cost per prescription was 87.74(±35.67) Indian rupees. Seventeen (11.11%) prescriptions were rational in all the aspects based on standard guidelines.Conclusions: The present study showed that paracetamol and amoxicillin with or without clavulanate were mostly commonly prescribed in children with AOM. Irrational prescribing was seen in maximum number of cases

Holomorphic anomaly equations and the Igusa cusp form conjecture

Let $S$ be a K3 surface and let $E$ be an elliptic curve. We solve the reduced Gromov-Witten theory of the Calabi-Yau threefold $S \times E$ for all curve classes which are primitive in the K3 factor. In particular, we deduce the Igusa cusp form conjecture. The proof relies on new results in the Gromov-Witten theory of elliptic curves and K3 surfaces. We show the generating series of Gromov-Witten classes of an elliptic curve are cycle-valued quasimodular forms and satisfy a holomorphic anomaly equation. The quasimodularity generalizes a result by Okounkov and Pandharipande, and the holomorphic anomaly equation proves a conjecture of Milanov, Ruan and Shen. We further conjecture quasimodularity and holomorphic anomaly equations for the cycle-valued Gromov-Witten theory of every elliptic fibration with section. The conjecture generalizes the holomorphic anomaly equations for ellliptic Calabi-Yau threefolds predicted by Bershadsky, Cecotti, Ooguri, and Vafa. We show a modified conjecture holds numerically for the reduced Gromov-Witten theory of K3 surfaces in primitive classes.Comment: 68 page

Topological wave functions and heat equations

It is generally known that the holomorphic anomaly equations in topological string theory reflect the quantum mechanical nature of the topological string partition function. We present two new results which make this assertion more precise: (i) we give a new, purely holomorphic version of the holomorphic anomaly equations, clarifying their relation to the heat equation satisfied by the Jacobi theta series; (ii) in cases where the moduli space is a Hermitian symmetric tube domain $G/K$, we show that the general solution of the anomaly equations is a matrix element \IP{\Psi | g | \Omega} of the Schr\"odinger-Weil representation of a Heisenberg extension of $G$, between an arbitrary state $\bra{\Psi}$ and a particular vacuum state $\ket{\Omega}$. Based on these results, we speculate on the existence of a one-parameter generalization of the usual topological amplitude, which in symmetric cases transforms in the smallest unitary representation of the duality group $G'$ in three dimensions, and on its relations to hypermultiplet couplings, nonabelian Donaldson-Thomas theory and black hole degeneracies.Comment: 50 pages; v2: small typos fixed, references added; v3: cosmetic changes, published version; v4: typos fixed, small clarification adde

Strengthening the Cohomological Crepant Resolution Conjecture for Hilbert-Chow morphisms

Given any smooth toric surface S, we prove a SYM-HILB correspondence which relates the 3-point, degree zero, extended Gromov-Witten invariants of the n-fold symmetric product stack [Sym^n(S)] of S to the 3-point extremal Gromov-Witten invariants of the Hilbert scheme Hilb^n(S) of n points on S. As we do not specialize the values of the quantum parameters involved, this result proves a strengthening of Ruan's Cohomological Crepant Resolution Conjecture for the Hilbert-Chow morphism from Hilb^n(S) to Sym^n(S) and yields a method of reconstructing the cup product for Hilb^n(S) from the orbifold invariants of [Sym^n(S)].Comment: Revised versio